a from scale-rotated-ellipse

Percentage Accurate: 2.8% → 20.0%

Time: 46.1s

Alternatives: 19

Speedup: 6.3×

• 30.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ t_1 := \sin t\_0\\ t_2 := \cos t\_0\\ t_3 := \frac{\frac{{\left(a \cdot t\_2\right)}^{2} + {\left(b \cdot t\_1\right)}^{2}}{y-scale}}{y-scale}\\ t_4 := \frac{\frac{{\left(a \cdot t\_1\right)}^{2} + {\left(b \cdot t\_2\right)}^{2}}{x-scale}}{x-scale}\\ t_5 := \left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\\ t_6 := \frac{4 \cdot t\_5}{{\left(x-scale \cdot y-scale\right)}^{2}}\\ \frac{-\sqrt{\left(\left(2 \cdot t\_6\right) \cdot t\_5\right) \cdot \left(\left(t\_4 + t\_3\right) + \sqrt{{\left(t\_4 - t\_3\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot t\_1\right) \cdot t\_2}{x-scale}}{y-scale}\right)}^{2}}\right)}}{t\_6} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (/ angle 180.0) PI))
        (t_1 (sin t_0))
        (t_2 (cos t_0))
        (t_3
         (/ (/ (+ (pow (* a t_2) 2.0) (pow (* b t_1) 2.0)) y-scale) y-scale))
        (t_4
         (/ (/ (+ (pow (* a t_1) 2.0) (pow (* b t_2) 2.0)) x-scale) x-scale))
        (t_5 (* (* b a) (* b (- a))))
        (t_6 (/ (* 4.0 t_5) (pow (* x-scale y-scale) 2.0))))
   (/
    (-
     (sqrt
      (*
       (* (* 2.0 t_6) t_5)
       (+
        (+ t_4 t_3)
        (sqrt
         (+
          (pow (- t_4 t_3) 2.0)
          (pow
           (/
            (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) t_1) t_2) x-scale)
            y-scale)
           2.0)))))))
    t_6)))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * ((double) M_PI);
	double t_1 = sin(t_0);
	double t_2 = cos(t_0);
	double t_3 = ((pow((a * t_2), 2.0) + pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale;
	double t_4 = ((pow((a * t_1), 2.0) + pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale;
	double t_5 = (b * a) * (b * -a);
	double t_6 = (4.0 * t_5) / pow((x_45_scale * y_45_scale), 2.0);
	return -sqrt((((2.0 * t_6) * t_5) * ((t_4 + t_3) + sqrt((pow((t_4 - t_3), 2.0) + pow((((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale), 2.0)))))) / t_6;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * Math.PI;
	double t_1 = Math.sin(t_0);
	double t_2 = Math.cos(t_0);
	double t_3 = ((Math.pow((a * t_2), 2.0) + Math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale;
	double t_4 = ((Math.pow((a * t_1), 2.0) + Math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale;
	double t_5 = (b * a) * (b * -a);
	double t_6 = (4.0 * t_5) / Math.pow((x_45_scale * y_45_scale), 2.0);
	return -Math.sqrt((((2.0 * t_6) * t_5) * ((t_4 + t_3) + Math.sqrt((Math.pow((t_4 - t_3), 2.0) + Math.pow((((((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale), 2.0)))))) / t_6;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (angle / 180.0) * math.pi
	t_1 = math.sin(t_0)
	t_2 = math.cos(t_0)
	t_3 = ((math.pow((a * t_2), 2.0) + math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale
	t_4 = ((math.pow((a * t_1), 2.0) + math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale
	t_5 = (b * a) * (b * -a)
	t_6 = (4.0 * t_5) / math.pow((x_45_scale * y_45_scale), 2.0)
	return -math.sqrt((((2.0 * t_6) * t_5) * ((t_4 + t_3) + math.sqrt((math.pow((t_4 - t_3), 2.0) + math.pow((((((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale), 2.0)))))) / t_6

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(angle / 180.0) * pi)
	t_1 = sin(t_0)
	t_2 = cos(t_0)
	t_3 = Float64(Float64(Float64((Float64(a * t_2) ^ 2.0) + (Float64(b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale)
	t_4 = Float64(Float64(Float64((Float64(a * t_1) ^ 2.0) + (Float64(b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale)
	t_5 = Float64(Float64(b * a) * Float64(b * Float64(-a)))
	t_6 = Float64(Float64(4.0 * t_5) / (Float64(x_45_scale * y_45_scale) ^ 2.0))
	return Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * t_6) * t_5) * Float64(Float64(t_4 + t_3) + sqrt(Float64((Float64(t_4 - t_3) ^ 2.0) + (Float64(Float64(Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale) ^ 2.0))))))) / t_6)
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (angle / 180.0) * pi;
	t_1 = sin(t_0);
	t_2 = cos(t_0);
	t_3 = ((((a * t_2) ^ 2.0) + ((b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale;
	t_4 = ((((a * t_1) ^ 2.0) + ((b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale;
	t_5 = (b * a) * (b * -a);
	t_6 = (4.0 * t_5) / ((x_45_scale * y_45_scale) ^ 2.0);
	tmp = -sqrt((((2.0 * t_6) * t_5) * ((t_4 + t_3) + sqrt((((t_4 - t_3) ^ 2.0) + ((((((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale) ^ 2.0)))))) / t_6;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[Power[N[(a * t$95$2), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[Power[N[(a * t$95$1), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$2), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]}, Block[{t$95$5 = N[(N[(b * a), $MachinePrecision] * N[(b * (-a)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(4.0 * t$95$5), $MachinePrecision] / N[Power[N[(x$45$scale * y$45$scale), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, N[((-N[Sqrt[N[(N[(N[(2.0 * t$95$6), $MachinePrecision] * t$95$5), $MachinePrecision] * N[(N[(t$95$4 + t$95$3), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(t$95$4 - t$95$3), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(N[(N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision] / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$6), $MachinePrecision]]]]]]]]

Initial Program: 2.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ t_1 := \sin t\_0\\ t_2 := \cos t\_0\\ t_3 := \frac{\frac{{\left(a \cdot t\_2\right)}^{2} + {\left(b \cdot t\_1\right)}^{2}}{y-scale}}{y-scale}\\ t_4 := \frac{\frac{{\left(a \cdot t\_1\right)}^{2} + {\left(b \cdot t\_2\right)}^{2}}{x-scale}}{x-scale}\\ t_5 := \left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\\ t_6 := \frac{4 \cdot t\_5}{{\left(x-scale \cdot y-scale\right)}^{2}}\\ \frac{-\sqrt{\left(\left(2 \cdot t\_6\right) \cdot t\_5\right) \cdot \left(\left(t\_4 + t\_3\right) + \sqrt{{\left(t\_4 - t\_3\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot t\_1\right) \cdot t\_2}{x-scale}}{y-scale}\right)}^{2}}\right)}}{t\_6} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (/ angle 180.0) PI))
        (t_1 (sin t_0))
        (t_2 (cos t_0))
        (t_3
         (/ (/ (+ (pow (* a t_2) 2.0) (pow (* b t_1) 2.0)) y-scale) y-scale))
        (t_4
         (/ (/ (+ (pow (* a t_1) 2.0) (pow (* b t_2) 2.0)) x-scale) x-scale))
        (t_5 (* (* b a) (* b (- a))))
        (t_6 (/ (* 4.0 t_5) (pow (* x-scale y-scale) 2.0))))
   (/
    (-
     (sqrt
      (*
       (* (* 2.0 t_6) t_5)
       (+
        (+ t_4 t_3)
        (sqrt
         (+
          (pow (- t_4 t_3) 2.0)
          (pow
           (/
            (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) t_1) t_2) x-scale)
            y-scale)
           2.0)))))))
    t_6)))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * ((double) M_PI);
	double t_1 = sin(t_0);
	double t_2 = cos(t_0);
	double t_3 = ((pow((a * t_2), 2.0) + pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale;
	double t_4 = ((pow((a * t_1), 2.0) + pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale;
	double t_5 = (b * a) * (b * -a);
	double t_6 = (4.0 * t_5) / pow((x_45_scale * y_45_scale), 2.0);
	return -sqrt((((2.0 * t_6) * t_5) * ((t_4 + t_3) + sqrt((pow((t_4 - t_3), 2.0) + pow((((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale), 2.0)))))) / t_6;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * Math.PI;
	double t_1 = Math.sin(t_0);
	double t_2 = Math.cos(t_0);
	double t_3 = ((Math.pow((a * t_2), 2.0) + Math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale;
	double t_4 = ((Math.pow((a * t_1), 2.0) + Math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale;
	double t_5 = (b * a) * (b * -a);
	double t_6 = (4.0 * t_5) / Math.pow((x_45_scale * y_45_scale), 2.0);
	return -Math.sqrt((((2.0 * t_6) * t_5) * ((t_4 + t_3) + Math.sqrt((Math.pow((t_4 - t_3), 2.0) + Math.pow((((((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale), 2.0)))))) / t_6;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (angle / 180.0) * math.pi
	t_1 = math.sin(t_0)
	t_2 = math.cos(t_0)
	t_3 = ((math.pow((a * t_2), 2.0) + math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale
	t_4 = ((math.pow((a * t_1), 2.0) + math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale
	t_5 = (b * a) * (b * -a)
	t_6 = (4.0 * t_5) / math.pow((x_45_scale * y_45_scale), 2.0)
	return -math.sqrt((((2.0 * t_6) * t_5) * ((t_4 + t_3) + math.sqrt((math.pow((t_4 - t_3), 2.0) + math.pow((((((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale), 2.0)))))) / t_6

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(angle / 180.0) * pi)
	t_1 = sin(t_0)
	t_2 = cos(t_0)
	t_3 = Float64(Float64(Float64((Float64(a * t_2) ^ 2.0) + (Float64(b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale)
	t_4 = Float64(Float64(Float64((Float64(a * t_1) ^ 2.0) + (Float64(b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale)
	t_5 = Float64(Float64(b * a) * Float64(b * Float64(-a)))
	t_6 = Float64(Float64(4.0 * t_5) / (Float64(x_45_scale * y_45_scale) ^ 2.0))
	return Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * t_6) * t_5) * Float64(Float64(t_4 + t_3) + sqrt(Float64((Float64(t_4 - t_3) ^ 2.0) + (Float64(Float64(Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale) ^ 2.0))))))) / t_6)
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (angle / 180.0) * pi;
	t_1 = sin(t_0);
	t_2 = cos(t_0);
	t_3 = ((((a * t_2) ^ 2.0) + ((b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale;
	t_4 = ((((a * t_1) ^ 2.0) + ((b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale;
	t_5 = (b * a) * (b * -a);
	t_6 = (4.0 * t_5) / ((x_45_scale * y_45_scale) ^ 2.0);
	tmp = -sqrt((((2.0 * t_6) * t_5) * ((t_4 + t_3) + sqrt((((t_4 - t_3) ^ 2.0) + ((((((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale) ^ 2.0)))))) / t_6;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[Power[N[(a * t$95$2), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[Power[N[(a * t$95$1), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$2), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]}, Block[{t$95$5 = N[(N[(b * a), $MachinePrecision] * N[(b * (-a)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(4.0 * t$95$5), $MachinePrecision] / N[Power[N[(x$45$scale * y$45$scale), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, N[((-N[Sqrt[N[(N[(N[(2.0 * t$95$6), $MachinePrecision] * t$95$5), $MachinePrecision] * N[(N[(t$95$4 + t$95$3), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(t$95$4 - t$95$3), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(N[(N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision] / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$6), $MachinePrecision]]]]]]]]

Alternative 1: 20.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := b \cdot \left|a\right|\\ t_1 := \sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right) \cdot \left|a\right|\\ t_2 := \frac{\mathsf{fma}\left(t\_1, t\_1, \left(\mathsf{fma}\left(\cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \left(\pi + \pi\right)\right), 0.5, 0.5\right) \cdot b\right) \cdot b\right)}{\left|x-scale\right| \cdot \left|x-scale\right|}\\ t_3 := \left|x-scale\right| \cdot \left|y-scale\right|\\ t_4 := {\left(\left|x-scale\right|\right)}^{2}\\ t_5 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_6 := \cos t\_5\\ t_7 := \sin t\_5\\ t_8 := \cos \left(\left(\left(\pi + \pi\right) \cdot angle\right) \cdot 0.005555555555555556\right)\\ t_9 := \left|y-scale\right| \cdot \left|x-scale\right|\\ t_10 := {t\_6}^{2}\\ t_11 := \left(t\_0 \cdot b\right) \cdot \left(-\left|a\right|\right)\\ t_12 := {t\_7}^{2}\\ t_13 := \frac{t\_10 \cdot t\_12}{t\_4}\\ t_14 := {\left(\left|y-scale\right|\right)}^{2}\\ t_15 := \sqrt{\frac{{t\_7}^{4}}{{\left(\left|x-scale\right|\right)}^{4}}}\\ t_16 := \sqrt{8 \cdot \frac{t\_15 + \frac{t\_12}{t\_4}}{t\_4}}\\ t_17 := \frac{\frac{\mathsf{fma}\left(\left|a\right| \cdot \left|a\right|, \mathsf{fma}\left(0.5, t\_8, 0.5\right), \left(\left(0.5 - t\_8 \cdot 0.5\right) \cdot b\right) \cdot b\right)}{\left|y-scale\right|}}{\left|y-scale\right|}\\ \mathbf{if}\;\left|y-scale\right| \leq 1.55 \cdot 10^{-162}:\\ \;\;\;\;-0.25 \cdot \left(\left|a\right| \cdot \left(t\_4 \cdot \left(-1 \cdot \left(\left|y-scale\right| \cdot \left(t\_16 + 4 \cdot \frac{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-2, t\_13, 4 \cdot t\_13\right)}{t\_4 \cdot t\_15}, \frac{t\_10}{t\_4}\right)}{t\_14 \cdot t\_16}\right)\right)\right)\right)\right)\\ \mathbf{elif}\;\left|y-scale\right| \leq 7.8 \cdot 10^{+76}:\\ \;\;\;\;-0.25 \cdot \left(\left|a\right| \cdot \left(-1 \cdot \left(\left|x-scale\right| \cdot \left(t\_14 \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{{t\_6}^{4}}{{\left(\left|y-scale\right|\right)}^{4}}} + \frac{t\_10}{t\_14}}{t\_14}}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\frac{\sqrt{\left(t\_11 \cdot 8\right) \cdot \left(t\_11 \cdot \left(\left(\mathsf{hypot}\left(t\_17 - t\_2, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - \left|a\right|\right) \cdot \left(b + \left|a\right|\right)\right)}{t\_3}\right) + t\_2\right) + t\_17\right)\right)}}{\left|t\_3\right|}}{4 \cdot t\_0}}{t\_0} \cdot t\_9\right) \cdot t\_9\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* b (fabs a)))
        (t_1 (* (sin (* (* angle PI) 0.005555555555555556)) (fabs a)))
        (t_2
         (/
          (fma
           t_1
           t_1
           (*
            (*
             (fma (cos (* (* 0.005555555555555556 angle) (+ PI PI))) 0.5 0.5)
             b)
            b))
          (* (fabs x-scale) (fabs x-scale))))
        (t_3 (* (fabs x-scale) (fabs y-scale)))
        (t_4 (pow (fabs x-scale) 2.0))
        (t_5 (* 0.005555555555555556 (* angle PI)))
        (t_6 (cos t_5))
        (t_7 (sin t_5))
        (t_8 (cos (* (* (+ PI PI) angle) 0.005555555555555556)))
        (t_9 (* (fabs y-scale) (fabs x-scale)))
        (t_10 (pow t_6 2.0))
        (t_11 (* (* t_0 b) (- (fabs a))))
        (t_12 (pow t_7 2.0))
        (t_13 (/ (* t_10 t_12) t_4))
        (t_14 (pow (fabs y-scale) 2.0))
        (t_15 (sqrt (/ (pow t_7 4.0) (pow (fabs x-scale) 4.0))))
        (t_16 (sqrt (* 8.0 (/ (+ t_15 (/ t_12 t_4)) t_4))))
        (t_17
         (/
          (/
           (fma
            (* (fabs a) (fabs a))
            (fma 0.5 t_8 0.5)
            (* (* (- 0.5 (* t_8 0.5)) b) b))
           (fabs y-scale))
          (fabs y-scale))))
   (if (<= (fabs y-scale) 1.55e-162)
     (*
      -0.25
      (*
       (fabs a)
       (*
        t_4
        (*
         -1.0
         (*
          (fabs y-scale)
          (+
           t_16
           (*
            4.0
            (/
             (fma
              0.5
              (/ (fma -2.0 t_13 (* 4.0 t_13)) (* t_4 t_15))
              (/ t_10 t_4))
             (* t_14 t_16)))))))))
     (if (<= (fabs y-scale) 7.8e+76)
       (*
        -0.25
        (*
         (fabs a)
         (*
          -1.0
          (*
           (fabs x-scale)
           (*
            t_14
            (sqrt
             (*
              8.0
              (/
               (+
                (sqrt (/ (pow t_6 4.0) (pow (fabs y-scale) 4.0)))
                (/ t_10 t_14))
               t_14))))))))
       (*
        (*
         (/
          (/
           (/
            (sqrt
             (*
              (* t_11 8.0)
              (*
               t_11
               (+
                (+
                 (hypot
                  (- t_17 t_2)
                  (/
                   (*
                    (sin (* (* 2.0 PI) (* angle 0.005555555555555556)))
                    (* (- b (fabs a)) (+ b (fabs a))))
                   t_3))
                 t_2)
                t_17))))
            (fabs t_3))
           (* 4.0 t_0))
          t_0)
         t_9)
        t_9)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = b * fabs(a);
	double t_1 = sin(((angle * ((double) M_PI)) * 0.005555555555555556)) * fabs(a);
	double t_2 = fma(t_1, t_1, ((fma(cos(((0.005555555555555556 * angle) * (((double) M_PI) + ((double) M_PI)))), 0.5, 0.5) * b) * b)) / (fabs(x_45_scale) * fabs(x_45_scale));
	double t_3 = fabs(x_45_scale) * fabs(y_45_scale);
	double t_4 = pow(fabs(x_45_scale), 2.0);
	double t_5 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_6 = cos(t_5);
	double t_7 = sin(t_5);
	double t_8 = cos((((((double) M_PI) + ((double) M_PI)) * angle) * 0.005555555555555556));
	double t_9 = fabs(y_45_scale) * fabs(x_45_scale);
	double t_10 = pow(t_6, 2.0);
	double t_11 = (t_0 * b) * -fabs(a);
	double t_12 = pow(t_7, 2.0);
	double t_13 = (t_10 * t_12) / t_4;
	double t_14 = pow(fabs(y_45_scale), 2.0);
	double t_15 = sqrt((pow(t_7, 4.0) / pow(fabs(x_45_scale), 4.0)));
	double t_16 = sqrt((8.0 * ((t_15 + (t_12 / t_4)) / t_4)));
	double t_17 = (fma((fabs(a) * fabs(a)), fma(0.5, t_8, 0.5), (((0.5 - (t_8 * 0.5)) * b) * b)) / fabs(y_45_scale)) / fabs(y_45_scale);
	double tmp;
	if (fabs(y_45_scale) <= 1.55e-162) {
		tmp = -0.25 * (fabs(a) * (t_4 * (-1.0 * (fabs(y_45_scale) * (t_16 + (4.0 * (fma(0.5, (fma(-2.0, t_13, (4.0 * t_13)) / (t_4 * t_15)), (t_10 / t_4)) / (t_14 * t_16))))))));
	} else if (fabs(y_45_scale) <= 7.8e+76) {
		tmp = -0.25 * (fabs(a) * (-1.0 * (fabs(x_45_scale) * (t_14 * sqrt((8.0 * ((sqrt((pow(t_6, 4.0) / pow(fabs(y_45_scale), 4.0))) + (t_10 / t_14)) / t_14)))))));
	} else {
		tmp = ((((sqrt(((t_11 * 8.0) * (t_11 * ((hypot((t_17 - t_2), ((sin(((2.0 * ((double) M_PI)) * (angle * 0.005555555555555556))) * ((b - fabs(a)) * (b + fabs(a)))) / t_3)) + t_2) + t_17)))) / fabs(t_3)) / (4.0 * t_0)) / t_0) * t_9) * t_9;
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(b * abs(a))
	t_1 = Float64(sin(Float64(Float64(angle * pi) * 0.005555555555555556)) * abs(a))
	t_2 = Float64(fma(t_1, t_1, Float64(Float64(fma(cos(Float64(Float64(0.005555555555555556 * angle) * Float64(pi + pi))), 0.5, 0.5) * b) * b)) / Float64(abs(x_45_scale) * abs(x_45_scale)))
	t_3 = Float64(abs(x_45_scale) * abs(y_45_scale))
	t_4 = abs(x_45_scale) ^ 2.0
	t_5 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_6 = cos(t_5)
	t_7 = sin(t_5)
	t_8 = cos(Float64(Float64(Float64(pi + pi) * angle) * 0.005555555555555556))
	t_9 = Float64(abs(y_45_scale) * abs(x_45_scale))
	t_10 = t_6 ^ 2.0
	t_11 = Float64(Float64(t_0 * b) * Float64(-abs(a)))
	t_12 = t_7 ^ 2.0
	t_13 = Float64(Float64(t_10 * t_12) / t_4)
	t_14 = abs(y_45_scale) ^ 2.0
	t_15 = sqrt(Float64((t_7 ^ 4.0) / (abs(x_45_scale) ^ 4.0)))
	t_16 = sqrt(Float64(8.0 * Float64(Float64(t_15 + Float64(t_12 / t_4)) / t_4)))
	t_17 = Float64(Float64(fma(Float64(abs(a) * abs(a)), fma(0.5, t_8, 0.5), Float64(Float64(Float64(0.5 - Float64(t_8 * 0.5)) * b) * b)) / abs(y_45_scale)) / abs(y_45_scale))
	tmp = 0.0
	if (abs(y_45_scale) <= 1.55e-162)
		tmp = Float64(-0.25 * Float64(abs(a) * Float64(t_4 * Float64(-1.0 * Float64(abs(y_45_scale) * Float64(t_16 + Float64(4.0 * Float64(fma(0.5, Float64(fma(-2.0, t_13, Float64(4.0 * t_13)) / Float64(t_4 * t_15)), Float64(t_10 / t_4)) / Float64(t_14 * t_16)))))))));
	elseif (abs(y_45_scale) <= 7.8e+76)
		tmp = Float64(-0.25 * Float64(abs(a) * Float64(-1.0 * Float64(abs(x_45_scale) * Float64(t_14 * sqrt(Float64(8.0 * Float64(Float64(sqrt(Float64((t_6 ^ 4.0) / (abs(y_45_scale) ^ 4.0))) + Float64(t_10 / t_14)) / t_14))))))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(sqrt(Float64(Float64(t_11 * 8.0) * Float64(t_11 * Float64(Float64(hypot(Float64(t_17 - t_2), Float64(Float64(sin(Float64(Float64(2.0 * pi) * Float64(angle * 0.005555555555555556))) * Float64(Float64(b - abs(a)) * Float64(b + abs(a)))) / t_3)) + t_2) + t_17)))) / abs(t_3)) / Float64(4.0 * t_0)) / t_0) * t_9) * t_9);
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(b * N[Abs[a], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[N[(N[(angle * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * t$95$1 + N[(N[(N[(N[Cos[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * N[(Pi + Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / N[(N[Abs[x$45$scale], $MachinePrecision] * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Abs[x$45$scale], $MachinePrecision] * N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Power[N[Abs[x$45$scale], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$5 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Cos[t$95$5], $MachinePrecision]}, Block[{t$95$7 = N[Sin[t$95$5], $MachinePrecision]}, Block[{t$95$8 = N[Cos[N[(N[(N[(Pi + Pi), $MachinePrecision] * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$9 = N[(N[Abs[y$45$scale], $MachinePrecision] * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$10 = N[Power[t$95$6, 2.0], $MachinePrecision]}, Block[{t$95$11 = N[(N[(t$95$0 * b), $MachinePrecision] * (-N[Abs[a], $MachinePrecision])), $MachinePrecision]}, Block[{t$95$12 = N[Power[t$95$7, 2.0], $MachinePrecision]}, Block[{t$95$13 = N[(N[(t$95$10 * t$95$12), $MachinePrecision] / t$95$4), $MachinePrecision]}, Block[{t$95$14 = N[Power[N[Abs[y$45$scale], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$15 = N[Sqrt[N[(N[Power[t$95$7, 4.0], $MachinePrecision] / N[Power[N[Abs[x$45$scale], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$16 = N[Sqrt[N[(8.0 * N[(N[(t$95$15 + N[(t$95$12 / t$95$4), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$17 = N[(N[(N[(N[(N[Abs[a], $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision] * N[(0.5 * t$95$8 + 0.5), $MachinePrecision] + N[(N[(N[(0.5 - N[(t$95$8 * 0.5), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision] / N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[y$45$scale], $MachinePrecision], 1.55e-162], N[(-0.25 * N[(N[Abs[a], $MachinePrecision] * N[(t$95$4 * N[(-1.0 * N[(N[Abs[y$45$scale], $MachinePrecision] * N[(t$95$16 + N[(4.0 * N[(N[(0.5 * N[(N[(-2.0 * t$95$13 + N[(4.0 * t$95$13), $MachinePrecision]), $MachinePrecision] / N[(t$95$4 * t$95$15), $MachinePrecision]), $MachinePrecision] + N[(t$95$10 / t$95$4), $MachinePrecision]), $MachinePrecision] / N[(t$95$14 * t$95$16), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[y$45$scale], $MachinePrecision], 7.8e+76], N[(-0.25 * N[(N[Abs[a], $MachinePrecision] * N[(-1.0 * N[(N[Abs[x$45$scale], $MachinePrecision] * N[(t$95$14 * N[Sqrt[N[(8.0 * N[(N[(N[Sqrt[N[(N[Power[t$95$6, 4.0], $MachinePrecision] / N[Power[N[Abs[y$45$scale], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(t$95$10 / t$95$14), $MachinePrecision]), $MachinePrecision] / t$95$14), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[Sqrt[N[(N[(t$95$11 * 8.0), $MachinePrecision] * N[(t$95$11 * N[(N[(N[Sqrt[N[(t$95$17 - t$95$2), $MachinePrecision] ^ 2 + N[(N[(N[Sin[N[(N[(2.0 * Pi), $MachinePrecision] * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(b - N[Abs[a], $MachinePrecision]), $MachinePrecision] * N[(b + N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision] ^ 2], $MachinePrecision] + t$95$2), $MachinePrecision] + t$95$17), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[t$95$3], $MachinePrecision]), $MachinePrecision] / N[(4.0 * t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] * t$95$9), $MachinePrecision] * t$95$9), $MachinePrecision]]]]]]]]]]]]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y-scale < 1.5499999999999999e-162

   1. Initial program 2.8%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   2. Taylor expanded in a around -inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \frac{a \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}{{b}^{2}}} \]

   4. Applied rewrites 0.7%

      \[\leadsto \color{blue}{-0.25 \cdot \frac{a \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}{{b}^{2}}} \]

   5. Taylor expanded in b around 0

      \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \color{blue}{\left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}\right) \]

   7. Applied rewrites 2.5%

      \[\leadsto -0.25 \cdot \left(a \cdot \color{blue}{\left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}\right) \]

   8. Taylor expanded in y-scale around -inf

      \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(-1 \cdot \left(y-scale \cdot \color{blue}{\left(\sqrt{8 \cdot \frac{\sqrt{\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{x-scale}^{4}}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}} + 4 \cdot \frac{\frac{1}{2} \cdot \frac{-2 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} + 4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}}{{x-scale}^{2} \cdot \sqrt{\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{x-scale}^{4}}}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}}{{y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{x-scale}^{4}}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}}\right)}\right)\right)\right)\right) \]

   10. Applied rewrites 4.1%

       \[\leadsto -0.25 \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(-1 \cdot \left(y-scale \cdot \color{blue}{\left(\sqrt{8 \cdot \frac{\sqrt{\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}{{x-scale}^{4}}} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}} + 4 \cdot \frac{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-2, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}, 4 \cdot \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)}{{x-scale}^{2} \cdot \sqrt{\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}{{x-scale}^{4}}}}, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)}{{y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}{{x-scale}^{4}}} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}}\right)}\right)\right)\right)\right) \]

   if 1.5499999999999999e-162 < y-scale < 7.79999999999999979e76

   1. Initial program 2.8%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   2. Taylor expanded in a around -inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \frac{a \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}{{b}^{2}}} \]

   4. Applied rewrites 0.7%

      \[\leadsto \color{blue}{-0.25 \cdot \frac{a \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}{{b}^{2}}} \]

   5. Taylor expanded in b around 0

      \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \color{blue}{\left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}\right) \]

   7. Applied rewrites 2.5%

      \[\leadsto -0.25 \cdot \left(a \cdot \color{blue}{\left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}\right) \]

   8. Taylor expanded in x-scale around -inf

      \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \left(-1 \cdot \left(x-scale \cdot \color{blue}{\left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{y-scale}^{4}}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}}{{y-scale}^{2}}}\right)}\right)\right)\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \left(-1 \cdot \left(x-scale \cdot \left({y-scale}^{2} \cdot \color{blue}{\sqrt{8 \cdot \frac{\sqrt{\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{y-scale}^{4}}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}}{{y-scale}^{2}}}}\right)\right)\right)\right) \]

   10. Applied rewrites 5.7%

       \[\leadsto -0.25 \cdot \left(a \cdot \left(-1 \cdot \left(x-scale \cdot \color{blue}{\left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}{{y-scale}^{4}}} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}}{{y-scale}^{2}}}\right)}\right)\right)\right) \]

   if 7.79999999999999979e76 < y-scale

   1. Initial program 2.8%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   3. Applied rewrites 6.3%

      \[\leadsto \color{blue}{\left(\frac{\frac{\sqrt{\left(8 \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \left(\left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right) \cdot \left(\left(\mathsf{hypot}\left(\frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}{y-scale \cdot x-scale}, \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), a \cdot a, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale} - \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), b \cdot b, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(a \cdot a\right)\right)}{y-scale \cdot y-scale}\right) + \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), a \cdot a, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}\right) + \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), b \cdot b, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(a \cdot a\right)\right)}{y-scale \cdot y-scale}\right)\right)}}{\left|y-scale \cdot x-scale\right|}}{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(a \cdot b\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right)} \]

   5. Applied rewrites 12.4%

      \[\leadsto \left(\color{blue}{\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot \left(\left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right) \cdot a, a, \left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b\right) \cdot b\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(\mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right) \cdot b\right) \cdot b\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(\mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right) \cdot b\right) \cdot b\right)}{x-scale \cdot x-scale}\right) + \frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right) \cdot a, a, \left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b\right) \cdot b\right)}{y-scale \cdot y-scale}\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a}} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   7. Applied rewrites 12.4%

      \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot \left(\left(\mathsf{hypot}\left(\color{blue}{\frac{\frac{\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(0.5, \cos \left(\left(\left(\pi + \pi\right) \cdot angle\right) \cdot 0.005555555555555556\right), 0.5\right), \left(\left(0.5 - \cos \left(\left(\left(\pi + \pi\right) \cdot angle\right) \cdot 0.005555555555555556\right) \cdot 0.5\right) \cdot b\right) \cdot b\right)}{y-scale}}{y-scale}} - \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(\mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right) \cdot b\right) \cdot b\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(\mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right) \cdot b\right) \cdot b\right)}{x-scale \cdot x-scale}\right) + \frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right) \cdot a, a, \left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b\right) \cdot b\right)}{y-scale \cdot y-scale}\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   9. Applied rewrites 15.2%

      \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot \left(\left(\mathsf{hypot}\left(\frac{\frac{\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(0.5, \cos \left(\left(\left(\pi + \pi\right) \cdot angle\right) \cdot 0.005555555555555556\right), 0.5\right), \left(\left(0.5 - \cos \left(\left(\left(\pi + \pi\right) \cdot angle\right) \cdot 0.005555555555555556\right) \cdot 0.5\right) \cdot b\right) \cdot b\right)}{y-scale}}{y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(\mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right) \cdot b\right) \cdot b\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(\mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right) \cdot b\right) \cdot b\right)}{x-scale \cdot x-scale}\right) + \color{blue}{\frac{\frac{\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(0.5, \cos \left(\left(\left(\pi + \pi\right) \cdot angle\right) \cdot 0.005555555555555556\right), 0.5\right), \left(\left(0.5 - \cos \left(\left(\left(\pi + \pi\right) \cdot angle\right) \cdot 0.005555555555555556\right) \cdot 0.5\right) \cdot b\right) \cdot b\right)}{y-scale}}{y-scale}}\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   11. Applied rewrites 15.3%

       \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot \left(\left(\mathsf{hypot}\left(\frac{\frac{\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(0.5, \cos \left(\left(\left(\pi + \pi\right) \cdot angle\right) \cdot 0.005555555555555556\right), 0.5\right), \left(\left(0.5 - \cos \left(\left(\left(\pi + \pi\right) \cdot angle\right) \cdot 0.005555555555555556\right) \cdot 0.5\right) \cdot b\right) \cdot b\right)}{y-scale}}{y-scale} - \frac{\color{blue}{\mathsf{fma}\left(\sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right) \cdot a, \sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right) \cdot a, \left(\mathsf{fma}\left(\cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \left(\pi + \pi\right)\right), 0.5, 0.5\right) \cdot b\right) \cdot b\right)}}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(\mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right) \cdot b\right) \cdot b\right)}{x-scale \cdot x-scale}\right) + \frac{\frac{\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(0.5, \cos \left(\left(\left(\pi + \pi\right) \cdot angle\right) \cdot 0.005555555555555556\right), 0.5\right), \left(\left(0.5 - \cos \left(\left(\left(\pi + \pi\right) \cdot angle\right) \cdot 0.005555555555555556\right) \cdot 0.5\right) \cdot b\right) \cdot b\right)}{y-scale}}{y-scale}\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   13. Applied rewrites 15.3%

       \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot \left(\left(\mathsf{hypot}\left(\frac{\frac{\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(0.5, \cos \left(\left(\left(\pi + \pi\right) \cdot angle\right) \cdot 0.005555555555555556\right), 0.5\right), \left(\left(0.5 - \cos \left(\left(\left(\pi + \pi\right) \cdot angle\right) \cdot 0.005555555555555556\right) \cdot 0.5\right) \cdot b\right) \cdot b\right)}{y-scale}}{y-scale} - \frac{\mathsf{fma}\left(\sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right) \cdot a, \sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right) \cdot a, \left(\mathsf{fma}\left(\cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \left(\pi + \pi\right)\right), 0.5, 0.5\right) \cdot b\right) \cdot b\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \frac{\color{blue}{\mathsf{fma}\left(\sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right) \cdot a, \sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right) \cdot a, \left(\mathsf{fma}\left(\cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \left(\pi + \pi\right)\right), 0.5, 0.5\right) \cdot b\right) \cdot b\right)}}{x-scale \cdot x-scale}\right) + \frac{\frac{\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(0.5, \cos \left(\left(\left(\pi + \pi\right) \cdot angle\right) \cdot 0.005555555555555556\right), 0.5\right), \left(\left(0.5 - \cos \left(\left(\left(\pi + \pi\right) \cdot angle\right) \cdot 0.005555555555555556\right) \cdot 0.5\right) \cdot b\right) \cdot b\right)}{y-scale}}{y-scale}\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 18.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := b \cdot \left|a\right|\\ t_1 := {\left(\left|y-scale\right|\right)}^{2}\\ t_2 := \left|x-scale\right| \cdot \left|y-scale\right|\\ t_3 := \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ t_4 := \left(t\_0 \cdot b\right) \cdot \left(-\left|a\right|\right)\\ t_5 := \cos \left(\left(\left(\pi + \pi\right) \cdot angle\right) \cdot 0.005555555555555556\right)\\ t_6 := \left|y-scale\right| \cdot \left|x-scale\right|\\ t_7 := \frac{\frac{\mathsf{fma}\left(\left|a\right| \cdot \left|a\right|, \mathsf{fma}\left(0.5, t\_5, 0.5\right), \left(\left(0.5 - t\_5 \cdot 0.5\right) \cdot b\right) \cdot b\right)}{\left|y-scale\right|}}{\left|y-scale\right|}\\ t_8 := \sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right) \cdot \left|a\right|\\ t_9 := \frac{\mathsf{fma}\left(t\_8, t\_8, \left(\mathsf{fma}\left(\cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \left(\pi + \pi\right)\right), 0.5, 0.5\right) \cdot b\right) \cdot b\right)}{\left|x-scale\right| \cdot \left|x-scale\right|}\\ \mathbf{if}\;\left|y-scale\right| \leq 7.8 \cdot 10^{+76}:\\ \;\;\;\;-0.25 \cdot \left(\left|a\right| \cdot \left(-1 \cdot \left(\left|x-scale\right| \cdot \left(t\_1 \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{{t\_3}^{4}}{{\left(\left|y-scale\right|\right)}^{4}}} + \frac{{t\_3}^{2}}{t\_1}}{t\_1}}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\frac{\sqrt{\left(t\_4 \cdot 8\right) \cdot \left(t\_4 \cdot \left(\left(\mathsf{hypot}\left(t\_7 - t\_9, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - \left|a\right|\right) \cdot \left(b + \left|a\right|\right)\right)}{t\_2}\right) + t\_9\right) + t\_7\right)\right)}}{\left|t\_2\right|}}{4 \cdot t\_0}}{t\_0} \cdot t\_6\right) \cdot t\_6\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* b (fabs a)))
        (t_1 (pow (fabs y-scale) 2.0))
        (t_2 (* (fabs x-scale) (fabs y-scale)))
        (t_3 (cos (* 0.005555555555555556 (* angle PI))))
        (t_4 (* (* t_0 b) (- (fabs a))))
        (t_5 (cos (* (* (+ PI PI) angle) 0.005555555555555556)))
        (t_6 (* (fabs y-scale) (fabs x-scale)))
        (t_7
         (/
          (/
           (fma
            (* (fabs a) (fabs a))
            (fma 0.5 t_5 0.5)
            (* (* (- 0.5 (* t_5 0.5)) b) b))
           (fabs y-scale))
          (fabs y-scale)))
        (t_8 (* (sin (* (* angle PI) 0.005555555555555556)) (fabs a)))
        (t_9
         (/
          (fma
           t_8
           t_8
           (*
            (*
             (fma (cos (* (* 0.005555555555555556 angle) (+ PI PI))) 0.5 0.5)
             b)
            b))
          (* (fabs x-scale) (fabs x-scale)))))
   (if (<= (fabs y-scale) 7.8e+76)
     (*
      -0.25
      (*
       (fabs a)
       (*
        -1.0
        (*
         (fabs x-scale)
         (*
          t_1
          (sqrt
           (*
            8.0
            (/
             (+
              (sqrt (/ (pow t_3 4.0) (pow (fabs y-scale) 4.0)))
              (/ (pow t_3 2.0) t_1))
             t_1))))))))
     (*
      (*
       (/
        (/
         (/
          (sqrt
           (*
            (* t_4 8.0)
            (*
             t_4
             (+
              (+
               (hypot
                (- t_7 t_9)
                (/
                 (*
                  (sin (* (* 2.0 PI) (* angle 0.005555555555555556)))
                  (* (- b (fabs a)) (+ b (fabs a))))
                 t_2))
               t_9)
              t_7))))
          (fabs t_2))
         (* 4.0 t_0))
        t_0)
       t_6)
      t_6))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = b * fabs(a);
	double t_1 = pow(fabs(y_45_scale), 2.0);
	double t_2 = fabs(x_45_scale) * fabs(y_45_scale);
	double t_3 = cos((0.005555555555555556 * (angle * ((double) M_PI))));
	double t_4 = (t_0 * b) * -fabs(a);
	double t_5 = cos((((((double) M_PI) + ((double) M_PI)) * angle) * 0.005555555555555556));
	double t_6 = fabs(y_45_scale) * fabs(x_45_scale);
	double t_7 = (fma((fabs(a) * fabs(a)), fma(0.5, t_5, 0.5), (((0.5 - (t_5 * 0.5)) * b) * b)) / fabs(y_45_scale)) / fabs(y_45_scale);
	double t_8 = sin(((angle * ((double) M_PI)) * 0.005555555555555556)) * fabs(a);
	double t_9 = fma(t_8, t_8, ((fma(cos(((0.005555555555555556 * angle) * (((double) M_PI) + ((double) M_PI)))), 0.5, 0.5) * b) * b)) / (fabs(x_45_scale) * fabs(x_45_scale));
	double tmp;
	if (fabs(y_45_scale) <= 7.8e+76) {
		tmp = -0.25 * (fabs(a) * (-1.0 * (fabs(x_45_scale) * (t_1 * sqrt((8.0 * ((sqrt((pow(t_3, 4.0) / pow(fabs(y_45_scale), 4.0))) + (pow(t_3, 2.0) / t_1)) / t_1)))))));
	} else {
		tmp = ((((sqrt(((t_4 * 8.0) * (t_4 * ((hypot((t_7 - t_9), ((sin(((2.0 * ((double) M_PI)) * (angle * 0.005555555555555556))) * ((b - fabs(a)) * (b + fabs(a)))) / t_2)) + t_9) + t_7)))) / fabs(t_2)) / (4.0 * t_0)) / t_0) * t_6) * t_6;
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(b * abs(a))
	t_1 = abs(y_45_scale) ^ 2.0
	t_2 = Float64(abs(x_45_scale) * abs(y_45_scale))
	t_3 = cos(Float64(0.005555555555555556 * Float64(angle * pi)))
	t_4 = Float64(Float64(t_0 * b) * Float64(-abs(a)))
	t_5 = cos(Float64(Float64(Float64(pi + pi) * angle) * 0.005555555555555556))
	t_6 = Float64(abs(y_45_scale) * abs(x_45_scale))
	t_7 = Float64(Float64(fma(Float64(abs(a) * abs(a)), fma(0.5, t_5, 0.5), Float64(Float64(Float64(0.5 - Float64(t_5 * 0.5)) * b) * b)) / abs(y_45_scale)) / abs(y_45_scale))
	t_8 = Float64(sin(Float64(Float64(angle * pi) * 0.005555555555555556)) * abs(a))
	t_9 = Float64(fma(t_8, t_8, Float64(Float64(fma(cos(Float64(Float64(0.005555555555555556 * angle) * Float64(pi + pi))), 0.5, 0.5) * b) * b)) / Float64(abs(x_45_scale) * abs(x_45_scale)))
	tmp = 0.0
	if (abs(y_45_scale) <= 7.8e+76)
		tmp = Float64(-0.25 * Float64(abs(a) * Float64(-1.0 * Float64(abs(x_45_scale) * Float64(t_1 * sqrt(Float64(8.0 * Float64(Float64(sqrt(Float64((t_3 ^ 4.0) / (abs(y_45_scale) ^ 4.0))) + Float64((t_3 ^ 2.0) / t_1)) / t_1))))))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(sqrt(Float64(Float64(t_4 * 8.0) * Float64(t_4 * Float64(Float64(hypot(Float64(t_7 - t_9), Float64(Float64(sin(Float64(Float64(2.0 * pi) * Float64(angle * 0.005555555555555556))) * Float64(Float64(b - abs(a)) * Float64(b + abs(a)))) / t_2)) + t_9) + t_7)))) / abs(t_2)) / Float64(4.0 * t_0)) / t_0) * t_6) * t_6);
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(b * N[Abs[a], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Abs[y$45$scale], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[x$45$scale], $MachinePrecision] * N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(t$95$0 * b), $MachinePrecision] * (-N[Abs[a], $MachinePrecision])), $MachinePrecision]}, Block[{t$95$5 = N[Cos[N[(N[(N[(Pi + Pi), $MachinePrecision] * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[(N[Abs[y$45$scale], $MachinePrecision] * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(N[(N[(N[(N[Abs[a], $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision] * N[(0.5 * t$95$5 + 0.5), $MachinePrecision] + N[(N[(N[(0.5 - N[(t$95$5 * 0.5), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision] / N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$8 = N[(N[Sin[N[(N[(angle * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$9 = N[(N[(t$95$8 * t$95$8 + N[(N[(N[(N[Cos[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * N[(Pi + Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / N[(N[Abs[x$45$scale], $MachinePrecision] * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[y$45$scale], $MachinePrecision], 7.8e+76], N[(-0.25 * N[(N[Abs[a], $MachinePrecision] * N[(-1.0 * N[(N[Abs[x$45$scale], $MachinePrecision] * N[(t$95$1 * N[Sqrt[N[(8.0 * N[(N[(N[Sqrt[N[(N[Power[t$95$3, 4.0], $MachinePrecision] / N[Power[N[Abs[y$45$scale], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[Power[t$95$3, 2.0], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[Sqrt[N[(N[(t$95$4 * 8.0), $MachinePrecision] * N[(t$95$4 * N[(N[(N[Sqrt[N[(t$95$7 - t$95$9), $MachinePrecision] ^ 2 + N[(N[(N[Sin[N[(N[(2.0 * Pi), $MachinePrecision] * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(b - N[Abs[a], $MachinePrecision]), $MachinePrecision] * N[(b + N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] ^ 2], $MachinePrecision] + t$95$9), $MachinePrecision] + t$95$7), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[t$95$2], $MachinePrecision]), $MachinePrecision] / N[(4.0 * t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] * t$95$6), $MachinePrecision] * t$95$6), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 7.79999999999999979e76

   1. Initial program 2.8%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   2. Taylor expanded in a around -inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \frac{a \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}{{b}^{2}}} \]

   4. Applied rewrites 0.7%

      \[\leadsto \color{blue}{-0.25 \cdot \frac{a \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}{{b}^{2}}} \]

   5. Taylor expanded in b around 0

      \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \color{blue}{\left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}\right) \]

   7. Applied rewrites 2.5%

      \[\leadsto -0.25 \cdot \left(a \cdot \color{blue}{\left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}\right) \]

   8. Taylor expanded in x-scale around -inf

      \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \left(-1 \cdot \left(x-scale \cdot \color{blue}{\left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{y-scale}^{4}}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}}{{y-scale}^{2}}}\right)}\right)\right)\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \left(-1 \cdot \left(x-scale \cdot \left({y-scale}^{2} \cdot \color{blue}{\sqrt{8 \cdot \frac{\sqrt{\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{y-scale}^{4}}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}}{{y-scale}^{2}}}}\right)\right)\right)\right) \]

   10. Applied rewrites 5.7%

       \[\leadsto -0.25 \cdot \left(a \cdot \left(-1 \cdot \left(x-scale \cdot \color{blue}{\left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}{{y-scale}^{4}}} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}}{{y-scale}^{2}}}\right)}\right)\right)\right) \]

   if 7.79999999999999979e76 < y-scale

   1. Initial program 2.8%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   3. Applied rewrites 6.3%

      \[\leadsto \color{blue}{\left(\frac{\frac{\sqrt{\left(8 \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \left(\left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right) \cdot \left(\left(\mathsf{hypot}\left(\frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}{y-scale \cdot x-scale}, \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), a \cdot a, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale} - \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), b \cdot b, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(a \cdot a\right)\right)}{y-scale \cdot y-scale}\right) + \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), a \cdot a, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}\right) + \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), b \cdot b, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(a \cdot a\right)\right)}{y-scale \cdot y-scale}\right)\right)}}{\left|y-scale \cdot x-scale\right|}}{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(a \cdot b\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right)} \]

   5. Applied rewrites 12.4%

      \[\leadsto \left(\color{blue}{\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot \left(\left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right) \cdot a, a, \left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b\right) \cdot b\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(\mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right) \cdot b\right) \cdot b\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(\mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right) \cdot b\right) \cdot b\right)}{x-scale \cdot x-scale}\right) + \frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right) \cdot a, a, \left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b\right) \cdot b\right)}{y-scale \cdot y-scale}\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a}} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   7. Applied rewrites 12.4%

      \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot \left(\left(\mathsf{hypot}\left(\color{blue}{\frac{\frac{\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(0.5, \cos \left(\left(\left(\pi + \pi\right) \cdot angle\right) \cdot 0.005555555555555556\right), 0.5\right), \left(\left(0.5 - \cos \left(\left(\left(\pi + \pi\right) \cdot angle\right) \cdot 0.005555555555555556\right) \cdot 0.5\right) \cdot b\right) \cdot b\right)}{y-scale}}{y-scale}} - \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(\mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right) \cdot b\right) \cdot b\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(\mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right) \cdot b\right) \cdot b\right)}{x-scale \cdot x-scale}\right) + \frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right) \cdot a, a, \left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b\right) \cdot b\right)}{y-scale \cdot y-scale}\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   9. Applied rewrites 15.2%

      \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot \left(\left(\mathsf{hypot}\left(\frac{\frac{\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(0.5, \cos \left(\left(\left(\pi + \pi\right) \cdot angle\right) \cdot 0.005555555555555556\right), 0.5\right), \left(\left(0.5 - \cos \left(\left(\left(\pi + \pi\right) \cdot angle\right) \cdot 0.005555555555555556\right) \cdot 0.5\right) \cdot b\right) \cdot b\right)}{y-scale}}{y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(\mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right) \cdot b\right) \cdot b\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(\mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right) \cdot b\right) \cdot b\right)}{x-scale \cdot x-scale}\right) + \color{blue}{\frac{\frac{\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(0.5, \cos \left(\left(\left(\pi + \pi\right) \cdot angle\right) \cdot 0.005555555555555556\right), 0.5\right), \left(\left(0.5 - \cos \left(\left(\left(\pi + \pi\right) \cdot angle\right) \cdot 0.005555555555555556\right) \cdot 0.5\right) \cdot b\right) \cdot b\right)}{y-scale}}{y-scale}}\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   11. Applied rewrites 15.3%

       \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot \left(\left(\mathsf{hypot}\left(\frac{\frac{\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(0.5, \cos \left(\left(\left(\pi + \pi\right) \cdot angle\right) \cdot 0.005555555555555556\right), 0.5\right), \left(\left(0.5 - \cos \left(\left(\left(\pi + \pi\right) \cdot angle\right) \cdot 0.005555555555555556\right) \cdot 0.5\right) \cdot b\right) \cdot b\right)}{y-scale}}{y-scale} - \frac{\color{blue}{\mathsf{fma}\left(\sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right) \cdot a, \sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right) \cdot a, \left(\mathsf{fma}\left(\cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \left(\pi + \pi\right)\right), 0.5, 0.5\right) \cdot b\right) \cdot b\right)}}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(\mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right) \cdot b\right) \cdot b\right)}{x-scale \cdot x-scale}\right) + \frac{\frac{\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(0.5, \cos \left(\left(\left(\pi + \pi\right) \cdot angle\right) \cdot 0.005555555555555556\right), 0.5\right), \left(\left(0.5 - \cos \left(\left(\left(\pi + \pi\right) \cdot angle\right) \cdot 0.005555555555555556\right) \cdot 0.5\right) \cdot b\right) \cdot b\right)}{y-scale}}{y-scale}\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   13. Applied rewrites 15.3%

       \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot \left(\left(\mathsf{hypot}\left(\frac{\frac{\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(0.5, \cos \left(\left(\left(\pi + \pi\right) \cdot angle\right) \cdot 0.005555555555555556\right), 0.5\right), \left(\left(0.5 - \cos \left(\left(\left(\pi + \pi\right) \cdot angle\right) \cdot 0.005555555555555556\right) \cdot 0.5\right) \cdot b\right) \cdot b\right)}{y-scale}}{y-scale} - \frac{\mathsf{fma}\left(\sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right) \cdot a, \sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right) \cdot a, \left(\mathsf{fma}\left(\cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \left(\pi + \pi\right)\right), 0.5, 0.5\right) \cdot b\right) \cdot b\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \frac{\color{blue}{\mathsf{fma}\left(\sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right) \cdot a, \sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right) \cdot a, \left(\mathsf{fma}\left(\cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \left(\pi + \pi\right)\right), 0.5, 0.5\right) \cdot b\right) \cdot b\right)}}{x-scale \cdot x-scale}\right) + \frac{\frac{\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(0.5, \cos \left(\left(\left(\pi + \pi\right) \cdot angle\right) \cdot 0.005555555555555556\right), 0.5\right), \left(\left(0.5 - \cos \left(\left(\left(\pi + \pi\right) \cdot angle\right) \cdot 0.005555555555555556\right) \cdot 0.5\right) \cdot b\right) \cdot b\right)}{y-scale}}{y-scale}\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 18.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} t_0 := \left|x-scale\right| \cdot \left|y-scale\right|\\ t_1 := b \cdot \left|a\right|\\ t_2 := \left(t\_1 \cdot b\right) \cdot \left(-\left|a\right|\right)\\ t_3 := \frac{\frac{{\left(\left|a\right|\right)}^{2}}{\left|y-scale\right|}}{\left|y-scale\right|}\\ t_4 := \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ t_5 := {\left(\left|y-scale\right|\right)}^{2}\\ t_6 := \left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\\ t_7 := \cos t\_6\\ t_8 := \frac{\mathsf{fma}\left(\left(0.5 - t\_7 \cdot 0.5\right) \cdot \left|a\right|, \left|a\right|, \left(\mathsf{fma}\left(t\_7, 0.5, 0.5\right) \cdot b\right) \cdot b\right)}{\left|x-scale\right| \cdot \left|x-scale\right|}\\ t_9 := \left|y-scale\right| \cdot \left|x-scale\right|\\ \mathbf{if}\;\left|y-scale\right| \leq 7.8 \cdot 10^{+76}:\\ \;\;\;\;-0.25 \cdot \left(\left|a\right| \cdot \left(-1 \cdot \left(\left|x-scale\right| \cdot \left(t\_5 \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{{t\_4}^{4}}{{\left(\left|y-scale\right|\right)}^{4}}} + \frac{{t\_4}^{2}}{t\_5}}{t\_5}}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\frac{\sqrt{\left(t\_2 \cdot 8\right) \cdot \left(t\_2 \cdot \left(\left(\mathsf{hypot}\left(t\_3 - t\_8, \frac{\sin t\_6 \cdot \left(\left(b - \left|a\right|\right) \cdot \left(b + \left|a\right|\right)\right)}{t\_0}\right) + t\_8\right) + t\_3\right)\right)}}{\left|t\_0\right|}}{4 \cdot t\_1}}{t\_1} \cdot t\_9\right) \cdot t\_9\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (fabs x-scale) (fabs y-scale)))
        (t_1 (* b (fabs a)))
        (t_2 (* (* t_1 b) (- (fabs a))))
        (t_3 (/ (/ (pow (fabs a) 2.0) (fabs y-scale)) (fabs y-scale)))
        (t_4 (cos (* 0.005555555555555556 (* angle PI))))
        (t_5 (pow (fabs y-scale) 2.0))
        (t_6 (* (* 2.0 PI) (* angle 0.005555555555555556)))
        (t_7 (cos t_6))
        (t_8
         (/
          (fma
           (* (- 0.5 (* t_7 0.5)) (fabs a))
           (fabs a)
           (* (* (fma t_7 0.5 0.5) b) b))
          (* (fabs x-scale) (fabs x-scale))))
        (t_9 (* (fabs y-scale) (fabs x-scale))))
   (if (<= (fabs y-scale) 7.8e+76)
     (*
      -0.25
      (*
       (fabs a)
       (*
        -1.0
        (*
         (fabs x-scale)
         (*
          t_5
          (sqrt
           (*
            8.0
            (/
             (+
              (sqrt (/ (pow t_4 4.0) (pow (fabs y-scale) 4.0)))
              (/ (pow t_4 2.0) t_5))
             t_5))))))))
     (*
      (*
       (/
        (/
         (/
          (sqrt
           (*
            (* t_2 8.0)
            (*
             t_2
             (+
              (+
               (hypot
                (- t_3 t_8)
                (/ (* (sin t_6) (* (- b (fabs a)) (+ b (fabs a)))) t_0))
               t_8)
              t_3))))
          (fabs t_0))
         (* 4.0 t_1))
        t_1)
       t_9)
      t_9))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = fabs(x_45_scale) * fabs(y_45_scale);
	double t_1 = b * fabs(a);
	double t_2 = (t_1 * b) * -fabs(a);
	double t_3 = (pow(fabs(a), 2.0) / fabs(y_45_scale)) / fabs(y_45_scale);
	double t_4 = cos((0.005555555555555556 * (angle * ((double) M_PI))));
	double t_5 = pow(fabs(y_45_scale), 2.0);
	double t_6 = (2.0 * ((double) M_PI)) * (angle * 0.005555555555555556);
	double t_7 = cos(t_6);
	double t_8 = fma(((0.5 - (t_7 * 0.5)) * fabs(a)), fabs(a), ((fma(t_7, 0.5, 0.5) * b) * b)) / (fabs(x_45_scale) * fabs(x_45_scale));
	double t_9 = fabs(y_45_scale) * fabs(x_45_scale);
	double tmp;
	if (fabs(y_45_scale) <= 7.8e+76) {
		tmp = -0.25 * (fabs(a) * (-1.0 * (fabs(x_45_scale) * (t_5 * sqrt((8.0 * ((sqrt((pow(t_4, 4.0) / pow(fabs(y_45_scale), 4.0))) + (pow(t_4, 2.0) / t_5)) / t_5)))))));
	} else {
		tmp = ((((sqrt(((t_2 * 8.0) * (t_2 * ((hypot((t_3 - t_8), ((sin(t_6) * ((b - fabs(a)) * (b + fabs(a)))) / t_0)) + t_8) + t_3)))) / fabs(t_0)) / (4.0 * t_1)) / t_1) * t_9) * t_9;
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(abs(x_45_scale) * abs(y_45_scale))
	t_1 = Float64(b * abs(a))
	t_2 = Float64(Float64(t_1 * b) * Float64(-abs(a)))
	t_3 = Float64(Float64((abs(a) ^ 2.0) / abs(y_45_scale)) / abs(y_45_scale))
	t_4 = cos(Float64(0.005555555555555556 * Float64(angle * pi)))
	t_5 = abs(y_45_scale) ^ 2.0
	t_6 = Float64(Float64(2.0 * pi) * Float64(angle * 0.005555555555555556))
	t_7 = cos(t_6)
	t_8 = Float64(fma(Float64(Float64(0.5 - Float64(t_7 * 0.5)) * abs(a)), abs(a), Float64(Float64(fma(t_7, 0.5, 0.5) * b) * b)) / Float64(abs(x_45_scale) * abs(x_45_scale)))
	t_9 = Float64(abs(y_45_scale) * abs(x_45_scale))
	tmp = 0.0
	if (abs(y_45_scale) <= 7.8e+76)
		tmp = Float64(-0.25 * Float64(abs(a) * Float64(-1.0 * Float64(abs(x_45_scale) * Float64(t_5 * sqrt(Float64(8.0 * Float64(Float64(sqrt(Float64((t_4 ^ 4.0) / (abs(y_45_scale) ^ 4.0))) + Float64((t_4 ^ 2.0) / t_5)) / t_5))))))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(sqrt(Float64(Float64(t_2 * 8.0) * Float64(t_2 * Float64(Float64(hypot(Float64(t_3 - t_8), Float64(Float64(sin(t_6) * Float64(Float64(b - abs(a)) * Float64(b + abs(a)))) / t_0)) + t_8) + t_3)))) / abs(t_0)) / Float64(4.0 * t_1)) / t_1) * t_9) * t_9);
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[Abs[x$45$scale], $MachinePrecision] * N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b * N[Abs[a], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * b), $MachinePrecision] * (-N[Abs[a], $MachinePrecision])), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Power[N[Abs[a], $MachinePrecision], 2.0], $MachinePrecision] / N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision] / N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Cos[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[Power[N[Abs[y$45$scale], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$6 = N[(N[(2.0 * Pi), $MachinePrecision] * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[Cos[t$95$6], $MachinePrecision]}, Block[{t$95$8 = N[(N[(N[(N[(0.5 - N[(t$95$7 * 0.5), $MachinePrecision]), $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision] * N[Abs[a], $MachinePrecision] + N[(N[(N[(t$95$7 * 0.5 + 0.5), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / N[(N[Abs[x$45$scale], $MachinePrecision] * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$9 = N[(N[Abs[y$45$scale], $MachinePrecision] * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[y$45$scale], $MachinePrecision], 7.8e+76], N[(-0.25 * N[(N[Abs[a], $MachinePrecision] * N[(-1.0 * N[(N[Abs[x$45$scale], $MachinePrecision] * N[(t$95$5 * N[Sqrt[N[(8.0 * N[(N[(N[Sqrt[N[(N[Power[t$95$4, 4.0], $MachinePrecision] / N[Power[N[Abs[y$45$scale], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[Power[t$95$4, 2.0], $MachinePrecision] / t$95$5), $MachinePrecision]), $MachinePrecision] / t$95$5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[Sqrt[N[(N[(t$95$2 * 8.0), $MachinePrecision] * N[(t$95$2 * N[(N[(N[Sqrt[N[(t$95$3 - t$95$8), $MachinePrecision] ^ 2 + N[(N[(N[Sin[t$95$6], $MachinePrecision] * N[(N[(b - N[Abs[a], $MachinePrecision]), $MachinePrecision] * N[(b + N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] ^ 2], $MachinePrecision] + t$95$8), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[t$95$0], $MachinePrecision]), $MachinePrecision] / N[(4.0 * t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] * t$95$9), $MachinePrecision] * t$95$9), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 7.79999999999999979e76

   1. Initial program 2.8%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   2. Taylor expanded in a around -inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \frac{a \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}{{b}^{2}}} \]

   4. Applied rewrites 0.7%

      \[\leadsto \color{blue}{-0.25 \cdot \frac{a \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}{{b}^{2}}} \]

   5. Taylor expanded in b around 0

      \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \color{blue}{\left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}\right) \]

   7. Applied rewrites 2.5%

      \[\leadsto -0.25 \cdot \left(a \cdot \color{blue}{\left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}\right) \]

   8. Taylor expanded in x-scale around -inf

      \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \left(-1 \cdot \left(x-scale \cdot \color{blue}{\left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{y-scale}^{4}}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}}{{y-scale}^{2}}}\right)}\right)\right)\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \left(-1 \cdot \left(x-scale \cdot \left({y-scale}^{2} \cdot \color{blue}{\sqrt{8 \cdot \frac{\sqrt{\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{y-scale}^{4}}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}}{{y-scale}^{2}}}}\right)\right)\right)\right) \]

   10. Applied rewrites 5.7%

       \[\leadsto -0.25 \cdot \left(a \cdot \left(-1 \cdot \left(x-scale \cdot \color{blue}{\left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}{{y-scale}^{4}}} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}}{{y-scale}^{2}}}\right)}\right)\right)\right) \]

   if 7.79999999999999979e76 < y-scale

   1. Initial program 2.8%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   3. Applied rewrites 6.3%

      \[\leadsto \color{blue}{\left(\frac{\frac{\sqrt{\left(8 \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \left(\left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right) \cdot \left(\left(\mathsf{hypot}\left(\frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}{y-scale \cdot x-scale}, \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), a \cdot a, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale} - \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), b \cdot b, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(a \cdot a\right)\right)}{y-scale \cdot y-scale}\right) + \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), a \cdot a, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}\right) + \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), b \cdot b, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(a \cdot a\right)\right)}{y-scale \cdot y-scale}\right)\right)}}{\left|y-scale \cdot x-scale\right|}}{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(a \cdot b\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right)} \]

   5. Applied rewrites 12.4%

      \[\leadsto \left(\color{blue}{\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot \left(\left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right) \cdot a, a, \left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b\right) \cdot b\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(\mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right) \cdot b\right) \cdot b\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(\mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right) \cdot b\right) \cdot b\right)}{x-scale \cdot x-scale}\right) + \frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right) \cdot a, a, \left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b\right) \cdot b\right)}{y-scale \cdot y-scale}\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a}} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   7. Applied rewrites 12.4%

      \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot \left(\left(\mathsf{hypot}\left(\color{blue}{\frac{\frac{\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(0.5, \cos \left(\left(\left(\pi + \pi\right) \cdot angle\right) \cdot 0.005555555555555556\right), 0.5\right), \left(\left(0.5 - \cos \left(\left(\left(\pi + \pi\right) \cdot angle\right) \cdot 0.005555555555555556\right) \cdot 0.5\right) \cdot b\right) \cdot b\right)}{y-scale}}{y-scale}} - \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(\mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right) \cdot b\right) \cdot b\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(\mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right) \cdot b\right) \cdot b\right)}{x-scale \cdot x-scale}\right) + \frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right) \cdot a, a, \left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b\right) \cdot b\right)}{y-scale \cdot y-scale}\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   9. Applied rewrites 15.2%

      \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot \left(\left(\mathsf{hypot}\left(\frac{\frac{\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(0.5, \cos \left(\left(\left(\pi + \pi\right) \cdot angle\right) \cdot 0.005555555555555556\right), 0.5\right), \left(\left(0.5 - \cos \left(\left(\left(\pi + \pi\right) \cdot angle\right) \cdot 0.005555555555555556\right) \cdot 0.5\right) \cdot b\right) \cdot b\right)}{y-scale}}{y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(\mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right) \cdot b\right) \cdot b\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(\mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right) \cdot b\right) \cdot b\right)}{x-scale \cdot x-scale}\right) + \color{blue}{\frac{\frac{\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(0.5, \cos \left(\left(\left(\pi + \pi\right) \cdot angle\right) \cdot 0.005555555555555556\right), 0.5\right), \left(\left(0.5 - \cos \left(\left(\left(\pi + \pi\right) \cdot angle\right) \cdot 0.005555555555555556\right) \cdot 0.5\right) \cdot b\right) \cdot b\right)}{y-scale}}{y-scale}}\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   10. Taylor expanded in angle around 0

       \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot \left(\left(\mathsf{hypot}\left(\frac{\color{blue}{\frac{{a}^{2}}{y-scale}}}{y-scale} - \frac{\mathsf{fma}\left(\left(\frac{1}{2} - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right) \cdot \frac{1}{2}\right) \cdot a, a, \left(\mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right), \frac{1}{2}, \frac{1}{2}\right) \cdot b\right) \cdot b\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \frac{\mathsf{fma}\left(\left(\frac{1}{2} - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right) \cdot \frac{1}{2}\right) \cdot a, a, \left(\mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right), \frac{1}{2}, \frac{1}{2}\right) \cdot b\right) \cdot b\right)}{x-scale \cdot x-scale}\right) + \frac{\frac{\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(\frac{1}{2}, \cos \left(\left(\left(\pi + \pi\right) \cdot angle\right) \cdot \frac{1}{180}\right), \frac{1}{2}\right), \left(\left(\frac{1}{2} - \cos \left(\left(\left(\pi + \pi\right) \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \frac{1}{2}\right) \cdot b\right) \cdot b\right)}{y-scale}}{y-scale}\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]
   11. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot \left(\left(\mathsf{hypot}\left(\frac{\frac{{a}^{2}}{\color{blue}{y-scale}}}{y-scale} - \frac{\mathsf{fma}\left(\left(\frac{1}{2} - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right) \cdot \frac{1}{2}\right) \cdot a, a, \left(\mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right), \frac{1}{2}, \frac{1}{2}\right) \cdot b\right) \cdot b\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \frac{\mathsf{fma}\left(\left(\frac{1}{2} - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right) \cdot \frac{1}{2}\right) \cdot a, a, \left(\mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right), \frac{1}{2}, \frac{1}{2}\right) \cdot b\right) \cdot b\right)}{x-scale \cdot x-scale}\right) + \frac{\frac{\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(\frac{1}{2}, \cos \left(\left(\left(\pi + \pi\right) \cdot angle\right) \cdot \frac{1}{180}\right), \frac{1}{2}\right), \left(\left(\frac{1}{2} - \cos \left(\left(\left(\pi + \pi\right) \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \frac{1}{2}\right) \cdot b\right) \cdot b\right)}{y-scale}}{y-scale}\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

       2. lower-pow.f64 15.2

          \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot \left(\left(\mathsf{hypot}\left(\frac{\frac{{a}^{2}}{y-scale}}{y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(\mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right) \cdot b\right) \cdot b\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(\mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right) \cdot b\right) \cdot b\right)}{x-scale \cdot x-scale}\right) + \frac{\frac{\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(0.5, \cos \left(\left(\left(\pi + \pi\right) \cdot angle\right) \cdot 0.005555555555555556\right), 0.5\right), \left(\left(0.5 - \cos \left(\left(\left(\pi + \pi\right) \cdot angle\right) \cdot 0.005555555555555556\right) \cdot 0.5\right) \cdot b\right) \cdot b\right)}{y-scale}}{y-scale}\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   12. Applied rewrites 15.2%

       \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot \left(\left(\mathsf{hypot}\left(\frac{\color{blue}{\frac{{a}^{2}}{y-scale}}}{y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(\mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right) \cdot b\right) \cdot b\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(\mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right) \cdot b\right) \cdot b\right)}{x-scale \cdot x-scale}\right) + \frac{\frac{\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(0.5, \cos \left(\left(\left(\pi + \pi\right) \cdot angle\right) \cdot 0.005555555555555556\right), 0.5\right), \left(\left(0.5 - \cos \left(\left(\left(\pi + \pi\right) \cdot angle\right) \cdot 0.005555555555555556\right) \cdot 0.5\right) \cdot b\right) \cdot b\right)}{y-scale}}{y-scale}\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   13. Taylor expanded in angle around 0

       \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot \left(\left(\mathsf{hypot}\left(\frac{\frac{{a}^{2}}{y-scale}}{y-scale} - \frac{\mathsf{fma}\left(\left(\frac{1}{2} - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right) \cdot \frac{1}{2}\right) \cdot a, a, \left(\mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right), \frac{1}{2}, \frac{1}{2}\right) \cdot b\right) \cdot b\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \frac{\mathsf{fma}\left(\left(\frac{1}{2} - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right) \cdot \frac{1}{2}\right) \cdot a, a, \left(\mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right), \frac{1}{2}, \frac{1}{2}\right) \cdot b\right) \cdot b\right)}{x-scale \cdot x-scale}\right) + \frac{\color{blue}{\frac{{a}^{2}}{y-scale}}}{y-scale}\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]
   14. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot \left(\left(\mathsf{hypot}\left(\frac{\frac{{a}^{2}}{y-scale}}{y-scale} - \frac{\mathsf{fma}\left(\left(\frac{1}{2} - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right) \cdot \frac{1}{2}\right) \cdot a, a, \left(\mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right), \frac{1}{2}, \frac{1}{2}\right) \cdot b\right) \cdot b\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \frac{\mathsf{fma}\left(\left(\frac{1}{2} - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right) \cdot \frac{1}{2}\right) \cdot a, a, \left(\mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right), \frac{1}{2}, \frac{1}{2}\right) \cdot b\right) \cdot b\right)}{x-scale \cdot x-scale}\right) + \frac{\frac{{a}^{2}}{\color{blue}{y-scale}}}{y-scale}\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

       2. lower-pow.f64 15.0

          \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot \left(\left(\mathsf{hypot}\left(\frac{\frac{{a}^{2}}{y-scale}}{y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(\mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right) \cdot b\right) \cdot b\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(\mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right) \cdot b\right) \cdot b\right)}{x-scale \cdot x-scale}\right) + \frac{\frac{{a}^{2}}{y-scale}}{y-scale}\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   15. Applied rewrites 15.0%

       \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot \left(\left(\mathsf{hypot}\left(\frac{\frac{{a}^{2}}{y-scale}}{y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(\mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right) \cdot b\right) \cdot b\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(\mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right) \cdot b\right) \cdot b\right)}{x-scale \cdot x-scale}\right) + \frac{\color{blue}{\frac{{a}^{2}}{y-scale}}}{y-scale}\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 17.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} t_0 := \left(\left(-\left|a\right|\right) \cdot b\right) \cdot b\\ t_1 := {\left(\left|y-scale\right|\right)}^{2}\\ t_2 := \left|a\right| \cdot b\\ t_3 := \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\\ t_4 := \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ t_5 := \mathsf{fma}\left(\mathsf{fma}\left(t\_3, 0.5, 0.5\right), b \cdot b, \left(\left(0.5 - t\_3 \cdot 0.5\right) \cdot \left|a\right|\right) \cdot \left|a\right|\right)\\ t_6 := \left|y-scale\right| \cdot \left|x-scale\right|\\ \mathbf{if}\;\left|y-scale\right| \leq 7.8 \cdot 10^{+76}:\\ \;\;\;\;-0.25 \cdot \left(\left|a\right| \cdot \left(-1 \cdot \left(\left|x-scale\right| \cdot \left(t\_1 \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{{t\_4}^{4}}{{\left(\left|y-scale\right|\right)}^{4}}} + \frac{{t\_4}^{2}}{t\_1}}{t\_1}}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\frac{\sqrt{\left(\left(\left(8 \cdot t\_0\right) \cdot \left|a\right|\right) \cdot \left(t\_0 \cdot \left|a\right|\right)\right) \cdot \frac{\left|t\_5\right| + t\_5}{\left|x-scale\right| \cdot \left|x-scale\right|}}}{\left|\left|x-scale\right| \cdot \left|y-scale\right|\right|}}{4 \cdot t\_2}}{t\_2} \cdot t\_6\right) \cdot t\_6\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (* (- (fabs a)) b) b))
        (t_1 (pow (fabs y-scale) 2.0))
        (t_2 (* (fabs a) b))
        (t_3 (cos (* 0.011111111111111112 (* angle PI))))
        (t_4 (cos (* 0.005555555555555556 (* angle PI))))
        (t_5
         (fma
          (fma t_3 0.5 0.5)
          (* b b)
          (* (* (- 0.5 (* t_3 0.5)) (fabs a)) (fabs a))))
        (t_6 (* (fabs y-scale) (fabs x-scale))))
   (if (<= (fabs y-scale) 7.8e+76)
     (*
      -0.25
      (*
       (fabs a)
       (*
        -1.0
        (*
         (fabs x-scale)
         (*
          t_1
          (sqrt
           (*
            8.0
            (/
             (+
              (sqrt (/ (pow t_4 4.0) (pow (fabs y-scale) 4.0)))
              (/ (pow t_4 2.0) t_1))
             t_1))))))))
     (*
      (*
       (/
        (/
         (/
          (sqrt
           (*
            (* (* (* 8.0 t_0) (fabs a)) (* t_0 (fabs a)))
            (/ (+ (fabs t_5) t_5) (* (fabs x-scale) (fabs x-scale)))))
          (fabs (* (fabs x-scale) (fabs y-scale))))
         (* 4.0 t_2))
        t_2)
       t_6)
      t_6))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (-fabs(a) * b) * b;
	double t_1 = pow(fabs(y_45_scale), 2.0);
	double t_2 = fabs(a) * b;
	double t_3 = cos((0.011111111111111112 * (angle * ((double) M_PI))));
	double t_4 = cos((0.005555555555555556 * (angle * ((double) M_PI))));
	double t_5 = fma(fma(t_3, 0.5, 0.5), (b * b), (((0.5 - (t_3 * 0.5)) * fabs(a)) * fabs(a)));
	double t_6 = fabs(y_45_scale) * fabs(x_45_scale);
	double tmp;
	if (fabs(y_45_scale) <= 7.8e+76) {
		tmp = -0.25 * (fabs(a) * (-1.0 * (fabs(x_45_scale) * (t_1 * sqrt((8.0 * ((sqrt((pow(t_4, 4.0) / pow(fabs(y_45_scale), 4.0))) + (pow(t_4, 2.0) / t_1)) / t_1)))))));
	} else {
		tmp = ((((sqrt(((((8.0 * t_0) * fabs(a)) * (t_0 * fabs(a))) * ((fabs(t_5) + t_5) / (fabs(x_45_scale) * fabs(x_45_scale))))) / fabs((fabs(x_45_scale) * fabs(y_45_scale)))) / (4.0 * t_2)) / t_2) * t_6) * t_6;
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(Float64(-abs(a)) * b) * b)
	t_1 = abs(y_45_scale) ^ 2.0
	t_2 = Float64(abs(a) * b)
	t_3 = cos(Float64(0.011111111111111112 * Float64(angle * pi)))
	t_4 = cos(Float64(0.005555555555555556 * Float64(angle * pi)))
	t_5 = fma(fma(t_3, 0.5, 0.5), Float64(b * b), Float64(Float64(Float64(0.5 - Float64(t_3 * 0.5)) * abs(a)) * abs(a)))
	t_6 = Float64(abs(y_45_scale) * abs(x_45_scale))
	tmp = 0.0
	if (abs(y_45_scale) <= 7.8e+76)
		tmp = Float64(-0.25 * Float64(abs(a) * Float64(-1.0 * Float64(abs(x_45_scale) * Float64(t_1 * sqrt(Float64(8.0 * Float64(Float64(sqrt(Float64((t_4 ^ 4.0) / (abs(y_45_scale) ^ 4.0))) + Float64((t_4 ^ 2.0) / t_1)) / t_1))))))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(sqrt(Float64(Float64(Float64(Float64(8.0 * t_0) * abs(a)) * Float64(t_0 * abs(a))) * Float64(Float64(abs(t_5) + t_5) / Float64(abs(x_45_scale) * abs(x_45_scale))))) / abs(Float64(abs(x_45_scale) * abs(y_45_scale)))) / Float64(4.0 * t_2)) / t_2) * t_6) * t_6);
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[((-N[Abs[a], $MachinePrecision]) * b), $MachinePrecision] * b), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Abs[y$45$scale], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[a], $MachinePrecision] * b), $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Cos[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(N[(t$95$3 * 0.5 + 0.5), $MachinePrecision] * N[(b * b), $MachinePrecision] + N[(N[(N[(0.5 - N[(t$95$3 * 0.5), $MachinePrecision]), $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[Abs[y$45$scale], $MachinePrecision] * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[y$45$scale], $MachinePrecision], 7.8e+76], N[(-0.25 * N[(N[Abs[a], $MachinePrecision] * N[(-1.0 * N[(N[Abs[x$45$scale], $MachinePrecision] * N[(t$95$1 * N[Sqrt[N[(8.0 * N[(N[(N[Sqrt[N[(N[Power[t$95$4, 4.0], $MachinePrecision] / N[Power[N[Abs[y$45$scale], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[Power[t$95$4, 2.0], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[Sqrt[N[(N[(N[(N[(8.0 * t$95$0), $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Abs[t$95$5], $MachinePrecision] + t$95$5), $MachinePrecision] / N[(N[Abs[x$45$scale], $MachinePrecision] * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[N[(N[Abs[x$45$scale], $MachinePrecision] * N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(4.0 * t$95$2), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] * t$95$6), $MachinePrecision] * t$95$6), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 7.79999999999999979e76

   1. Initial program 2.8%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   2. Taylor expanded in a around -inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \frac{a \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}{{b}^{2}}} \]

   4. Applied rewrites 0.7%

      \[\leadsto \color{blue}{-0.25 \cdot \frac{a \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}{{b}^{2}}} \]

   5. Taylor expanded in b around 0

      \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \color{blue}{\left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}\right) \]

   7. Applied rewrites 2.5%

      \[\leadsto -0.25 \cdot \left(a \cdot \color{blue}{\left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}\right) \]

   8. Taylor expanded in x-scale around -inf

      \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \left(-1 \cdot \left(x-scale \cdot \color{blue}{\left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{y-scale}^{4}}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}}{{y-scale}^{2}}}\right)}\right)\right)\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \left(-1 \cdot \left(x-scale \cdot \left({y-scale}^{2} \cdot \color{blue}{\sqrt{8 \cdot \frac{\sqrt{\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{y-scale}^{4}}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}}{{y-scale}^{2}}}}\right)\right)\right)\right) \]

   10. Applied rewrites 5.7%

       \[\leadsto -0.25 \cdot \left(a \cdot \left(-1 \cdot \left(x-scale \cdot \color{blue}{\left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}{{y-scale}^{4}}} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}}{{y-scale}^{2}}}\right)}\right)\right)\right) \]

   if 7.79999999999999979e76 < y-scale

   1. Initial program 2.8%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   3. Applied rewrites 6.3%

      \[\leadsto \color{blue}{\left(\frac{\frac{\sqrt{\left(8 \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \left(\left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right) \cdot \left(\left(\mathsf{hypot}\left(\frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}{y-scale \cdot x-scale}, \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), a \cdot a, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale} - \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), b \cdot b, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(a \cdot a\right)\right)}{y-scale \cdot y-scale}\right) + \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), a \cdot a, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}\right) + \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), b \cdot b, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(a \cdot a\right)\right)}{y-scale \cdot y-scale}\right)\right)}}{\left|y-scale \cdot x-scale\right|}}{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(a \cdot b\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right)} \]

   4. Taylor expanded in x-scale around 0

      \[\leadsto \left(\frac{\frac{\sqrt{\left(8 \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \left(\left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right) \cdot \color{blue}{\frac{\sqrt{{\left({a}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) + {b}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}^{2}} + \left({a}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) + {b}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}{{x-scale}^{2}}}\right)}}{\left|y-scale \cdot x-scale\right|}}{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(a \cdot b\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 6.7%

      \[\leadsto \left(\frac{\frac{\sqrt{\left(8 \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \left(\left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right) \cdot \color{blue}{\frac{\sqrt{{\left(\mathsf{fma}\left({a}^{2}, 0.5 - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right), {b}^{2} \cdot \left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)}^{2}} + \mathsf{fma}\left({a}^{2}, 0.5 - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right), {b}^{2} \cdot \left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}{{x-scale}^{2}}}\right)}}{\left|y-scale \cdot x-scale\right|}}{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(a \cdot b\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   8. Applied rewrites 9.9%

      \[\leadsto \left(\color{blue}{\frac{\frac{\frac{\sqrt{\left(\left(\left(8 \cdot \left(\left(\left(-a\right) \cdot b\right) \cdot b\right)\right) \cdot a\right) \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \frac{\left|\mathsf{fma}\left(0.5 - \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot 0.5, a \cdot a, \mathsf{fma}\left(\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right), 0.5, 0.5\right) \cdot \left(b \cdot b\right)\right)\right| + \mathsf{fma}\left(0.5 - \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot 0.5, a \cdot a, \mathsf{fma}\left(\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right), 0.5, 0.5\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(a \cdot b\right)}}{a \cdot b}} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]
   9. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(8 \cdot \left(\left(\left(-a\right) \cdot b\right) \cdot b\right)\right) \cdot a\right) \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \frac{\left|\left(\frac{1}{2} - \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right) \cdot \frac{1}{2}\right) \cdot \left(a \cdot a\right) + \mathsf{fma}\left(\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), \frac{1}{2}, \frac{1}{2}\right) \cdot \left(b \cdot b\right)\right| + \mathsf{fma}\left(\frac{1}{2} - \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right) \cdot \frac{1}{2}, a \cdot a, \mathsf{fma}\left(\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), \frac{1}{2}, \frac{1}{2}\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(a \cdot b\right)}}{a \cdot b} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(8 \cdot \left(\left(\left(-a\right) \cdot b\right) \cdot b\right)\right) \cdot a\right) \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \frac{\left|\mathsf{fma}\left(\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), \frac{1}{2}, \frac{1}{2}\right) \cdot \left(b \cdot b\right) + \left(\frac{1}{2} - \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right) \cdot \frac{1}{2}\right) \cdot \left(a \cdot a\right)\right| + \mathsf{fma}\left(\frac{1}{2} - \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right) \cdot \frac{1}{2}, a \cdot a, \mathsf{fma}\left(\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), \frac{1}{2}, \frac{1}{2}\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(a \cdot b\right)}}{a \cdot b} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(8 \cdot \left(\left(\left(-a\right) \cdot b\right) \cdot b\right)\right) \cdot a\right) \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \frac{\left|\mathsf{fma}\left(\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), \frac{1}{2}, \frac{1}{2}\right) \cdot \left(b \cdot b\right) + \left(\frac{1}{2} - \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right) \cdot \frac{1}{2}\right) \cdot \left(a \cdot a\right)\right| + \mathsf{fma}\left(\frac{1}{2} - \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right) \cdot \frac{1}{2}, a \cdot a, \mathsf{fma}\left(\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), \frac{1}{2}, \frac{1}{2}\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(a \cdot b\right)}}{a \cdot b} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(8 \cdot \left(\left(\left(-a\right) \cdot b\right) \cdot b\right)\right) \cdot a\right) \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \frac{\left|\mathsf{fma}\left(\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), \frac{1}{2}, \frac{1}{2}\right) \cdot \left(b \cdot b\right) + \left(\frac{1}{2} - \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right) \cdot \frac{1}{2}\right) \cdot \left(a \cdot a\right)\right| + \mathsf{fma}\left(\frac{1}{2} - \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right) \cdot \frac{1}{2}, a \cdot a, \mathsf{fma}\left(\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), \frac{1}{2}, \frac{1}{2}\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(a \cdot b\right)}}{a \cdot b} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

      5. pow2 N/A

         \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(8 \cdot \left(\left(\left(-a\right) \cdot b\right) \cdot b\right)\right) \cdot a\right) \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \frac{\left|\mathsf{fma}\left(\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), \frac{1}{2}, \frac{1}{2}\right) \cdot {b}^{2} + \left(\frac{1}{2} - \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right) \cdot \frac{1}{2}\right) \cdot \left(a \cdot a\right)\right| + \mathsf{fma}\left(\frac{1}{2} - \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right) \cdot \frac{1}{2}, a \cdot a, \mathsf{fma}\left(\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), \frac{1}{2}, \frac{1}{2}\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(a \cdot b\right)}}{a \cdot b} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(8 \cdot \left(\left(\left(-a\right) \cdot b\right) \cdot b\right)\right) \cdot a\right) \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \frac{\left|\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), \frac{1}{2}, \frac{1}{2}\right), {b}^{2}, \left(\frac{1}{2} - \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right) \cdot \frac{1}{2}\right) \cdot \left(a \cdot a\right)\right)\right| + \mathsf{fma}\left(\frac{1}{2} - \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right) \cdot \frac{1}{2}, a \cdot a, \mathsf{fma}\left(\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), \frac{1}{2}, \frac{1}{2}\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(a \cdot b\right)}}{a \cdot b} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

      7. pow2 N/A

         \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(8 \cdot \left(\left(\left(-a\right) \cdot b\right) \cdot b\right)\right) \cdot a\right) \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \frac{\left|\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), \frac{1}{2}, \frac{1}{2}\right), b \cdot b, \left(\frac{1}{2} - \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right) \cdot \frac{1}{2}\right) \cdot \left(a \cdot a\right)\right)\right| + \mathsf{fma}\left(\frac{1}{2} - \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right) \cdot \frac{1}{2}, a \cdot a, \mathsf{fma}\left(\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), \frac{1}{2}, \frac{1}{2}\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(a \cdot b\right)}}{a \cdot b} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(8 \cdot \left(\left(\left(-a\right) \cdot b\right) \cdot b\right)\right) \cdot a\right) \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \frac{\left|\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), \frac{1}{2}, \frac{1}{2}\right), b \cdot b, \left(\frac{1}{2} - \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right) \cdot \frac{1}{2}\right) \cdot \left(a \cdot a\right)\right)\right| + \mathsf{fma}\left(\frac{1}{2} - \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right) \cdot \frac{1}{2}, a \cdot a, \mathsf{fma}\left(\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), \frac{1}{2}, \frac{1}{2}\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(a \cdot b\right)}}{a \cdot b} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(8 \cdot \left(\left(\left(-a\right) \cdot b\right) \cdot b\right)\right) \cdot a\right) \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \frac{\left|\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), \frac{1}{2}, \frac{1}{2}\right), b \cdot b, \left(\frac{1}{2} - \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right) \cdot \frac{1}{2}\right) \cdot \left(a \cdot a\right)\right)\right| + \mathsf{fma}\left(\frac{1}{2} - \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right) \cdot \frac{1}{2}, a \cdot a, \mathsf{fma}\left(\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), \frac{1}{2}, \frac{1}{2}\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(a \cdot b\right)}}{a \cdot b} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

      10. associate-*r* N/A

          \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(8 \cdot \left(\left(\left(-a\right) \cdot b\right) \cdot b\right)\right) \cdot a\right) \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \frac{\left|\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), \frac{1}{2}, \frac{1}{2}\right), b \cdot b, \left(\left(\frac{1}{2} - \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right) \cdot \frac{1}{2}\right) \cdot a\right) \cdot a\right)\right| + \mathsf{fma}\left(\frac{1}{2} - \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right) \cdot \frac{1}{2}, a \cdot a, \mathsf{fma}\left(\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), \frac{1}{2}, \frac{1}{2}\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(a \cdot b\right)}}{a \cdot b} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(8 \cdot \left(\left(\left(-a\right) \cdot b\right) \cdot b\right)\right) \cdot a\right) \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \frac{\left|\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), \frac{1}{2}, \frac{1}{2}\right), b \cdot b, \left(\left(\frac{1}{2} - \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right) \cdot \frac{1}{2}\right) \cdot a\right) \cdot a\right)\right| + \mathsf{fma}\left(\frac{1}{2} - \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right) \cdot \frac{1}{2}, a \cdot a, \mathsf{fma}\left(\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), \frac{1}{2}, \frac{1}{2}\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(a \cdot b\right)}}{a \cdot b} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

      12. lower-*.f64 9.9

          \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(8 \cdot \left(\left(\left(-a\right) \cdot b\right) \cdot b\right)\right) \cdot a\right) \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \frac{\left|\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right), 0.5, 0.5\right), b \cdot b, \left(\left(0.5 - \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot 0.5\right) \cdot a\right) \cdot a\right)\right| + \mathsf{fma}\left(0.5 - \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot 0.5, a \cdot a, \mathsf{fma}\left(\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right), 0.5, 0.5\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(a \cdot b\right)}}{a \cdot b} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   10. Applied rewrites 9.9%

       \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(8 \cdot \left(\left(\left(-a\right) \cdot b\right) \cdot b\right)\right) \cdot a\right) \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \frac{\left|\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right), 0.5, 0.5\right), b \cdot b, \left(\left(0.5 - \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot 0.5\right) \cdot a\right) \cdot a\right)\right| + \mathsf{fma}\left(0.5 - \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot 0.5, a \cdot a, \mathsf{fma}\left(\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right), 0.5, 0.5\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(a \cdot b\right)}}{a \cdot b} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]
   11. Step-by-step derivation

       1. lift-fma.f64 N/A

          \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(8 \cdot \left(\left(\left(-a\right) \cdot b\right) \cdot b\right)\right) \cdot a\right) \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \frac{\left|\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), \frac{1}{2}, \frac{1}{2}\right), b \cdot b, \left(\left(\frac{1}{2} - \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right) \cdot \frac{1}{2}\right) \cdot a\right) \cdot a\right)\right| + \left(\left(\frac{1}{2} - \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right) \cdot \frac{1}{2}\right) \cdot \left(a \cdot a\right) + \mathsf{fma}\left(\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), \frac{1}{2}, \frac{1}{2}\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(a \cdot b\right)}}{a \cdot b} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

       2. +-commutative N/A

          \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(8 \cdot \left(\left(\left(-a\right) \cdot b\right) \cdot b\right)\right) \cdot a\right) \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \frac{\left|\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), \frac{1}{2}, \frac{1}{2}\right), b \cdot b, \left(\left(\frac{1}{2} - \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right) \cdot \frac{1}{2}\right) \cdot a\right) \cdot a\right)\right| + \left(\mathsf{fma}\left(\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), \frac{1}{2}, \frac{1}{2}\right) \cdot \left(b \cdot b\right) + \left(\frac{1}{2} - \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right) \cdot \frac{1}{2}\right) \cdot \left(a \cdot a\right)\right)}{x-scale \cdot x-scale}}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(a \cdot b\right)}}{a \cdot b} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(8 \cdot \left(\left(\left(-a\right) \cdot b\right) \cdot b\right)\right) \cdot a\right) \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \frac{\left|\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), \frac{1}{2}, \frac{1}{2}\right), b \cdot b, \left(\left(\frac{1}{2} - \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right) \cdot \frac{1}{2}\right) \cdot a\right) \cdot a\right)\right| + \left(\mathsf{fma}\left(\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), \frac{1}{2}, \frac{1}{2}\right) \cdot \left(b \cdot b\right) + \left(\frac{1}{2} - \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right) \cdot \frac{1}{2}\right) \cdot \left(a \cdot a\right)\right)}{x-scale \cdot x-scale}}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(a \cdot b\right)}}{a \cdot b} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

       4. lift-*.f64 N/A

          \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(8 \cdot \left(\left(\left(-a\right) \cdot b\right) \cdot b\right)\right) \cdot a\right) \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \frac{\left|\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), \frac{1}{2}, \frac{1}{2}\right), b \cdot b, \left(\left(\frac{1}{2} - \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right) \cdot \frac{1}{2}\right) \cdot a\right) \cdot a\right)\right| + \left(\mathsf{fma}\left(\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), \frac{1}{2}, \frac{1}{2}\right) \cdot \left(b \cdot b\right) + \left(\frac{1}{2} - \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right) \cdot \frac{1}{2}\right) \cdot \left(a \cdot a\right)\right)}{x-scale \cdot x-scale}}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(a \cdot b\right)}}{a \cdot b} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

       5. pow2 N/A

          \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(8 \cdot \left(\left(\left(-a\right) \cdot b\right) \cdot b\right)\right) \cdot a\right) \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \frac{\left|\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), \frac{1}{2}, \frac{1}{2}\right), b \cdot b, \left(\left(\frac{1}{2} - \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right) \cdot \frac{1}{2}\right) \cdot a\right) \cdot a\right)\right| + \left(\mathsf{fma}\left(\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), \frac{1}{2}, \frac{1}{2}\right) \cdot {b}^{2} + \left(\frac{1}{2} - \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right) \cdot \frac{1}{2}\right) \cdot \left(a \cdot a\right)\right)}{x-scale \cdot x-scale}}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(a \cdot b\right)}}{a \cdot b} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

       6. lower-fma.f64 N/A

          \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(8 \cdot \left(\left(\left(-a\right) \cdot b\right) \cdot b\right)\right) \cdot a\right) \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \frac{\left|\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), \frac{1}{2}, \frac{1}{2}\right), b \cdot b, \left(\left(\frac{1}{2} - \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right) \cdot \frac{1}{2}\right) \cdot a\right) \cdot a\right)\right| + \mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), \frac{1}{2}, \frac{1}{2}\right), {b}^{2}, \left(\frac{1}{2} - \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right) \cdot \frac{1}{2}\right) \cdot \left(a \cdot a\right)\right)}{x-scale \cdot x-scale}}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(a \cdot b\right)}}{a \cdot b} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

       7. pow2 N/A

          \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(8 \cdot \left(\left(\left(-a\right) \cdot b\right) \cdot b\right)\right) \cdot a\right) \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \frac{\left|\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), \frac{1}{2}, \frac{1}{2}\right), b \cdot b, \left(\left(\frac{1}{2} - \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right) \cdot \frac{1}{2}\right) \cdot a\right) \cdot a\right)\right| + \mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), \frac{1}{2}, \frac{1}{2}\right), b \cdot b, \left(\frac{1}{2} - \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right) \cdot \frac{1}{2}\right) \cdot \left(a \cdot a\right)\right)}{x-scale \cdot x-scale}}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(a \cdot b\right)}}{a \cdot b} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

       8. lift-*.f64 N/A

          \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(8 \cdot \left(\left(\left(-a\right) \cdot b\right) \cdot b\right)\right) \cdot a\right) \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \frac{\left|\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), \frac{1}{2}, \frac{1}{2}\right), b \cdot b, \left(\left(\frac{1}{2} - \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right) \cdot \frac{1}{2}\right) \cdot a\right) \cdot a\right)\right| + \mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), \frac{1}{2}, \frac{1}{2}\right), b \cdot b, \left(\frac{1}{2} - \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right) \cdot \frac{1}{2}\right) \cdot \left(a \cdot a\right)\right)}{x-scale \cdot x-scale}}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(a \cdot b\right)}}{a \cdot b} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

       9. lift-*.f64 N/A

          \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(8 \cdot \left(\left(\left(-a\right) \cdot b\right) \cdot b\right)\right) \cdot a\right) \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \frac{\left|\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), \frac{1}{2}, \frac{1}{2}\right), b \cdot b, \left(\left(\frac{1}{2} - \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right) \cdot \frac{1}{2}\right) \cdot a\right) \cdot a\right)\right| + \mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), \frac{1}{2}, \frac{1}{2}\right), b \cdot b, \left(\frac{1}{2} - \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right) \cdot \frac{1}{2}\right) \cdot \left(a \cdot a\right)\right)}{x-scale \cdot x-scale}}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(a \cdot b\right)}}{a \cdot b} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

       10. associate-*r* N/A

           \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(8 \cdot \left(\left(\left(-a\right) \cdot b\right) \cdot b\right)\right) \cdot a\right) \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \frac{\left|\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), \frac{1}{2}, \frac{1}{2}\right), b \cdot b, \left(\left(\frac{1}{2} - \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right) \cdot \frac{1}{2}\right) \cdot a\right) \cdot a\right)\right| + \mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), \frac{1}{2}, \frac{1}{2}\right), b \cdot b, \left(\left(\frac{1}{2} - \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right) \cdot \frac{1}{2}\right) \cdot a\right) \cdot a\right)}{x-scale \cdot x-scale}}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(a \cdot b\right)}}{a \cdot b} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

       11. lower-*.f64 N/A

           \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(8 \cdot \left(\left(\left(-a\right) \cdot b\right) \cdot b\right)\right) \cdot a\right) \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \frac{\left|\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), \frac{1}{2}, \frac{1}{2}\right), b \cdot b, \left(\left(\frac{1}{2} - \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right) \cdot \frac{1}{2}\right) \cdot a\right) \cdot a\right)\right| + \mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), \frac{1}{2}, \frac{1}{2}\right), b \cdot b, \left(\left(\frac{1}{2} - \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right) \cdot \frac{1}{2}\right) \cdot a\right) \cdot a\right)}{x-scale \cdot x-scale}}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(a \cdot b\right)}}{a \cdot b} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

       12. lower-*.f64 10.6

           \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(8 \cdot \left(\left(\left(-a\right) \cdot b\right) \cdot b\right)\right) \cdot a\right) \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \frac{\left|\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right), 0.5, 0.5\right), b \cdot b, \left(\left(0.5 - \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot 0.5\right) \cdot a\right) \cdot a\right)\right| + \mathsf{fma}\left(\mathsf{fma}\left(\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right), 0.5, 0.5\right), b \cdot b, \left(\left(0.5 - \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot 0.5\right) \cdot a\right) \cdot a\right)}{x-scale \cdot x-scale}}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(a \cdot b\right)}}{a \cdot b} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   12. Applied rewrites 10.6%

       \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(8 \cdot \left(\left(\left(-a\right) \cdot b\right) \cdot b\right)\right) \cdot a\right) \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \frac{\left|\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right), 0.5, 0.5\right), b \cdot b, \left(\left(0.5 - \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot 0.5\right) \cdot a\right) \cdot a\right)\right| + \mathsf{fma}\left(\mathsf{fma}\left(\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right), 0.5, 0.5\right), b \cdot b, \left(\left(0.5 - \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot 0.5\right) \cdot a\right) \cdot a\right)}{x-scale \cdot x-scale}}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(a \cdot b\right)}}{a \cdot b} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 17.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} t_0 := {\left(\left|y-scale\right|\right)}^{2}\\ t_1 := \left|a\right| \cdot b\\ t_2 := \left|y-scale\right| \cdot \left|x-scale\right|\\ t_3 := \left(\left(-\left|a\right|\right) \cdot b\right) \cdot b\\ t_4 := \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ t_5 := 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\\ t_6 := {b}^{2} \cdot \left(0.5 + t\_5\right)\\ \mathbf{if}\;\left|y-scale\right| \leq 7.8 \cdot 10^{+76}:\\ \;\;\;\;-0.25 \cdot \left(\left|a\right| \cdot \left(-1 \cdot \left(\left|x-scale\right| \cdot \left(t\_0 \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{{t\_4}^{4}}{{\left(\left|y-scale\right|\right)}^{4}}} + \frac{{t\_4}^{2}}{t\_0}}{t\_0}}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\frac{\sqrt{\left(\left(\left(8 \cdot t\_3\right) \cdot \left|a\right|\right) \cdot \left(t\_3 \cdot \left|a\right|\right)\right) \cdot \frac{\left|\mathsf{fma}\left({\left(\left|a\right|\right)}^{2}, 0.5 - t\_5, t\_6\right)\right| + t\_6}{\left|x-scale\right| \cdot \left|x-scale\right|}}}{\left|\left|x-scale\right| \cdot \left|y-scale\right|\right|}}{4 \cdot t\_1}}{t\_1} \cdot t\_2\right) \cdot t\_2\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (pow (fabs y-scale) 2.0))
        (t_1 (* (fabs a) b))
        (t_2 (* (fabs y-scale) (fabs x-scale)))
        (t_3 (* (* (- (fabs a)) b) b))
        (t_4 (cos (* 0.005555555555555556 (* angle PI))))
        (t_5 (* 0.5 (cos (* 0.011111111111111112 (* angle PI)))))
        (t_6 (* (pow b 2.0) (+ 0.5 t_5))))
   (if (<= (fabs y-scale) 7.8e+76)
     (*
      -0.25
      (*
       (fabs a)
       (*
        -1.0
        (*
         (fabs x-scale)
         (*
          t_0
          (sqrt
           (*
            8.0
            (/
             (+
              (sqrt (/ (pow t_4 4.0) (pow (fabs y-scale) 4.0)))
              (/ (pow t_4 2.0) t_0))
             t_0))))))))
     (*
      (*
       (/
        (/
         (/
          (sqrt
           (*
            (* (* (* 8.0 t_3) (fabs a)) (* t_3 (fabs a)))
            (/
             (+ (fabs (fma (pow (fabs a) 2.0) (- 0.5 t_5) t_6)) t_6)
             (* (fabs x-scale) (fabs x-scale)))))
          (fabs (* (fabs x-scale) (fabs y-scale))))
         (* 4.0 t_1))
        t_1)
       t_2)
      t_2))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = pow(fabs(y_45_scale), 2.0);
	double t_1 = fabs(a) * b;
	double t_2 = fabs(y_45_scale) * fabs(x_45_scale);
	double t_3 = (-fabs(a) * b) * b;
	double t_4 = cos((0.005555555555555556 * (angle * ((double) M_PI))));
	double t_5 = 0.5 * cos((0.011111111111111112 * (angle * ((double) M_PI))));
	double t_6 = pow(b, 2.0) * (0.5 + t_5);
	double tmp;
	if (fabs(y_45_scale) <= 7.8e+76) {
		tmp = -0.25 * (fabs(a) * (-1.0 * (fabs(x_45_scale) * (t_0 * sqrt((8.0 * ((sqrt((pow(t_4, 4.0) / pow(fabs(y_45_scale), 4.0))) + (pow(t_4, 2.0) / t_0)) / t_0)))))));
	} else {
		tmp = ((((sqrt(((((8.0 * t_3) * fabs(a)) * (t_3 * fabs(a))) * ((fabs(fma(pow(fabs(a), 2.0), (0.5 - t_5), t_6)) + t_6) / (fabs(x_45_scale) * fabs(x_45_scale))))) / fabs((fabs(x_45_scale) * fabs(y_45_scale)))) / (4.0 * t_1)) / t_1) * t_2) * t_2;
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = abs(y_45_scale) ^ 2.0
	t_1 = Float64(abs(a) * b)
	t_2 = Float64(abs(y_45_scale) * abs(x_45_scale))
	t_3 = Float64(Float64(Float64(-abs(a)) * b) * b)
	t_4 = cos(Float64(0.005555555555555556 * Float64(angle * pi)))
	t_5 = Float64(0.5 * cos(Float64(0.011111111111111112 * Float64(angle * pi))))
	t_6 = Float64((b ^ 2.0) * Float64(0.5 + t_5))
	tmp = 0.0
	if (abs(y_45_scale) <= 7.8e+76)
		tmp = Float64(-0.25 * Float64(abs(a) * Float64(-1.0 * Float64(abs(x_45_scale) * Float64(t_0 * sqrt(Float64(8.0 * Float64(Float64(sqrt(Float64((t_4 ^ 4.0) / (abs(y_45_scale) ^ 4.0))) + Float64((t_4 ^ 2.0) / t_0)) / t_0))))))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(sqrt(Float64(Float64(Float64(Float64(8.0 * t_3) * abs(a)) * Float64(t_3 * abs(a))) * Float64(Float64(abs(fma((abs(a) ^ 2.0), Float64(0.5 - t_5), t_6)) + t_6) / Float64(abs(x_45_scale) * abs(x_45_scale))))) / abs(Float64(abs(x_45_scale) * abs(y_45_scale)))) / Float64(4.0 * t_1)) / t_1) * t_2) * t_2);
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[Power[N[Abs[y$45$scale], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[a], $MachinePrecision] * b), $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[y$45$scale], $MachinePrecision] * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[((-N[Abs[a], $MachinePrecision]) * b), $MachinePrecision] * b), $MachinePrecision]}, Block[{t$95$4 = N[Cos[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(0.5 * N[Cos[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[Power[b, 2.0], $MachinePrecision] * N[(0.5 + t$95$5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[y$45$scale], $MachinePrecision], 7.8e+76], N[(-0.25 * N[(N[Abs[a], $MachinePrecision] * N[(-1.0 * N[(N[Abs[x$45$scale], $MachinePrecision] * N[(t$95$0 * N[Sqrt[N[(8.0 * N[(N[(N[Sqrt[N[(N[Power[t$95$4, 4.0], $MachinePrecision] / N[Power[N[Abs[y$45$scale], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[Power[t$95$4, 2.0], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[Sqrt[N[(N[(N[(N[(8.0 * t$95$3), $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision] * N[(t$95$3 * N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Abs[N[(N[Power[N[Abs[a], $MachinePrecision], 2.0], $MachinePrecision] * N[(0.5 - t$95$5), $MachinePrecision] + t$95$6), $MachinePrecision]], $MachinePrecision] + t$95$6), $MachinePrecision] / N[(N[Abs[x$45$scale], $MachinePrecision] * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[N[(N[Abs[x$45$scale], $MachinePrecision] * N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(4.0 * t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$2), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 7.79999999999999979e76

   1. Initial program 2.8%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   2. Taylor expanded in a around -inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \frac{a \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}{{b}^{2}}} \]

   4. Applied rewrites 0.7%

      \[\leadsto \color{blue}{-0.25 \cdot \frac{a \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}{{b}^{2}}} \]

   5. Taylor expanded in b around 0

      \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \color{blue}{\left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}\right) \]

   7. Applied rewrites 2.5%

      \[\leadsto -0.25 \cdot \left(a \cdot \color{blue}{\left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}\right) \]

   8. Taylor expanded in x-scale around -inf

      \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \left(-1 \cdot \left(x-scale \cdot \color{blue}{\left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{y-scale}^{4}}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}}{{y-scale}^{2}}}\right)}\right)\right)\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \left(-1 \cdot \left(x-scale \cdot \left({y-scale}^{2} \cdot \color{blue}{\sqrt{8 \cdot \frac{\sqrt{\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{y-scale}^{4}}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}}{{y-scale}^{2}}}}\right)\right)\right)\right) \]

   10. Applied rewrites 5.7%

       \[\leadsto -0.25 \cdot \left(a \cdot \left(-1 \cdot \left(x-scale \cdot \color{blue}{\left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}{{y-scale}^{4}}} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}}{{y-scale}^{2}}}\right)}\right)\right)\right) \]

   if 7.79999999999999979e76 < y-scale

   1. Initial program 2.8%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   3. Applied rewrites 6.3%

      \[\leadsto \color{blue}{\left(\frac{\frac{\sqrt{\left(8 \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \left(\left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right) \cdot \left(\left(\mathsf{hypot}\left(\frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}{y-scale \cdot x-scale}, \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), a \cdot a, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale} - \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), b \cdot b, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(a \cdot a\right)\right)}{y-scale \cdot y-scale}\right) + \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), a \cdot a, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}\right) + \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), b \cdot b, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(a \cdot a\right)\right)}{y-scale \cdot y-scale}\right)\right)}}{\left|y-scale \cdot x-scale\right|}}{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(a \cdot b\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right)} \]

   4. Taylor expanded in x-scale around 0

      \[\leadsto \left(\frac{\frac{\sqrt{\left(8 \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \left(\left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right) \cdot \color{blue}{\frac{\sqrt{{\left({a}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) + {b}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}^{2}} + \left({a}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) + {b}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}{{x-scale}^{2}}}\right)}}{\left|y-scale \cdot x-scale\right|}}{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(a \cdot b\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 6.7%

      \[\leadsto \left(\frac{\frac{\sqrt{\left(8 \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \left(\left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right) \cdot \color{blue}{\frac{\sqrt{{\left(\mathsf{fma}\left({a}^{2}, 0.5 - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right), {b}^{2} \cdot \left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)}^{2}} + \mathsf{fma}\left({a}^{2}, 0.5 - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right), {b}^{2} \cdot \left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}{{x-scale}^{2}}}\right)}}{\left|y-scale \cdot x-scale\right|}}{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(a \cdot b\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   8. Applied rewrites 9.9%

      \[\leadsto \left(\color{blue}{\frac{\frac{\frac{\sqrt{\left(\left(\left(8 \cdot \left(\left(\left(-a\right) \cdot b\right) \cdot b\right)\right) \cdot a\right) \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \frac{\left|\mathsf{fma}\left(0.5 - \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot 0.5, a \cdot a, \mathsf{fma}\left(\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right), 0.5, 0.5\right) \cdot \left(b \cdot b\right)\right)\right| + \mathsf{fma}\left(0.5 - \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot 0.5, a \cdot a, \mathsf{fma}\left(\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right), 0.5, 0.5\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(a \cdot b\right)}}{a \cdot b}} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   9. Taylor expanded in a around 0

      \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(8 \cdot \left(\left(\left(-a\right) \cdot b\right) \cdot b\right)\right) \cdot a\right) \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \frac{\left|{a}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) + {b}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right| + {b}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{\color{blue}{x-scale} \cdot x-scale}}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(a \cdot b\right)}}{a \cdot b} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]
   10. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(8 \cdot \left(\left(\left(-a\right) \cdot b\right) \cdot b\right)\right) \cdot a\right) \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \frac{\left|{a}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) + {b}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right| + {b}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{x-scale \cdot x-scale}}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(a \cdot b\right)}}{a \cdot b} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   11. Applied rewrites 9.9%

       \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(8 \cdot \left(\left(\left(-a\right) \cdot b\right) \cdot b\right)\right) \cdot a\right) \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \frac{\left|\mathsf{fma}\left({a}^{2}, 0.5 - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right), {b}^{2} \cdot \left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right| + {b}^{2} \cdot \left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{\color{blue}{x-scale} \cdot x-scale}}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(a \cdot b\right)}}{a \cdot b} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 17.3% accurate, 3.1× speedup

Math

\[\begin{array}{l} t_0 := \left(\left(-\left|a\right|\right) \cdot b\right) \cdot b\\ t_1 := {\left(\left|y-scale\right|\right)}^{2}\\ t_2 := \left|a\right| \cdot b\\ t_3 := \left|y-scale\right| \cdot \left|x-scale\right|\\ t_4 := \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ t_5 := 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{if}\;\left|y-scale\right| \leq 7.8 \cdot 10^{+76}:\\ \;\;\;\;-0.25 \cdot \left(\left|a\right| \cdot \left(-1 \cdot \left(\left|x-scale\right| \cdot \left(t\_1 \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{{t\_4}^{4}}{{\left(\left|y-scale\right|\right)}^{4}}} + \frac{{t\_4}^{2}}{t\_1}}{t\_1}}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\frac{\sqrt{\left(\left(\left(8 \cdot t\_0\right) \cdot \left|a\right|\right) \cdot \left(t\_0 \cdot \left|a\right|\right)\right) \cdot \frac{\left|\mathsf{fma}\left({\left(\left|a\right|\right)}^{2}, 0.5 - t\_5, {b}^{2} \cdot \left(0.5 + t\_5\right)\right)\right| + {b}^{2}}{\left|x-scale\right| \cdot \left|x-scale\right|}}}{\left|\left|x-scale\right| \cdot \left|y-scale\right|\right|}}{4 \cdot t\_2}}{t\_2} \cdot t\_3\right) \cdot t\_3\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (* (- (fabs a)) b) b))
        (t_1 (pow (fabs y-scale) 2.0))
        (t_2 (* (fabs a) b))
        (t_3 (* (fabs y-scale) (fabs x-scale)))
        (t_4 (cos (* 0.005555555555555556 (* angle PI))))
        (t_5 (* 0.5 (cos (* 0.011111111111111112 (* angle PI))))))
   (if (<= (fabs y-scale) 7.8e+76)
     (*
      -0.25
      (*
       (fabs a)
       (*
        -1.0
        (*
         (fabs x-scale)
         (*
          t_1
          (sqrt
           (*
            8.0
            (/
             (+
              (sqrt (/ (pow t_4 4.0) (pow (fabs y-scale) 4.0)))
              (/ (pow t_4 2.0) t_1))
             t_1))))))))
     (*
      (*
       (/
        (/
         (/
          (sqrt
           (*
            (* (* (* 8.0 t_0) (fabs a)) (* t_0 (fabs a)))
            (/
             (+
              (fabs
               (fma
                (pow (fabs a) 2.0)
                (- 0.5 t_5)
                (* (pow b 2.0) (+ 0.5 t_5))))
              (pow b 2.0))
             (* (fabs x-scale) (fabs x-scale)))))
          (fabs (* (fabs x-scale) (fabs y-scale))))
         (* 4.0 t_2))
        t_2)
       t_3)
      t_3))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (-fabs(a) * b) * b;
	double t_1 = pow(fabs(y_45_scale), 2.0);
	double t_2 = fabs(a) * b;
	double t_3 = fabs(y_45_scale) * fabs(x_45_scale);
	double t_4 = cos((0.005555555555555556 * (angle * ((double) M_PI))));
	double t_5 = 0.5 * cos((0.011111111111111112 * (angle * ((double) M_PI))));
	double tmp;
	if (fabs(y_45_scale) <= 7.8e+76) {
		tmp = -0.25 * (fabs(a) * (-1.0 * (fabs(x_45_scale) * (t_1 * sqrt((8.0 * ((sqrt((pow(t_4, 4.0) / pow(fabs(y_45_scale), 4.0))) + (pow(t_4, 2.0) / t_1)) / t_1)))))));
	} else {
		tmp = ((((sqrt(((((8.0 * t_0) * fabs(a)) * (t_0 * fabs(a))) * ((fabs(fma(pow(fabs(a), 2.0), (0.5 - t_5), (pow(b, 2.0) * (0.5 + t_5)))) + pow(b, 2.0)) / (fabs(x_45_scale) * fabs(x_45_scale))))) / fabs((fabs(x_45_scale) * fabs(y_45_scale)))) / (4.0 * t_2)) / t_2) * t_3) * t_3;
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(Float64(-abs(a)) * b) * b)
	t_1 = abs(y_45_scale) ^ 2.0
	t_2 = Float64(abs(a) * b)
	t_3 = Float64(abs(y_45_scale) * abs(x_45_scale))
	t_4 = cos(Float64(0.005555555555555556 * Float64(angle * pi)))
	t_5 = Float64(0.5 * cos(Float64(0.011111111111111112 * Float64(angle * pi))))
	tmp = 0.0
	if (abs(y_45_scale) <= 7.8e+76)
		tmp = Float64(-0.25 * Float64(abs(a) * Float64(-1.0 * Float64(abs(x_45_scale) * Float64(t_1 * sqrt(Float64(8.0 * Float64(Float64(sqrt(Float64((t_4 ^ 4.0) / (abs(y_45_scale) ^ 4.0))) + Float64((t_4 ^ 2.0) / t_1)) / t_1))))))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(sqrt(Float64(Float64(Float64(Float64(8.0 * t_0) * abs(a)) * Float64(t_0 * abs(a))) * Float64(Float64(abs(fma((abs(a) ^ 2.0), Float64(0.5 - t_5), Float64((b ^ 2.0) * Float64(0.5 + t_5)))) + (b ^ 2.0)) / Float64(abs(x_45_scale) * abs(x_45_scale))))) / abs(Float64(abs(x_45_scale) * abs(y_45_scale)))) / Float64(4.0 * t_2)) / t_2) * t_3) * t_3);
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[((-N[Abs[a], $MachinePrecision]) * b), $MachinePrecision] * b), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Abs[y$45$scale], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[a], $MachinePrecision] * b), $MachinePrecision]}, Block[{t$95$3 = N[(N[Abs[y$45$scale], $MachinePrecision] * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Cos[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(0.5 * N[Cos[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[y$45$scale], $MachinePrecision], 7.8e+76], N[(-0.25 * N[(N[Abs[a], $MachinePrecision] * N[(-1.0 * N[(N[Abs[x$45$scale], $MachinePrecision] * N[(t$95$1 * N[Sqrt[N[(8.0 * N[(N[(N[Sqrt[N[(N[Power[t$95$4, 4.0], $MachinePrecision] / N[Power[N[Abs[y$45$scale], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[Power[t$95$4, 2.0], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[Sqrt[N[(N[(N[(N[(8.0 * t$95$0), $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Abs[N[(N[Power[N[Abs[a], $MachinePrecision], 2.0], $MachinePrecision] * N[(0.5 - t$95$5), $MachinePrecision] + N[(N[Power[b, 2.0], $MachinePrecision] * N[(0.5 + t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Abs[x$45$scale], $MachinePrecision] * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[N[(N[Abs[x$45$scale], $MachinePrecision] * N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(4.0 * t$95$2), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] * t$95$3), $MachinePrecision] * t$95$3), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 7.79999999999999979e76

   1. Initial program 2.8%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   2. Taylor expanded in a around -inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \frac{a \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}{{b}^{2}}} \]

   4. Applied rewrites 0.7%

      \[\leadsto \color{blue}{-0.25 \cdot \frac{a \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}{{b}^{2}}} \]

   5. Taylor expanded in b around 0

      \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \color{blue}{\left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}\right) \]

   7. Applied rewrites 2.5%

      \[\leadsto -0.25 \cdot \left(a \cdot \color{blue}{\left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}\right) \]

   8. Taylor expanded in x-scale around -inf

      \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \left(-1 \cdot \left(x-scale \cdot \color{blue}{\left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{y-scale}^{4}}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}}{{y-scale}^{2}}}\right)}\right)\right)\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \left(-1 \cdot \left(x-scale \cdot \left({y-scale}^{2} \cdot \color{blue}{\sqrt{8 \cdot \frac{\sqrt{\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{y-scale}^{4}}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}}{{y-scale}^{2}}}}\right)\right)\right)\right) \]

   10. Applied rewrites 5.7%

       \[\leadsto -0.25 \cdot \left(a \cdot \left(-1 \cdot \left(x-scale \cdot \color{blue}{\left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}{{y-scale}^{4}}} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}}{{y-scale}^{2}}}\right)}\right)\right)\right) \]

   if 7.79999999999999979e76 < y-scale

   1. Initial program 2.8%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   3. Applied rewrites 6.3%

      \[\leadsto \color{blue}{\left(\frac{\frac{\sqrt{\left(8 \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \left(\left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right) \cdot \left(\left(\mathsf{hypot}\left(\frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}{y-scale \cdot x-scale}, \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), a \cdot a, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale} - \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), b \cdot b, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(a \cdot a\right)\right)}{y-scale \cdot y-scale}\right) + \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), a \cdot a, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}\right) + \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), b \cdot b, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(a \cdot a\right)\right)}{y-scale \cdot y-scale}\right)\right)}}{\left|y-scale \cdot x-scale\right|}}{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(a \cdot b\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right)} \]

   4. Taylor expanded in x-scale around 0

      \[\leadsto \left(\frac{\frac{\sqrt{\left(8 \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \left(\left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right) \cdot \color{blue}{\frac{\sqrt{{\left({a}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) + {b}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}^{2}} + \left({a}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) + {b}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}{{x-scale}^{2}}}\right)}}{\left|y-scale \cdot x-scale\right|}}{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(a \cdot b\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 6.7%

      \[\leadsto \left(\frac{\frac{\sqrt{\left(8 \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \left(\left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right) \cdot \color{blue}{\frac{\sqrt{{\left(\mathsf{fma}\left({a}^{2}, 0.5 - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right), {b}^{2} \cdot \left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)}^{2}} + \mathsf{fma}\left({a}^{2}, 0.5 - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right), {b}^{2} \cdot \left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}{{x-scale}^{2}}}\right)}}{\left|y-scale \cdot x-scale\right|}}{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(a \cdot b\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   8. Applied rewrites 9.9%

      \[\leadsto \left(\color{blue}{\frac{\frac{\frac{\sqrt{\left(\left(\left(8 \cdot \left(\left(\left(-a\right) \cdot b\right) \cdot b\right)\right) \cdot a\right) \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \frac{\left|\mathsf{fma}\left(0.5 - \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot 0.5, a \cdot a, \mathsf{fma}\left(\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right), 0.5, 0.5\right) \cdot \left(b \cdot b\right)\right)\right| + \mathsf{fma}\left(0.5 - \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot 0.5, a \cdot a, \mathsf{fma}\left(\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right), 0.5, 0.5\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(a \cdot b\right)}}{a \cdot b}} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   9. Taylor expanded in angle around 0

      \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(8 \cdot \left(\left(\left(-a\right) \cdot b\right) \cdot b\right)\right) \cdot a\right) \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \frac{\left|{a}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) + {b}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right| + {b}^{2}}{\color{blue}{x-scale} \cdot x-scale}}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(a \cdot b\right)}}{a \cdot b} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]
   10. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(8 \cdot \left(\left(\left(-a\right) \cdot b\right) \cdot b\right)\right) \cdot a\right) \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \frac{\left|{a}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) + {b}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right| + {b}^{2}}{x-scale \cdot x-scale}}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(a \cdot b\right)}}{a \cdot b} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   11. Applied rewrites 9.9%

       \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(8 \cdot \left(\left(\left(-a\right) \cdot b\right) \cdot b\right)\right) \cdot a\right) \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \frac{\left|\mathsf{fma}\left({a}^{2}, 0.5 - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right), {b}^{2} \cdot \left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right| + {b}^{2}}{\color{blue}{x-scale} \cdot x-scale}}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(a \cdot b\right)}}{a \cdot b} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 15.1% accurate, 3.5× speedup

Math

\[\begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := {\left(\left|y-scale\right|\right)}^{2}\\ t_2 := \sin t\_0\\ t_3 := {\left(\left|x-scale\right|\right)}^{2}\\ t_4 := \cos t\_0\\ \mathbf{if}\;\left|y-scale\right| \leq 7.8 \cdot 10^{+76}:\\ \;\;\;\;-0.25 \cdot \left(\left|a\right| \cdot \left(-1 \cdot \left(\left|x-scale\right| \cdot \left(t\_1 \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{{t\_4}^{4}}{{\left(\left|y-scale\right|\right)}^{4}}} + \frac{{t\_4}^{2}}{t\_1}}{t\_1}}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(\left|a\right| \cdot \left(t\_3 \cdot \left(-1 \cdot \left(\left|y-scale\right| \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{{t\_2}^{4}}{{\left(\left|x-scale\right|\right)}^{4}}} + \frac{{t\_2}^{2}}{t\_3}}{t\_3}}\right)\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI)))
        (t_1 (pow (fabs y-scale) 2.0))
        (t_2 (sin t_0))
        (t_3 (pow (fabs x-scale) 2.0))
        (t_4 (cos t_0)))
   (if (<= (fabs y-scale) 7.8e+76)
     (*
      -0.25
      (*
       (fabs a)
       (*
        -1.0
        (*
         (fabs x-scale)
         (*
          t_1
          (sqrt
           (*
            8.0
            (/
             (+
              (sqrt (/ (pow t_4 4.0) (pow (fabs y-scale) 4.0)))
              (/ (pow t_4 2.0) t_1))
             t_1))))))))
     (*
      -0.25
      (*
       (fabs a)
       (*
        t_3
        (*
         -1.0
         (*
          (fabs y-scale)
          (sqrt
           (*
            8.0
            (/
             (+
              (sqrt (/ (pow t_2 4.0) (pow (fabs x-scale) 4.0)))
              (/ (pow t_2 2.0) t_3))
             t_3)))))))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_1 = pow(fabs(y_45_scale), 2.0);
	double t_2 = sin(t_0);
	double t_3 = pow(fabs(x_45_scale), 2.0);
	double t_4 = cos(t_0);
	double tmp;
	if (fabs(y_45_scale) <= 7.8e+76) {
		tmp = -0.25 * (fabs(a) * (-1.0 * (fabs(x_45_scale) * (t_1 * sqrt((8.0 * ((sqrt((pow(t_4, 4.0) / pow(fabs(y_45_scale), 4.0))) + (pow(t_4, 2.0) / t_1)) / t_1)))))));
	} else {
		tmp = -0.25 * (fabs(a) * (t_3 * (-1.0 * (fabs(y_45_scale) * sqrt((8.0 * ((sqrt((pow(t_2, 4.0) / pow(fabs(x_45_scale), 4.0))) + (pow(t_2, 2.0) / t_3)) / t_3)))))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * Math.PI);
	double t_1 = Math.pow(Math.abs(y_45_scale), 2.0);
	double t_2 = Math.sin(t_0);
	double t_3 = Math.pow(Math.abs(x_45_scale), 2.0);
	double t_4 = Math.cos(t_0);
	double tmp;
	if (Math.abs(y_45_scale) <= 7.8e+76) {
		tmp = -0.25 * (Math.abs(a) * (-1.0 * (Math.abs(x_45_scale) * (t_1 * Math.sqrt((8.0 * ((Math.sqrt((Math.pow(t_4, 4.0) / Math.pow(Math.abs(y_45_scale), 4.0))) + (Math.pow(t_4, 2.0) / t_1)) / t_1)))))));
	} else {
		tmp = -0.25 * (Math.abs(a) * (t_3 * (-1.0 * (Math.abs(y_45_scale) * Math.sqrt((8.0 * ((Math.sqrt((Math.pow(t_2, 4.0) / Math.pow(Math.abs(x_45_scale), 4.0))) + (Math.pow(t_2, 2.0) / t_3)) / t_3)))))));
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = 0.005555555555555556 * (angle * math.pi)
	t_1 = math.pow(math.fabs(y_45_scale), 2.0)
	t_2 = math.sin(t_0)
	t_3 = math.pow(math.fabs(x_45_scale), 2.0)
	t_4 = math.cos(t_0)
	tmp = 0
	if math.fabs(y_45_scale) <= 7.8e+76:
		tmp = -0.25 * (math.fabs(a) * (-1.0 * (math.fabs(x_45_scale) * (t_1 * math.sqrt((8.0 * ((math.sqrt((math.pow(t_4, 4.0) / math.pow(math.fabs(y_45_scale), 4.0))) + (math.pow(t_4, 2.0) / t_1)) / t_1)))))))
	else:
		tmp = -0.25 * (math.fabs(a) * (t_3 * (-1.0 * (math.fabs(y_45_scale) * math.sqrt((8.0 * ((math.sqrt((math.pow(t_2, 4.0) / math.pow(math.fabs(x_45_scale), 4.0))) + (math.pow(t_2, 2.0) / t_3)) / t_3)))))))
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_1 = abs(y_45_scale) ^ 2.0
	t_2 = sin(t_0)
	t_3 = abs(x_45_scale) ^ 2.0
	t_4 = cos(t_0)
	tmp = 0.0
	if (abs(y_45_scale) <= 7.8e+76)
		tmp = Float64(-0.25 * Float64(abs(a) * Float64(-1.0 * Float64(abs(x_45_scale) * Float64(t_1 * sqrt(Float64(8.0 * Float64(Float64(sqrt(Float64((t_4 ^ 4.0) / (abs(y_45_scale) ^ 4.0))) + Float64((t_4 ^ 2.0) / t_1)) / t_1))))))));
	else
		tmp = Float64(-0.25 * Float64(abs(a) * Float64(t_3 * Float64(-1.0 * Float64(abs(y_45_scale) * sqrt(Float64(8.0 * Float64(Float64(sqrt(Float64((t_2 ^ 4.0) / (abs(x_45_scale) ^ 4.0))) + Float64((t_2 ^ 2.0) / t_3)) / t_3))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = 0.005555555555555556 * (angle * pi);
	t_1 = abs(y_45_scale) ^ 2.0;
	t_2 = sin(t_0);
	t_3 = abs(x_45_scale) ^ 2.0;
	t_4 = cos(t_0);
	tmp = 0.0;
	if (abs(y_45_scale) <= 7.8e+76)
		tmp = -0.25 * (abs(a) * (-1.0 * (abs(x_45_scale) * (t_1 * sqrt((8.0 * ((sqrt(((t_4 ^ 4.0) / (abs(y_45_scale) ^ 4.0))) + ((t_4 ^ 2.0) / t_1)) / t_1)))))));
	else
		tmp = -0.25 * (abs(a) * (t_3 * (-1.0 * (abs(y_45_scale) * sqrt((8.0 * ((sqrt(((t_2 ^ 4.0) / (abs(x_45_scale) ^ 4.0))) + ((t_2 ^ 2.0) / t_3)) / t_3)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Abs[y$45$scale], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Abs[x$45$scale], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[Cos[t$95$0], $MachinePrecision]}, If[LessEqual[N[Abs[y$45$scale], $MachinePrecision], 7.8e+76], N[(-0.25 * N[(N[Abs[a], $MachinePrecision] * N[(-1.0 * N[(N[Abs[x$45$scale], $MachinePrecision] * N[(t$95$1 * N[Sqrt[N[(8.0 * N[(N[(N[Sqrt[N[(N[Power[t$95$4, 4.0], $MachinePrecision] / N[Power[N[Abs[y$45$scale], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[Power[t$95$4, 2.0], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.25 * N[(N[Abs[a], $MachinePrecision] * N[(t$95$3 * N[(-1.0 * N[(N[Abs[y$45$scale], $MachinePrecision] * N[Sqrt[N[(8.0 * N[(N[(N[Sqrt[N[(N[Power[t$95$2, 4.0], $MachinePrecision] / N[Power[N[Abs[x$45$scale], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[Power[t$95$2, 2.0], $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 7.79999999999999979e76

   1. Initial program 2.8%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   2. Taylor expanded in a around -inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \frac{a \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}{{b}^{2}}} \]

   4. Applied rewrites 0.7%

      \[\leadsto \color{blue}{-0.25 \cdot \frac{a \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}{{b}^{2}}} \]

   5. Taylor expanded in b around 0

      \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \color{blue}{\left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}\right) \]

   7. Applied rewrites 2.5%

      \[\leadsto -0.25 \cdot \left(a \cdot \color{blue}{\left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}\right) \]

   8. Taylor expanded in x-scale around -inf

      \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \left(-1 \cdot \left(x-scale \cdot \color{blue}{\left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{y-scale}^{4}}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}}{{y-scale}^{2}}}\right)}\right)\right)\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \left(-1 \cdot \left(x-scale \cdot \left({y-scale}^{2} \cdot \color{blue}{\sqrt{8 \cdot \frac{\sqrt{\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{y-scale}^{4}}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}}{{y-scale}^{2}}}}\right)\right)\right)\right) \]

   10. Applied rewrites 5.7%

       \[\leadsto -0.25 \cdot \left(a \cdot \left(-1 \cdot \left(x-scale \cdot \color{blue}{\left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}{{y-scale}^{4}}} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}}{{y-scale}^{2}}}\right)}\right)\right)\right) \]

   if 7.79999999999999979e76 < y-scale

   1. Initial program 2.8%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   2. Taylor expanded in a around -inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \frac{a \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}{{b}^{2}}} \]

   4. Applied rewrites 0.7%

      \[\leadsto \color{blue}{-0.25 \cdot \frac{a \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}{{b}^{2}}} \]

   5. Taylor expanded in b around 0

      \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \color{blue}{\left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}\right) \]

   7. Applied rewrites 2.5%

      \[\leadsto -0.25 \cdot \left(a \cdot \color{blue}{\left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}\right) \]

   8. Taylor expanded in y-scale around -inf

      \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(-1 \cdot \left(y-scale \cdot \color{blue}{\sqrt{8 \cdot \frac{\sqrt{\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{x-scale}^{4}}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}}\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(-1 \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{x-scale}^{4}}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right)\right) \]

   10. Applied rewrites 3.4%

       \[\leadsto -0.25 \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(-1 \cdot \left(y-scale \cdot \color{blue}{\sqrt{8 \cdot \frac{\sqrt{\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}{{x-scale}^{4}}} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}}\right)\right)\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 8.5% accurate, 3.6× speedup

Math

\[\begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := {\left(\left|y-scale\right|\right)}^{2}\\ t_2 := \sin t\_0\\ t_3 := \cos t\_0\\ \mathbf{if}\;\left|y-scale\right| \leq 1.55 \cdot 10^{+66}:\\ \;\;\;\;0.25 \cdot \frac{\left|b\right| \cdot \left(t\_1 \cdot \sqrt{8 \cdot \frac{{\left(\left|a\right|\right)}^{4} \cdot \left(\sqrt{{t\_3}^{4}} + {t\_3}^{2}\right)}{t\_1}}\right)}{{\left(\left|a\right|\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(\left|a\right| \cdot \left({x-scale}^{2} \cdot \left(-1 \cdot \left(\left|y-scale\right| \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{{t\_2}^{4}}{{x-scale}^{4}}} + \frac{{t\_2}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI)))
        (t_1 (pow (fabs y-scale) 2.0))
        (t_2 (sin t_0))
        (t_3 (cos t_0)))
   (if (<= (fabs y-scale) 1.55e+66)
     (*
      0.25
      (/
       (*
        (fabs b)
        (*
         t_1
         (sqrt
          (*
           8.0
           (/
            (* (pow (fabs a) 4.0) (+ (sqrt (pow t_3 4.0)) (pow t_3 2.0)))
            t_1)))))
       (pow (fabs a) 2.0)))
     (*
      -0.25
      (*
       (fabs a)
       (*
        (pow x-scale 2.0)
        (*
         -1.0
         (*
          (fabs y-scale)
          (sqrt
           (*
            8.0
            (/
             (+
              (sqrt (/ (pow t_2 4.0) (pow x-scale 4.0)))
              (/ (pow t_2 2.0) (pow x-scale 2.0)))
             (pow x-scale 2.0))))))))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_1 = pow(fabs(y_45_scale), 2.0);
	double t_2 = sin(t_0);
	double t_3 = cos(t_0);
	double tmp;
	if (fabs(y_45_scale) <= 1.55e+66) {
		tmp = 0.25 * ((fabs(b) * (t_1 * sqrt((8.0 * ((pow(fabs(a), 4.0) * (sqrt(pow(t_3, 4.0)) + pow(t_3, 2.0))) / t_1))))) / pow(fabs(a), 2.0));
	} else {
		tmp = -0.25 * (fabs(a) * (pow(x_45_scale, 2.0) * (-1.0 * (fabs(y_45_scale) * sqrt((8.0 * ((sqrt((pow(t_2, 4.0) / pow(x_45_scale, 4.0))) + (pow(t_2, 2.0) / pow(x_45_scale, 2.0))) / pow(x_45_scale, 2.0))))))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * Math.PI);
	double t_1 = Math.pow(Math.abs(y_45_scale), 2.0);
	double t_2 = Math.sin(t_0);
	double t_3 = Math.cos(t_0);
	double tmp;
	if (Math.abs(y_45_scale) <= 1.55e+66) {
		tmp = 0.25 * ((Math.abs(b) * (t_1 * Math.sqrt((8.0 * ((Math.pow(Math.abs(a), 4.0) * (Math.sqrt(Math.pow(t_3, 4.0)) + Math.pow(t_3, 2.0))) / t_1))))) / Math.pow(Math.abs(a), 2.0));
	} else {
		tmp = -0.25 * (Math.abs(a) * (Math.pow(x_45_scale, 2.0) * (-1.0 * (Math.abs(y_45_scale) * Math.sqrt((8.0 * ((Math.sqrt((Math.pow(t_2, 4.0) / Math.pow(x_45_scale, 4.0))) + (Math.pow(t_2, 2.0) / Math.pow(x_45_scale, 2.0))) / Math.pow(x_45_scale, 2.0))))))));
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = 0.005555555555555556 * (angle * math.pi)
	t_1 = math.pow(math.fabs(y_45_scale), 2.0)
	t_2 = math.sin(t_0)
	t_3 = math.cos(t_0)
	tmp = 0
	if math.fabs(y_45_scale) <= 1.55e+66:
		tmp = 0.25 * ((math.fabs(b) * (t_1 * math.sqrt((8.0 * ((math.pow(math.fabs(a), 4.0) * (math.sqrt(math.pow(t_3, 4.0)) + math.pow(t_3, 2.0))) / t_1))))) / math.pow(math.fabs(a), 2.0))
	else:
		tmp = -0.25 * (math.fabs(a) * (math.pow(x_45_scale, 2.0) * (-1.0 * (math.fabs(y_45_scale) * math.sqrt((8.0 * ((math.sqrt((math.pow(t_2, 4.0) / math.pow(x_45_scale, 4.0))) + (math.pow(t_2, 2.0) / math.pow(x_45_scale, 2.0))) / math.pow(x_45_scale, 2.0))))))))
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_1 = abs(y_45_scale) ^ 2.0
	t_2 = sin(t_0)
	t_3 = cos(t_0)
	tmp = 0.0
	if (abs(y_45_scale) <= 1.55e+66)
		tmp = Float64(0.25 * Float64(Float64(abs(b) * Float64(t_1 * sqrt(Float64(8.0 * Float64(Float64((abs(a) ^ 4.0) * Float64(sqrt((t_3 ^ 4.0)) + (t_3 ^ 2.0))) / t_1))))) / (abs(a) ^ 2.0)));
	else
		tmp = Float64(-0.25 * Float64(abs(a) * Float64((x_45_scale ^ 2.0) * Float64(-1.0 * Float64(abs(y_45_scale) * sqrt(Float64(8.0 * Float64(Float64(sqrt(Float64((t_2 ^ 4.0) / (x_45_scale ^ 4.0))) + Float64((t_2 ^ 2.0) / (x_45_scale ^ 2.0))) / (x_45_scale ^ 2.0)))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = 0.005555555555555556 * (angle * pi);
	t_1 = abs(y_45_scale) ^ 2.0;
	t_2 = sin(t_0);
	t_3 = cos(t_0);
	tmp = 0.0;
	if (abs(y_45_scale) <= 1.55e+66)
		tmp = 0.25 * ((abs(b) * (t_1 * sqrt((8.0 * (((abs(a) ^ 4.0) * (sqrt((t_3 ^ 4.0)) + (t_3 ^ 2.0))) / t_1))))) / (abs(a) ^ 2.0));
	else
		tmp = -0.25 * (abs(a) * ((x_45_scale ^ 2.0) * (-1.0 * (abs(y_45_scale) * sqrt((8.0 * ((sqrt(((t_2 ^ 4.0) / (x_45_scale ^ 4.0))) + ((t_2 ^ 2.0) / (x_45_scale ^ 2.0))) / (x_45_scale ^ 2.0))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Abs[y$45$scale], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$3 = N[Cos[t$95$0], $MachinePrecision]}, If[LessEqual[N[Abs[y$45$scale], $MachinePrecision], 1.55e+66], N[(0.25 * N[(N[(N[Abs[b], $MachinePrecision] * N[(t$95$1 * N[Sqrt[N[(8.0 * N[(N[(N[Power[N[Abs[a], $MachinePrecision], 4.0], $MachinePrecision] * N[(N[Sqrt[N[Power[t$95$3, 4.0], $MachinePrecision]], $MachinePrecision] + N[Power[t$95$3, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[Abs[a], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.25 * N[(N[Abs[a], $MachinePrecision] * N[(N[Power[x$45$scale, 2.0], $MachinePrecision] * N[(-1.0 * N[(N[Abs[y$45$scale], $MachinePrecision] * N[Sqrt[N[(8.0 * N[(N[(N[Sqrt[N[(N[Power[t$95$2, 4.0], $MachinePrecision] / N[Power[x$45$scale, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[Power[t$95$2, 2.0], $MachinePrecision] / N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 1.55000000000000009e66

   1. Initial program 2.8%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   2. Taylor expanded in x-scale around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left({b}^{4} \cdot \left(\sqrt{{\left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}^{2}} + \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)}{{y-scale}^{2}}}}{{a}^{2} \cdot {b}^{2}}} \]

   4. Applied rewrites 1.2%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(\mathsf{fma}\left({a}^{2}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}^{2}} + \mathsf{fma}\left({a}^{2}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)\right)}{{y-scale}^{2}}}}{{a}^{2} \cdot {b}^{2}}} \]

   5. Taylor expanded in b around inf

      \[\leadsto \frac{1}{4} \cdot \frac{b \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{{y-scale}^{2}}}\right)}{\color{blue}{{a}^{2}}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{b \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{{y-scale}^{2}}}\right)}{{a}^{\color{blue}{2}}} \]

   7. Applied rewrites 2.8%

      \[\leadsto 0.25 \cdot \frac{b \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{y-scale}^{2}}}\right)}{\color{blue}{{a}^{2}}} \]

   if 1.55000000000000009e66 < y-scale

   1. Initial program 2.8%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   2. Taylor expanded in a around -inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \frac{a \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}{{b}^{2}}} \]

   4. Applied rewrites 0.7%

      \[\leadsto \color{blue}{-0.25 \cdot \frac{a \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}{{b}^{2}}} \]

   5. Taylor expanded in b around 0

      \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \color{blue}{\left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}\right) \]

   7. Applied rewrites 2.5%

      \[\leadsto -0.25 \cdot \left(a \cdot \color{blue}{\left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}\right) \]

   8. Taylor expanded in y-scale around -inf

      \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(-1 \cdot \left(y-scale \cdot \color{blue}{\sqrt{8 \cdot \frac{\sqrt{\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{x-scale}^{4}}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}}\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(-1 \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{x-scale}^{4}}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right)\right) \]

   10. Applied rewrites 3.4%

       \[\leadsto -0.25 \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(-1 \cdot \left(y-scale \cdot \color{blue}{\sqrt{8 \cdot \frac{\sqrt{\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}{{x-scale}^{4}}} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}}\right)\right)\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 7.5% accurate, 3.6× speedup

Math

\[\begin{array}{l} t_0 := {\left(\left|y-scale\right|\right)}^{2}\\ t_1 := \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{if}\;\left|y-scale\right| \leq 7.6 \cdot 10^{+111}:\\ \;\;\;\;0.25 \cdot \frac{\left|b\right| \cdot \left(t\_0 \cdot \sqrt{8 \cdot \frac{{\left(\left|a\right|\right)}^{4} \cdot \left(\sqrt{{t\_1}^{4}} + {t\_1}^{2}\right)}{t\_0}}\right)}{{\left(\left|a\right|\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(\left|a\right| \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \left(\left|y-scale\right| \cdot \sqrt{8 \cdot \frac{\sqrt{9.525986892242036 \cdot 10^{-10} \cdot \frac{{\pi}^{4}}{{x-scale}^{4}}} + 3.08641975308642 \cdot 10^{-5} \cdot \frac{{\pi}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (pow (fabs y-scale) 2.0))
        (t_1 (cos (* 0.005555555555555556 (* angle PI)))))
   (if (<= (fabs y-scale) 7.6e+111)
     (*
      0.25
      (/
       (*
        (fabs b)
        (*
         t_0
         (sqrt
          (*
           8.0
           (/
            (* (pow (fabs a) 4.0) (+ (sqrt (pow t_1 4.0)) (pow t_1 2.0)))
            t_0)))))
       (pow (fabs a) 2.0)))
     (*
      -0.25
      (*
       (fabs a)
       (*
        (pow x-scale 2.0)
        (*
         angle
         (*
          (fabs y-scale)
          (sqrt
           (*
            8.0
            (/
             (+
              (sqrt
               (* 9.525986892242036e-10 (/ (pow PI 4.0) (pow x-scale 4.0))))
              (* 3.08641975308642e-5 (/ (pow PI 2.0) (pow x-scale 2.0))))
             (pow x-scale 2.0))))))))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = pow(fabs(y_45_scale), 2.0);
	double t_1 = cos((0.005555555555555556 * (angle * ((double) M_PI))));
	double tmp;
	if (fabs(y_45_scale) <= 7.6e+111) {
		tmp = 0.25 * ((fabs(b) * (t_0 * sqrt((8.0 * ((pow(fabs(a), 4.0) * (sqrt(pow(t_1, 4.0)) + pow(t_1, 2.0))) / t_0))))) / pow(fabs(a), 2.0));
	} else {
		tmp = -0.25 * (fabs(a) * (pow(x_45_scale, 2.0) * (angle * (fabs(y_45_scale) * sqrt((8.0 * ((sqrt((9.525986892242036e-10 * (pow(((double) M_PI), 4.0) / pow(x_45_scale, 4.0)))) + (3.08641975308642e-5 * (pow(((double) M_PI), 2.0) / pow(x_45_scale, 2.0)))) / pow(x_45_scale, 2.0))))))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = Math.pow(Math.abs(y_45_scale), 2.0);
	double t_1 = Math.cos((0.005555555555555556 * (angle * Math.PI)));
	double tmp;
	if (Math.abs(y_45_scale) <= 7.6e+111) {
		tmp = 0.25 * ((Math.abs(b) * (t_0 * Math.sqrt((8.0 * ((Math.pow(Math.abs(a), 4.0) * (Math.sqrt(Math.pow(t_1, 4.0)) + Math.pow(t_1, 2.0))) / t_0))))) / Math.pow(Math.abs(a), 2.0));
	} else {
		tmp = -0.25 * (Math.abs(a) * (Math.pow(x_45_scale, 2.0) * (angle * (Math.abs(y_45_scale) * Math.sqrt((8.0 * ((Math.sqrt((9.525986892242036e-10 * (Math.pow(Math.PI, 4.0) / Math.pow(x_45_scale, 4.0)))) + (3.08641975308642e-5 * (Math.pow(Math.PI, 2.0) / Math.pow(x_45_scale, 2.0)))) / Math.pow(x_45_scale, 2.0))))))));
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = math.pow(math.fabs(y_45_scale), 2.0)
	t_1 = math.cos((0.005555555555555556 * (angle * math.pi)))
	tmp = 0
	if math.fabs(y_45_scale) <= 7.6e+111:
		tmp = 0.25 * ((math.fabs(b) * (t_0 * math.sqrt((8.0 * ((math.pow(math.fabs(a), 4.0) * (math.sqrt(math.pow(t_1, 4.0)) + math.pow(t_1, 2.0))) / t_0))))) / math.pow(math.fabs(a), 2.0))
	else:
		tmp = -0.25 * (math.fabs(a) * (math.pow(x_45_scale, 2.0) * (angle * (math.fabs(y_45_scale) * math.sqrt((8.0 * ((math.sqrt((9.525986892242036e-10 * (math.pow(math.pi, 4.0) / math.pow(x_45_scale, 4.0)))) + (3.08641975308642e-5 * (math.pow(math.pi, 2.0) / math.pow(x_45_scale, 2.0)))) / math.pow(x_45_scale, 2.0))))))))
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = abs(y_45_scale) ^ 2.0
	t_1 = cos(Float64(0.005555555555555556 * Float64(angle * pi)))
	tmp = 0.0
	if (abs(y_45_scale) <= 7.6e+111)
		tmp = Float64(0.25 * Float64(Float64(abs(b) * Float64(t_0 * sqrt(Float64(8.0 * Float64(Float64((abs(a) ^ 4.0) * Float64(sqrt((t_1 ^ 4.0)) + (t_1 ^ 2.0))) / t_0))))) / (abs(a) ^ 2.0)));
	else
		tmp = Float64(-0.25 * Float64(abs(a) * Float64((x_45_scale ^ 2.0) * Float64(angle * Float64(abs(y_45_scale) * sqrt(Float64(8.0 * Float64(Float64(sqrt(Float64(9.525986892242036e-10 * Float64((pi ^ 4.0) / (x_45_scale ^ 4.0)))) + Float64(3.08641975308642e-5 * Float64((pi ^ 2.0) / (x_45_scale ^ 2.0)))) / (x_45_scale ^ 2.0)))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = abs(y_45_scale) ^ 2.0;
	t_1 = cos((0.005555555555555556 * (angle * pi)));
	tmp = 0.0;
	if (abs(y_45_scale) <= 7.6e+111)
		tmp = 0.25 * ((abs(b) * (t_0 * sqrt((8.0 * (((abs(a) ^ 4.0) * (sqrt((t_1 ^ 4.0)) + (t_1 ^ 2.0))) / t_0))))) / (abs(a) ^ 2.0));
	else
		tmp = -0.25 * (abs(a) * ((x_45_scale ^ 2.0) * (angle * (abs(y_45_scale) * sqrt((8.0 * ((sqrt((9.525986892242036e-10 * ((pi ^ 4.0) / (x_45_scale ^ 4.0)))) + (3.08641975308642e-5 * ((pi ^ 2.0) / (x_45_scale ^ 2.0)))) / (x_45_scale ^ 2.0))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[Power[N[Abs[y$45$scale], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[y$45$scale], $MachinePrecision], 7.6e+111], N[(0.25 * N[(N[(N[Abs[b], $MachinePrecision] * N[(t$95$0 * N[Sqrt[N[(8.0 * N[(N[(N[Power[N[Abs[a], $MachinePrecision], 4.0], $MachinePrecision] * N[(N[Sqrt[N[Power[t$95$1, 4.0], $MachinePrecision]], $MachinePrecision] + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[Abs[a], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.25 * N[(N[Abs[a], $MachinePrecision] * N[(N[Power[x$45$scale, 2.0], $MachinePrecision] * N[(angle * N[(N[Abs[y$45$scale], $MachinePrecision] * N[Sqrt[N[(8.0 * N[(N[(N[Sqrt[N[(9.525986892242036e-10 * N[(N[Power[Pi, 4.0], $MachinePrecision] / N[Power[x$45$scale, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(3.08641975308642e-5 * N[(N[Power[Pi, 2.0], $MachinePrecision] / N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 7.59999999999999951e111

   1. Initial program 2.8%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   2. Taylor expanded in x-scale around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left({b}^{4} \cdot \left(\sqrt{{\left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}^{2}} + \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)}{{y-scale}^{2}}}}{{a}^{2} \cdot {b}^{2}}} \]

   4. Applied rewrites 1.2%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(\mathsf{fma}\left({a}^{2}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}^{2}} + \mathsf{fma}\left({a}^{2}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)\right)}{{y-scale}^{2}}}}{{a}^{2} \cdot {b}^{2}}} \]

   5. Taylor expanded in b around inf

      \[\leadsto \frac{1}{4} \cdot \frac{b \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{{y-scale}^{2}}}\right)}{\color{blue}{{a}^{2}}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{b \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{{y-scale}^{2}}}\right)}{{a}^{\color{blue}{2}}} \]

   7. Applied rewrites 2.8%

      \[\leadsto 0.25 \cdot \frac{b \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{y-scale}^{2}}}\right)}{\color{blue}{{a}^{2}}} \]

   if 7.59999999999999951e111 < y-scale

   1. Initial program 2.8%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   2. Taylor expanded in a around -inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \frac{a \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}{{b}^{2}}} \]

   4. Applied rewrites 0.7%

      \[\leadsto \color{blue}{-0.25 \cdot \frac{a \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}{{b}^{2}}} \]

   5. Taylor expanded in b around 0

      \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \color{blue}{\left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}\right) \]

   7. Applied rewrites 2.5%

      \[\leadsto -0.25 \cdot \left(a \cdot \color{blue}{\left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}\right) \]

   8. Taylor expanded in y-scale around inf

      \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{x-scale}^{4}}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{x-scale}^{4}}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right) \]

   10. Applied rewrites 3.5%

       \[\leadsto -0.25 \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}{{x-scale}^{4}}} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right) \]

   11. Taylor expanded in angle around 0

       \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{1}{1049760000} \cdot \frac{{\mathsf{PI}\left(\right)}^{4}}{{x-scale}^{4}}} + \frac{1}{32400} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right)\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{1}{1049760000} \cdot \frac{{\mathsf{PI}\left(\right)}^{4}}{{x-scale}^{4}}} + \frac{1}{32400} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right)\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{1}{1049760000} \cdot \frac{{\mathsf{PI}\left(\right)}^{4}}{{x-scale}^{4}}} + \frac{1}{32400} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right)\right) \]

       3. lower-sqrt.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{1}{1049760000} \cdot \frac{{\mathsf{PI}\left(\right)}^{4}}{{x-scale}^{4}}} + \frac{1}{32400} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right)\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{1}{1049760000} \cdot \frac{{\mathsf{PI}\left(\right)}^{4}}{{x-scale}^{4}}} + \frac{1}{32400} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right)\right) \]

   13. Applied rewrites 4.5%

       \[\leadsto -0.25 \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{\sqrt{9.525986892242036 \cdot 10^{-10} \cdot \frac{{\pi}^{4}}{{x-scale}^{4}}} + 3.08641975308642 \cdot 10^{-5} \cdot \frac{{\pi}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 6.8% accurate, 3.6× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{if}\;angle \leq -2.15 \cdot 10^{-188}:\\ \;\;\;\;-0.25 \cdot \left(\left|a\right| \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \left(\left|y-scale\right| \cdot \sqrt{8 \cdot \frac{\sqrt{9.525986892242036 \cdot 10^{-10} \cdot \frac{{\pi}^{4}}{{x-scale}^{4}}} + 3.08641975308642 \cdot 10^{-5} \cdot \frac{{\pi}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{{\left(\left|b\right|\right)}^{3} \cdot \left(\left|y-scale\right| \cdot \sqrt{8 \cdot \left({\left(\left|a\right|\right)}^{4} \cdot \left(\sqrt{{t\_0}^{4}} + {t\_0}^{2}\right)\right)}\right)}{{\left(\left|a\right|\right)}^{2} \cdot {\left(\left|b\right|\right)}^{2}}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (cos (* 0.005555555555555556 (* angle PI)))))
   (if (<= angle -2.15e-188)
     (*
      -0.25
      (*
       (fabs a)
       (*
        (pow x-scale 2.0)
        (*
         angle
         (*
          (fabs y-scale)
          (sqrt
           (*
            8.0
            (/
             (+
              (sqrt
               (* 9.525986892242036e-10 (/ (pow PI 4.0) (pow x-scale 4.0))))
              (* 3.08641975308642e-5 (/ (pow PI 2.0) (pow x-scale 2.0))))
             (pow x-scale 2.0)))))))))
     (*
      0.25
      (/
       (*
        (pow (fabs b) 3.0)
        (*
         (fabs y-scale)
         (sqrt
          (*
           8.0
           (* (pow (fabs a) 4.0) (+ (sqrt (pow t_0 4.0)) (pow t_0 2.0)))))))
       (* (pow (fabs a) 2.0) (pow (fabs b) 2.0)))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = cos((0.005555555555555556 * (angle * ((double) M_PI))));
	double tmp;
	if (angle <= -2.15e-188) {
		tmp = -0.25 * (fabs(a) * (pow(x_45_scale, 2.0) * (angle * (fabs(y_45_scale) * sqrt((8.0 * ((sqrt((9.525986892242036e-10 * (pow(((double) M_PI), 4.0) / pow(x_45_scale, 4.0)))) + (3.08641975308642e-5 * (pow(((double) M_PI), 2.0) / pow(x_45_scale, 2.0)))) / pow(x_45_scale, 2.0))))))));
	} else {
		tmp = 0.25 * ((pow(fabs(b), 3.0) * (fabs(y_45_scale) * sqrt((8.0 * (pow(fabs(a), 4.0) * (sqrt(pow(t_0, 4.0)) + pow(t_0, 2.0))))))) / (pow(fabs(a), 2.0) * pow(fabs(b), 2.0)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = Math.cos((0.005555555555555556 * (angle * Math.PI)));
	double tmp;
	if (angle <= -2.15e-188) {
		tmp = -0.25 * (Math.abs(a) * (Math.pow(x_45_scale, 2.0) * (angle * (Math.abs(y_45_scale) * Math.sqrt((8.0 * ((Math.sqrt((9.525986892242036e-10 * (Math.pow(Math.PI, 4.0) / Math.pow(x_45_scale, 4.0)))) + (3.08641975308642e-5 * (Math.pow(Math.PI, 2.0) / Math.pow(x_45_scale, 2.0)))) / Math.pow(x_45_scale, 2.0))))))));
	} else {
		tmp = 0.25 * ((Math.pow(Math.abs(b), 3.0) * (Math.abs(y_45_scale) * Math.sqrt((8.0 * (Math.pow(Math.abs(a), 4.0) * (Math.sqrt(Math.pow(t_0, 4.0)) + Math.pow(t_0, 2.0))))))) / (Math.pow(Math.abs(a), 2.0) * Math.pow(Math.abs(b), 2.0)));
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = math.cos((0.005555555555555556 * (angle * math.pi)))
	tmp = 0
	if angle <= -2.15e-188:
		tmp = -0.25 * (math.fabs(a) * (math.pow(x_45_scale, 2.0) * (angle * (math.fabs(y_45_scale) * math.sqrt((8.0 * ((math.sqrt((9.525986892242036e-10 * (math.pow(math.pi, 4.0) / math.pow(x_45_scale, 4.0)))) + (3.08641975308642e-5 * (math.pow(math.pi, 2.0) / math.pow(x_45_scale, 2.0)))) / math.pow(x_45_scale, 2.0))))))))
	else:
		tmp = 0.25 * ((math.pow(math.fabs(b), 3.0) * (math.fabs(y_45_scale) * math.sqrt((8.0 * (math.pow(math.fabs(a), 4.0) * (math.sqrt(math.pow(t_0, 4.0)) + math.pow(t_0, 2.0))))))) / (math.pow(math.fabs(a), 2.0) * math.pow(math.fabs(b), 2.0)))
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = cos(Float64(0.005555555555555556 * Float64(angle * pi)))
	tmp = 0.0
	if (angle <= -2.15e-188)
		tmp = Float64(-0.25 * Float64(abs(a) * Float64((x_45_scale ^ 2.0) * Float64(angle * Float64(abs(y_45_scale) * sqrt(Float64(8.0 * Float64(Float64(sqrt(Float64(9.525986892242036e-10 * Float64((pi ^ 4.0) / (x_45_scale ^ 4.0)))) + Float64(3.08641975308642e-5 * Float64((pi ^ 2.0) / (x_45_scale ^ 2.0)))) / (x_45_scale ^ 2.0)))))))));
	else
		tmp = Float64(0.25 * Float64(Float64((abs(b) ^ 3.0) * Float64(abs(y_45_scale) * sqrt(Float64(8.0 * Float64((abs(a) ^ 4.0) * Float64(sqrt((t_0 ^ 4.0)) + (t_0 ^ 2.0))))))) / Float64((abs(a) ^ 2.0) * (abs(b) ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = cos((0.005555555555555556 * (angle * pi)));
	tmp = 0.0;
	if (angle <= -2.15e-188)
		tmp = -0.25 * (abs(a) * ((x_45_scale ^ 2.0) * (angle * (abs(y_45_scale) * sqrt((8.0 * ((sqrt((9.525986892242036e-10 * ((pi ^ 4.0) / (x_45_scale ^ 4.0)))) + (3.08641975308642e-5 * ((pi ^ 2.0) / (x_45_scale ^ 2.0)))) / (x_45_scale ^ 2.0))))))));
	else
		tmp = 0.25 * (((abs(b) ^ 3.0) * (abs(y_45_scale) * sqrt((8.0 * ((abs(a) ^ 4.0) * (sqrt((t_0 ^ 4.0)) + (t_0 ^ 2.0))))))) / ((abs(a) ^ 2.0) * (abs(b) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[Cos[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[angle, -2.15e-188], N[(-0.25 * N[(N[Abs[a], $MachinePrecision] * N[(N[Power[x$45$scale, 2.0], $MachinePrecision] * N[(angle * N[(N[Abs[y$45$scale], $MachinePrecision] * N[Sqrt[N[(8.0 * N[(N[(N[Sqrt[N[(9.525986892242036e-10 * N[(N[Power[Pi, 4.0], $MachinePrecision] / N[Power[x$45$scale, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(3.08641975308642e-5 * N[(N[Power[Pi, 2.0], $MachinePrecision] / N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(N[Power[N[Abs[b], $MachinePrecision], 3.0], $MachinePrecision] * N[(N[Abs[y$45$scale], $MachinePrecision] * N[Sqrt[N[(8.0 * N[(N[Power[N[Abs[a], $MachinePrecision], 4.0], $MachinePrecision] * N[(N[Sqrt[N[Power[t$95$0, 4.0], $MachinePrecision]], $MachinePrecision] + N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[Abs[a], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[N[Abs[b], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if angle < -2.14999999999999994e-188

   1. Initial program 2.8%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   2. Taylor expanded in a around -inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \frac{a \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}{{b}^{2}}} \]

   4. Applied rewrites 0.7%

      \[\leadsto \color{blue}{-0.25 \cdot \frac{a \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}{{b}^{2}}} \]

   5. Taylor expanded in b around 0

      \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \color{blue}{\left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}\right) \]

   7. Applied rewrites 2.5%

      \[\leadsto -0.25 \cdot \left(a \cdot \color{blue}{\left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}\right) \]

   8. Taylor expanded in y-scale around inf

      \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{x-scale}^{4}}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{x-scale}^{4}}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right) \]

   10. Applied rewrites 3.5%

       \[\leadsto -0.25 \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}{{x-scale}^{4}}} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right) \]

   11. Taylor expanded in angle around 0

       \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{1}{1049760000} \cdot \frac{{\mathsf{PI}\left(\right)}^{4}}{{x-scale}^{4}}} + \frac{1}{32400} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right)\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{1}{1049760000} \cdot \frac{{\mathsf{PI}\left(\right)}^{4}}{{x-scale}^{4}}} + \frac{1}{32400} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right)\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{1}{1049760000} \cdot \frac{{\mathsf{PI}\left(\right)}^{4}}{{x-scale}^{4}}} + \frac{1}{32400} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right)\right) \]

       3. lower-sqrt.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{1}{1049760000} \cdot \frac{{\mathsf{PI}\left(\right)}^{4}}{{x-scale}^{4}}} + \frac{1}{32400} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right)\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{1}{1049760000} \cdot \frac{{\mathsf{PI}\left(\right)}^{4}}{{x-scale}^{4}}} + \frac{1}{32400} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right)\right) \]

   13. Applied rewrites 4.5%

       \[\leadsto -0.25 \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{\sqrt{9.525986892242036 \cdot 10^{-10} \cdot \frac{{\pi}^{4}}{{x-scale}^{4}}} + 3.08641975308642 \cdot 10^{-5} \cdot \frac{{\pi}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right)\right) \]

   if -2.14999999999999994e-188 < angle

   1. Initial program 2.8%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   2. Taylor expanded in x-scale around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left({b}^{4} \cdot \left(\sqrt{{\left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}^{2}} + \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)}{{y-scale}^{2}}}}{{a}^{2} \cdot {b}^{2}}} \]

   4. Applied rewrites 1.2%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(\mathsf{fma}\left({a}^{2}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}^{2}} + \mathsf{fma}\left({a}^{2}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)\right)}{{y-scale}^{2}}}}{{a}^{2} \cdot {b}^{2}}} \]

   5. Taylor expanded in b around inf

      \[\leadsto \frac{1}{4} \cdot \frac{{b}^{3} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{{y-scale}^{2}}}\right)}{\color{blue}{{a}^{2}} \cdot {b}^{2}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{{b}^{3} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{{y-scale}^{2}}}\right)}{{a}^{\color{blue}{2}} \cdot {b}^{2}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{{b}^{3} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{{y-scale}^{2}}}\right)}{{a}^{2} \cdot {b}^{2}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{{b}^{3} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{{y-scale}^{2}}}\right)}{{a}^{2} \cdot {b}^{2}} \]

   7. Applied rewrites 0.8%

      \[\leadsto 0.25 \cdot \frac{{b}^{3} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{y-scale}^{2}}}\right)}{\color{blue}{{a}^{2}} \cdot {b}^{2}} \]

   8. Taylor expanded in y-scale around 0

      \[\leadsto \frac{1}{4} \cdot \frac{{b}^{3} \cdot \left(y-scale \cdot \sqrt{8 \cdot \left({a}^{4} \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)}\right)}{{a}^{2} \cdot {b}^{2}} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{{b}^{3} \cdot \left(y-scale \cdot \sqrt{8 \cdot \left({a}^{4} \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)}\right)}{{a}^{2} \cdot {b}^{2}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{{b}^{3} \cdot \left(y-scale \cdot \sqrt{8 \cdot \left({a}^{4} \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)}\right)}{{a}^{2} \cdot {b}^{2}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{{b}^{3} \cdot \left(y-scale \cdot \sqrt{8 \cdot \left({a}^{4} \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)}\right)}{{a}^{2} \cdot {b}^{2}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{{b}^{3} \cdot \left(y-scale \cdot \sqrt{8 \cdot \left({a}^{4} \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)}\right)}{{a}^{2} \cdot {b}^{2}} \]

   10. Applied rewrites 2.2%

       \[\leadsto 0.25 \cdot \frac{{b}^{3} \cdot \left(y-scale \cdot \sqrt{8 \cdot \left({a}^{4} \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)}{{a}^{2} \cdot {b}^{2}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 6.6% accurate, 3.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;angle \leq -2.15 \cdot 10^{-188}:\\ \;\;\;\;-0.25 \cdot \left(\left|a\right| \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \left(\left|y-scale\right| \cdot \sqrt{8 \cdot \frac{\sqrt{9.525986892242036 \cdot 10^{-10} \cdot \frac{{\pi}^{4}}{{x-scale}^{4}}} + 3.08641975308642 \cdot 10^{-5} \cdot \frac{{\pi}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{\left(\frac{\sqrt{8 \cdot \left(\mathsf{fma}\left(\cos \left(\left(\left(\pi + \pi\right) \cdot angle\right) \cdot 0.005555555555555556\right) + 1, 0.5, \sqrt{{\cos \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}^{4}}\right) \cdot {\left(\left|a\right|\right)}^{4}\right)}}{\left|\left|y-scale\right|\right|} \cdot \left(\left|y-scale\right| \cdot \left|y-scale\right|\right)\right) \cdot \left(\left(\left|b\right| \cdot \left|b\right|\right) \cdot \left|b\right|\right)}{{\left(\left|a\right|\right)}^{2} \cdot {\left(\left|b\right|\right)}^{2}}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= angle -2.15e-188)
   (*
    -0.25
    (*
     (fabs a)
     (*
      (pow x-scale 2.0)
      (*
       angle
       (*
        (fabs y-scale)
        (sqrt
         (*
          8.0
          (/
           (+
            (sqrt (* 9.525986892242036e-10 (/ (pow PI 4.0) (pow x-scale 4.0))))
            (* 3.08641975308642e-5 (/ (pow PI 2.0) (pow x-scale 2.0))))
           (pow x-scale 2.0)))))))))
   (*
    0.25
    (/
     (*
      (*
       (/
        (sqrt
         (*
          8.0
          (*
           (fma
            (+ (cos (* (* (+ PI PI) angle) 0.005555555555555556)) 1.0)
            0.5
            (sqrt (pow (cos (* (* angle PI) 0.005555555555555556)) 4.0)))
           (pow (fabs a) 4.0))))
        (fabs (fabs y-scale)))
       (* (fabs y-scale) (fabs y-scale)))
      (* (* (fabs b) (fabs b)) (fabs b)))
     (* (pow (fabs a) 2.0) (pow (fabs b) 2.0))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (angle <= -2.15e-188) {
		tmp = -0.25 * (fabs(a) * (pow(x_45_scale, 2.0) * (angle * (fabs(y_45_scale) * sqrt((8.0 * ((sqrt((9.525986892242036e-10 * (pow(((double) M_PI), 4.0) / pow(x_45_scale, 4.0)))) + (3.08641975308642e-5 * (pow(((double) M_PI), 2.0) / pow(x_45_scale, 2.0)))) / pow(x_45_scale, 2.0))))))));
	} else {
		tmp = 0.25 * ((((sqrt((8.0 * (fma((cos((((((double) M_PI) + ((double) M_PI)) * angle) * 0.005555555555555556)) + 1.0), 0.5, sqrt(pow(cos(((angle * ((double) M_PI)) * 0.005555555555555556)), 4.0))) * pow(fabs(a), 4.0)))) / fabs(fabs(y_45_scale))) * (fabs(y_45_scale) * fabs(y_45_scale))) * ((fabs(b) * fabs(b)) * fabs(b))) / (pow(fabs(a), 2.0) * pow(fabs(b), 2.0)));
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (angle <= -2.15e-188)
		tmp = Float64(-0.25 * Float64(abs(a) * Float64((x_45_scale ^ 2.0) * Float64(angle * Float64(abs(y_45_scale) * sqrt(Float64(8.0 * Float64(Float64(sqrt(Float64(9.525986892242036e-10 * Float64((pi ^ 4.0) / (x_45_scale ^ 4.0)))) + Float64(3.08641975308642e-5 * Float64((pi ^ 2.0) / (x_45_scale ^ 2.0)))) / (x_45_scale ^ 2.0)))))))));
	else
		tmp = Float64(0.25 * Float64(Float64(Float64(Float64(sqrt(Float64(8.0 * Float64(fma(Float64(cos(Float64(Float64(Float64(pi + pi) * angle) * 0.005555555555555556)) + 1.0), 0.5, sqrt((cos(Float64(Float64(angle * pi) * 0.005555555555555556)) ^ 4.0))) * (abs(a) ^ 4.0)))) / abs(abs(y_45_scale))) * Float64(abs(y_45_scale) * abs(y_45_scale))) * Float64(Float64(abs(b) * abs(b)) * abs(b))) / Float64((abs(a) ^ 2.0) * (abs(b) ^ 2.0))));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[angle, -2.15e-188], N[(-0.25 * N[(N[Abs[a], $MachinePrecision] * N[(N[Power[x$45$scale, 2.0], $MachinePrecision] * N[(angle * N[(N[Abs[y$45$scale], $MachinePrecision] * N[Sqrt[N[(8.0 * N[(N[(N[Sqrt[N[(9.525986892242036e-10 * N[(N[Power[Pi, 4.0], $MachinePrecision] / N[Power[x$45$scale, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(3.08641975308642e-5 * N[(N[Power[Pi, 2.0], $MachinePrecision] / N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(N[(N[(N[Sqrt[N[(8.0 * N[(N[(N[(N[Cos[N[(N[(N[(Pi + Pi), $MachinePrecision] * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] * 0.5 + N[Sqrt[N[Power[N[Cos[N[(N[(angle * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision], 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Power[N[Abs[a], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[N[Abs[y$45$scale], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[y$45$scale], $MachinePrecision] * N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Abs[b], $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[Abs[a], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[N[Abs[b], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < -2.14999999999999994e-188

   1. Initial program 2.8%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   2. Taylor expanded in a around -inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \frac{a \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}{{b}^{2}}} \]

   4. Applied rewrites 0.7%

      \[\leadsto \color{blue}{-0.25 \cdot \frac{a \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}{{b}^{2}}} \]

   5. Taylor expanded in b around 0

      \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \color{blue}{\left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}\right) \]

   7. Applied rewrites 2.5%

      \[\leadsto -0.25 \cdot \left(a \cdot \color{blue}{\left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}\right) \]

   8. Taylor expanded in y-scale around inf

      \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{x-scale}^{4}}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{x-scale}^{4}}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right) \]

   10. Applied rewrites 3.5%

       \[\leadsto -0.25 \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}{{x-scale}^{4}}} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right) \]

   11. Taylor expanded in angle around 0

       \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{1}{1049760000} \cdot \frac{{\mathsf{PI}\left(\right)}^{4}}{{x-scale}^{4}}} + \frac{1}{32400} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right)\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{1}{1049760000} \cdot \frac{{\mathsf{PI}\left(\right)}^{4}}{{x-scale}^{4}}} + \frac{1}{32400} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right)\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{1}{1049760000} \cdot \frac{{\mathsf{PI}\left(\right)}^{4}}{{x-scale}^{4}}} + \frac{1}{32400} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right)\right) \]

       3. lower-sqrt.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{1}{1049760000} \cdot \frac{{\mathsf{PI}\left(\right)}^{4}}{{x-scale}^{4}}} + \frac{1}{32400} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right)\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{1}{1049760000} \cdot \frac{{\mathsf{PI}\left(\right)}^{4}}{{x-scale}^{4}}} + \frac{1}{32400} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right)\right) \]

   13. Applied rewrites 4.5%

       \[\leadsto -0.25 \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{\sqrt{9.525986892242036 \cdot 10^{-10} \cdot \frac{{\pi}^{4}}{{x-scale}^{4}}} + 3.08641975308642 \cdot 10^{-5} \cdot \frac{{\pi}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right)\right) \]

   if -2.14999999999999994e-188 < angle

   1. Initial program 2.8%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   2. Taylor expanded in x-scale around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left({b}^{4} \cdot \left(\sqrt{{\left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}^{2}} + \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)}{{y-scale}^{2}}}}{{a}^{2} \cdot {b}^{2}}} \]

   4. Applied rewrites 1.2%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(\mathsf{fma}\left({a}^{2}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}^{2}} + \mathsf{fma}\left({a}^{2}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)\right)}{{y-scale}^{2}}}}{{a}^{2} \cdot {b}^{2}}} \]

   5. Taylor expanded in b around inf

      \[\leadsto \frac{1}{4} \cdot \frac{{b}^{3} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{{y-scale}^{2}}}\right)}{\color{blue}{{a}^{2}} \cdot {b}^{2}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{{b}^{3} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{{y-scale}^{2}}}\right)}{{a}^{\color{blue}{2}} \cdot {b}^{2}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{{b}^{3} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{{y-scale}^{2}}}\right)}{{a}^{2} \cdot {b}^{2}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{{b}^{3} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{{y-scale}^{2}}}\right)}{{a}^{2} \cdot {b}^{2}} \]

   7. Applied rewrites 0.8%

      \[\leadsto 0.25 \cdot \frac{{b}^{3} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{y-scale}^{2}}}\right)}{\color{blue}{{a}^{2}} \cdot {b}^{2}} \]

   9. Applied rewrites 1.5%

      \[\leadsto 0.25 \cdot \frac{\left(\frac{\sqrt{8 \cdot \left(\mathsf{fma}\left(\cos \left(\left(\left(\pi + \pi\right) \cdot angle\right) \cdot 0.005555555555555556\right) + 1, 0.5, \sqrt{{\cos \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}^{4}}\right) \cdot {a}^{4}\right)}}{\left|y-scale\right|} \cdot \left(y-scale \cdot y-scale\right)\right) \cdot \left(\left(b \cdot b\right) \cdot b\right)}{{a}^{\color{blue}{2}} \cdot {b}^{2}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 6.1% accurate, 4.2× speedup

Math

\[\begin{array}{l} t_0 := \left|b\right| \cdot \left|b\right|\\ \mathbf{if}\;angle \leq -1.3 \cdot 10^{-187}:\\ \;\;\;\;-0.25 \cdot \left(\left|a\right| \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \left(\left|y-scale\right| \cdot \sqrt{8 \cdot \frac{\sqrt{9.525986892242036 \cdot 10^{-10} \cdot \frac{{\pi}^{4}}{{x-scale}^{4}}} + 3.08641975308642 \cdot 10^{-5} \cdot \frac{{\pi}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{\frac{\left(\left(t\_0 \cdot \left|b\right|\right) \cdot \left(\left|y-scale\right| \cdot \left|y-scale\right|\right)\right) \cdot \frac{\sqrt{8 \cdot \left(\mathsf{fma}\left(\cos \left(\left(\left(\pi + \pi\right) \cdot angle\right) \cdot 0.005555555555555556\right) + 1, 0.5, \sqrt{{\cos \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}^{4}}\right) \cdot {\left(\left|a\right|\right)}^{4}\right)}}{\left|\left|y-scale\right|\right|}}{\left|a\right| \cdot \left|a\right|}}{t\_0}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (fabs b) (fabs b))))
   (if (<= angle -1.3e-187)
     (*
      -0.25
      (*
       (fabs a)
       (*
        (pow x-scale 2.0)
        (*
         angle
         (*
          (fabs y-scale)
          (sqrt
           (*
            8.0
            (/
             (+
              (sqrt
               (* 9.525986892242036e-10 (/ (pow PI 4.0) (pow x-scale 4.0))))
              (* 3.08641975308642e-5 (/ (pow PI 2.0) (pow x-scale 2.0))))
             (pow x-scale 2.0)))))))))
     (*
      0.25
      (/
       (/
        (*
         (* (* t_0 (fabs b)) (* (fabs y-scale) (fabs y-scale)))
         (/
          (sqrt
           (*
            8.0
            (*
             (fma
              (+ (cos (* (* (+ PI PI) angle) 0.005555555555555556)) 1.0)
              0.5
              (sqrt (pow (cos (* (* angle PI) 0.005555555555555556)) 4.0)))
             (pow (fabs a) 4.0))))
          (fabs (fabs y-scale))))
        (* (fabs a) (fabs a)))
       t_0)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = fabs(b) * fabs(b);
	double tmp;
	if (angle <= -1.3e-187) {
		tmp = -0.25 * (fabs(a) * (pow(x_45_scale, 2.0) * (angle * (fabs(y_45_scale) * sqrt((8.0 * ((sqrt((9.525986892242036e-10 * (pow(((double) M_PI), 4.0) / pow(x_45_scale, 4.0)))) + (3.08641975308642e-5 * (pow(((double) M_PI), 2.0) / pow(x_45_scale, 2.0)))) / pow(x_45_scale, 2.0))))))));
	} else {
		tmp = 0.25 * (((((t_0 * fabs(b)) * (fabs(y_45_scale) * fabs(y_45_scale))) * (sqrt((8.0 * (fma((cos((((((double) M_PI) + ((double) M_PI)) * angle) * 0.005555555555555556)) + 1.0), 0.5, sqrt(pow(cos(((angle * ((double) M_PI)) * 0.005555555555555556)), 4.0))) * pow(fabs(a), 4.0)))) / fabs(fabs(y_45_scale)))) / (fabs(a) * fabs(a))) / t_0);
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(abs(b) * abs(b))
	tmp = 0.0
	if (angle <= -1.3e-187)
		tmp = Float64(-0.25 * Float64(abs(a) * Float64((x_45_scale ^ 2.0) * Float64(angle * Float64(abs(y_45_scale) * sqrt(Float64(8.0 * Float64(Float64(sqrt(Float64(9.525986892242036e-10 * Float64((pi ^ 4.0) / (x_45_scale ^ 4.0)))) + Float64(3.08641975308642e-5 * Float64((pi ^ 2.0) / (x_45_scale ^ 2.0)))) / (x_45_scale ^ 2.0)))))))));
	else
		tmp = Float64(0.25 * Float64(Float64(Float64(Float64(Float64(t_0 * abs(b)) * Float64(abs(y_45_scale) * abs(y_45_scale))) * Float64(sqrt(Float64(8.0 * Float64(fma(Float64(cos(Float64(Float64(Float64(pi + pi) * angle) * 0.005555555555555556)) + 1.0), 0.5, sqrt((cos(Float64(Float64(angle * pi) * 0.005555555555555556)) ^ 4.0))) * (abs(a) ^ 4.0)))) / abs(abs(y_45_scale)))) / Float64(abs(a) * abs(a))) / t_0));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[Abs[b], $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[angle, -1.3e-187], N[(-0.25 * N[(N[Abs[a], $MachinePrecision] * N[(N[Power[x$45$scale, 2.0], $MachinePrecision] * N[(angle * N[(N[Abs[y$45$scale], $MachinePrecision] * N[Sqrt[N[(8.0 * N[(N[(N[Sqrt[N[(9.525986892242036e-10 * N[(N[Power[Pi, 4.0], $MachinePrecision] / N[Power[x$45$scale, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(3.08641975308642e-5 * N[(N[Power[Pi, 2.0], $MachinePrecision] / N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(N[(N[(N[(t$95$0 * N[Abs[b], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[y$45$scale], $MachinePrecision] * N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(8.0 * N[(N[(N[(N[Cos[N[(N[(N[(Pi + Pi), $MachinePrecision] * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] * 0.5 + N[Sqrt[N[Power[N[Cos[N[(N[(angle * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision], 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Power[N[Abs[a], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[N[Abs[y$45$scale], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Abs[a], $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if angle < -1.3e-187

   1. Initial program 2.8%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   2. Taylor expanded in a around -inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \frac{a \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}{{b}^{2}}} \]

   4. Applied rewrites 0.7%

      \[\leadsto \color{blue}{-0.25 \cdot \frac{a \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}{{b}^{2}}} \]

   5. Taylor expanded in b around 0

      \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \color{blue}{\left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}\right) \]

   7. Applied rewrites 2.5%

      \[\leadsto -0.25 \cdot \left(a \cdot \color{blue}{\left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}\right) \]

   8. Taylor expanded in y-scale around inf

      \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{x-scale}^{4}}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{x-scale}^{4}}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right) \]

   10. Applied rewrites 3.5%

       \[\leadsto -0.25 \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}{{x-scale}^{4}}} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right) \]

   11. Taylor expanded in angle around 0

       \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{1}{1049760000} \cdot \frac{{\mathsf{PI}\left(\right)}^{4}}{{x-scale}^{4}}} + \frac{1}{32400} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right)\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{1}{1049760000} \cdot \frac{{\mathsf{PI}\left(\right)}^{4}}{{x-scale}^{4}}} + \frac{1}{32400} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right)\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{1}{1049760000} \cdot \frac{{\mathsf{PI}\left(\right)}^{4}}{{x-scale}^{4}}} + \frac{1}{32400} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right)\right) \]

       3. lower-sqrt.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{1}{1049760000} \cdot \frac{{\mathsf{PI}\left(\right)}^{4}}{{x-scale}^{4}}} + \frac{1}{32400} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right)\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{1}{1049760000} \cdot \frac{{\mathsf{PI}\left(\right)}^{4}}{{x-scale}^{4}}} + \frac{1}{32400} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right)\right) \]

   13. Applied rewrites 4.5%

       \[\leadsto -0.25 \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{\sqrt{9.525986892242036 \cdot 10^{-10} \cdot \frac{{\pi}^{4}}{{x-scale}^{4}}} + 3.08641975308642 \cdot 10^{-5} \cdot \frac{{\pi}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right)\right) \]

   if -1.3e-187 < angle

   1. Initial program 2.8%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   2. Taylor expanded in x-scale around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left({b}^{4} \cdot \left(\sqrt{{\left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}^{2}} + \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)}{{y-scale}^{2}}}}{{a}^{2} \cdot {b}^{2}}} \]

   4. Applied rewrites 1.2%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(\mathsf{fma}\left({a}^{2}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}^{2}} + \mathsf{fma}\left({a}^{2}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)\right)}{{y-scale}^{2}}}}{{a}^{2} \cdot {b}^{2}}} \]

   5. Taylor expanded in b around inf

      \[\leadsto \frac{1}{4} \cdot \frac{{b}^{3} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{{y-scale}^{2}}}\right)}{\color{blue}{{a}^{2}} \cdot {b}^{2}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{{b}^{3} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{{y-scale}^{2}}}\right)}{{a}^{\color{blue}{2}} \cdot {b}^{2}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{{b}^{3} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{{y-scale}^{2}}}\right)}{{a}^{2} \cdot {b}^{2}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{{b}^{3} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{{y-scale}^{2}}}\right)}{{a}^{2} \cdot {b}^{2}} \]

   7. Applied rewrites 0.8%

      \[\leadsto 0.25 \cdot \frac{{b}^{3} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{y-scale}^{2}}}\right)}{\color{blue}{{a}^{2}} \cdot {b}^{2}} \]

   9. Applied rewrites 2.0%

      \[\leadsto 0.25 \cdot \frac{\frac{\left(\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(y-scale \cdot y-scale\right)\right) \cdot \frac{\sqrt{8 \cdot \left(\mathsf{fma}\left(\cos \left(\left(\left(\pi + \pi\right) \cdot angle\right) \cdot 0.005555555555555556\right) + 1, 0.5, \sqrt{{\cos \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}^{4}}\right) \cdot {a}^{4}\right)}}{\left|y-scale\right|}}{a \cdot a}}{\color{blue}{b \cdot b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 6.1% accurate, 4.2× speedup

Math

\[\begin{array}{l} t_0 := \left|b\right| \cdot \left|b\right|\\ \mathbf{if}\;angle \leq -2.15 \cdot 10^{-188}:\\ \;\;\;\;-0.25 \cdot \left(\left|a\right| \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \left(\left|y-scale\right| \cdot \sqrt{8 \cdot \frac{\sqrt{9.525986892242036 \cdot 10^{-10} \cdot \frac{{\pi}^{4}}{{x-scale}^{4}}} + 3.08641975308642 \cdot 10^{-5} \cdot \frac{{\pi}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25 \cdot \left(\left(\left(t\_0 \cdot \left|b\right|\right) \cdot \left(\left|y-scale\right| \cdot \left|y-scale\right|\right)\right) \cdot \frac{\sqrt{8 \cdot \left(\mathsf{fma}\left(\cos \left(\left(\left(\pi + \pi\right) \cdot angle\right) \cdot 0.005555555555555556\right) + 1, 0.5, \sqrt{{\cos \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}^{4}}\right) \cdot {\left(\left|a\right|\right)}^{4}\right)}}{\left|\left|y-scale\right|\right|}\right)}{\left(\left|a\right| \cdot \left|a\right|\right) \cdot t\_0}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (fabs b) (fabs b))))
   (if (<= angle -2.15e-188)
     (*
      -0.25
      (*
       (fabs a)
       (*
        (pow x-scale 2.0)
        (*
         angle
         (*
          (fabs y-scale)
          (sqrt
           (*
            8.0
            (/
             (+
              (sqrt
               (* 9.525986892242036e-10 (/ (pow PI 4.0) (pow x-scale 4.0))))
              (* 3.08641975308642e-5 (/ (pow PI 2.0) (pow x-scale 2.0))))
             (pow x-scale 2.0)))))))))
     (/
      (*
       0.25
       (*
        (* (* t_0 (fabs b)) (* (fabs y-scale) (fabs y-scale)))
        (/
         (sqrt
          (*
           8.0
           (*
            (fma
             (+ (cos (* (* (+ PI PI) angle) 0.005555555555555556)) 1.0)
             0.5
             (sqrt (pow (cos (* (* angle PI) 0.005555555555555556)) 4.0)))
            (pow (fabs a) 4.0))))
         (fabs (fabs y-scale)))))
      (* (* (fabs a) (fabs a)) t_0)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = fabs(b) * fabs(b);
	double tmp;
	if (angle <= -2.15e-188) {
		tmp = -0.25 * (fabs(a) * (pow(x_45_scale, 2.0) * (angle * (fabs(y_45_scale) * sqrt((8.0 * ((sqrt((9.525986892242036e-10 * (pow(((double) M_PI), 4.0) / pow(x_45_scale, 4.0)))) + (3.08641975308642e-5 * (pow(((double) M_PI), 2.0) / pow(x_45_scale, 2.0)))) / pow(x_45_scale, 2.0))))))));
	} else {
		tmp = (0.25 * (((t_0 * fabs(b)) * (fabs(y_45_scale) * fabs(y_45_scale))) * (sqrt((8.0 * (fma((cos((((((double) M_PI) + ((double) M_PI)) * angle) * 0.005555555555555556)) + 1.0), 0.5, sqrt(pow(cos(((angle * ((double) M_PI)) * 0.005555555555555556)), 4.0))) * pow(fabs(a), 4.0)))) / fabs(fabs(y_45_scale))))) / ((fabs(a) * fabs(a)) * t_0);
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(abs(b) * abs(b))
	tmp = 0.0
	if (angle <= -2.15e-188)
		tmp = Float64(-0.25 * Float64(abs(a) * Float64((x_45_scale ^ 2.0) * Float64(angle * Float64(abs(y_45_scale) * sqrt(Float64(8.0 * Float64(Float64(sqrt(Float64(9.525986892242036e-10 * Float64((pi ^ 4.0) / (x_45_scale ^ 4.0)))) + Float64(3.08641975308642e-5 * Float64((pi ^ 2.0) / (x_45_scale ^ 2.0)))) / (x_45_scale ^ 2.0)))))))));
	else
		tmp = Float64(Float64(0.25 * Float64(Float64(Float64(t_0 * abs(b)) * Float64(abs(y_45_scale) * abs(y_45_scale))) * Float64(sqrt(Float64(8.0 * Float64(fma(Float64(cos(Float64(Float64(Float64(pi + pi) * angle) * 0.005555555555555556)) + 1.0), 0.5, sqrt((cos(Float64(Float64(angle * pi) * 0.005555555555555556)) ^ 4.0))) * (abs(a) ^ 4.0)))) / abs(abs(y_45_scale))))) / Float64(Float64(abs(a) * abs(a)) * t_0));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[Abs[b], $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[angle, -2.15e-188], N[(-0.25 * N[(N[Abs[a], $MachinePrecision] * N[(N[Power[x$45$scale, 2.0], $MachinePrecision] * N[(angle * N[(N[Abs[y$45$scale], $MachinePrecision] * N[Sqrt[N[(8.0 * N[(N[(N[Sqrt[N[(9.525986892242036e-10 * N[(N[Power[Pi, 4.0], $MachinePrecision] / N[Power[x$45$scale, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(3.08641975308642e-5 * N[(N[Power[Pi, 2.0], $MachinePrecision] / N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.25 * N[(N[(N[(t$95$0 * N[Abs[b], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[y$45$scale], $MachinePrecision] * N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(8.0 * N[(N[(N[(N[Cos[N[(N[(N[(Pi + Pi), $MachinePrecision] * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] * 0.5 + N[Sqrt[N[Power[N[Cos[N[(N[(angle * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision], 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Power[N[Abs[a], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[N[Abs[y$45$scale], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Abs[a], $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if angle < -2.14999999999999994e-188

   1. Initial program 2.8%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   2. Taylor expanded in a around -inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \frac{a \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}{{b}^{2}}} \]

   4. Applied rewrites 0.7%

      \[\leadsto \color{blue}{-0.25 \cdot \frac{a \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}{{b}^{2}}} \]

   5. Taylor expanded in b around 0

      \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \color{blue}{\left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}\right) \]

   7. Applied rewrites 2.5%

      \[\leadsto -0.25 \cdot \left(a \cdot \color{blue}{\left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}\right) \]

   8. Taylor expanded in y-scale around inf

      \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{x-scale}^{4}}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{x-scale}^{4}}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right) \]

   10. Applied rewrites 3.5%

       \[\leadsto -0.25 \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}{{x-scale}^{4}}} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right) \]

   11. Taylor expanded in angle around 0

       \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{1}{1049760000} \cdot \frac{{\mathsf{PI}\left(\right)}^{4}}{{x-scale}^{4}}} + \frac{1}{32400} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right)\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{1}{1049760000} \cdot \frac{{\mathsf{PI}\left(\right)}^{4}}{{x-scale}^{4}}} + \frac{1}{32400} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right)\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{1}{1049760000} \cdot \frac{{\mathsf{PI}\left(\right)}^{4}}{{x-scale}^{4}}} + \frac{1}{32400} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right)\right) \]

       3. lower-sqrt.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{1}{1049760000} \cdot \frac{{\mathsf{PI}\left(\right)}^{4}}{{x-scale}^{4}}} + \frac{1}{32400} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right)\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{1}{1049760000} \cdot \frac{{\mathsf{PI}\left(\right)}^{4}}{{x-scale}^{4}}} + \frac{1}{32400} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right)\right) \]

   13. Applied rewrites 4.5%

       \[\leadsto -0.25 \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{\sqrt{9.525986892242036 \cdot 10^{-10} \cdot \frac{{\pi}^{4}}{{x-scale}^{4}}} + 3.08641975308642 \cdot 10^{-5} \cdot \frac{{\pi}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right)\right) \]

   if -2.14999999999999994e-188 < angle

   1. Initial program 2.8%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   2. Taylor expanded in x-scale around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left({b}^{4} \cdot \left(\sqrt{{\left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}^{2}} + \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)}{{y-scale}^{2}}}}{{a}^{2} \cdot {b}^{2}}} \]

   4. Applied rewrites 1.2%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(\mathsf{fma}\left({a}^{2}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}^{2}} + \mathsf{fma}\left({a}^{2}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)\right)}{{y-scale}^{2}}}}{{a}^{2} \cdot {b}^{2}}} \]

   5. Taylor expanded in b around inf

      \[\leadsto \frac{1}{4} \cdot \frac{{b}^{3} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{{y-scale}^{2}}}\right)}{\color{blue}{{a}^{2}} \cdot {b}^{2}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{{b}^{3} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{{y-scale}^{2}}}\right)}{{a}^{\color{blue}{2}} \cdot {b}^{2}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{{b}^{3} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{{y-scale}^{2}}}\right)}{{a}^{2} \cdot {b}^{2}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{{b}^{3} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{{y-scale}^{2}}}\right)}{{a}^{2} \cdot {b}^{2}} \]

   7. Applied rewrites 0.8%

      \[\leadsto 0.25 \cdot \frac{{b}^{3} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{y-scale}^{2}}}\right)}{\color{blue}{{a}^{2}} \cdot {b}^{2}} \]

   9. Applied rewrites 1.5%

      \[\leadsto \frac{0.25 \cdot \left(\left(\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(y-scale \cdot y-scale\right)\right) \cdot \frac{\sqrt{8 \cdot \left(\mathsf{fma}\left(\cos \left(\left(\left(\pi + \pi\right) \cdot angle\right) \cdot 0.005555555555555556\right) + 1, 0.5, \sqrt{{\cos \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}^{4}}\right) \cdot {a}^{4}\right)}}{\left|y-scale\right|}\right)}{\color{blue}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 5.6% accurate, 4.6× speedup

Math

\[\begin{array}{l} t_0 := \frac{\left|a\right|}{\left|y-scale\right|}\\ t_1 := \frac{b \cdot b}{x-scale \cdot x-scale}\\ t_2 := \left(b \cdot \left|a\right|\right) \cdot \left(b \cdot \left(-\left|a\right|\right)\right)\\ t_3 := \frac{4 \cdot t\_2}{{\left(x-scale \cdot \left|y-scale\right|\right)}^{2}}\\ \mathbf{if}\;angle \leq -6.8 \cdot 10^{-270}:\\ \;\;\;\;-0.25 \cdot \left(\left|a\right| \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \left(\left|y-scale\right| \cdot \sqrt{8 \cdot \frac{\sqrt{9.525986892242036 \cdot 10^{-10} \cdot \frac{{\pi}^{4}}{{x-scale}^{4}}} + 3.08641975308642 \cdot 10^{-5} \cdot \frac{{\pi}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(2 \cdot t\_3\right) \cdot t\_2\right) \cdot \left(\mathsf{fma}\left(t\_0, t\_0, t\_1\right) + \left|t\_1 - \frac{\left|a\right| \cdot \left|a\right|}{\left|y-scale\right| \cdot \left|y-scale\right|}\right|\right)}}{t\_3}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ (fabs a) (fabs y-scale)))
        (t_1 (/ (* b b) (* x-scale x-scale)))
        (t_2 (* (* b (fabs a)) (* b (- (fabs a)))))
        (t_3 (/ (* 4.0 t_2) (pow (* x-scale (fabs y-scale)) 2.0))))
   (if (<= angle -6.8e-270)
     (*
      -0.25
      (*
       (fabs a)
       (*
        (pow x-scale 2.0)
        (*
         angle
         (*
          (fabs y-scale)
          (sqrt
           (*
            8.0
            (/
             (+
              (sqrt
               (* 9.525986892242036e-10 (/ (pow PI 4.0) (pow x-scale 4.0))))
              (* 3.08641975308642e-5 (/ (pow PI 2.0) (pow x-scale 2.0))))
             (pow x-scale 2.0)))))))))
     (/
      (-
       (sqrt
        (*
         (* (* 2.0 t_3) t_2)
         (+
          (fma t_0 t_0 t_1)
          (fabs
           (-
            t_1
            (/ (* (fabs a) (fabs a)) (* (fabs y-scale) (fabs y-scale)))))))))
      t_3))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = fabs(a) / fabs(y_45_scale);
	double t_1 = (b * b) / (x_45_scale * x_45_scale);
	double t_2 = (b * fabs(a)) * (b * -fabs(a));
	double t_3 = (4.0 * t_2) / pow((x_45_scale * fabs(y_45_scale)), 2.0);
	double tmp;
	if (angle <= -6.8e-270) {
		tmp = -0.25 * (fabs(a) * (pow(x_45_scale, 2.0) * (angle * (fabs(y_45_scale) * sqrt((8.0 * ((sqrt((9.525986892242036e-10 * (pow(((double) M_PI), 4.0) / pow(x_45_scale, 4.0)))) + (3.08641975308642e-5 * (pow(((double) M_PI), 2.0) / pow(x_45_scale, 2.0)))) / pow(x_45_scale, 2.0))))))));
	} else {
		tmp = -sqrt((((2.0 * t_3) * t_2) * (fma(t_0, t_0, t_1) + fabs((t_1 - ((fabs(a) * fabs(a)) / (fabs(y_45_scale) * fabs(y_45_scale)))))))) / t_3;
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(abs(a) / abs(y_45_scale))
	t_1 = Float64(Float64(b * b) / Float64(x_45_scale * x_45_scale))
	t_2 = Float64(Float64(b * abs(a)) * Float64(b * Float64(-abs(a))))
	t_3 = Float64(Float64(4.0 * t_2) / (Float64(x_45_scale * abs(y_45_scale)) ^ 2.0))
	tmp = 0.0
	if (angle <= -6.8e-270)
		tmp = Float64(-0.25 * Float64(abs(a) * Float64((x_45_scale ^ 2.0) * Float64(angle * Float64(abs(y_45_scale) * sqrt(Float64(8.0 * Float64(Float64(sqrt(Float64(9.525986892242036e-10 * Float64((pi ^ 4.0) / (x_45_scale ^ 4.0)))) + Float64(3.08641975308642e-5 * Float64((pi ^ 2.0) / (x_45_scale ^ 2.0)))) / (x_45_scale ^ 2.0)))))))));
	else
		tmp = Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * t_3) * t_2) * Float64(fma(t_0, t_0, t_1) + abs(Float64(t_1 - Float64(Float64(abs(a) * abs(a)) / Float64(abs(y_45_scale) * abs(y_45_scale))))))))) / t_3);
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[Abs[a], $MachinePrecision] / N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b * b), $MachinePrecision] / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(b * N[Abs[a], $MachinePrecision]), $MachinePrecision] * N[(b * (-N[Abs[a], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(4.0 * t$95$2), $MachinePrecision] / N[Power[N[(x$45$scale * N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[angle, -6.8e-270], N[(-0.25 * N[(N[Abs[a], $MachinePrecision] * N[(N[Power[x$45$scale, 2.0], $MachinePrecision] * N[(angle * N[(N[Abs[y$45$scale], $MachinePrecision] * N[Sqrt[N[(8.0 * N[(N[(N[Sqrt[N[(9.525986892242036e-10 * N[(N[Power[Pi, 4.0], $MachinePrecision] / N[Power[x$45$scale, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(3.08641975308642e-5 * N[(N[Power[Pi, 2.0], $MachinePrecision] / N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[N[(N[(N[(2.0 * t$95$3), $MachinePrecision] * t$95$2), $MachinePrecision] * N[(N[(t$95$0 * t$95$0 + t$95$1), $MachinePrecision] + N[Abs[N[(t$95$1 - N[(N[(N[Abs[a], $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision] / N[(N[Abs[y$45$scale], $MachinePrecision] * N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$3), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if angle < -6.8000000000000001e-270

   1. Initial program 2.8%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   2. Taylor expanded in a around -inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \frac{a \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}{{b}^{2}}} \]

   4. Applied rewrites 0.7%

      \[\leadsto \color{blue}{-0.25 \cdot \frac{a \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}{{b}^{2}}} \]

   5. Taylor expanded in b around 0

      \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \color{blue}{\left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}\right) \]

   7. Applied rewrites 2.5%

      \[\leadsto -0.25 \cdot \left(a \cdot \color{blue}{\left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}\right) \]

   8. Taylor expanded in y-scale around inf

      \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{x-scale}^{4}}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{x-scale}^{4}}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right) \]

   10. Applied rewrites 3.5%

       \[\leadsto -0.25 \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}{{x-scale}^{4}}} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right) \]

   11. Taylor expanded in angle around 0

       \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{1}{1049760000} \cdot \frac{{\mathsf{PI}\left(\right)}^{4}}{{x-scale}^{4}}} + \frac{1}{32400} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right)\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{1}{1049760000} \cdot \frac{{\mathsf{PI}\left(\right)}^{4}}{{x-scale}^{4}}} + \frac{1}{32400} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right)\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{1}{1049760000} \cdot \frac{{\mathsf{PI}\left(\right)}^{4}}{{x-scale}^{4}}} + \frac{1}{32400} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right)\right) \]

       3. lower-sqrt.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{1}{1049760000} \cdot \frac{{\mathsf{PI}\left(\right)}^{4}}{{x-scale}^{4}}} + \frac{1}{32400} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right)\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{1}{1049760000} \cdot \frac{{\mathsf{PI}\left(\right)}^{4}}{{x-scale}^{4}}} + \frac{1}{32400} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right)\right) \]

   13. Applied rewrites 4.5%

       \[\leadsto -0.25 \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{\sqrt{9.525986892242036 \cdot 10^{-10} \cdot \frac{{\pi}^{4}}{{x-scale}^{4}}} + 3.08641975308642 \cdot 10^{-5} \cdot \frac{{\pi}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right)\right) \]

   if -6.8000000000000001e-270 < angle

   1. Initial program 2.8%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   2. Taylor expanded in angle around 0

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \color{blue}{\left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   3. Step-by-step derivation

   4. Applied rewrites 4.6%

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \color{blue}{\left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \color{blue}{\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) + \color{blue}{\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      3. lower-+.f64 4.6

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right) + \color{blue}{\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   6. Applied rewrites 4.8%

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\mathsf{fma}\left(\frac{a}{y-scale}, \frac{a}{y-scale}, \frac{b \cdot b}{x-scale \cdot x-scale}\right) + \color{blue}{\left|\frac{b \cdot b}{x-scale \cdot x-scale} - \frac{a \cdot a}{y-scale \cdot y-scale}\right|}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 5.5% accurate, 5.2× speedup

Math

\[\begin{array}{l} t_0 := x-scale \cdot \left|y-scale\right|\\ t_1 := t\_0 \cdot t\_0\\ t_2 := \frac{\left|a\right|}{\left|y-scale\right|}\\ t_3 := b \cdot \left|a\right|\\ t_4 := \frac{b \cdot b}{x-scale \cdot x-scale}\\ t_5 := -\left|a\right|\\ t_6 := \left(t\_3 \cdot b\right) \cdot t\_5\\ \mathbf{if}\;angle \leq -6.8 \cdot 10^{-270}:\\ \;\;\;\;-0.25 \cdot \left(\left|a\right| \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \left(\left|y-scale\right| \cdot \sqrt{8 \cdot \frac{\sqrt{9.525986892242036 \cdot 10^{-10} \cdot \frac{{\pi}^{4}}{{x-scale}^{4}}} + 3.08641975308642 \cdot 10^{-5} \cdot \frac{{\pi}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(\mathsf{fma}\left(t\_2, t\_2, t\_4\right) + \left|t\_4 - \frac{\left|a\right| \cdot \left|a\right|}{\left|y-scale\right| \cdot \left|y-scale\right|}\right|\right) \cdot \left(\left(\left(4 \cdot \frac{t\_6}{t\_1}\right) \cdot 2\right) \cdot t\_6\right)}}{\left(4 \cdot t\_3\right) \cdot \left(b \cdot t\_5\right)} \cdot t\_1\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* x-scale (fabs y-scale)))
        (t_1 (* t_0 t_0))
        (t_2 (/ (fabs a) (fabs y-scale)))
        (t_3 (* b (fabs a)))
        (t_4 (/ (* b b) (* x-scale x-scale)))
        (t_5 (- (fabs a)))
        (t_6 (* (* t_3 b) t_5)))
   (if (<= angle -6.8e-270)
     (*
      -0.25
      (*
       (fabs a)
       (*
        (pow x-scale 2.0)
        (*
         angle
         (*
          (fabs y-scale)
          (sqrt
           (*
            8.0
            (/
             (+
              (sqrt
               (* 9.525986892242036e-10 (/ (pow PI 4.0) (pow x-scale 4.0))))
              (* 3.08641975308642e-5 (/ (pow PI 2.0) (pow x-scale 2.0))))
             (pow x-scale 2.0)))))))))
     (*
      (/
       (-
        (sqrt
         (*
          (+
           (fma t_2 t_2 t_4)
           (fabs
            (-
             t_4
             (/ (* (fabs a) (fabs a)) (* (fabs y-scale) (fabs y-scale))))))
          (* (* (* 4.0 (/ t_6 t_1)) 2.0) t_6))))
       (* (* 4.0 t_3) (* b t_5)))
      t_1))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = x_45_scale * fabs(y_45_scale);
	double t_1 = t_0 * t_0;
	double t_2 = fabs(a) / fabs(y_45_scale);
	double t_3 = b * fabs(a);
	double t_4 = (b * b) / (x_45_scale * x_45_scale);
	double t_5 = -fabs(a);
	double t_6 = (t_3 * b) * t_5;
	double tmp;
	if (angle <= -6.8e-270) {
		tmp = -0.25 * (fabs(a) * (pow(x_45_scale, 2.0) * (angle * (fabs(y_45_scale) * sqrt((8.0 * ((sqrt((9.525986892242036e-10 * (pow(((double) M_PI), 4.0) / pow(x_45_scale, 4.0)))) + (3.08641975308642e-5 * (pow(((double) M_PI), 2.0) / pow(x_45_scale, 2.0)))) / pow(x_45_scale, 2.0))))))));
	} else {
		tmp = (-sqrt(((fma(t_2, t_2, t_4) + fabs((t_4 - ((fabs(a) * fabs(a)) / (fabs(y_45_scale) * fabs(y_45_scale)))))) * (((4.0 * (t_6 / t_1)) * 2.0) * t_6))) / ((4.0 * t_3) * (b * t_5))) * t_1;
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(x_45_scale * abs(y_45_scale))
	t_1 = Float64(t_0 * t_0)
	t_2 = Float64(abs(a) / abs(y_45_scale))
	t_3 = Float64(b * abs(a))
	t_4 = Float64(Float64(b * b) / Float64(x_45_scale * x_45_scale))
	t_5 = Float64(-abs(a))
	t_6 = Float64(Float64(t_3 * b) * t_5)
	tmp = 0.0
	if (angle <= -6.8e-270)
		tmp = Float64(-0.25 * Float64(abs(a) * Float64((x_45_scale ^ 2.0) * Float64(angle * Float64(abs(y_45_scale) * sqrt(Float64(8.0 * Float64(Float64(sqrt(Float64(9.525986892242036e-10 * Float64((pi ^ 4.0) / (x_45_scale ^ 4.0)))) + Float64(3.08641975308642e-5 * Float64((pi ^ 2.0) / (x_45_scale ^ 2.0)))) / (x_45_scale ^ 2.0)))))))));
	else
		tmp = Float64(Float64(Float64(-sqrt(Float64(Float64(fma(t_2, t_2, t_4) + abs(Float64(t_4 - Float64(Float64(abs(a) * abs(a)) / Float64(abs(y_45_scale) * abs(y_45_scale)))))) * Float64(Float64(Float64(4.0 * Float64(t_6 / t_1)) * 2.0) * t_6)))) / Float64(Float64(4.0 * t_3) * Float64(b * t_5))) * t_1);
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(x$45$scale * N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[a], $MachinePrecision] / N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(b * N[Abs[a], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(b * b), $MachinePrecision] / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = (-N[Abs[a], $MachinePrecision])}, Block[{t$95$6 = N[(N[(t$95$3 * b), $MachinePrecision] * t$95$5), $MachinePrecision]}, If[LessEqual[angle, -6.8e-270], N[(-0.25 * N[(N[Abs[a], $MachinePrecision] * N[(N[Power[x$45$scale, 2.0], $MachinePrecision] * N[(angle * N[(N[Abs[y$45$scale], $MachinePrecision] * N[Sqrt[N[(8.0 * N[(N[(N[Sqrt[N[(9.525986892242036e-10 * N[(N[Power[Pi, 4.0], $MachinePrecision] / N[Power[x$45$scale, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(3.08641975308642e-5 * N[(N[Power[Pi, 2.0], $MachinePrecision] / N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-N[Sqrt[N[(N[(N[(t$95$2 * t$95$2 + t$95$4), $MachinePrecision] + N[Abs[N[(t$95$4 - N[(N[(N[Abs[a], $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision] / N[(N[Abs[y$45$scale], $MachinePrecision] * N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(4.0 * N[(t$95$6 / t$95$1), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * t$95$6), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[(4.0 * t$95$3), $MachinePrecision] * N[(b * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if angle < -6.8000000000000001e-270

   1. Initial program 2.8%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   2. Taylor expanded in a around -inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \frac{a \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}{{b}^{2}}} \]

   4. Applied rewrites 0.7%

      \[\leadsto \color{blue}{-0.25 \cdot \frac{a \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}{{b}^{2}}} \]

   5. Taylor expanded in b around 0

      \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \color{blue}{\left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}\right) \]

   7. Applied rewrites 2.5%

      \[\leadsto -0.25 \cdot \left(a \cdot \color{blue}{\left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}\right) \]

   8. Taylor expanded in y-scale around inf

      \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{x-scale}^{4}}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{x-scale}^{4}}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right) \]

   10. Applied rewrites 3.5%

       \[\leadsto -0.25 \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}{{x-scale}^{4}}} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right) \]

   11. Taylor expanded in angle around 0

       \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{1}{1049760000} \cdot \frac{{\mathsf{PI}\left(\right)}^{4}}{{x-scale}^{4}}} + \frac{1}{32400} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right)\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{1}{1049760000} \cdot \frac{{\mathsf{PI}\left(\right)}^{4}}{{x-scale}^{4}}} + \frac{1}{32400} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right)\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{1}{1049760000} \cdot \frac{{\mathsf{PI}\left(\right)}^{4}}{{x-scale}^{4}}} + \frac{1}{32400} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right)\right) \]

       3. lower-sqrt.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{1}{1049760000} \cdot \frac{{\mathsf{PI}\left(\right)}^{4}}{{x-scale}^{4}}} + \frac{1}{32400} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right)\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{1}{1049760000} \cdot \frac{{\mathsf{PI}\left(\right)}^{4}}{{x-scale}^{4}}} + \frac{1}{32400} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right)\right) \]

   13. Applied rewrites 4.5%

       \[\leadsto -0.25 \cdot \left(a \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \left(y-scale \cdot \sqrt{8 \cdot \frac{\sqrt{9.525986892242036 \cdot 10^{-10} \cdot \frac{{\pi}^{4}}{{x-scale}^{4}}} + 3.08641975308642 \cdot 10^{-5} \cdot \frac{{\pi}^{2}}{{x-scale}^{2}}}{{x-scale}^{2}}}\right)\right)\right)\right) \]

   if -6.8000000000000001e-270 < angle

   1. Initial program 2.8%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   2. Taylor expanded in angle around 0

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \color{blue}{\left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   3. Step-by-step derivation

   4. Applied rewrites 4.6%

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \color{blue}{\left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   6. Applied rewrites 2.1%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(\mathsf{fma}\left(\frac{a}{y-scale}, \frac{a}{y-scale}, \frac{b \cdot b}{x-scale \cdot x-scale}\right) + \left|\frac{b \cdot b}{x-scale \cdot x-scale} - \frac{a \cdot a}{y-scale \cdot y-scale}\right|\right) \cdot \left(\left(\left(4 \cdot \frac{\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)}{\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)}\right) \cdot 2\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)\right)}}{\left(4 \cdot \left(b \cdot a\right)\right) \cdot \left(b \cdot \left(-a\right)\right)} \cdot \left(\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(\mathsf{fma}\left(\frac{a}{y-scale}, \frac{a}{y-scale}, \frac{b \cdot b}{x-scale \cdot x-scale}\right) + \left|\frac{b \cdot b}{x-scale \cdot x-scale} - \frac{a \cdot a}{y-scale \cdot y-scale}\right|\right) \cdot \left(\left(\left(4 \cdot \frac{\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)}{\color{blue}{\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)}}\right) \cdot 2\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)\right)}}{\left(4 \cdot \left(b \cdot a\right)\right) \cdot \left(b \cdot \left(-a\right)\right)} \cdot \left(\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(\mathsf{fma}\left(\frac{a}{y-scale}, \frac{a}{y-scale}, \frac{b \cdot b}{x-scale \cdot x-scale}\right) + \left|\frac{b \cdot b}{x-scale \cdot x-scale} - \frac{a \cdot a}{y-scale \cdot y-scale}\right|\right) \cdot \left(\left(\left(4 \cdot \frac{\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)}{\color{blue}{\left(y-scale \cdot y-scale\right)} \cdot \left(x-scale \cdot x-scale\right)}\right) \cdot 2\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)\right)}}{\left(4 \cdot \left(b \cdot a\right)\right) \cdot \left(b \cdot \left(-a\right)\right)} \cdot \left(\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(\mathsf{fma}\left(\frac{a}{y-scale}, \frac{a}{y-scale}, \frac{b \cdot b}{x-scale \cdot x-scale}\right) + \left|\frac{b \cdot b}{x-scale \cdot x-scale} - \frac{a \cdot a}{y-scale \cdot y-scale}\right|\right) \cdot \left(\left(\left(4 \cdot \frac{\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)}{\left(y-scale \cdot y-scale\right) \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}}\right) \cdot 2\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)\right)}}{\left(4 \cdot \left(b \cdot a\right)\right) \cdot \left(b \cdot \left(-a\right)\right)} \cdot \left(\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)\right) \]

      4. unswap-sqr N/A

         \[\leadsto \frac{-\sqrt{\left(\mathsf{fma}\left(\frac{a}{y-scale}, \frac{a}{y-scale}, \frac{b \cdot b}{x-scale \cdot x-scale}\right) + \left|\frac{b \cdot b}{x-scale \cdot x-scale} - \frac{a \cdot a}{y-scale \cdot y-scale}\right|\right) \cdot \left(\left(\left(4 \cdot \frac{\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)}{\color{blue}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}}\right) \cdot 2\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)\right)}}{\left(4 \cdot \left(b \cdot a\right)\right) \cdot \left(b \cdot \left(-a\right)\right)} \cdot \left(\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(\mathsf{fma}\left(\frac{a}{y-scale}, \frac{a}{y-scale}, \frac{b \cdot b}{x-scale \cdot x-scale}\right) + \left|\frac{b \cdot b}{x-scale \cdot x-scale} - \frac{a \cdot a}{y-scale \cdot y-scale}\right|\right) \cdot \left(\left(\left(4 \cdot \frac{\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)}{\color{blue}{\left(y-scale \cdot x-scale\right)} \cdot \left(y-scale \cdot x-scale\right)}\right) \cdot 2\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)\right)}}{\left(4 \cdot \left(b \cdot a\right)\right) \cdot \left(b \cdot \left(-a\right)\right)} \cdot \left(\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(\mathsf{fma}\left(\frac{a}{y-scale}, \frac{a}{y-scale}, \frac{b \cdot b}{x-scale \cdot x-scale}\right) + \left|\frac{b \cdot b}{x-scale \cdot x-scale} - \frac{a \cdot a}{y-scale \cdot y-scale}\right|\right) \cdot \left(\left(\left(4 \cdot \frac{\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)}{\left(y-scale \cdot x-scale\right) \cdot \color{blue}{\left(y-scale \cdot x-scale\right)}}\right) \cdot 2\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)\right)}}{\left(4 \cdot \left(b \cdot a\right)\right) \cdot \left(b \cdot \left(-a\right)\right)} \cdot \left(\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)\right) \]

      7. lower-*.f64 2.9

         \[\leadsto \frac{-\sqrt{\left(\mathsf{fma}\left(\frac{a}{y-scale}, \frac{a}{y-scale}, \frac{b \cdot b}{x-scale \cdot x-scale}\right) + \left|\frac{b \cdot b}{x-scale \cdot x-scale} - \frac{a \cdot a}{y-scale \cdot y-scale}\right|\right) \cdot \left(\left(\left(4 \cdot \frac{\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)}{\color{blue}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}}\right) \cdot 2\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)\right)}}{\left(4 \cdot \left(b \cdot a\right)\right) \cdot \left(b \cdot \left(-a\right)\right)} \cdot \left(\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(\mathsf{fma}\left(\frac{a}{y-scale}, \frac{a}{y-scale}, \frac{b \cdot b}{x-scale \cdot x-scale}\right) + \left|\frac{b \cdot b}{x-scale \cdot x-scale} - \frac{a \cdot a}{y-scale \cdot y-scale}\right|\right) \cdot \left(\left(\left(4 \cdot \frac{\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)}{\color{blue}{\left(y-scale \cdot x-scale\right)} \cdot \left(y-scale \cdot x-scale\right)}\right) \cdot 2\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)\right)}}{\left(4 \cdot \left(b \cdot a\right)\right) \cdot \left(b \cdot \left(-a\right)\right)} \cdot \left(\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \frac{-\sqrt{\left(\mathsf{fma}\left(\frac{a}{y-scale}, \frac{a}{y-scale}, \frac{b \cdot b}{x-scale \cdot x-scale}\right) + \left|\frac{b \cdot b}{x-scale \cdot x-scale} - \frac{a \cdot a}{y-scale \cdot y-scale}\right|\right) \cdot \left(\left(\left(4 \cdot \frac{\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)}{\color{blue}{\left(x-scale \cdot y-scale\right)} \cdot \left(y-scale \cdot x-scale\right)}\right) \cdot 2\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)\right)}}{\left(4 \cdot \left(b \cdot a\right)\right) \cdot \left(b \cdot \left(-a\right)\right)} \cdot \left(\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)\right) \]

      10. lift-*.f64 2.9

          \[\leadsto \frac{-\sqrt{\left(\mathsf{fma}\left(\frac{a}{y-scale}, \frac{a}{y-scale}, \frac{b \cdot b}{x-scale \cdot x-scale}\right) + \left|\frac{b \cdot b}{x-scale \cdot x-scale} - \frac{a \cdot a}{y-scale \cdot y-scale}\right|\right) \cdot \left(\left(\left(4 \cdot \frac{\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)}{\color{blue}{\left(x-scale \cdot y-scale\right)} \cdot \left(y-scale \cdot x-scale\right)}\right) \cdot 2\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)\right)}}{\left(4 \cdot \left(b \cdot a\right)\right) \cdot \left(b \cdot \left(-a\right)\right)} \cdot \left(\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{-\sqrt{\left(\mathsf{fma}\left(\frac{a}{y-scale}, \frac{a}{y-scale}, \frac{b \cdot b}{x-scale \cdot x-scale}\right) + \left|\frac{b \cdot b}{x-scale \cdot x-scale} - \frac{a \cdot a}{y-scale \cdot y-scale}\right|\right) \cdot \left(\left(\left(4 \cdot \frac{\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)}{\left(x-scale \cdot y-scale\right) \cdot \color{blue}{\left(y-scale \cdot x-scale\right)}}\right) \cdot 2\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)\right)}}{\left(4 \cdot \left(b \cdot a\right)\right) \cdot \left(b \cdot \left(-a\right)\right)} \cdot \left(\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \frac{-\sqrt{\left(\mathsf{fma}\left(\frac{a}{y-scale}, \frac{a}{y-scale}, \frac{b \cdot b}{x-scale \cdot x-scale}\right) + \left|\frac{b \cdot b}{x-scale \cdot x-scale} - \frac{a \cdot a}{y-scale \cdot y-scale}\right|\right) \cdot \left(\left(\left(4 \cdot \frac{\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)}{\left(x-scale \cdot y-scale\right) \cdot \color{blue}{\left(x-scale \cdot y-scale\right)}}\right) \cdot 2\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)\right)}}{\left(4 \cdot \left(b \cdot a\right)\right) \cdot \left(b \cdot \left(-a\right)\right)} \cdot \left(\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)\right) \]

      13. lift-*.f64 2.9

          \[\leadsto \frac{-\sqrt{\left(\mathsf{fma}\left(\frac{a}{y-scale}, \frac{a}{y-scale}, \frac{b \cdot b}{x-scale \cdot x-scale}\right) + \left|\frac{b \cdot b}{x-scale \cdot x-scale} - \frac{a \cdot a}{y-scale \cdot y-scale}\right|\right) \cdot \left(\left(\left(4 \cdot \frac{\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)}{\left(x-scale \cdot y-scale\right) \cdot \color{blue}{\left(x-scale \cdot y-scale\right)}}\right) \cdot 2\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)\right)}}{\left(4 \cdot \left(b \cdot a\right)\right) \cdot \left(b \cdot \left(-a\right)\right)} \cdot \left(\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)\right) \]

   8. Applied rewrites 2.9%

      \[\leadsto \frac{-\sqrt{\left(\mathsf{fma}\left(\frac{a}{y-scale}, \frac{a}{y-scale}, \frac{b \cdot b}{x-scale \cdot x-scale}\right) + \left|\frac{b \cdot b}{x-scale \cdot x-scale} - \frac{a \cdot a}{y-scale \cdot y-scale}\right|\right) \cdot \left(\left(\left(4 \cdot \frac{\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)}{\color{blue}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}}\right) \cdot 2\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)\right)}}{\left(4 \cdot \left(b \cdot a\right)\right) \cdot \left(b \cdot \left(-a\right)\right)} \cdot \left(\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)\right) \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(\mathsf{fma}\left(\frac{a}{y-scale}, \frac{a}{y-scale}, \frac{b \cdot b}{x-scale \cdot x-scale}\right) + \left|\frac{b \cdot b}{x-scale \cdot x-scale} - \frac{a \cdot a}{y-scale \cdot y-scale}\right|\right) \cdot \left(\left(\left(4 \cdot \frac{\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}\right) \cdot 2\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)\right)}}{\left(4 \cdot \left(b \cdot a\right)\right) \cdot \left(b \cdot \left(-a\right)\right)} \cdot \color{blue}{\left(\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(\mathsf{fma}\left(\frac{a}{y-scale}, \frac{a}{y-scale}, \frac{b \cdot b}{x-scale \cdot x-scale}\right) + \left|\frac{b \cdot b}{x-scale \cdot x-scale} - \frac{a \cdot a}{y-scale \cdot y-scale}\right|\right) \cdot \left(\left(\left(4 \cdot \frac{\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}\right) \cdot 2\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)\right)}}{\left(4 \cdot \left(b \cdot a\right)\right) \cdot \left(b \cdot \left(-a\right)\right)} \cdot \left(\color{blue}{\left(y-scale \cdot y-scale\right)} \cdot \left(x-scale \cdot x-scale\right)\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(\mathsf{fma}\left(\frac{a}{y-scale}, \frac{a}{y-scale}, \frac{b \cdot b}{x-scale \cdot x-scale}\right) + \left|\frac{b \cdot b}{x-scale \cdot x-scale} - \frac{a \cdot a}{y-scale \cdot y-scale}\right|\right) \cdot \left(\left(\left(4 \cdot \frac{\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}\right) \cdot 2\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)\right)}}{\left(4 \cdot \left(b \cdot a\right)\right) \cdot \left(b \cdot \left(-a\right)\right)} \cdot \left(\left(y-scale \cdot y-scale\right) \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}\right) \]

      4. unswap-sqr N/A

         \[\leadsto \frac{-\sqrt{\left(\mathsf{fma}\left(\frac{a}{y-scale}, \frac{a}{y-scale}, \frac{b \cdot b}{x-scale \cdot x-scale}\right) + \left|\frac{b \cdot b}{x-scale \cdot x-scale} - \frac{a \cdot a}{y-scale \cdot y-scale}\right|\right) \cdot \left(\left(\left(4 \cdot \frac{\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}\right) \cdot 2\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)\right)}}{\left(4 \cdot \left(b \cdot a\right)\right) \cdot \left(b \cdot \left(-a\right)\right)} \cdot \color{blue}{\left(\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(\mathsf{fma}\left(\frac{a}{y-scale}, \frac{a}{y-scale}, \frac{b \cdot b}{x-scale \cdot x-scale}\right) + \left|\frac{b \cdot b}{x-scale \cdot x-scale} - \frac{a \cdot a}{y-scale \cdot y-scale}\right|\right) \cdot \left(\left(\left(4 \cdot \frac{\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}\right) \cdot 2\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)\right)}}{\left(4 \cdot \left(b \cdot a\right)\right) \cdot \left(b \cdot \left(-a\right)\right)} \cdot \left(\color{blue}{\left(y-scale \cdot x-scale\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(\mathsf{fma}\left(\frac{a}{y-scale}, \frac{a}{y-scale}, \frac{b \cdot b}{x-scale \cdot x-scale}\right) + \left|\frac{b \cdot b}{x-scale \cdot x-scale} - \frac{a \cdot a}{y-scale \cdot y-scale}\right|\right) \cdot \left(\left(\left(4 \cdot \frac{\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}\right) \cdot 2\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)\right)}}{\left(4 \cdot \left(b \cdot a\right)\right) \cdot \left(b \cdot \left(-a\right)\right)} \cdot \left(\left(y-scale \cdot x-scale\right) \cdot \color{blue}{\left(y-scale \cdot x-scale\right)}\right) \]

      7. lower-*.f64 5.2

         \[\leadsto \frac{-\sqrt{\left(\mathsf{fma}\left(\frac{a}{y-scale}, \frac{a}{y-scale}, \frac{b \cdot b}{x-scale \cdot x-scale}\right) + \left|\frac{b \cdot b}{x-scale \cdot x-scale} - \frac{a \cdot a}{y-scale \cdot y-scale}\right|\right) \cdot \left(\left(\left(4 \cdot \frac{\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}\right) \cdot 2\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)\right)}}{\left(4 \cdot \left(b \cdot a\right)\right) \cdot \left(b \cdot \left(-a\right)\right)} \cdot \color{blue}{\left(\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(\mathsf{fma}\left(\frac{a}{y-scale}, \frac{a}{y-scale}, \frac{b \cdot b}{x-scale \cdot x-scale}\right) + \left|\frac{b \cdot b}{x-scale \cdot x-scale} - \frac{a \cdot a}{y-scale \cdot y-scale}\right|\right) \cdot \left(\left(\left(4 \cdot \frac{\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}\right) \cdot 2\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)\right)}}{\left(4 \cdot \left(b \cdot a\right)\right) \cdot \left(b \cdot \left(-a\right)\right)} \cdot \left(\color{blue}{\left(y-scale \cdot x-scale\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \frac{-\sqrt{\left(\mathsf{fma}\left(\frac{a}{y-scale}, \frac{a}{y-scale}, \frac{b \cdot b}{x-scale \cdot x-scale}\right) + \left|\frac{b \cdot b}{x-scale \cdot x-scale} - \frac{a \cdot a}{y-scale \cdot y-scale}\right|\right) \cdot \left(\left(\left(4 \cdot \frac{\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}\right) \cdot 2\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)\right)}}{\left(4 \cdot \left(b \cdot a\right)\right) \cdot \left(b \cdot \left(-a\right)\right)} \cdot \left(\color{blue}{\left(x-scale \cdot y-scale\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \]

      10. lift-*.f64 5.2

          \[\leadsto \frac{-\sqrt{\left(\mathsf{fma}\left(\frac{a}{y-scale}, \frac{a}{y-scale}, \frac{b \cdot b}{x-scale \cdot x-scale}\right) + \left|\frac{b \cdot b}{x-scale \cdot x-scale} - \frac{a \cdot a}{y-scale \cdot y-scale}\right|\right) \cdot \left(\left(\left(4 \cdot \frac{\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}\right) \cdot 2\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)\right)}}{\left(4 \cdot \left(b \cdot a\right)\right) \cdot \left(b \cdot \left(-a\right)\right)} \cdot \left(\color{blue}{\left(x-scale \cdot y-scale\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{-\sqrt{\left(\mathsf{fma}\left(\frac{a}{y-scale}, \frac{a}{y-scale}, \frac{b \cdot b}{x-scale \cdot x-scale}\right) + \left|\frac{b \cdot b}{x-scale \cdot x-scale} - \frac{a \cdot a}{y-scale \cdot y-scale}\right|\right) \cdot \left(\left(\left(4 \cdot \frac{\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}\right) \cdot 2\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)\right)}}{\left(4 \cdot \left(b \cdot a\right)\right) \cdot \left(b \cdot \left(-a\right)\right)} \cdot \left(\left(x-scale \cdot y-scale\right) \cdot \color{blue}{\left(y-scale \cdot x-scale\right)}\right) \]

      12. *-commutative N/A

          \[\leadsto \frac{-\sqrt{\left(\mathsf{fma}\left(\frac{a}{y-scale}, \frac{a}{y-scale}, \frac{b \cdot b}{x-scale \cdot x-scale}\right) + \left|\frac{b \cdot b}{x-scale \cdot x-scale} - \frac{a \cdot a}{y-scale \cdot y-scale}\right|\right) \cdot \left(\left(\left(4 \cdot \frac{\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}\right) \cdot 2\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)\right)}}{\left(4 \cdot \left(b \cdot a\right)\right) \cdot \left(b \cdot \left(-a\right)\right)} \cdot \left(\left(x-scale \cdot y-scale\right) \cdot \color{blue}{\left(x-scale \cdot y-scale\right)}\right) \]

      13. lift-*.f64 5.2

          \[\leadsto \frac{-\sqrt{\left(\mathsf{fma}\left(\frac{a}{y-scale}, \frac{a}{y-scale}, \frac{b \cdot b}{x-scale \cdot x-scale}\right) + \left|\frac{b \cdot b}{x-scale \cdot x-scale} - \frac{a \cdot a}{y-scale \cdot y-scale}\right|\right) \cdot \left(\left(\left(4 \cdot \frac{\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}\right) \cdot 2\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)\right)}}{\left(4 \cdot \left(b \cdot a\right)\right) \cdot \left(b \cdot \left(-a\right)\right)} \cdot \left(\left(x-scale \cdot y-scale\right) \cdot \color{blue}{\left(x-scale \cdot y-scale\right)}\right) \]

   10. Applied rewrites 5.2%

       \[\leadsto \frac{-\sqrt{\left(\mathsf{fma}\left(\frac{a}{y-scale}, \frac{a}{y-scale}, \frac{b \cdot b}{x-scale \cdot x-scale}\right) + \left|\frac{b \cdot b}{x-scale \cdot x-scale} - \frac{a \cdot a}{y-scale \cdot y-scale}\right|\right) \cdot \left(\left(\left(4 \cdot \frac{\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}\right) \cdot 2\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)\right)}}{\left(4 \cdot \left(b \cdot a\right)\right) \cdot \left(b \cdot \left(-a\right)\right)} \cdot \color{blue}{\left(\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 5.4% accurate, 6.3× speedup

Math

\[\begin{array}{l} t_0 := \left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\\ \left(\frac{\frac{\sqrt{\left(8 \cdot t\_0\right) \cdot \left(t\_0 \cdot \frac{\sqrt{{b}^{4}} + {b}^{2}}{{x-scale}^{2}}\right)}}{\left|y-scale \cdot x-scale\right|}}{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(a \cdot b\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (* (* (- a) b) b) a)))
   (*
    (*
     (/
      (/
       (sqrt
        (*
         (* 8.0 t_0)
         (* t_0 (/ (+ (sqrt (pow b 4.0)) (pow b 2.0)) (pow x-scale 2.0)))))
       (fabs (* y-scale x-scale)))
      (* (* (* a b) 4.0) (* a b)))
     (* y-scale x-scale))
    (* y-scale x-scale))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = ((-a * b) * b) * a;
	return (((sqrt(((8.0 * t_0) * (t_0 * ((sqrt(pow(b, 4.0)) + pow(b, 2.0)) / pow(x_45_scale, 2.0))))) / fabs((y_45_scale * x_45_scale))) / (((a * b) * 4.0) * (a * b))) * (y_45_scale * x_45_scale)) * (y_45_scale * x_45_scale);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    t_0 = ((-a * b) * b) * a
    code = (((sqrt(((8.0d0 * t_0) * (t_0 * ((sqrt((b ** 4.0d0)) + (b ** 2.0d0)) / (x_45scale ** 2.0d0))))) / abs((y_45scale * x_45scale))) / (((a * b) * 4.0d0) * (a * b))) * (y_45scale * x_45scale)) * (y_45scale * x_45scale)
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = ((-a * b) * b) * a;
	return (((Math.sqrt(((8.0 * t_0) * (t_0 * ((Math.sqrt(Math.pow(b, 4.0)) + Math.pow(b, 2.0)) / Math.pow(x_45_scale, 2.0))))) / Math.abs((y_45_scale * x_45_scale))) / (((a * b) * 4.0) * (a * b))) * (y_45_scale * x_45_scale)) * (y_45_scale * x_45_scale);
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = ((-a * b) * b) * a
	return (((math.sqrt(((8.0 * t_0) * (t_0 * ((math.sqrt(math.pow(b, 4.0)) + math.pow(b, 2.0)) / math.pow(x_45_scale, 2.0))))) / math.fabs((y_45_scale * x_45_scale))) / (((a * b) * 4.0) * (a * b))) * (y_45_scale * x_45_scale)) * (y_45_scale * x_45_scale)

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(Float64(Float64(-a) * b) * b) * a)
	return Float64(Float64(Float64(Float64(sqrt(Float64(Float64(8.0 * t_0) * Float64(t_0 * Float64(Float64(sqrt((b ^ 4.0)) + (b ^ 2.0)) / (x_45_scale ^ 2.0))))) / abs(Float64(y_45_scale * x_45_scale))) / Float64(Float64(Float64(a * b) * 4.0) * Float64(a * b))) * Float64(y_45_scale * x_45_scale)) * Float64(y_45_scale * x_45_scale))
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = ((-a * b) * b) * a;
	tmp = (((sqrt(((8.0 * t_0) * (t_0 * ((sqrt((b ^ 4.0)) + (b ^ 2.0)) / (x_45_scale ^ 2.0))))) / abs((y_45_scale * x_45_scale))) / (((a * b) * 4.0) * (a * b))) * (y_45_scale * x_45_scale)) * (y_45_scale * x_45_scale);
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(N[((-a) * b), $MachinePrecision] * b), $MachinePrecision] * a), $MachinePrecision]}, N[(N[(N[(N[(N[Sqrt[N[(N[(8.0 * t$95$0), $MachinePrecision] * N[(t$95$0 * N[(N[(N[Sqrt[N[Power[b, 4.0], $MachinePrecision]], $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[N[(y$45$scale * x$45$scale), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(a * b), $MachinePrecision] * 4.0), $MachinePrecision] * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 2.8%

   \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

3. Applied rewrites 6.3%

   \[\leadsto \color{blue}{\left(\frac{\frac{\sqrt{\left(8 \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \left(\left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right) \cdot \left(\left(\mathsf{hypot}\left(\frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}{y-scale \cdot x-scale}, \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), a \cdot a, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale} - \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), b \cdot b, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(a \cdot a\right)\right)}{y-scale \cdot y-scale}\right) + \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), a \cdot a, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}\right) + \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), b \cdot b, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(a \cdot a\right)\right)}{y-scale \cdot y-scale}\right)\right)}}{\left|y-scale \cdot x-scale\right|}}{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(a \cdot b\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right)} \]

4. Taylor expanded in x-scale around 0

   \[\leadsto \left(\frac{\frac{\sqrt{\left(8 \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \left(\left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right) \cdot \color{blue}{\frac{\sqrt{{\left({a}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) + {b}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}^{2}} + \left({a}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) + {b}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}{{x-scale}^{2}}}\right)}}{\left|y-scale \cdot x-scale\right|}}{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(a \cdot b\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]
5. Step-by-step derivation

6. Applied rewrites 6.7%

   \[\leadsto \left(\frac{\frac{\sqrt{\left(8 \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \left(\left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right) \cdot \color{blue}{\frac{\sqrt{{\left(\mathsf{fma}\left({a}^{2}, 0.5 - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right), {b}^{2} \cdot \left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)}^{2}} + \mathsf{fma}\left({a}^{2}, 0.5 - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right), {b}^{2} \cdot \left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}{{x-scale}^{2}}}\right)}}{\left|y-scale \cdot x-scale\right|}}{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(a \cdot b\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

7. Taylor expanded in angle around 0

   \[\leadsto \left(\frac{\frac{\sqrt{\left(8 \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \left(\left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right) \cdot \frac{\sqrt{{b}^{4}} + {b}^{2}}{{\color{blue}{x-scale}}^{2}}\right)}}{\left|y-scale \cdot x-scale\right|}}{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(a \cdot b\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]
8. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \left(\frac{\frac{\sqrt{\left(8 \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \left(\left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right) \cdot \frac{\sqrt{{b}^{4}} + {b}^{2}}{{x-scale}^{2}}\right)}}{\left|y-scale \cdot x-scale\right|}}{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(a \cdot b\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   2. lower-sqrt.f64 N/A

      \[\leadsto \left(\frac{\frac{\sqrt{\left(8 \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \left(\left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right) \cdot \frac{\sqrt{{b}^{4}} + {b}^{2}}{{x-scale}^{2}}\right)}}{\left|y-scale \cdot x-scale\right|}}{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(a \cdot b\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   3. lower-pow.f64 N/A

      \[\leadsto \left(\frac{\frac{\sqrt{\left(8 \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \left(\left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right) \cdot \frac{\sqrt{{b}^{4}} + {b}^{2}}{{x-scale}^{2}}\right)}}{\left|y-scale \cdot x-scale\right|}}{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(a \cdot b\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   4. lower-pow.f64 5.5

      \[\leadsto \left(\frac{\frac{\sqrt{\left(8 \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \left(\left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right) \cdot \frac{\sqrt{{b}^{4}} + {b}^{2}}{{x-scale}^{2}}\right)}}{\left|y-scale \cdot x-scale\right|}}{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(a \cdot b\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

9. Applied rewrites 5.5%

   \[\leadsto \left(\frac{\frac{\sqrt{\left(8 \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \left(\left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right) \cdot \frac{\sqrt{{b}^{4}} + {b}^{2}}{{\color{blue}{x-scale}}^{2}}\right)}}{\left|y-scale \cdot x-scale\right|}}{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(a \cdot b\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]
10. Add Preprocessing

Alternative 17: 3.7% accurate, 6.3× speedup

Math

\[\begin{array}{l} t_0 := \frac{a}{y-scale \cdot y-scale}\\ t_1 := \frac{b}{x-scale \cdot x-scale}\\ t_2 := \left(-a\right) \cdot b\\ \frac{\frac{-\sqrt{\left(\left(\left(\left(\left(\left(a \cdot b\right) \cdot b\right) \cdot \frac{-a}{\left(\left(y-scale \cdot y-scale\right) \cdot x-scale\right) \cdot x-scale}\right) \cdot 4\right) \cdot 2\right) \cdot \left(\left(t\_2 \cdot b\right) \cdot a\right)\right) \cdot \mathsf{fma}\left(a, t\_0, \mathsf{fma}\left(b, t\_1, \left|a \cdot t\_0 - b \cdot t\_1\right|\right)\right)}}{4 \cdot \left(a \cdot b\right)}}{t\_2} \cdot \left(\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)\right) \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ a (* y-scale y-scale)))
        (t_1 (/ b (* x-scale x-scale)))
        (t_2 (* (- a) b)))
   (*
    (/
     (/
      (-
       (sqrt
        (*
         (*
          (*
           (*
            (*
             (* (* a b) b)
             (/ (- a) (* (* (* y-scale y-scale) x-scale) x-scale)))
            4.0)
           2.0)
          (* (* t_2 b) a))
         (fma a t_0 (fma b t_1 (fabs (- (* a t_0) (* b t_1))))))))
      (* 4.0 (* a b)))
     t_2)
    (* (* y-scale y-scale) (* x-scale x-scale)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = a / (y_45_scale * y_45_scale);
	double t_1 = b / (x_45_scale * x_45_scale);
	double t_2 = -a * b;
	return ((-sqrt((((((((a * b) * b) * (-a / (((y_45_scale * y_45_scale) * x_45_scale) * x_45_scale))) * 4.0) * 2.0) * ((t_2 * b) * a)) * fma(a, t_0, fma(b, t_1, fabs(((a * t_0) - (b * t_1))))))) / (4.0 * (a * b))) / t_2) * ((y_45_scale * y_45_scale) * (x_45_scale * x_45_scale));
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(a / Float64(y_45_scale * y_45_scale))
	t_1 = Float64(b / Float64(x_45_scale * x_45_scale))
	t_2 = Float64(Float64(-a) * b)
	return Float64(Float64(Float64(Float64(-sqrt(Float64(Float64(Float64(Float64(Float64(Float64(Float64(a * b) * b) * Float64(Float64(-a) / Float64(Float64(Float64(y_45_scale * y_45_scale) * x_45_scale) * x_45_scale))) * 4.0) * 2.0) * Float64(Float64(t_2 * b) * a)) * fma(a, t_0, fma(b, t_1, abs(Float64(Float64(a * t_0) - Float64(b * t_1)))))))) / Float64(4.0 * Float64(a * b))) / t_2) * Float64(Float64(y_45_scale * y_45_scale) * Float64(x_45_scale * x_45_scale)))
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(a / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[((-a) * b), $MachinePrecision]}, N[(N[(N[((-N[Sqrt[N[(N[(N[(N[(N[(N[(N[(a * b), $MachinePrecision] * b), $MachinePrecision] * N[((-a) / N[(N[(N[(y$45$scale * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[(t$95$2 * b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * N[(a * t$95$0 + N[(b * t$95$1 + N[Abs[N[(N[(a * t$95$0), $MachinePrecision] - N[(b * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(4.0 * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] * N[(N[(y$45$scale * y$45$scale), $MachinePrecision] * N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 2.8%

   \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

2. Taylor expanded in angle around 0

   \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \color{blue}{\left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
3. Step-by-step derivation

4. Applied rewrites 4.6%

   \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \color{blue}{\left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

6. Applied rewrites 2.1%

   \[\leadsto \color{blue}{\frac{-\sqrt{\left(\mathsf{fma}\left(\frac{a}{y-scale}, \frac{a}{y-scale}, \frac{b \cdot b}{x-scale \cdot x-scale}\right) + \left|\frac{b \cdot b}{x-scale \cdot x-scale} - \frac{a \cdot a}{y-scale \cdot y-scale}\right|\right) \cdot \left(\left(\left(4 \cdot \frac{\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)}{\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)}\right) \cdot 2\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)\right)}}{\left(4 \cdot \left(b \cdot a\right)\right) \cdot \left(b \cdot \left(-a\right)\right)} \cdot \left(\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)\right)} \]

8. Applied rewrites 3.7%

   \[\leadsto \color{blue}{\frac{\frac{-\sqrt{\left(\left(\left(\left(\left(\left(a \cdot b\right) \cdot b\right) \cdot \frac{-a}{\left(\left(y-scale \cdot y-scale\right) \cdot x-scale\right) \cdot x-scale}\right) \cdot 4\right) \cdot 2\right) \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \mathsf{fma}\left(a, \frac{a}{y-scale \cdot y-scale}, \mathsf{fma}\left(b, \frac{b}{x-scale \cdot x-scale}, \left|a \cdot \frac{a}{y-scale \cdot y-scale} - b \cdot \frac{b}{x-scale \cdot x-scale}\right|\right)\right)}}{4 \cdot \left(a \cdot b\right)}}{\left(-a\right) \cdot b}} \cdot \left(\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)\right) \]
9. Add Preprocessing

Alternative 18: 1.9% accurate, 11.8× speedup

Math

\[\begin{array}{l} t_0 := \left|b\right| \cdot \left|b\right|\\ 0.25 \cdot \frac{\frac{\left(\left(y-scale \cdot y-scale\right) \cdot \sqrt{\frac{16 \cdot {a}^{4}}{y-scale \cdot y-scale}}\right) \cdot \left(t\_0 \cdot \left|b\right|\right)}{a \cdot a}}{t\_0} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (fabs b) (fabs b))))
   (*
    0.25
    (/
     (/
      (*
       (*
        (* y-scale y-scale)
        (sqrt (/ (* 16.0 (pow a 4.0)) (* y-scale y-scale))))
       (* t_0 (fabs b)))
      (* a a))
     t_0))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = fabs(b) * fabs(b);
	return 0.25 * (((((y_45_scale * y_45_scale) * sqrt(((16.0 * pow(a, 4.0)) / (y_45_scale * y_45_scale)))) * (t_0 * fabs(b))) / (a * a)) / t_0);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    t_0 = abs(b) * abs(b)
    code = 0.25d0 * (((((y_45scale * y_45scale) * sqrt(((16.0d0 * (a ** 4.0d0)) / (y_45scale * y_45scale)))) * (t_0 * abs(b))) / (a * a)) / t_0)
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = Math.abs(b) * Math.abs(b);
	return 0.25 * (((((y_45_scale * y_45_scale) * Math.sqrt(((16.0 * Math.pow(a, 4.0)) / (y_45_scale * y_45_scale)))) * (t_0 * Math.abs(b))) / (a * a)) / t_0);
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = math.fabs(b) * math.fabs(b)
	return 0.25 * (((((y_45_scale * y_45_scale) * math.sqrt(((16.0 * math.pow(a, 4.0)) / (y_45_scale * y_45_scale)))) * (t_0 * math.fabs(b))) / (a * a)) / t_0)

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(abs(b) * abs(b))
	return Float64(0.25 * Float64(Float64(Float64(Float64(Float64(y_45_scale * y_45_scale) * sqrt(Float64(Float64(16.0 * (a ^ 4.0)) / Float64(y_45_scale * y_45_scale)))) * Float64(t_0 * abs(b))) / Float64(a * a)) / t_0))
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = abs(b) * abs(b);
	tmp = 0.25 * (((((y_45_scale * y_45_scale) * sqrt(((16.0 * (a ^ 4.0)) / (y_45_scale * y_45_scale)))) * (t_0 * abs(b))) / (a * a)) / t_0);
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[Abs[b], $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]}, N[(0.25 * N[(N[(N[(N[(N[(y$45$scale * y$45$scale), $MachinePrecision] * N[Sqrt[N[(N[(16.0 * N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 2.8%

   \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

2. Taylor expanded in x-scale around 0

   \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left({b}^{4} \cdot \left(\sqrt{{\left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}^{2}} + \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)}{{y-scale}^{2}}}}{{a}^{2} \cdot {b}^{2}}} \]

4. Applied rewrites 1.2%

   \[\leadsto \color{blue}{0.25 \cdot \frac{{y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(\mathsf{fma}\left({a}^{2}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}^{2}} + \mathsf{fma}\left({a}^{2}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)\right)}{{y-scale}^{2}}}}{{a}^{2} \cdot {b}^{2}}} \]

5. Taylor expanded in b around inf

   \[\leadsto \frac{1}{4} \cdot \frac{{b}^{3} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{{y-scale}^{2}}}\right)}{\color{blue}{{a}^{2}} \cdot {b}^{2}} \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{4} \cdot \frac{{b}^{3} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{{y-scale}^{2}}}\right)}{{a}^{\color{blue}{2}} \cdot {b}^{2}} \]

   2. lower-pow.f64 N/A

      \[\leadsto \frac{1}{4} \cdot \frac{{b}^{3} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{{y-scale}^{2}}}\right)}{{a}^{2} \cdot {b}^{2}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{1}{4} \cdot \frac{{b}^{3} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{{y-scale}^{2}}}\right)}{{a}^{2} \cdot {b}^{2}} \]

7. Applied rewrites 0.8%

   \[\leadsto 0.25 \cdot \frac{{b}^{3} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{y-scale}^{2}}}\right)}{\color{blue}{{a}^{2}} \cdot {b}^{2}} \]

8. Taylor expanded in angle around 0

   \[\leadsto \frac{1}{4} \cdot \frac{{b}^{3} \cdot \left({y-scale}^{2} \cdot \sqrt{16 \cdot \frac{{a}^{4}}{{y-scale}^{2}}}\right)}{{a}^{2} \cdot {b}^{2}} \]
9. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{4} \cdot \frac{{b}^{3} \cdot \left({y-scale}^{2} \cdot \sqrt{16 \cdot \frac{{a}^{4}}{{y-scale}^{2}}}\right)}{{a}^{2} \cdot {b}^{2}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{1}{4} \cdot \frac{{b}^{3} \cdot \left({y-scale}^{2} \cdot \sqrt{16 \cdot \frac{{a}^{4}}{{y-scale}^{2}}}\right)}{{a}^{2} \cdot {b}^{2}} \]

   3. lower-pow.f64 N/A

      \[\leadsto \frac{1}{4} \cdot \frac{{b}^{3} \cdot \left({y-scale}^{2} \cdot \sqrt{16 \cdot \frac{{a}^{4}}{{y-scale}^{2}}}\right)}{{a}^{2} \cdot {b}^{2}} \]

   4. lower-pow.f64 0.8

      \[\leadsto 0.25 \cdot \frac{{b}^{3} \cdot \left({y-scale}^{2} \cdot \sqrt{16 \cdot \frac{{a}^{4}}{{y-scale}^{2}}}\right)}{{a}^{2} \cdot {b}^{2}} \]

10. Applied rewrites 0.8%

    \[\leadsto 0.25 \cdot \frac{{b}^{3} \cdot \left({y-scale}^{2} \cdot \sqrt{16 \cdot \frac{{a}^{4}}{{y-scale}^{2}}}\right)}{{a}^{2} \cdot {b}^{2}} \]
11. Step-by-step derivation

    1. lift-/.f64 N/A

       \[\leadsto \frac{1}{4} \cdot \frac{{b}^{3} \cdot \left({y-scale}^{2} \cdot \sqrt{16 \cdot \frac{{a}^{4}}{{y-scale}^{2}}}\right)}{\color{blue}{{a}^{2} \cdot {b}^{2}}} \]

12. Applied rewrites 1.0%

    \[\leadsto 0.25 \cdot \frac{\frac{\left(\left(y-scale \cdot y-scale\right) \cdot \sqrt{\frac{16 \cdot {a}^{4}}{y-scale \cdot y-scale}}\right) \cdot \left(\left(b \cdot b\right) \cdot b\right)}{a \cdot a}}{\color{blue}{b \cdot b}} \]
13. Add Preprocessing

Alternative 19: 1.8% accurate, 11.9× speedup

Math

\[\begin{array}{l} t_0 := a \cdot \left|b\right|\\ \frac{\left(\left(y-scale \cdot y-scale\right) \cdot \sqrt{\frac{16 \cdot {a}^{4}}{y-scale \cdot y-scale}}\right) \cdot \left(\left(\left|b\right| \cdot \left|b\right|\right) \cdot \left|b\right|\right)}{t\_0 \cdot t\_0} \cdot 0.25 \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* a (fabs b))))
   (*
    (/
     (*
      (*
       (* y-scale y-scale)
       (sqrt (/ (* 16.0 (pow a 4.0)) (* y-scale y-scale))))
      (* (* (fabs b) (fabs b)) (fabs b)))
     (* t_0 t_0))
    0.25)))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = a * fabs(b);
	return ((((y_45_scale * y_45_scale) * sqrt(((16.0 * pow(a, 4.0)) / (y_45_scale * y_45_scale)))) * ((fabs(b) * fabs(b)) * fabs(b))) / (t_0 * t_0)) * 0.25;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    t_0 = a * abs(b)
    code = ((((y_45scale * y_45scale) * sqrt(((16.0d0 * (a ** 4.0d0)) / (y_45scale * y_45scale)))) * ((abs(b) * abs(b)) * abs(b))) / (t_0 * t_0)) * 0.25d0
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = a * Math.abs(b);
	return ((((y_45_scale * y_45_scale) * Math.sqrt(((16.0 * Math.pow(a, 4.0)) / (y_45_scale * y_45_scale)))) * ((Math.abs(b) * Math.abs(b)) * Math.abs(b))) / (t_0 * t_0)) * 0.25;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = a * math.fabs(b)
	return ((((y_45_scale * y_45_scale) * math.sqrt(((16.0 * math.pow(a, 4.0)) / (y_45_scale * y_45_scale)))) * ((math.fabs(b) * math.fabs(b)) * math.fabs(b))) / (t_0 * t_0)) * 0.25

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(a * abs(b))
	return Float64(Float64(Float64(Float64(Float64(y_45_scale * y_45_scale) * sqrt(Float64(Float64(16.0 * (a ^ 4.0)) / Float64(y_45_scale * y_45_scale)))) * Float64(Float64(abs(b) * abs(b)) * abs(b))) / Float64(t_0 * t_0)) * 0.25)
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = a * abs(b);
	tmp = ((((y_45_scale * y_45_scale) * sqrt(((16.0 * (a ^ 4.0)) / (y_45_scale * y_45_scale)))) * ((abs(b) * abs(b)) * abs(b))) / (t_0 * t_0)) * 0.25;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(a * N[Abs[b], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[(y$45$scale * y$45$scale), $MachinePrecision] * N[Sqrt[N[(N[(16.0 * N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Abs[b], $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]]

Derivation

1. Initial program 2.8%

   \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

2. Taylor expanded in x-scale around 0

   \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left({b}^{4} \cdot \left(\sqrt{{\left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}^{2}} + \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)}{{y-scale}^{2}}}}{{a}^{2} \cdot {b}^{2}}} \]

4. Applied rewrites 1.2%

   \[\leadsto \color{blue}{0.25 \cdot \frac{{y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(\mathsf{fma}\left({a}^{2}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}^{2}} + \mathsf{fma}\left({a}^{2}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)\right)}{{y-scale}^{2}}}}{{a}^{2} \cdot {b}^{2}}} \]

5. Taylor expanded in b around inf

   \[\leadsto \frac{1}{4} \cdot \frac{{b}^{3} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{{y-scale}^{2}}}\right)}{\color{blue}{{a}^{2}} \cdot {b}^{2}} \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{4} \cdot \frac{{b}^{3} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{{y-scale}^{2}}}\right)}{{a}^{\color{blue}{2}} \cdot {b}^{2}} \]

   2. lower-pow.f64 N/A

      \[\leadsto \frac{1}{4} \cdot \frac{{b}^{3} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{{y-scale}^{2}}}\right)}{{a}^{2} \cdot {b}^{2}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{1}{4} \cdot \frac{{b}^{3} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{{y-scale}^{2}}}\right)}{{a}^{2} \cdot {b}^{2}} \]

7. Applied rewrites 0.8%

   \[\leadsto 0.25 \cdot \frac{{b}^{3} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{y-scale}^{2}}}\right)}{\color{blue}{{a}^{2}} \cdot {b}^{2}} \]

8. Taylor expanded in angle around 0

   \[\leadsto \frac{1}{4} \cdot \frac{{b}^{3} \cdot \left({y-scale}^{2} \cdot \sqrt{16 \cdot \frac{{a}^{4}}{{y-scale}^{2}}}\right)}{{a}^{2} \cdot {b}^{2}} \]
9. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{4} \cdot \frac{{b}^{3} \cdot \left({y-scale}^{2} \cdot \sqrt{16 \cdot \frac{{a}^{4}}{{y-scale}^{2}}}\right)}{{a}^{2} \cdot {b}^{2}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{1}{4} \cdot \frac{{b}^{3} \cdot \left({y-scale}^{2} \cdot \sqrt{16 \cdot \frac{{a}^{4}}{{y-scale}^{2}}}\right)}{{a}^{2} \cdot {b}^{2}} \]

   3. lower-pow.f64 N/A

      \[\leadsto \frac{1}{4} \cdot \frac{{b}^{3} \cdot \left({y-scale}^{2} \cdot \sqrt{16 \cdot \frac{{a}^{4}}{{y-scale}^{2}}}\right)}{{a}^{2} \cdot {b}^{2}} \]

   4. lower-pow.f64 0.8

      \[\leadsto 0.25 \cdot \frac{{b}^{3} \cdot \left({y-scale}^{2} \cdot \sqrt{16 \cdot \frac{{a}^{4}}{{y-scale}^{2}}}\right)}{{a}^{2} \cdot {b}^{2}} \]

10. Applied rewrites 0.8%

    \[\leadsto 0.25 \cdot \frac{{b}^{3} \cdot \left({y-scale}^{2} \cdot \sqrt{16 \cdot \frac{{a}^{4}}{{y-scale}^{2}}}\right)}{{a}^{2} \cdot {b}^{2}} \]

12. Applied rewrites 1.0%

    \[\leadsto \frac{\left(\left(y-scale \cdot y-scale\right) \cdot \sqrt{\frac{16 \cdot {a}^{4}}{y-scale \cdot y-scale}}\right) \cdot \left(\left(b \cdot b\right) \cdot b\right)}{\left(a \cdot b\right) \cdot \left(a \cdot b\right)} \cdot \color{blue}{0.25} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025181 
(FPCore (a b angle x-scale y-scale)
  :name "a from scale-rotated-ellipse"
  :precision binary64
  (/ (- (sqrt (* (* (* 2.0 (/ (* 4.0 (* (* b a) (* b (- a)))) (pow (* x-scale y-scale) 2.0))) (* (* b a) (* b (- a)))) (+ (+ (/ (/ (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)) x-scale) x-scale) (/ (/ (+ (pow (* a (cos (* (/ angle 180.0) PI))) 2.0) (pow (* b (sin (* (/ angle 180.0) PI))) 2.0)) y-scale) y-scale)) (sqrt (+ (pow (- (/ (/ (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)) x-scale) x-scale) (/ (/ (+ (pow (* a (cos (* (/ angle 180.0) PI))) 2.0) (pow (* b (sin (* (/ angle 180.0) PI))) 2.0)) y-scale) y-scale)) 2.0) (pow (/ (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin (* (/ angle 180.0) PI))) (cos (* (/ angle 180.0) PI))) x-scale) y-scale) 2.0))))))) (/ (* 4.0 (* (* b a) (* b (- a)))) (pow (* x-scale y-scale) 2.0))))