a from scale-rotated-ellipse

Percentage Accurate: 2.7% → 26.9%

Time: 39.7s

Alternatives: 9

Speedup: 19.1×

• 30.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 2.7% accurate, 1.0× speedup

Math

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

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: 26.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := 1 + -1.54320987654321 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\\ t_1 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_2 := \cos t\_1\\ t_3 := {t\_2}^{4}\\ t_4 := \sqrt{\frac{t\_3}{{x-scale\_m}^{4}}}\\ t_5 := {t\_2}^{2}\\ t_6 := \sqrt{8 \cdot \frac{t\_4 + \frac{t\_5}{x-scale\_m \cdot x-scale\_m}}{x-scale\_m \cdot x-scale\_m}}\\ t_7 := \sin t\_1\\ t_8 := \frac{{\left(t\_2 \cdot t\_7\right)}^{2}}{x-scale\_m \cdot x-scale\_m}\\ \mathbf{if}\;b\_m \leq 1.85 \cdot 10^{-143}:\\ \;\;\;\;-0.25 \cdot \left(a\_m \cdot \left(-1 \cdot \left(x-scale\_m \cdot \left(\left(y-scale\_m \cdot y-scale\_m\right) \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{t\_3}{{y-scale\_m}^{4}}} + \frac{t\_5}{y-scale\_m \cdot y-scale\_m}}{y-scale\_m \cdot y-scale\_m}}\right)\right)\right)\right)\\ \mathbf{elif}\;b\_m \leq 2.9 \cdot 10^{-65}:\\ \;\;\;\;-0.25 \cdot \frac{a\_m \cdot \left(-1 \cdot \left(x-scale\_m \cdot \left(\left(b\_m \cdot b\_m\right) \cdot \sqrt{8 \cdot \left(\sqrt{t\_3} + t\_5\right)}\right)\right)\right)}{b\_m \cdot b\_m}\\ \mathbf{elif}\;b\_m \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;-0.25 \cdot \frac{a\_m \cdot \left(-1 \cdot \left(x-scale\_m \cdot \sqrt{8 \cdot \left({b\_m}^{4} \cdot \left(\sqrt{{t\_0}^{4}} + {t\_0}^{2}\right)\right)}\right)\right)}{b\_m \cdot b\_m}\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(b\_m \cdot \left(-1 \cdot \left(y-scale\_m \cdot \mathsf{fma}\left(4, \frac{\left(x-scale\_m \cdot x-scale\_m\right) \cdot \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-2, t\_8, 4 \cdot t\_8\right)}{\left(x-scale\_m \cdot x-scale\_m\right) \cdot t\_4}, \frac{{t\_7}^{2}}{x-scale\_m \cdot x-scale\_m}\right)}{\left(y-scale\_m \cdot y-scale\_m\right) \cdot t\_6}, \left(x-scale\_m \cdot x-scale\_m\right) \cdot t\_6\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a_m b_m angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* -1.54320987654321e-5 (* (* angle angle) (* PI PI)))))
        (t_1 (* 0.005555555555555556 (* angle PI)))
        (t_2 (cos t_1))
        (t_3 (pow t_2 4.0))
        (t_4 (sqrt (/ t_3 (pow x-scale_m 4.0))))
        (t_5 (pow t_2 2.0))
        (t_6
         (sqrt
          (*
           8.0
           (/
            (+ t_4 (/ t_5 (* x-scale_m x-scale_m)))
            (* x-scale_m x-scale_m)))))
        (t_7 (sin t_1))
        (t_8 (/ (pow (* t_2 t_7) 2.0) (* x-scale_m x-scale_m))))
   (if (<= b_m 1.85e-143)
     (*
      -0.25
      (*
       a_m
       (*
        -1.0
        (*
         x-scale_m
         (*
          (* y-scale_m y-scale_m)
          (sqrt
           (*
            8.0
            (/
             (+
              (sqrt (/ t_3 (pow y-scale_m 4.0)))
              (/ t_5 (* y-scale_m y-scale_m)))
             (* y-scale_m y-scale_m)))))))))
     (if (<= b_m 2.9e-65)
       (*
        -0.25
        (/
         (*
          a_m
          (*
           -1.0
           (* x-scale_m (* (* b_m b_m) (sqrt (* 8.0 (+ (sqrt t_3) t_5)))))))
         (* b_m b_m)))
       (if (<= b_m 1.35e+154)
         (*
          -0.25
          (/
           (*
            a_m
            (*
             -1.0
             (*
              x-scale_m
              (sqrt
               (*
                8.0
                (* (pow b_m 4.0) (+ (sqrt (pow t_0 4.0)) (pow t_0 2.0))))))))
           (* b_m b_m)))
         (*
          -0.25
          (*
           b_m
           (*
            -1.0
            (*
             y-scale_m
             (fma
              4.0
              (/
               (*
                (* x-scale_m x-scale_m)
                (fma
                 0.5
                 (/ (fma -2.0 t_8 (* 4.0 t_8)) (* (* x-scale_m x-scale_m) t_4))
                 (/ (pow t_7 2.0) (* x-scale_m x-scale_m))))
               (* (* y-scale_m y-scale_m) t_6))
              (* (* x-scale_m x-scale_m) t_6)))))))))))

C

a_m = fabs(a);
b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = 1.0 + (-1.54320987654321e-5 * ((angle * angle) * (((double) M_PI) * ((double) M_PI))));
	double t_1 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_2 = cos(t_1);
	double t_3 = pow(t_2, 4.0);
	double t_4 = sqrt((t_3 / pow(x_45_scale_m, 4.0)));
	double t_5 = pow(t_2, 2.0);
	double t_6 = sqrt((8.0 * ((t_4 + (t_5 / (x_45_scale_m * x_45_scale_m))) / (x_45_scale_m * x_45_scale_m))));
	double t_7 = sin(t_1);
	double t_8 = pow((t_2 * t_7), 2.0) / (x_45_scale_m * x_45_scale_m);
	double tmp;
	if (b_m <= 1.85e-143) {
		tmp = -0.25 * (a_m * (-1.0 * (x_45_scale_m * ((y_45_scale_m * y_45_scale_m) * sqrt((8.0 * ((sqrt((t_3 / pow(y_45_scale_m, 4.0))) + (t_5 / (y_45_scale_m * y_45_scale_m))) / (y_45_scale_m * y_45_scale_m))))))));
	} else if (b_m <= 2.9e-65) {
		tmp = -0.25 * ((a_m * (-1.0 * (x_45_scale_m * ((b_m * b_m) * sqrt((8.0 * (sqrt(t_3) + t_5))))))) / (b_m * b_m));
	} else if (b_m <= 1.35e+154) {
		tmp = -0.25 * ((a_m * (-1.0 * (x_45_scale_m * sqrt((8.0 * (pow(b_m, 4.0) * (sqrt(pow(t_0, 4.0)) + pow(t_0, 2.0)))))))) / (b_m * b_m));
	} else {
		tmp = -0.25 * (b_m * (-1.0 * (y_45_scale_m * fma(4.0, (((x_45_scale_m * x_45_scale_m) * fma(0.5, (fma(-2.0, t_8, (4.0 * t_8)) / ((x_45_scale_m * x_45_scale_m) * t_4)), (pow(t_7, 2.0) / (x_45_scale_m * x_45_scale_m)))) / ((y_45_scale_m * y_45_scale_m) * t_6)), ((x_45_scale_m * x_45_scale_m) * t_6)))));
	}
	return tmp;
}

Julia

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(1.0 + Float64(-1.54320987654321e-5 * Float64(Float64(angle * angle) * Float64(pi * pi))))
	t_1 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_2 = cos(t_1)
	t_3 = t_2 ^ 4.0
	t_4 = sqrt(Float64(t_3 / (x_45_scale_m ^ 4.0)))
	t_5 = t_2 ^ 2.0
	t_6 = sqrt(Float64(8.0 * Float64(Float64(t_4 + Float64(t_5 / Float64(x_45_scale_m * x_45_scale_m))) / Float64(x_45_scale_m * x_45_scale_m))))
	t_7 = sin(t_1)
	t_8 = Float64((Float64(t_2 * t_7) ^ 2.0) / Float64(x_45_scale_m * x_45_scale_m))
	tmp = 0.0
	if (b_m <= 1.85e-143)
		tmp = Float64(-0.25 * Float64(a_m * Float64(-1.0 * Float64(x_45_scale_m * Float64(Float64(y_45_scale_m * y_45_scale_m) * sqrt(Float64(8.0 * Float64(Float64(sqrt(Float64(t_3 / (y_45_scale_m ^ 4.0))) + Float64(t_5 / Float64(y_45_scale_m * y_45_scale_m))) / Float64(y_45_scale_m * y_45_scale_m)))))))));
	elseif (b_m <= 2.9e-65)
		tmp = Float64(-0.25 * Float64(Float64(a_m * Float64(-1.0 * Float64(x_45_scale_m * Float64(Float64(b_m * b_m) * sqrt(Float64(8.0 * Float64(sqrt(t_3) + t_5))))))) / Float64(b_m * b_m)));
	elseif (b_m <= 1.35e+154)
		tmp = Float64(-0.25 * Float64(Float64(a_m * Float64(-1.0 * Float64(x_45_scale_m * sqrt(Float64(8.0 * Float64((b_m ^ 4.0) * Float64(sqrt((t_0 ^ 4.0)) + (t_0 ^ 2.0)))))))) / Float64(b_m * b_m)));
	else
		tmp = Float64(-0.25 * Float64(b_m * Float64(-1.0 * Float64(y_45_scale_m * fma(4.0, Float64(Float64(Float64(x_45_scale_m * x_45_scale_m) * fma(0.5, Float64(fma(-2.0, t_8, Float64(4.0 * t_8)) / Float64(Float64(x_45_scale_m * x_45_scale_m) * t_4)), Float64((t_7 ^ 2.0) / Float64(x_45_scale_m * x_45_scale_m)))) / Float64(Float64(y_45_scale_m * y_45_scale_m) * t_6)), Float64(Float64(x_45_scale_m * x_45_scale_m) * t_6))))));
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(1.0 + N[(-1.54320987654321e-5 * N[(N[(angle * angle), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[Power[t$95$2, 4.0], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(t$95$3 / N[Power[x$45$scale$95$m, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[Power[t$95$2, 2.0], $MachinePrecision]}, Block[{t$95$6 = N[Sqrt[N[(8.0 * N[(N[(t$95$4 + N[(t$95$5 / N[(x$45$scale$95$m * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale$95$m * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$7 = N[Sin[t$95$1], $MachinePrecision]}, Block[{t$95$8 = N[(N[Power[N[(t$95$2 * t$95$7), $MachinePrecision], 2.0], $MachinePrecision] / N[(x$45$scale$95$m * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b$95$m, 1.85e-143], N[(-0.25 * N[(a$95$m * N[(-1.0 * N[(x$45$scale$95$m * N[(N[(y$45$scale$95$m * y$45$scale$95$m), $MachinePrecision] * N[Sqrt[N[(8.0 * N[(N[(N[Sqrt[N[(t$95$3 / N[Power[y$45$scale$95$m, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(t$95$5 / N[(y$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 2.9e-65], N[(-0.25 * N[(N[(a$95$m * N[(-1.0 * N[(x$45$scale$95$m * N[(N[(b$95$m * b$95$m), $MachinePrecision] * N[Sqrt[N[(8.0 * N[(N[Sqrt[t$95$3], $MachinePrecision] + t$95$5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 1.35e+154], N[(-0.25 * N[(N[(a$95$m * N[(-1.0 * N[(x$45$scale$95$m * N[Sqrt[N[(8.0 * N[(N[Power[b$95$m, 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]), $MachinePrecision] / N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.25 * N[(b$95$m * N[(-1.0 * N[(y$45$scale$95$m * N[(4.0 * N[(N[(N[(x$45$scale$95$m * x$45$scale$95$m), $MachinePrecision] * N[(0.5 * N[(N[(-2.0 * t$95$8 + N[(4.0 * t$95$8), $MachinePrecision]), $MachinePrecision] / N[(N[(x$45$scale$95$m * x$45$scale$95$m), $MachinePrecision] * t$95$4), $MachinePrecision]), $MachinePrecision] + N[(N[Power[t$95$7, 2.0], $MachinePrecision] / N[(x$45$scale$95$m * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(y$45$scale$95$m * y$45$scale$95$m), $MachinePrecision] * t$95$6), $MachinePrecision]), $MachinePrecision] + N[(N[(x$45$scale$95$m * x$45$scale$95$m), $MachinePrecision] * t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < 1.85e-143

   1. Initial program 2.7%

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

   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 0.1%

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

   5. Taylor expanded in x-scale around -inf

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

   7. Applied rewrites 4.2%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 0.3%

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

   11. Taylor expanded in x-scale around -inf

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

   13. Applied rewrites 11.4%

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

   if 1.85e-143 < b < 2.8999999999999998e-65

   1. Initial program 2.7%

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

   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 0.1%

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

   5. Taylor expanded in x-scale around -inf

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

   7. Applied rewrites 4.2%

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

   8. Taylor expanded in y-scale around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

   10. Applied rewrites 13.6%

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

   11. Taylor expanded in b around 0

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

       1. lower-*.f64 N/A

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

       2. pow2 N/A

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

       3. lift-*.f64 N/A

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

       4. lower-sqrt.f64 N/A

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

       5. lower-*.f64 N/A

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

   13. Applied rewrites 16.1%

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

   if 2.8999999999999998e-65 < b < 1.35000000000000003e154

   1. Initial program 2.7%

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

   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 0.1%

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

   5. Taylor expanded in x-scale around -inf

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

   7. Applied rewrites 4.2%

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

   8. Taylor expanded in y-scale around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

   10. Applied rewrites 13.6%

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

   11. Taylor expanded in angle around 0

       \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]
   12. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       4. pow2 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       5. lift-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       6. pow2 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       7. lift-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       8. lift-PI.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       9. lift-PI.f64 14.4

          \[\leadsto -0.25 \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + -1.54320987654321 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

   13. Applied rewrites 14.4%

       \[\leadsto -0.25 \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + -1.54320987654321 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

   14. Taylor expanded in angle around 0

       \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]
   15. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       4. pow2 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       5. lift-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       6. pow2 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       7. lift-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       8. lift-PI.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       9. lift-PI.f64 14.4

          \[\leadsto -0.25 \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + -1.54320987654321 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + -1.54320987654321 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

   16. Applied rewrites 14.4%

       \[\leadsto -0.25 \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + -1.54320987654321 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + -1.54320987654321 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

   if 1.35000000000000003e154 < b

   1. Initial program 2.7%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 1.4%

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

   5. Taylor expanded in b around -inf

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

   7. Applied rewrites 0.3%

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

   8. Taylor expanded in y-scale around -inf

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

   10. Applied rewrites 13.2%

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

Alternative 2: 25.8% accurate, 4.3× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := 1 + -1.54320987654321 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\\ t_1 := \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ t_2 := {t\_1}^{2}\\ t_3 := {t\_1}^{4}\\ \mathbf{if}\;b\_m \leq 1.85 \cdot 10^{-143}:\\ \;\;\;\;-0.25 \cdot \left(a\_m \cdot \left(-1 \cdot \left(x-scale\_m \cdot \left(\left(y-scale\_m \cdot y-scale\_m\right) \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{t\_3}{{y-scale\_m}^{4}}} + \frac{t\_2}{y-scale\_m \cdot y-scale\_m}}{y-scale\_m \cdot y-scale\_m}}\right)\right)\right)\right)\\ \mathbf{elif}\;b\_m \leq 2.9 \cdot 10^{-65}:\\ \;\;\;\;-0.25 \cdot \frac{a\_m \cdot \left(-1 \cdot \left(x-scale\_m \cdot \left(\left(b\_m \cdot b\_m\right) \cdot \sqrt{8 \cdot \left(\sqrt{t\_3} + t\_2\right)}\right)\right)\right)}{b\_m \cdot b\_m}\\ \mathbf{elif}\;b\_m \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;-0.25 \cdot \frac{a\_m \cdot \left(-1 \cdot \left(x-scale\_m \cdot \sqrt{8 \cdot \left({b\_m}^{4} \cdot \left(\sqrt{{t\_0}^{4}} + {t\_0}^{2}\right)\right)}\right)\right)}{b\_m \cdot b\_m}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(b\_m \cdot \left(\left(x-scale\_m \cdot x-scale\_m\right) \cdot \left(y-scale\_m \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{t\_3}{{x-scale\_m}^{4}}} + \frac{t\_2}{x-scale\_m \cdot x-scale\_m}}{x-scale\_m \cdot x-scale\_m}}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a_m b_m angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* -1.54320987654321e-5 (* (* angle angle) (* PI PI)))))
        (t_1 (cos (* 0.005555555555555556 (* angle PI))))
        (t_2 (pow t_1 2.0))
        (t_3 (pow t_1 4.0)))
   (if (<= b_m 1.85e-143)
     (*
      -0.25
      (*
       a_m
       (*
        -1.0
        (*
         x-scale_m
         (*
          (* y-scale_m y-scale_m)
          (sqrt
           (*
            8.0
            (/
             (+
              (sqrt (/ t_3 (pow y-scale_m 4.0)))
              (/ t_2 (* y-scale_m y-scale_m)))
             (* y-scale_m y-scale_m)))))))))
     (if (<= b_m 2.9e-65)
       (*
        -0.25
        (/
         (*
          a_m
          (*
           -1.0
           (* x-scale_m (* (* b_m b_m) (sqrt (* 8.0 (+ (sqrt t_3) t_2)))))))
         (* b_m b_m)))
       (if (<= b_m 1.35e+154)
         (*
          -0.25
          (/
           (*
            a_m
            (*
             -1.0
             (*
              x-scale_m
              (sqrt
               (*
                8.0
                (* (pow b_m 4.0) (+ (sqrt (pow t_0 4.0)) (pow t_0 2.0))))))))
           (* b_m b_m)))
         (*
          0.25
          (*
           b_m
           (*
            (* x-scale_m x-scale_m)
            (*
             y-scale_m
             (sqrt
              (*
               8.0
               (/
                (+
                 (sqrt (/ t_3 (pow x-scale_m 4.0)))
                 (/ t_2 (* x-scale_m x-scale_m)))
                (* x-scale_m x-scale_m)))))))))))))

C

a_m = fabs(a);
b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = 1.0 + (-1.54320987654321e-5 * ((angle * angle) * (((double) M_PI) * ((double) M_PI))));
	double t_1 = cos((0.005555555555555556 * (angle * ((double) M_PI))));
	double t_2 = pow(t_1, 2.0);
	double t_3 = pow(t_1, 4.0);
	double tmp;
	if (b_m <= 1.85e-143) {
		tmp = -0.25 * (a_m * (-1.0 * (x_45_scale_m * ((y_45_scale_m * y_45_scale_m) * sqrt((8.0 * ((sqrt((t_3 / pow(y_45_scale_m, 4.0))) + (t_2 / (y_45_scale_m * y_45_scale_m))) / (y_45_scale_m * y_45_scale_m))))))));
	} else if (b_m <= 2.9e-65) {
		tmp = -0.25 * ((a_m * (-1.0 * (x_45_scale_m * ((b_m * b_m) * sqrt((8.0 * (sqrt(t_3) + t_2))))))) / (b_m * b_m));
	} else if (b_m <= 1.35e+154) {
		tmp = -0.25 * ((a_m * (-1.0 * (x_45_scale_m * sqrt((8.0 * (pow(b_m, 4.0) * (sqrt(pow(t_0, 4.0)) + pow(t_0, 2.0)))))))) / (b_m * b_m));
	} else {
		tmp = 0.25 * (b_m * ((x_45_scale_m * x_45_scale_m) * (y_45_scale_m * sqrt((8.0 * ((sqrt((t_3 / pow(x_45_scale_m, 4.0))) + (t_2 / (x_45_scale_m * x_45_scale_m))) / (x_45_scale_m * x_45_scale_m)))))));
	}
	return tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = 1.0 + (-1.54320987654321e-5 * ((angle * angle) * (Math.PI * Math.PI)));
	double t_1 = Math.cos((0.005555555555555556 * (angle * Math.PI)));
	double t_2 = Math.pow(t_1, 2.0);
	double t_3 = Math.pow(t_1, 4.0);
	double tmp;
	if (b_m <= 1.85e-143) {
		tmp = -0.25 * (a_m * (-1.0 * (x_45_scale_m * ((y_45_scale_m * y_45_scale_m) * Math.sqrt((8.0 * ((Math.sqrt((t_3 / Math.pow(y_45_scale_m, 4.0))) + (t_2 / (y_45_scale_m * y_45_scale_m))) / (y_45_scale_m * y_45_scale_m))))))));
	} else if (b_m <= 2.9e-65) {
		tmp = -0.25 * ((a_m * (-1.0 * (x_45_scale_m * ((b_m * b_m) * Math.sqrt((8.0 * (Math.sqrt(t_3) + t_2))))))) / (b_m * b_m));
	} else if (b_m <= 1.35e+154) {
		tmp = -0.25 * ((a_m * (-1.0 * (x_45_scale_m * Math.sqrt((8.0 * (Math.pow(b_m, 4.0) * (Math.sqrt(Math.pow(t_0, 4.0)) + Math.pow(t_0, 2.0)))))))) / (b_m * b_m));
	} else {
		tmp = 0.25 * (b_m * ((x_45_scale_m * x_45_scale_m) * (y_45_scale_m * Math.sqrt((8.0 * ((Math.sqrt((t_3 / Math.pow(x_45_scale_m, 4.0))) + (t_2 / (x_45_scale_m * x_45_scale_m))) / (x_45_scale_m * x_45_scale_m)))))));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m):
	t_0 = 1.0 + (-1.54320987654321e-5 * ((angle * angle) * (math.pi * math.pi)))
	t_1 = math.cos((0.005555555555555556 * (angle * math.pi)))
	t_2 = math.pow(t_1, 2.0)
	t_3 = math.pow(t_1, 4.0)
	tmp = 0
	if b_m <= 1.85e-143:
		tmp = -0.25 * (a_m * (-1.0 * (x_45_scale_m * ((y_45_scale_m * y_45_scale_m) * math.sqrt((8.0 * ((math.sqrt((t_3 / math.pow(y_45_scale_m, 4.0))) + (t_2 / (y_45_scale_m * y_45_scale_m))) / (y_45_scale_m * y_45_scale_m))))))))
	elif b_m <= 2.9e-65:
		tmp = -0.25 * ((a_m * (-1.0 * (x_45_scale_m * ((b_m * b_m) * math.sqrt((8.0 * (math.sqrt(t_3) + t_2))))))) / (b_m * b_m))
	elif b_m <= 1.35e+154:
		tmp = -0.25 * ((a_m * (-1.0 * (x_45_scale_m * math.sqrt((8.0 * (math.pow(b_m, 4.0) * (math.sqrt(math.pow(t_0, 4.0)) + math.pow(t_0, 2.0)))))))) / (b_m * b_m))
	else:
		tmp = 0.25 * (b_m * ((x_45_scale_m * x_45_scale_m) * (y_45_scale_m * math.sqrt((8.0 * ((math.sqrt((t_3 / math.pow(x_45_scale_m, 4.0))) + (t_2 / (x_45_scale_m * x_45_scale_m))) / (x_45_scale_m * x_45_scale_m)))))))
	return tmp

Julia

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(1.0 + Float64(-1.54320987654321e-5 * Float64(Float64(angle * angle) * Float64(pi * pi))))
	t_1 = cos(Float64(0.005555555555555556 * Float64(angle * pi)))
	t_2 = t_1 ^ 2.0
	t_3 = t_1 ^ 4.0
	tmp = 0.0
	if (b_m <= 1.85e-143)
		tmp = Float64(-0.25 * Float64(a_m * Float64(-1.0 * Float64(x_45_scale_m * Float64(Float64(y_45_scale_m * y_45_scale_m) * sqrt(Float64(8.0 * Float64(Float64(sqrt(Float64(t_3 / (y_45_scale_m ^ 4.0))) + Float64(t_2 / Float64(y_45_scale_m * y_45_scale_m))) / Float64(y_45_scale_m * y_45_scale_m)))))))));
	elseif (b_m <= 2.9e-65)
		tmp = Float64(-0.25 * Float64(Float64(a_m * Float64(-1.0 * Float64(x_45_scale_m * Float64(Float64(b_m * b_m) * sqrt(Float64(8.0 * Float64(sqrt(t_3) + t_2))))))) / Float64(b_m * b_m)));
	elseif (b_m <= 1.35e+154)
		tmp = Float64(-0.25 * Float64(Float64(a_m * Float64(-1.0 * Float64(x_45_scale_m * sqrt(Float64(8.0 * Float64((b_m ^ 4.0) * Float64(sqrt((t_0 ^ 4.0)) + (t_0 ^ 2.0)))))))) / Float64(b_m * b_m)));
	else
		tmp = Float64(0.25 * Float64(b_m * Float64(Float64(x_45_scale_m * x_45_scale_m) * Float64(y_45_scale_m * sqrt(Float64(8.0 * Float64(Float64(sqrt(Float64(t_3 / (x_45_scale_m ^ 4.0))) + Float64(t_2 / Float64(x_45_scale_m * x_45_scale_m))) / Float64(x_45_scale_m * x_45_scale_m))))))));
	end
	return tmp
end

MATLAB

a_m = abs(a);
b_m = abs(b);
x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = 1.0 + (-1.54320987654321e-5 * ((angle * angle) * (pi * pi)));
	t_1 = cos((0.005555555555555556 * (angle * pi)));
	t_2 = t_1 ^ 2.0;
	t_3 = t_1 ^ 4.0;
	tmp = 0.0;
	if (b_m <= 1.85e-143)
		tmp = -0.25 * (a_m * (-1.0 * (x_45_scale_m * ((y_45_scale_m * y_45_scale_m) * sqrt((8.0 * ((sqrt((t_3 / (y_45_scale_m ^ 4.0))) + (t_2 / (y_45_scale_m * y_45_scale_m))) / (y_45_scale_m * y_45_scale_m))))))));
	elseif (b_m <= 2.9e-65)
		tmp = -0.25 * ((a_m * (-1.0 * (x_45_scale_m * ((b_m * b_m) * sqrt((8.0 * (sqrt(t_3) + t_2))))))) / (b_m * b_m));
	elseif (b_m <= 1.35e+154)
		tmp = -0.25 * ((a_m * (-1.0 * (x_45_scale_m * sqrt((8.0 * ((b_m ^ 4.0) * (sqrt((t_0 ^ 4.0)) + (t_0 ^ 2.0)))))))) / (b_m * b_m));
	else
		tmp = 0.25 * (b_m * ((x_45_scale_m * x_45_scale_m) * (y_45_scale_m * sqrt((8.0 * ((sqrt((t_3 / (x_45_scale_m ^ 4.0))) + (t_2 / (x_45_scale_m * x_45_scale_m))) / (x_45_scale_m * x_45_scale_m)))))));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(1.0 + N[(-1.54320987654321e-5 * N[(N[(angle * angle), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Power[t$95$1, 2.0], $MachinePrecision]}, Block[{t$95$3 = N[Power[t$95$1, 4.0], $MachinePrecision]}, If[LessEqual[b$95$m, 1.85e-143], N[(-0.25 * N[(a$95$m * N[(-1.0 * N[(x$45$scale$95$m * N[(N[(y$45$scale$95$m * y$45$scale$95$m), $MachinePrecision] * N[Sqrt[N[(8.0 * N[(N[(N[Sqrt[N[(t$95$3 / N[Power[y$45$scale$95$m, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(t$95$2 / N[(y$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 2.9e-65], N[(-0.25 * N[(N[(a$95$m * N[(-1.0 * N[(x$45$scale$95$m * N[(N[(b$95$m * b$95$m), $MachinePrecision] * N[Sqrt[N[(8.0 * N[(N[Sqrt[t$95$3], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 1.35e+154], N[(-0.25 * N[(N[(a$95$m * N[(-1.0 * N[(x$45$scale$95$m * N[Sqrt[N[(8.0 * N[(N[Power[b$95$m, 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]), $MachinePrecision] / N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(b$95$m * N[(N[(x$45$scale$95$m * x$45$scale$95$m), $MachinePrecision] * N[(y$45$scale$95$m * N[Sqrt[N[(8.0 * N[(N[(N[Sqrt[N[(t$95$3 / N[Power[x$45$scale$95$m, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(t$95$2 / N[(x$45$scale$95$m * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale$95$m * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < 1.85e-143

   1. Initial program 2.7%

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

   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 0.1%

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

   5. Taylor expanded in x-scale around -inf

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

   7. Applied rewrites 4.2%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 0.3%

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

   11. Taylor expanded in x-scale around -inf

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

   13. Applied rewrites 11.4%

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

   if 1.85e-143 < b < 2.8999999999999998e-65

   1. Initial program 2.7%

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

   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 0.1%

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

   5. Taylor expanded in x-scale around -inf

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

   7. Applied rewrites 4.2%

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

   8. Taylor expanded in y-scale around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

   10. Applied rewrites 13.6%

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

   11. Taylor expanded in b around 0

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

       1. lower-*.f64 N/A

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

       2. pow2 N/A

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

       3. lift-*.f64 N/A

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

       4. lower-sqrt.f64 N/A

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

       5. lower-*.f64 N/A

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

   13. Applied rewrites 16.1%

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

   if 2.8999999999999998e-65 < b < 1.35000000000000003e154

   1. Initial program 2.7%

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

   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 0.1%

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

   5. Taylor expanded in x-scale around -inf

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

   7. Applied rewrites 4.2%

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

   8. Taylor expanded in y-scale around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

   10. Applied rewrites 13.6%

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

   11. Taylor expanded in angle around 0

       \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]
   12. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       4. pow2 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       5. lift-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       6. pow2 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       7. lift-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       8. lift-PI.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       9. lift-PI.f64 14.4

          \[\leadsto -0.25 \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + -1.54320987654321 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

   13. Applied rewrites 14.4%

       \[\leadsto -0.25 \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + -1.54320987654321 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

   14. Taylor expanded in angle around 0

       \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]
   15. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       4. pow2 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       5. lift-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       6. pow2 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       7. lift-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       8. lift-PI.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       9. lift-PI.f64 14.4

          \[\leadsto -0.25 \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + -1.54320987654321 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + -1.54320987654321 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

   16. Applied rewrites 14.4%

       \[\leadsto -0.25 \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + -1.54320987654321 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + -1.54320987654321 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

   if 1.35000000000000003e154 < b

   1. Initial program 2.7%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 1.4%

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

   5. Taylor expanded in b around -inf

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

   7. Applied rewrites 0.3%

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

   8. Taylor expanded in y-scale around -inf

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

   10. Applied rewrites 11.4%

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

Alternative 3: 21.7% accurate, 4.4× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ t_1 := {t\_0}^{4}\\ t_2 := 1 + -1.54320987654321 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\\ t_3 := {t\_0}^{2}\\ \mathbf{if}\;x-scale\_m \leq 4 \cdot 10^{-98}:\\ \;\;\;\;-0.25 \cdot \frac{a\_m \cdot \left(-1 \cdot \left(x-scale\_m \cdot \sqrt{8 \cdot \left({b\_m}^{4} \cdot \left(\sqrt{{t\_2}^{4}} + {t\_2}^{2}\right)\right)}\right)\right)}{b\_m \cdot b\_m}\\ \mathbf{elif}\;x-scale\_m \leq 1.55 \cdot 10^{+63}:\\ \;\;\;\;0.25 \cdot \left(b\_m \cdot \left(\left(x-scale\_m \cdot x-scale\_m\right) \cdot \left(y-scale\_m \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{t\_1}{{x-scale\_m}^{4}}} + \frac{t\_3}{x-scale\_m \cdot x-scale\_m}}{x-scale\_m \cdot x-scale\_m}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \frac{a\_m \cdot \left(-1 \cdot \left(x-scale\_m \cdot \left(\left(b\_m \cdot b\_m\right) \cdot \sqrt{8 \cdot \left(\sqrt{t\_1} + t\_3\right)}\right)\right)\right)}{b\_m \cdot b\_m}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a_m b_m angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (cos (* 0.005555555555555556 (* angle PI))))
        (t_1 (pow t_0 4.0))
        (t_2 (+ 1.0 (* -1.54320987654321e-5 (* (* angle angle) (* PI PI)))))
        (t_3 (pow t_0 2.0)))
   (if (<= x-scale_m 4e-98)
     (*
      -0.25
      (/
       (*
        a_m
        (*
         -1.0
         (*
          x-scale_m
          (sqrt
           (* 8.0 (* (pow b_m 4.0) (+ (sqrt (pow t_2 4.0)) (pow t_2 2.0))))))))
       (* b_m b_m)))
     (if (<= x-scale_m 1.55e+63)
       (*
        0.25
        (*
         b_m
         (*
          (* x-scale_m x-scale_m)
          (*
           y-scale_m
           (sqrt
            (*
             8.0
             (/
              (+
               (sqrt (/ t_1 (pow x-scale_m 4.0)))
               (/ t_3 (* x-scale_m x-scale_m)))
              (* x-scale_m x-scale_m))))))))
       (*
        -0.25
        (/
         (*
          a_m
          (*
           -1.0
           (* x-scale_m (* (* b_m b_m) (sqrt (* 8.0 (+ (sqrt t_1) t_3)))))))
         (* b_m b_m)))))))

C

a_m = fabs(a);
b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = cos((0.005555555555555556 * (angle * ((double) M_PI))));
	double t_1 = pow(t_0, 4.0);
	double t_2 = 1.0 + (-1.54320987654321e-5 * ((angle * angle) * (((double) M_PI) * ((double) M_PI))));
	double t_3 = pow(t_0, 2.0);
	double tmp;
	if (x_45_scale_m <= 4e-98) {
		tmp = -0.25 * ((a_m * (-1.0 * (x_45_scale_m * sqrt((8.0 * (pow(b_m, 4.0) * (sqrt(pow(t_2, 4.0)) + pow(t_2, 2.0)))))))) / (b_m * b_m));
	} else if (x_45_scale_m <= 1.55e+63) {
		tmp = 0.25 * (b_m * ((x_45_scale_m * x_45_scale_m) * (y_45_scale_m * sqrt((8.0 * ((sqrt((t_1 / pow(x_45_scale_m, 4.0))) + (t_3 / (x_45_scale_m * x_45_scale_m))) / (x_45_scale_m * x_45_scale_m)))))));
	} else {
		tmp = -0.25 * ((a_m * (-1.0 * (x_45_scale_m * ((b_m * b_m) * sqrt((8.0 * (sqrt(t_1) + t_3))))))) / (b_m * b_m));
	}
	return tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = Math.cos((0.005555555555555556 * (angle * Math.PI)));
	double t_1 = Math.pow(t_0, 4.0);
	double t_2 = 1.0 + (-1.54320987654321e-5 * ((angle * angle) * (Math.PI * Math.PI)));
	double t_3 = Math.pow(t_0, 2.0);
	double tmp;
	if (x_45_scale_m <= 4e-98) {
		tmp = -0.25 * ((a_m * (-1.0 * (x_45_scale_m * Math.sqrt((8.0 * (Math.pow(b_m, 4.0) * (Math.sqrt(Math.pow(t_2, 4.0)) + Math.pow(t_2, 2.0)))))))) / (b_m * b_m));
	} else if (x_45_scale_m <= 1.55e+63) {
		tmp = 0.25 * (b_m * ((x_45_scale_m * x_45_scale_m) * (y_45_scale_m * Math.sqrt((8.0 * ((Math.sqrt((t_1 / Math.pow(x_45_scale_m, 4.0))) + (t_3 / (x_45_scale_m * x_45_scale_m))) / (x_45_scale_m * x_45_scale_m)))))));
	} else {
		tmp = -0.25 * ((a_m * (-1.0 * (x_45_scale_m * ((b_m * b_m) * Math.sqrt((8.0 * (Math.sqrt(t_1) + t_3))))))) / (b_m * b_m));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m):
	t_0 = math.cos((0.005555555555555556 * (angle * math.pi)))
	t_1 = math.pow(t_0, 4.0)
	t_2 = 1.0 + (-1.54320987654321e-5 * ((angle * angle) * (math.pi * math.pi)))
	t_3 = math.pow(t_0, 2.0)
	tmp = 0
	if x_45_scale_m <= 4e-98:
		tmp = -0.25 * ((a_m * (-1.0 * (x_45_scale_m * math.sqrt((8.0 * (math.pow(b_m, 4.0) * (math.sqrt(math.pow(t_2, 4.0)) + math.pow(t_2, 2.0)))))))) / (b_m * b_m))
	elif x_45_scale_m <= 1.55e+63:
		tmp = 0.25 * (b_m * ((x_45_scale_m * x_45_scale_m) * (y_45_scale_m * math.sqrt((8.0 * ((math.sqrt((t_1 / math.pow(x_45_scale_m, 4.0))) + (t_3 / (x_45_scale_m * x_45_scale_m))) / (x_45_scale_m * x_45_scale_m)))))))
	else:
		tmp = -0.25 * ((a_m * (-1.0 * (x_45_scale_m * ((b_m * b_m) * math.sqrt((8.0 * (math.sqrt(t_1) + t_3))))))) / (b_m * b_m))
	return tmp

Julia

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = cos(Float64(0.005555555555555556 * Float64(angle * pi)))
	t_1 = t_0 ^ 4.0
	t_2 = Float64(1.0 + Float64(-1.54320987654321e-5 * Float64(Float64(angle * angle) * Float64(pi * pi))))
	t_3 = t_0 ^ 2.0
	tmp = 0.0
	if (x_45_scale_m <= 4e-98)
		tmp = Float64(-0.25 * Float64(Float64(a_m * Float64(-1.0 * Float64(x_45_scale_m * sqrt(Float64(8.0 * Float64((b_m ^ 4.0) * Float64(sqrt((t_2 ^ 4.0)) + (t_2 ^ 2.0)))))))) / Float64(b_m * b_m)));
	elseif (x_45_scale_m <= 1.55e+63)
		tmp = Float64(0.25 * Float64(b_m * Float64(Float64(x_45_scale_m * x_45_scale_m) * Float64(y_45_scale_m * sqrt(Float64(8.0 * Float64(Float64(sqrt(Float64(t_1 / (x_45_scale_m ^ 4.0))) + Float64(t_3 / Float64(x_45_scale_m * x_45_scale_m))) / Float64(x_45_scale_m * x_45_scale_m))))))));
	else
		tmp = Float64(-0.25 * Float64(Float64(a_m * Float64(-1.0 * Float64(x_45_scale_m * Float64(Float64(b_m * b_m) * sqrt(Float64(8.0 * Float64(sqrt(t_1) + t_3))))))) / Float64(b_m * b_m)));
	end
	return tmp
end

MATLAB

a_m = abs(a);
b_m = abs(b);
x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = cos((0.005555555555555556 * (angle * pi)));
	t_1 = t_0 ^ 4.0;
	t_2 = 1.0 + (-1.54320987654321e-5 * ((angle * angle) * (pi * pi)));
	t_3 = t_0 ^ 2.0;
	tmp = 0.0;
	if (x_45_scale_m <= 4e-98)
		tmp = -0.25 * ((a_m * (-1.0 * (x_45_scale_m * sqrt((8.0 * ((b_m ^ 4.0) * (sqrt((t_2 ^ 4.0)) + (t_2 ^ 2.0)))))))) / (b_m * b_m));
	elseif (x_45_scale_m <= 1.55e+63)
		tmp = 0.25 * (b_m * ((x_45_scale_m * x_45_scale_m) * (y_45_scale_m * sqrt((8.0 * ((sqrt((t_1 / (x_45_scale_m ^ 4.0))) + (t_3 / (x_45_scale_m * x_45_scale_m))) / (x_45_scale_m * x_45_scale_m)))))));
	else
		tmp = -0.25 * ((a_m * (-1.0 * (x_45_scale_m * ((b_m * b_m) * sqrt((8.0 * (sqrt(t_1) + t_3))))))) / (b_m * b_m));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[Cos[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[t$95$0, 4.0], $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + N[(-1.54320987654321e-5 * N[(N[(angle * angle), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[t$95$0, 2.0], $MachinePrecision]}, If[LessEqual[x$45$scale$95$m, 4e-98], N[(-0.25 * N[(N[(a$95$m * N[(-1.0 * N[(x$45$scale$95$m * N[Sqrt[N[(8.0 * N[(N[Power[b$95$m, 4.0], $MachinePrecision] * N[(N[Sqrt[N[Power[t$95$2, 4.0], $MachinePrecision]], $MachinePrecision] + N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$45$scale$95$m, 1.55e+63], N[(0.25 * N[(b$95$m * N[(N[(x$45$scale$95$m * x$45$scale$95$m), $MachinePrecision] * N[(y$45$scale$95$m * N[Sqrt[N[(8.0 * N[(N[(N[Sqrt[N[(t$95$1 / N[Power[x$45$scale$95$m, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(t$95$3 / N[(x$45$scale$95$m * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale$95$m * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.25 * N[(N[(a$95$m * N[(-1.0 * N[(x$45$scale$95$m * N[(N[(b$95$m * b$95$m), $MachinePrecision] * N[Sqrt[N[(8.0 * N[(N[Sqrt[t$95$1], $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x-scale < 3.99999999999999976e-98

   1. Initial program 2.7%

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

   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 0.1%

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

   5. Taylor expanded in x-scale around -inf

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

   7. Applied rewrites 4.2%

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

   8. Taylor expanded in y-scale around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

   10. Applied rewrites 13.6%

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

   11. Taylor expanded in angle around 0

       \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]
   12. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       4. pow2 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       5. lift-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       6. pow2 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       7. lift-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       8. lift-PI.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       9. lift-PI.f64 14.4

          \[\leadsto -0.25 \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + -1.54320987654321 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

   13. Applied rewrites 14.4%

       \[\leadsto -0.25 \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + -1.54320987654321 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

   14. Taylor expanded in angle around 0

       \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]
   15. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       4. pow2 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       5. lift-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       6. pow2 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       7. lift-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       8. lift-PI.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       9. lift-PI.f64 14.4

          \[\leadsto -0.25 \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + -1.54320987654321 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + -1.54320987654321 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

   16. Applied rewrites 14.4%

       \[\leadsto -0.25 \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + -1.54320987654321 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + -1.54320987654321 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

   if 3.99999999999999976e-98 < x-scale < 1.55e63

   1. Initial program 2.7%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 1.4%

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

   5. Taylor expanded in b around -inf

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

   7. Applied rewrites 0.3%

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

   8. Taylor expanded in y-scale around -inf

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

   10. Applied rewrites 11.4%

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

   if 1.55e63 < x-scale

   1. Initial program 2.7%

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

   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 0.1%

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

   5. Taylor expanded in x-scale around -inf

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

   7. Applied rewrites 4.2%

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

   8. Taylor expanded in y-scale around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

   10. Applied rewrites 13.6%

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

   11. Taylor expanded in b around 0

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

       1. lower-*.f64 N/A

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

       2. pow2 N/A

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

       3. lift-*.f64 N/A

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

       4. lower-sqrt.f64 N/A

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

       5. lower-*.f64 N/A

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

   13. Applied rewrites 16.1%

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

Alternative 4: 20.9% accurate, 4.4× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := 1 + -1.54320987654321 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\\ t_1 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_2 := \sin t\_1\\ t_3 := \cos t\_1\\ \mathbf{if}\;b\_m \leq 2.9 \cdot 10^{-65}:\\ \;\;\;\;-0.25 \cdot \frac{a\_m \cdot \left(-1 \cdot \left(x-scale\_m \cdot \left(\left(b\_m \cdot b\_m\right) \cdot \sqrt{8 \cdot \left(\sqrt{{t\_3}^{4}} + {t\_3}^{2}\right)}\right)\right)\right)}{b\_m \cdot b\_m}\\ \mathbf{elif}\;b\_m \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;-0.25 \cdot \frac{a\_m \cdot \left(-1 \cdot \left(x-scale\_m \cdot \sqrt{8 \cdot \left({b\_m}^{4} \cdot \left(\sqrt{{t\_0}^{4}} + {t\_0}^{2}\right)\right)}\right)\right)}{b\_m \cdot b\_m}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(b\_m \cdot \left(x-scale\_m \cdot \left(\left(y-scale\_m \cdot y-scale\_m\right) \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{{t\_2}^{4}}{{y-scale\_m}^{4}}} + \frac{{t\_2}^{2}}{y-scale\_m \cdot y-scale\_m}}{y-scale\_m \cdot y-scale\_m}}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a_m b_m angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* -1.54320987654321e-5 (* (* angle angle) (* PI PI)))))
        (t_1 (* 0.005555555555555556 (* angle PI)))
        (t_2 (sin t_1))
        (t_3 (cos t_1)))
   (if (<= b_m 2.9e-65)
     (*
      -0.25
      (/
       (*
        a_m
        (*
         -1.0
         (*
          x-scale_m
          (*
           (* b_m b_m)
           (sqrt (* 8.0 (+ (sqrt (pow t_3 4.0)) (pow t_3 2.0))))))))
       (* b_m b_m)))
     (if (<= b_m 1.35e+154)
       (*
        -0.25
        (/
         (*
          a_m
          (*
           -1.0
           (*
            x-scale_m
            (sqrt
             (*
              8.0
              (* (pow b_m 4.0) (+ (sqrt (pow t_0 4.0)) (pow t_0 2.0))))))))
         (* b_m b_m)))
       (*
        0.25
        (*
         b_m
         (*
          x-scale_m
          (*
           (* y-scale_m y-scale_m)
           (sqrt
            (*
             8.0
             (/
              (+
               (sqrt (/ (pow t_2 4.0) (pow y-scale_m 4.0)))
               (/ (pow t_2 2.0) (* y-scale_m y-scale_m)))
              (* y-scale_m y-scale_m))))))))))))

C

a_m = fabs(a);
b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = 1.0 + (-1.54320987654321e-5 * ((angle * angle) * (((double) M_PI) * ((double) M_PI))));
	double t_1 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_2 = sin(t_1);
	double t_3 = cos(t_1);
	double tmp;
	if (b_m <= 2.9e-65) {
		tmp = -0.25 * ((a_m * (-1.0 * (x_45_scale_m * ((b_m * b_m) * sqrt((8.0 * (sqrt(pow(t_3, 4.0)) + pow(t_3, 2.0)))))))) / (b_m * b_m));
	} else if (b_m <= 1.35e+154) {
		tmp = -0.25 * ((a_m * (-1.0 * (x_45_scale_m * sqrt((8.0 * (pow(b_m, 4.0) * (sqrt(pow(t_0, 4.0)) + pow(t_0, 2.0)))))))) / (b_m * b_m));
	} else {
		tmp = 0.25 * (b_m * (x_45_scale_m * ((y_45_scale_m * y_45_scale_m) * sqrt((8.0 * ((sqrt((pow(t_2, 4.0) / pow(y_45_scale_m, 4.0))) + (pow(t_2, 2.0) / (y_45_scale_m * y_45_scale_m))) / (y_45_scale_m * y_45_scale_m)))))));
	}
	return tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = 1.0 + (-1.54320987654321e-5 * ((angle * angle) * (Math.PI * Math.PI)));
	double t_1 = 0.005555555555555556 * (angle * Math.PI);
	double t_2 = Math.sin(t_1);
	double t_3 = Math.cos(t_1);
	double tmp;
	if (b_m <= 2.9e-65) {
		tmp = -0.25 * ((a_m * (-1.0 * (x_45_scale_m * ((b_m * b_m) * Math.sqrt((8.0 * (Math.sqrt(Math.pow(t_3, 4.0)) + Math.pow(t_3, 2.0)))))))) / (b_m * b_m));
	} else if (b_m <= 1.35e+154) {
		tmp = -0.25 * ((a_m * (-1.0 * (x_45_scale_m * Math.sqrt((8.0 * (Math.pow(b_m, 4.0) * (Math.sqrt(Math.pow(t_0, 4.0)) + Math.pow(t_0, 2.0)))))))) / (b_m * b_m));
	} else {
		tmp = 0.25 * (b_m * (x_45_scale_m * ((y_45_scale_m * y_45_scale_m) * Math.sqrt((8.0 * ((Math.sqrt((Math.pow(t_2, 4.0) / Math.pow(y_45_scale_m, 4.0))) + (Math.pow(t_2, 2.0) / (y_45_scale_m * y_45_scale_m))) / (y_45_scale_m * y_45_scale_m)))))));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m):
	t_0 = 1.0 + (-1.54320987654321e-5 * ((angle * angle) * (math.pi * math.pi)))
	t_1 = 0.005555555555555556 * (angle * math.pi)
	t_2 = math.sin(t_1)
	t_3 = math.cos(t_1)
	tmp = 0
	if b_m <= 2.9e-65:
		tmp = -0.25 * ((a_m * (-1.0 * (x_45_scale_m * ((b_m * b_m) * math.sqrt((8.0 * (math.sqrt(math.pow(t_3, 4.0)) + math.pow(t_3, 2.0)))))))) / (b_m * b_m))
	elif b_m <= 1.35e+154:
		tmp = -0.25 * ((a_m * (-1.0 * (x_45_scale_m * math.sqrt((8.0 * (math.pow(b_m, 4.0) * (math.sqrt(math.pow(t_0, 4.0)) + math.pow(t_0, 2.0)))))))) / (b_m * b_m))
	else:
		tmp = 0.25 * (b_m * (x_45_scale_m * ((y_45_scale_m * y_45_scale_m) * math.sqrt((8.0 * ((math.sqrt((math.pow(t_2, 4.0) / math.pow(y_45_scale_m, 4.0))) + (math.pow(t_2, 2.0) / (y_45_scale_m * y_45_scale_m))) / (y_45_scale_m * y_45_scale_m)))))))
	return tmp

Julia

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(1.0 + Float64(-1.54320987654321e-5 * Float64(Float64(angle * angle) * Float64(pi * pi))))
	t_1 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_2 = sin(t_1)
	t_3 = cos(t_1)
	tmp = 0.0
	if (b_m <= 2.9e-65)
		tmp = Float64(-0.25 * Float64(Float64(a_m * Float64(-1.0 * Float64(x_45_scale_m * Float64(Float64(b_m * b_m) * sqrt(Float64(8.0 * Float64(sqrt((t_3 ^ 4.0)) + (t_3 ^ 2.0)))))))) / Float64(b_m * b_m)));
	elseif (b_m <= 1.35e+154)
		tmp = Float64(-0.25 * Float64(Float64(a_m * Float64(-1.0 * Float64(x_45_scale_m * sqrt(Float64(8.0 * Float64((b_m ^ 4.0) * Float64(sqrt((t_0 ^ 4.0)) + (t_0 ^ 2.0)))))))) / Float64(b_m * b_m)));
	else
		tmp = Float64(0.25 * Float64(b_m * Float64(x_45_scale_m * Float64(Float64(y_45_scale_m * y_45_scale_m) * sqrt(Float64(8.0 * Float64(Float64(sqrt(Float64((t_2 ^ 4.0) / (y_45_scale_m ^ 4.0))) + Float64((t_2 ^ 2.0) / Float64(y_45_scale_m * y_45_scale_m))) / Float64(y_45_scale_m * y_45_scale_m))))))));
	end
	return tmp
end

MATLAB

a_m = abs(a);
b_m = abs(b);
x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = 1.0 + (-1.54320987654321e-5 * ((angle * angle) * (pi * pi)));
	t_1 = 0.005555555555555556 * (angle * pi);
	t_2 = sin(t_1);
	t_3 = cos(t_1);
	tmp = 0.0;
	if (b_m <= 2.9e-65)
		tmp = -0.25 * ((a_m * (-1.0 * (x_45_scale_m * ((b_m * b_m) * sqrt((8.0 * (sqrt((t_3 ^ 4.0)) + (t_3 ^ 2.0)))))))) / (b_m * b_m));
	elseif (b_m <= 1.35e+154)
		tmp = -0.25 * ((a_m * (-1.0 * (x_45_scale_m * sqrt((8.0 * ((b_m ^ 4.0) * (sqrt((t_0 ^ 4.0)) + (t_0 ^ 2.0)))))))) / (b_m * b_m));
	else
		tmp = 0.25 * (b_m * (x_45_scale_m * ((y_45_scale_m * y_45_scale_m) * sqrt((8.0 * ((sqrt(((t_2 ^ 4.0) / (y_45_scale_m ^ 4.0))) + ((t_2 ^ 2.0) / (y_45_scale_m * y_45_scale_m))) / (y_45_scale_m * y_45_scale_m)))))));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(1.0 + N[(-1.54320987654321e-5 * N[(N[(angle * angle), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[Cos[t$95$1], $MachinePrecision]}, If[LessEqual[b$95$m, 2.9e-65], N[(-0.25 * N[(N[(a$95$m * N[(-1.0 * N[(x$45$scale$95$m * N[(N[(b$95$m * b$95$m), $MachinePrecision] * N[Sqrt[N[(8.0 * N[(N[Sqrt[N[Power[t$95$3, 4.0], $MachinePrecision]], $MachinePrecision] + N[Power[t$95$3, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 1.35e+154], N[(-0.25 * N[(N[(a$95$m * N[(-1.0 * N[(x$45$scale$95$m * N[Sqrt[N[(8.0 * N[(N[Power[b$95$m, 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]), $MachinePrecision] / N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(b$95$m * N[(x$45$scale$95$m * N[(N[(y$45$scale$95$m * y$45$scale$95$m), $MachinePrecision] * N[Sqrt[N[(8.0 * N[(N[(N[Sqrt[N[(N[Power[t$95$2, 4.0], $MachinePrecision] / N[Power[y$45$scale$95$m, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[Power[t$95$2, 2.0], $MachinePrecision] / N[(y$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 2.8999999999999998e-65

   1. Initial program 2.7%

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

   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 0.1%

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

   5. Taylor expanded in x-scale around -inf

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

   7. Applied rewrites 4.2%

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

   8. Taylor expanded in y-scale around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

   10. Applied rewrites 13.6%

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

   11. Taylor expanded in b around 0

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

       1. lower-*.f64 N/A

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

       2. pow2 N/A

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

       3. lift-*.f64 N/A

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

       4. lower-sqrt.f64 N/A

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

       5. lower-*.f64 N/A

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

   13. Applied rewrites 16.1%

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

   if 2.8999999999999998e-65 < b < 1.35000000000000003e154

   1. Initial program 2.7%

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

   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 0.1%

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

   5. Taylor expanded in x-scale around -inf

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

   7. Applied rewrites 4.2%

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

   8. Taylor expanded in y-scale around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

   10. Applied rewrites 13.6%

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

   11. Taylor expanded in angle around 0

       \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]
   12. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       4. pow2 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       5. lift-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       6. pow2 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       7. lift-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       8. lift-PI.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       9. lift-PI.f64 14.4

          \[\leadsto -0.25 \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + -1.54320987654321 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

   13. Applied rewrites 14.4%

       \[\leadsto -0.25 \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + -1.54320987654321 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

   14. Taylor expanded in angle around 0

       \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]
   15. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       4. pow2 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       5. lift-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       6. pow2 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       7. lift-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       8. lift-PI.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       9. lift-PI.f64 14.4

          \[\leadsto -0.25 \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + -1.54320987654321 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + -1.54320987654321 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

   16. Applied rewrites 14.4%

       \[\leadsto -0.25 \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + -1.54320987654321 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + -1.54320987654321 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

   if 1.35000000000000003e154 < b

   1. Initial program 2.7%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 1.4%

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

   5. Taylor expanded in b around -inf

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

   7. Applied rewrites 0.3%

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

   8. Taylor expanded in x-scale around -inf

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

   10. Applied rewrites 6.0%

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

Alternative 5: 17.9% accurate, 5.2× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ t_1 := 1 + -1.54320987654321 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\\ \mathbf{if}\;b\_m \leq 2.9 \cdot 10^{-65}:\\ \;\;\;\;-0.25 \cdot \frac{a\_m \cdot \left(-1 \cdot \left(x-scale\_m \cdot \left(\left(b\_m \cdot b\_m\right) \cdot \sqrt{8 \cdot \left(\sqrt{{t\_0}^{4}} + {t\_0}^{2}\right)}\right)\right)\right)}{b\_m \cdot b\_m}\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \frac{a\_m \cdot \left(-1 \cdot \left(x-scale\_m \cdot \sqrt{8 \cdot \left({b\_m}^{4} \cdot \left(\sqrt{{t\_1}^{4}} + {t\_1}^{2}\right)\right)}\right)\right)}{b\_m \cdot b\_m}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a_m b_m angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (cos (* 0.005555555555555556 (* angle PI))))
        (t_1 (+ 1.0 (* -1.54320987654321e-5 (* (* angle angle) (* PI PI))))))
   (if (<= b_m 2.9e-65)
     (*
      -0.25
      (/
       (*
        a_m
        (*
         -1.0
         (*
          x-scale_m
          (*
           (* b_m b_m)
           (sqrt (* 8.0 (+ (sqrt (pow t_0 4.0)) (pow t_0 2.0))))))))
       (* b_m b_m)))
     (*
      -0.25
      (/
       (*
        a_m
        (*
         -1.0
         (*
          x-scale_m
          (sqrt
           (* 8.0 (* (pow b_m 4.0) (+ (sqrt (pow t_1 4.0)) (pow t_1 2.0))))))))
       (* b_m b_m))))))

C

a_m = fabs(a);
b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = cos((0.005555555555555556 * (angle * ((double) M_PI))));
	double t_1 = 1.0 + (-1.54320987654321e-5 * ((angle * angle) * (((double) M_PI) * ((double) M_PI))));
	double tmp;
	if (b_m <= 2.9e-65) {
		tmp = -0.25 * ((a_m * (-1.0 * (x_45_scale_m * ((b_m * b_m) * sqrt((8.0 * (sqrt(pow(t_0, 4.0)) + pow(t_0, 2.0)))))))) / (b_m * b_m));
	} else {
		tmp = -0.25 * ((a_m * (-1.0 * (x_45_scale_m * sqrt((8.0 * (pow(b_m, 4.0) * (sqrt(pow(t_1, 4.0)) + pow(t_1, 2.0)))))))) / (b_m * b_m));
	}
	return tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = Math.cos((0.005555555555555556 * (angle * Math.PI)));
	double t_1 = 1.0 + (-1.54320987654321e-5 * ((angle * angle) * (Math.PI * Math.PI)));
	double tmp;
	if (b_m <= 2.9e-65) {
		tmp = -0.25 * ((a_m * (-1.0 * (x_45_scale_m * ((b_m * b_m) * Math.sqrt((8.0 * (Math.sqrt(Math.pow(t_0, 4.0)) + Math.pow(t_0, 2.0)))))))) / (b_m * b_m));
	} else {
		tmp = -0.25 * ((a_m * (-1.0 * (x_45_scale_m * Math.sqrt((8.0 * (Math.pow(b_m, 4.0) * (Math.sqrt(Math.pow(t_1, 4.0)) + Math.pow(t_1, 2.0)))))))) / (b_m * b_m));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m):
	t_0 = math.cos((0.005555555555555556 * (angle * math.pi)))
	t_1 = 1.0 + (-1.54320987654321e-5 * ((angle * angle) * (math.pi * math.pi)))
	tmp = 0
	if b_m <= 2.9e-65:
		tmp = -0.25 * ((a_m * (-1.0 * (x_45_scale_m * ((b_m * b_m) * math.sqrt((8.0 * (math.sqrt(math.pow(t_0, 4.0)) + math.pow(t_0, 2.0)))))))) / (b_m * b_m))
	else:
		tmp = -0.25 * ((a_m * (-1.0 * (x_45_scale_m * math.sqrt((8.0 * (math.pow(b_m, 4.0) * (math.sqrt(math.pow(t_1, 4.0)) + math.pow(t_1, 2.0)))))))) / (b_m * b_m))
	return tmp

Julia

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = cos(Float64(0.005555555555555556 * Float64(angle * pi)))
	t_1 = Float64(1.0 + Float64(-1.54320987654321e-5 * Float64(Float64(angle * angle) * Float64(pi * pi))))
	tmp = 0.0
	if (b_m <= 2.9e-65)
		tmp = Float64(-0.25 * Float64(Float64(a_m * Float64(-1.0 * Float64(x_45_scale_m * Float64(Float64(b_m * b_m) * sqrt(Float64(8.0 * Float64(sqrt((t_0 ^ 4.0)) + (t_0 ^ 2.0)))))))) / Float64(b_m * b_m)));
	else
		tmp = Float64(-0.25 * Float64(Float64(a_m * Float64(-1.0 * Float64(x_45_scale_m * sqrt(Float64(8.0 * Float64((b_m ^ 4.0) * Float64(sqrt((t_1 ^ 4.0)) + (t_1 ^ 2.0)))))))) / Float64(b_m * b_m)));
	end
	return tmp
end

MATLAB

a_m = abs(a);
b_m = abs(b);
x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = cos((0.005555555555555556 * (angle * pi)));
	t_1 = 1.0 + (-1.54320987654321e-5 * ((angle * angle) * (pi * pi)));
	tmp = 0.0;
	if (b_m <= 2.9e-65)
		tmp = -0.25 * ((a_m * (-1.0 * (x_45_scale_m * ((b_m * b_m) * sqrt((8.0 * (sqrt((t_0 ^ 4.0)) + (t_0 ^ 2.0)))))))) / (b_m * b_m));
	else
		tmp = -0.25 * ((a_m * (-1.0 * (x_45_scale_m * sqrt((8.0 * ((b_m ^ 4.0) * (sqrt((t_1 ^ 4.0)) + (t_1 ^ 2.0)))))))) / (b_m * b_m));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[Cos[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(-1.54320987654321e-5 * N[(N[(angle * angle), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b$95$m, 2.9e-65], N[(-0.25 * N[(N[(a$95$m * N[(-1.0 * N[(x$45$scale$95$m * N[(N[(b$95$m * b$95$m), $MachinePrecision] * N[Sqrt[N[(8.0 * 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]), $MachinePrecision] / N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.25 * N[(N[(a$95$m * N[(-1.0 * N[(x$45$scale$95$m * N[Sqrt[N[(8.0 * N[(N[Power[b$95$m, 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]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 2.8999999999999998e-65

   1. Initial program 2.7%

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

   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 0.1%

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

   5. Taylor expanded in x-scale around -inf

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

   7. Applied rewrites 4.2%

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

   8. Taylor expanded in y-scale around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

   10. Applied rewrites 13.6%

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

   11. Taylor expanded in b around 0

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

       1. lower-*.f64 N/A

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

       2. pow2 N/A

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

       3. lift-*.f64 N/A

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

       4. lower-sqrt.f64 N/A

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

       5. lower-*.f64 N/A

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

   13. Applied rewrites 16.1%

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

   if 2.8999999999999998e-65 < b

   1. Initial program 2.7%

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

   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 0.1%

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

   5. Taylor expanded in x-scale around -inf

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

   7. Applied rewrites 4.2%

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

   8. Taylor expanded in y-scale around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

   10. Applied rewrites 13.6%

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

   11. Taylor expanded in angle around 0

       \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]
   12. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       4. pow2 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       5. lift-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       6. pow2 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       7. lift-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       8. lift-PI.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       9. lift-PI.f64 14.4

          \[\leadsto -0.25 \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + -1.54320987654321 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

   13. Applied rewrites 14.4%

       \[\leadsto -0.25 \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + -1.54320987654321 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

   14. Taylor expanded in angle around 0

       \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]
   15. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       4. pow2 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       5. lift-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       6. pow2 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       7. lift-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       8. lift-PI.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       9. lift-PI.f64 14.4

          \[\leadsto -0.25 \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + -1.54320987654321 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + -1.54320987654321 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

   16. Applied rewrites 14.4%

       \[\leadsto -0.25 \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + -1.54320987654321 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + -1.54320987654321 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 15.6% accurate, 5.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := 1 + -1.54320987654321 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\\ t_1 := \left(b\_m \cdot a\_m\right) \cdot \left(b\_m \cdot \left(-a\_m\right)\right)\\ t_2 := \frac{4 \cdot t\_1}{{\left(x-scale\_m \cdot y-scale\_m\right)}^{2}}\\ \mathbf{if}\;b\_m \leq 1.35 \cdot 10^{-105}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(2 \cdot t\_2\right) \cdot t\_1\right) \cdot \frac{\sqrt{{a\_m}^{4}} + a\_m \cdot a\_m}{y-scale\_m \cdot y-scale\_m}}}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \frac{a\_m \cdot \left(-1 \cdot \left(x-scale\_m \cdot \sqrt{8 \cdot \left({b\_m}^{4} \cdot \left(\sqrt{{t\_0}^{4}} + {t\_0}^{2}\right)\right)}\right)\right)}{b\_m \cdot b\_m}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a_m b_m angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* -1.54320987654321e-5 (* (* angle angle) (* PI PI)))))
        (t_1 (* (* b_m a_m) (* b_m (- a_m))))
        (t_2 (/ (* 4.0 t_1) (pow (* x-scale_m y-scale_m) 2.0))))
   (if (<= b_m 1.35e-105)
     (/
      (-
       (sqrt
        (*
         (* (* 2.0 t_2) t_1)
         (/ (+ (sqrt (pow a_m 4.0)) (* a_m a_m)) (* y-scale_m y-scale_m)))))
      t_2)
     (*
      -0.25
      (/
       (*
        a_m
        (*
         -1.0
         (*
          x-scale_m
          (sqrt
           (* 8.0 (* (pow b_m 4.0) (+ (sqrt (pow t_0 4.0)) (pow t_0 2.0))))))))
       (* b_m b_m))))))

C

a_m = fabs(a);
b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = 1.0 + (-1.54320987654321e-5 * ((angle * angle) * (((double) M_PI) * ((double) M_PI))));
	double t_1 = (b_m * a_m) * (b_m * -a_m);
	double t_2 = (4.0 * t_1) / pow((x_45_scale_m * y_45_scale_m), 2.0);
	double tmp;
	if (b_m <= 1.35e-105) {
		tmp = -sqrt((((2.0 * t_2) * t_1) * ((sqrt(pow(a_m, 4.0)) + (a_m * a_m)) / (y_45_scale_m * y_45_scale_m)))) / t_2;
	} else {
		tmp = -0.25 * ((a_m * (-1.0 * (x_45_scale_m * sqrt((8.0 * (pow(b_m, 4.0) * (sqrt(pow(t_0, 4.0)) + pow(t_0, 2.0)))))))) / (b_m * b_m));
	}
	return tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = 1.0 + (-1.54320987654321e-5 * ((angle * angle) * (Math.PI * Math.PI)));
	double t_1 = (b_m * a_m) * (b_m * -a_m);
	double t_2 = (4.0 * t_1) / Math.pow((x_45_scale_m * y_45_scale_m), 2.0);
	double tmp;
	if (b_m <= 1.35e-105) {
		tmp = -Math.sqrt((((2.0 * t_2) * t_1) * ((Math.sqrt(Math.pow(a_m, 4.0)) + (a_m * a_m)) / (y_45_scale_m * y_45_scale_m)))) / t_2;
	} else {
		tmp = -0.25 * ((a_m * (-1.0 * (x_45_scale_m * Math.sqrt((8.0 * (Math.pow(b_m, 4.0) * (Math.sqrt(Math.pow(t_0, 4.0)) + Math.pow(t_0, 2.0)))))))) / (b_m * b_m));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m):
	t_0 = 1.0 + (-1.54320987654321e-5 * ((angle * angle) * (math.pi * math.pi)))
	t_1 = (b_m * a_m) * (b_m * -a_m)
	t_2 = (4.0 * t_1) / math.pow((x_45_scale_m * y_45_scale_m), 2.0)
	tmp = 0
	if b_m <= 1.35e-105:
		tmp = -math.sqrt((((2.0 * t_2) * t_1) * ((math.sqrt(math.pow(a_m, 4.0)) + (a_m * a_m)) / (y_45_scale_m * y_45_scale_m)))) / t_2
	else:
		tmp = -0.25 * ((a_m * (-1.0 * (x_45_scale_m * math.sqrt((8.0 * (math.pow(b_m, 4.0) * (math.sqrt(math.pow(t_0, 4.0)) + math.pow(t_0, 2.0)))))))) / (b_m * b_m))
	return tmp

Julia

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(1.0 + Float64(-1.54320987654321e-5 * Float64(Float64(angle * angle) * Float64(pi * pi))))
	t_1 = Float64(Float64(b_m * a_m) * Float64(b_m * Float64(-a_m)))
	t_2 = Float64(Float64(4.0 * t_1) / (Float64(x_45_scale_m * y_45_scale_m) ^ 2.0))
	tmp = 0.0
	if (b_m <= 1.35e-105)
		tmp = Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * t_2) * t_1) * Float64(Float64(sqrt((a_m ^ 4.0)) + Float64(a_m * a_m)) / Float64(y_45_scale_m * y_45_scale_m))))) / t_2);
	else
		tmp = Float64(-0.25 * Float64(Float64(a_m * Float64(-1.0 * Float64(x_45_scale_m * sqrt(Float64(8.0 * Float64((b_m ^ 4.0) * Float64(sqrt((t_0 ^ 4.0)) + (t_0 ^ 2.0)))))))) / Float64(b_m * b_m)));
	end
	return tmp
end

MATLAB

a_m = abs(a);
b_m = abs(b);
x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = 1.0 + (-1.54320987654321e-5 * ((angle * angle) * (pi * pi)));
	t_1 = (b_m * a_m) * (b_m * -a_m);
	t_2 = (4.0 * t_1) / ((x_45_scale_m * y_45_scale_m) ^ 2.0);
	tmp = 0.0;
	if (b_m <= 1.35e-105)
		tmp = -sqrt((((2.0 * t_2) * t_1) * ((sqrt((a_m ^ 4.0)) + (a_m * a_m)) / (y_45_scale_m * y_45_scale_m)))) / t_2;
	else
		tmp = -0.25 * ((a_m * (-1.0 * (x_45_scale_m * sqrt((8.0 * ((b_m ^ 4.0) * (sqrt((t_0 ^ 4.0)) + (t_0 ^ 2.0)))))))) / (b_m * b_m));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(1.0 + N[(-1.54320987654321e-5 * N[(N[(angle * angle), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b$95$m * a$95$m), $MachinePrecision] * N[(b$95$m * (-a$95$m)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(4.0 * t$95$1), $MachinePrecision] / N[Power[N[(x$45$scale$95$m * y$45$scale$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b$95$m, 1.35e-105], N[((-N[Sqrt[N[(N[(N[(2.0 * t$95$2), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(N[(N[Sqrt[N[Power[a$95$m, 4.0], $MachinePrecision]], $MachinePrecision] + N[(a$95$m * a$95$m), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision], N[(-0.25 * N[(N[(a$95$m * N[(-1.0 * N[(x$45$scale$95$m * N[Sqrt[N[(8.0 * N[(N[Power[b$95$m, 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]), $MachinePrecision] / N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.34999999999999996e-105

   1. Initial program 2.7%

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

   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 4.1%

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

   5. Taylor expanded in y-scale around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 2.4

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

   7. Applied rewrites 2.4%

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

   if 1.34999999999999996e-105 < b

   1. Initial program 2.7%

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

   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 0.1%

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

   5. Taylor expanded in x-scale around -inf

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

   7. Applied rewrites 4.2%

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

   8. Taylor expanded in y-scale around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

   10. Applied rewrites 13.6%

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

   11. Taylor expanded in angle around 0

       \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]
   12. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       4. pow2 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       5. lift-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       6. pow2 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       7. lift-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       8. lift-PI.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       9. lift-PI.f64 14.4

          \[\leadsto -0.25 \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + -1.54320987654321 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

   13. Applied rewrites 14.4%

       \[\leadsto -0.25 \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + -1.54320987654321 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

   14. Taylor expanded in angle around 0

       \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]
   15. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       4. pow2 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       5. lift-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       6. pow2 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       7. lift-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       8. lift-PI.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

       9. lift-PI.f64 14.4

          \[\leadsto -0.25 \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + -1.54320987654321 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + -1.54320987654321 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

   16. Applied rewrites 14.4%

       \[\leadsto -0.25 \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + -1.54320987654321 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + -1.54320987654321 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 14.4% accurate, 7.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := 1 + -1.54320987654321 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\\ -0.25 \cdot \frac{a\_m \cdot \left(-1 \cdot \left(x-scale\_m \cdot \sqrt{8 \cdot \left({b\_m}^{4} \cdot \left(\sqrt{{t\_0}^{4}} + {t\_0}^{2}\right)\right)}\right)\right)}{b\_m \cdot b\_m} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a_m b_m angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* -1.54320987654321e-5 (* (* angle angle) (* PI PI))))))
   (*
    -0.25
    (/
     (*
      a_m
      (*
       -1.0
       (*
        x-scale_m
        (sqrt
         (* 8.0 (* (pow b_m 4.0) (+ (sqrt (pow t_0 4.0)) (pow t_0 2.0))))))))
     (* b_m b_m)))))

C

a_m = fabs(a);
b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = 1.0 + (-1.54320987654321e-5 * ((angle * angle) * (((double) M_PI) * ((double) M_PI))));
	return -0.25 * ((a_m * (-1.0 * (x_45_scale_m * sqrt((8.0 * (pow(b_m, 4.0) * (sqrt(pow(t_0, 4.0)) + pow(t_0, 2.0)))))))) / (b_m * b_m));
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = 1.0 + (-1.54320987654321e-5 * ((angle * angle) * (Math.PI * Math.PI)));
	return -0.25 * ((a_m * (-1.0 * (x_45_scale_m * Math.sqrt((8.0 * (Math.pow(b_m, 4.0) * (Math.sqrt(Math.pow(t_0, 4.0)) + Math.pow(t_0, 2.0)))))))) / (b_m * b_m));
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m):
	t_0 = 1.0 + (-1.54320987654321e-5 * ((angle * angle) * (math.pi * math.pi)))
	return -0.25 * ((a_m * (-1.0 * (x_45_scale_m * math.sqrt((8.0 * (math.pow(b_m, 4.0) * (math.sqrt(math.pow(t_0, 4.0)) + math.pow(t_0, 2.0)))))))) / (b_m * b_m))

Julia

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(1.0 + Float64(-1.54320987654321e-5 * Float64(Float64(angle * angle) * Float64(pi * pi))))
	return Float64(-0.25 * Float64(Float64(a_m * Float64(-1.0 * Float64(x_45_scale_m * sqrt(Float64(8.0 * Float64((b_m ^ 4.0) * Float64(sqrt((t_0 ^ 4.0)) + (t_0 ^ 2.0)))))))) / Float64(b_m * b_m)))
end

MATLAB

a_m = abs(a);
b_m = abs(b);
x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp = code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = 1.0 + (-1.54320987654321e-5 * ((angle * angle) * (pi * pi)));
	tmp = -0.25 * ((a_m * (-1.0 * (x_45_scale_m * sqrt((8.0 * ((b_m ^ 4.0) * (sqrt((t_0 ^ 4.0)) + (t_0 ^ 2.0)))))))) / (b_m * b_m));
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(1.0 + N[(-1.54320987654321e-5 * N[(N[(angle * angle), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(-0.25 * N[(N[(a$95$m * N[(-1.0 * N[(x$45$scale$95$m * N[Sqrt[N[(8.0 * N[(N[Power[b$95$m, 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]), $MachinePrecision] / N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 2.7%

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

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 0.1%

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

5. Taylor expanded in x-scale around -inf

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

7. Applied rewrites 4.2%

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

8. Taylor expanded in y-scale around 0

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

   1. lower-sqrt.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lift-pow.f64 N/A

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

10. Applied rewrites 13.6%

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

11. Taylor expanded in angle around 0

    \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]
12. Step-by-step derivation

    1. lower-+.f64 N/A

       \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

    2. lower-*.f64 N/A

       \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

    3. lower-*.f64 N/A

       \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

    4. pow2 N/A

       \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

    5. lift-*.f64 N/A

       \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

    6. pow2 N/A

       \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

    7. lift-*.f64 N/A

       \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

    8. lift-PI.f64 N/A

       \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

    9. lift-PI.f64 14.4

       \[\leadsto -0.25 \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + -1.54320987654321 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

13. Applied rewrites 14.4%

    \[\leadsto -0.25 \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + -1.54320987654321 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

14. Taylor expanded in angle around 0

    \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]
15. Step-by-step derivation

    1. lower-+.f64 N/A

       \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

    2. lower-*.f64 N/A

       \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

    3. lower-*.f64 N/A

       \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

    4. pow2 N/A

       \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

    5. lift-*.f64 N/A

       \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

    6. pow2 N/A

       \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

    7. lift-*.f64 N/A

       \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

    8. lift-PI.f64 N/A

       \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + \frac{-1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

    9. lift-PI.f64 14.4

       \[\leadsto -0.25 \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + -1.54320987654321 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + -1.54320987654321 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]

16. Applied rewrites 14.4%

    \[\leadsto -0.25 \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(1 + -1.54320987654321 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{4}} + {\left(1 + -1.54320987654321 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]
17. Add Preprocessing

Alternative 8: 13.7% accurate, 7.7× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ -0.25 \cdot \frac{a\_m \cdot \left(-1 \cdot \left(x-scale\_m \cdot \sqrt{8 \cdot \left({b\_m}^{4} \cdot \left(1 + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b\_m \cdot b\_m} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a_m b_m angle x-scale_m y-scale_m)
 :precision binary64
 (*
  -0.25
  (/
   (*
    a_m
    (*
     -1.0
     (*
      x-scale_m
      (sqrt
       (*
        8.0
        (*
         (pow b_m 4.0)
         (+ 1.0 (pow (cos (* 0.005555555555555556 (* angle PI))) 2.0))))))))
   (* b_m b_m))))

C

a_m = fabs(a);
b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	return -0.25 * ((a_m * (-1.0 * (x_45_scale_m * sqrt((8.0 * (pow(b_m, 4.0) * (1.0 + pow(cos((0.005555555555555556 * (angle * ((double) M_PI)))), 2.0)))))))) / (b_m * b_m));
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	return -0.25 * ((a_m * (-1.0 * (x_45_scale_m * Math.sqrt((8.0 * (Math.pow(b_m, 4.0) * (1.0 + Math.pow(Math.cos((0.005555555555555556 * (angle * Math.PI))), 2.0)))))))) / (b_m * b_m));
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m):
	return -0.25 * ((a_m * (-1.0 * (x_45_scale_m * math.sqrt((8.0 * (math.pow(b_m, 4.0) * (1.0 + math.pow(math.cos((0.005555555555555556 * (angle * math.pi))), 2.0)))))))) / (b_m * b_m))

Julia

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	return Float64(-0.25 * Float64(Float64(a_m * Float64(-1.0 * Float64(x_45_scale_m * sqrt(Float64(8.0 * Float64((b_m ^ 4.0) * Float64(1.0 + (cos(Float64(0.005555555555555556 * Float64(angle * pi))) ^ 2.0)))))))) / Float64(b_m * b_m)))
end

MATLAB

a_m = abs(a);
b_m = abs(b);
x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp = code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = -0.25 * ((a_m * (-1.0 * (x_45_scale_m * sqrt((8.0 * ((b_m ^ 4.0) * (1.0 + (cos((0.005555555555555556 * (angle * pi))) ^ 2.0)))))))) / (b_m * b_m));
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := N[(-0.25 * N[(N[(a$95$m * N[(-1.0 * N[(x$45$scale$95$m * N[Sqrt[N[(8.0 * N[(N[Power[b$95$m, 4.0], $MachinePrecision] * N[(1.0 + N[Power[N[Cos[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 2.7%

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

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 0.1%

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

5. Taylor expanded in x-scale around -inf

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

7. Applied rewrites 4.2%

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

8. Taylor expanded in y-scale around 0

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

   1. lower-sqrt.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lift-pow.f64 N/A

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

10. Applied rewrites 13.6%

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

11. Taylor expanded in angle around 0

    \[\leadsto \frac{-1}{4} \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(1 + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]
12. Step-by-step derivation

13. Applied rewrites 13.6%

    \[\leadsto -0.25 \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(1 + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{b \cdot b} \]
14. Add Preprocessing

Alternative 9: 13.6% accurate, 19.1× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ -0.25 \cdot \frac{a\_m \cdot \left(-1 \cdot \left(x-scale\_m \cdot \sqrt{16 \cdot {b\_m}^{4}}\right)\right)}{b\_m \cdot b\_m} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a_m b_m angle x-scale_m y-scale_m)
 :precision binary64
 (*
  -0.25
  (/
   (* a_m (* -1.0 (* x-scale_m (sqrt (* 16.0 (pow b_m 4.0))))))
   (* b_m b_m))))

C

a_m = fabs(a);
b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	return -0.25 * ((a_m * (-1.0 * (x_45_scale_m * sqrt((16.0 * pow(b_m, 4.0)))))) / (b_m * b_m));
}

Fortran

a_m =     private
b_m =     private
x-scale_m =     private
y-scale_m =     private
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_m, b_m, angle, x_45scale_m, y_45scale_m)
use fmin_fmax_functions
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale_m
    code = (-0.25d0) * ((a_m * ((-1.0d0) * (x_45scale_m * sqrt((16.0d0 * (b_m ** 4.0d0)))))) / (b_m * b_m))
end function

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	return -0.25 * ((a_m * (-1.0 * (x_45_scale_m * Math.sqrt((16.0 * Math.pow(b_m, 4.0)))))) / (b_m * b_m));
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m):
	return -0.25 * ((a_m * (-1.0 * (x_45_scale_m * math.sqrt((16.0 * math.pow(b_m, 4.0)))))) / (b_m * b_m))

Julia

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	return Float64(-0.25 * Float64(Float64(a_m * Float64(-1.0 * Float64(x_45_scale_m * sqrt(Float64(16.0 * (b_m ^ 4.0)))))) / Float64(b_m * b_m)))
end

MATLAB

a_m = abs(a);
b_m = abs(b);
x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp = code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = -0.25 * ((a_m * (-1.0 * (x_45_scale_m * sqrt((16.0 * (b_m ^ 4.0)))))) / (b_m * b_m));
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := N[(-0.25 * N[(N[(a$95$m * N[(-1.0 * N[(x$45$scale$95$m * N[Sqrt[N[(16.0 * N[Power[b$95$m, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 2.7%

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

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 0.1%

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

5. Taylor expanded in x-scale around -inf

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

7. Applied rewrites 4.2%

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

8. Taylor expanded in y-scale around 0

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

   1. lower-sqrt.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lift-pow.f64 N/A

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

10. Applied rewrites 13.6%

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

11. Taylor expanded in angle around 0

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

    1. lower-*.f64 N/A

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

    2. lift-pow.f64 13.7

       \[\leadsto -0.25 \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{16 \cdot {b}^{4}}\right)\right)}{b \cdot b} \]

13. Applied rewrites 13.7%

    \[\leadsto -0.25 \cdot \frac{a \cdot \left(-1 \cdot \left(x-scale \cdot \sqrt{16 \cdot {b}^{4}}\right)\right)}{b \cdot b} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2025140 
(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))))