a from scale-rotated-ellipse

Percentage Accurate: 2.9% → 7.0%

Time: 31.0s

Alternatives: 13

Speedup: 6.3×

• 30.0% 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: 2.9% 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: 7.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := \left(b\_m \cdot a\right) \cdot \left(b\_m \cdot \left(-a\right)\right)\\ t_1 := \frac{4 \cdot t\_0}{{\left(x-scale\_m \cdot y-scale\_m\right)}^{2}}\\ t_2 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_3 := \cos t\_2\\ t_4 := \frac{angle}{180} \cdot \pi\\ t_5 := \sin t\_2\\ \mathbf{if}\;y-scale\_m \leq 8.8 \cdot 10^{-81}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(2 \cdot t\_1\right) \cdot t\_0\right) \cdot \left(\left(\frac{\frac{{b\_m}^{2}}{x-scale\_m}}{x-scale\_m} + \frac{\frac{{\left(a \cdot \cos t\_4\right)}^{2} + {\left(b\_m \cdot \sin t\_4\right)}^{2}}{y-scale\_m}}{y-scale\_m}\right) + \frac{\sqrt{{\left({\left(a \cdot t\_3\right)}^{2} + {\left(b\_m \cdot t\_5\right)}^{2}\right)}^{2}}}{y-scale\_m \cdot y-scale\_m}\right)}}{t\_1}\\ \mathbf{elif}\;y-scale\_m \leq 3.2 \cdot 10^{+151}:\\ \;\;\;\;0.25 \cdot \frac{b\_m \cdot \left(x-scale\_m \cdot \left(\left(y-scale\_m \cdot y-scale\_m\right) \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\sqrt{\frac{{t\_5}^{4}}{{y-scale\_m}^{4}}} + \frac{{t\_5}^{2}}{y-scale\_m \cdot y-scale\_m}\right)}{y-scale\_m \cdot y-scale\_m}}\right)\right)}{a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{b\_m \cdot \left(\left(x-scale\_m \cdot x-scale\_m\right) \cdot \left(y-scale\_m \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\sqrt{\frac{{t\_3}^{4}}{{x-scale\_m}^{4}}} + \frac{{t\_3}^{2}}{x-scale\_m \cdot x-scale\_m}\right)}{x-scale\_m \cdot x-scale\_m}}\right)\right)}{a \cdot a}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b_m angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (* (* b_m a) (* b_m (- a))))
        (t_1 (/ (* 4.0 t_0) (pow (* x-scale_m y-scale_m) 2.0)))
        (t_2 (* 0.005555555555555556 (* angle PI)))
        (t_3 (cos t_2))
        (t_4 (* (/ angle 180.0) PI))
        (t_5 (sin t_2)))
   (if (<= y-scale_m 8.8e-81)
     (/
      (-
       (sqrt
        (*
         (* (* 2.0 t_1) t_0)
         (+
          (+
           (/ (/ (pow b_m 2.0) x-scale_m) x-scale_m)
           (/
            (/
             (+ (pow (* a (cos t_4)) 2.0) (pow (* b_m (sin t_4)) 2.0))
             y-scale_m)
            y-scale_m))
          (/
           (sqrt (pow (+ (pow (* a t_3) 2.0) (pow (* b_m t_5) 2.0)) 2.0))
           (* y-scale_m y-scale_m))))))
      t_1)
     (if (<= y-scale_m 3.2e+151)
       (*
        0.25
        (/
         (*
          b_m
          (*
           x-scale_m
           (*
            (* y-scale_m y-scale_m)
            (sqrt
             (*
              8.0
              (/
               (*
                (pow a 4.0)
                (+
                 (sqrt (/ (pow t_5 4.0) (pow y-scale_m 4.0)))
                 (/ (pow t_5 2.0) (* y-scale_m y-scale_m))))
               (* y-scale_m y-scale_m)))))))
         (* a a)))
       (*
        0.25
        (/
         (*
          b_m
          (*
           (* x-scale_m x-scale_m)
           (*
            y-scale_m
            (sqrt
             (*
              8.0
              (/
               (*
                (pow a 4.0)
                (+
                 (sqrt (/ (pow t_3 4.0) (pow x-scale_m 4.0)))
                 (/ (pow t_3 2.0) (* x-scale_m x-scale_m))))
               (* x-scale_m x-scale_m)))))))
         (* a a)))))))

C

b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = (b_m * a) * (b_m * -a);
	double t_1 = (4.0 * t_0) / pow((x_45_scale_m * y_45_scale_m), 2.0);
	double t_2 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_3 = cos(t_2);
	double t_4 = (angle / 180.0) * ((double) M_PI);
	double t_5 = sin(t_2);
	double tmp;
	if (y_45_scale_m <= 8.8e-81) {
		tmp = -sqrt((((2.0 * t_1) * t_0) * ((((pow(b_m, 2.0) / x_45_scale_m) / x_45_scale_m) + (((pow((a * cos(t_4)), 2.0) + pow((b_m * sin(t_4)), 2.0)) / y_45_scale_m) / y_45_scale_m)) + (sqrt(pow((pow((a * t_3), 2.0) + pow((b_m * t_5), 2.0)), 2.0)) / (y_45_scale_m * y_45_scale_m))))) / t_1;
	} else if (y_45_scale_m <= 3.2e+151) {
		tmp = 0.25 * ((b_m * (x_45_scale_m * ((y_45_scale_m * y_45_scale_m) * sqrt((8.0 * ((pow(a, 4.0) * (sqrt((pow(t_5, 4.0) / pow(y_45_scale_m, 4.0))) + (pow(t_5, 2.0) / (y_45_scale_m * y_45_scale_m)))) / (y_45_scale_m * y_45_scale_m))))))) / (a * a));
	} else {
		tmp = 0.25 * ((b_m * ((x_45_scale_m * x_45_scale_m) * (y_45_scale_m * sqrt((8.0 * ((pow(a, 4.0) * (sqrt((pow(t_3, 4.0) / pow(x_45_scale_m, 4.0))) + (pow(t_3, 2.0) / (x_45_scale_m * x_45_scale_m)))) / (x_45_scale_m * x_45_scale_m))))))) / (a * a));
	}
	return tmp;
}

Java

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, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = (b_m * a) * (b_m * -a);
	double t_1 = (4.0 * t_0) / Math.pow((x_45_scale_m * y_45_scale_m), 2.0);
	double t_2 = 0.005555555555555556 * (angle * Math.PI);
	double t_3 = Math.cos(t_2);
	double t_4 = (angle / 180.0) * Math.PI;
	double t_5 = Math.sin(t_2);
	double tmp;
	if (y_45_scale_m <= 8.8e-81) {
		tmp = -Math.sqrt((((2.0 * t_1) * t_0) * ((((Math.pow(b_m, 2.0) / x_45_scale_m) / x_45_scale_m) + (((Math.pow((a * Math.cos(t_4)), 2.0) + Math.pow((b_m * Math.sin(t_4)), 2.0)) / y_45_scale_m) / y_45_scale_m)) + (Math.sqrt(Math.pow((Math.pow((a * t_3), 2.0) + Math.pow((b_m * t_5), 2.0)), 2.0)) / (y_45_scale_m * y_45_scale_m))))) / t_1;
	} else if (y_45_scale_m <= 3.2e+151) {
		tmp = 0.25 * ((b_m * (x_45_scale_m * ((y_45_scale_m * y_45_scale_m) * Math.sqrt((8.0 * ((Math.pow(a, 4.0) * (Math.sqrt((Math.pow(t_5, 4.0) / Math.pow(y_45_scale_m, 4.0))) + (Math.pow(t_5, 2.0) / (y_45_scale_m * y_45_scale_m)))) / (y_45_scale_m * y_45_scale_m))))))) / (a * a));
	} else {
		tmp = 0.25 * ((b_m * ((x_45_scale_m * x_45_scale_m) * (y_45_scale_m * Math.sqrt((8.0 * ((Math.pow(a, 4.0) * (Math.sqrt((Math.pow(t_3, 4.0) / Math.pow(x_45_scale_m, 4.0))) + (Math.pow(t_3, 2.0) / (x_45_scale_m * x_45_scale_m)))) / (x_45_scale_m * x_45_scale_m))))))) / (a * a));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b_m, angle, x_45_scale_m, y_45_scale_m):
	t_0 = (b_m * a) * (b_m * -a)
	t_1 = (4.0 * t_0) / math.pow((x_45_scale_m * y_45_scale_m), 2.0)
	t_2 = 0.005555555555555556 * (angle * math.pi)
	t_3 = math.cos(t_2)
	t_4 = (angle / 180.0) * math.pi
	t_5 = math.sin(t_2)
	tmp = 0
	if y_45_scale_m <= 8.8e-81:
		tmp = -math.sqrt((((2.0 * t_1) * t_0) * ((((math.pow(b_m, 2.0) / x_45_scale_m) / x_45_scale_m) + (((math.pow((a * math.cos(t_4)), 2.0) + math.pow((b_m * math.sin(t_4)), 2.0)) / y_45_scale_m) / y_45_scale_m)) + (math.sqrt(math.pow((math.pow((a * t_3), 2.0) + math.pow((b_m * t_5), 2.0)), 2.0)) / (y_45_scale_m * y_45_scale_m))))) / t_1
	elif y_45_scale_m <= 3.2e+151:
		tmp = 0.25 * ((b_m * (x_45_scale_m * ((y_45_scale_m * y_45_scale_m) * math.sqrt((8.0 * ((math.pow(a, 4.0) * (math.sqrt((math.pow(t_5, 4.0) / math.pow(y_45_scale_m, 4.0))) + (math.pow(t_5, 2.0) / (y_45_scale_m * y_45_scale_m)))) / (y_45_scale_m * y_45_scale_m))))))) / (a * a))
	else:
		tmp = 0.25 * ((b_m * ((x_45_scale_m * x_45_scale_m) * (y_45_scale_m * math.sqrt((8.0 * ((math.pow(a, 4.0) * (math.sqrt((math.pow(t_3, 4.0) / math.pow(x_45_scale_m, 4.0))) + (math.pow(t_3, 2.0) / (x_45_scale_m * x_45_scale_m)))) / (x_45_scale_m * x_45_scale_m))))))) / (a * a))
	return tmp

Julia

b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(Float64(b_m * a) * Float64(b_m * Float64(-a)))
	t_1 = Float64(Float64(4.0 * t_0) / (Float64(x_45_scale_m * y_45_scale_m) ^ 2.0))
	t_2 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_3 = cos(t_2)
	t_4 = Float64(Float64(angle / 180.0) * pi)
	t_5 = sin(t_2)
	tmp = 0.0
	if (y_45_scale_m <= 8.8e-81)
		tmp = Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * t_1) * t_0) * Float64(Float64(Float64(Float64((b_m ^ 2.0) / x_45_scale_m) / x_45_scale_m) + Float64(Float64(Float64((Float64(a * cos(t_4)) ^ 2.0) + (Float64(b_m * sin(t_4)) ^ 2.0)) / y_45_scale_m) / y_45_scale_m)) + Float64(sqrt((Float64((Float64(a * t_3) ^ 2.0) + (Float64(b_m * t_5) ^ 2.0)) ^ 2.0)) / Float64(y_45_scale_m * y_45_scale_m)))))) / t_1);
	elseif (y_45_scale_m <= 3.2e+151)
		tmp = Float64(0.25 * Float64(Float64(b_m * Float64(x_45_scale_m * Float64(Float64(y_45_scale_m * y_45_scale_m) * sqrt(Float64(8.0 * Float64(Float64((a ^ 4.0) * Float64(sqrt(Float64((t_5 ^ 4.0) / (y_45_scale_m ^ 4.0))) + Float64((t_5 ^ 2.0) / Float64(y_45_scale_m * y_45_scale_m)))) / Float64(y_45_scale_m * y_45_scale_m))))))) / Float64(a * a)));
	else
		tmp = Float64(0.25 * Float64(Float64(b_m * Float64(Float64(x_45_scale_m * x_45_scale_m) * Float64(y_45_scale_m * sqrt(Float64(8.0 * Float64(Float64((a ^ 4.0) * Float64(sqrt(Float64((t_3 ^ 4.0) / (x_45_scale_m ^ 4.0))) + Float64((t_3 ^ 2.0) / Float64(x_45_scale_m * x_45_scale_m)))) / Float64(x_45_scale_m * x_45_scale_m))))))) / Float64(a * a)));
	end
	return tmp
end

MATLAB

b_m = abs(b);
x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = (b_m * a) * (b_m * -a);
	t_1 = (4.0 * t_0) / ((x_45_scale_m * y_45_scale_m) ^ 2.0);
	t_2 = 0.005555555555555556 * (angle * pi);
	t_3 = cos(t_2);
	t_4 = (angle / 180.0) * pi;
	t_5 = sin(t_2);
	tmp = 0.0;
	if (y_45_scale_m <= 8.8e-81)
		tmp = -sqrt((((2.0 * t_1) * t_0) * (((((b_m ^ 2.0) / x_45_scale_m) / x_45_scale_m) + (((((a * cos(t_4)) ^ 2.0) + ((b_m * sin(t_4)) ^ 2.0)) / y_45_scale_m) / y_45_scale_m)) + (sqrt(((((a * t_3) ^ 2.0) + ((b_m * t_5) ^ 2.0)) ^ 2.0)) / (y_45_scale_m * y_45_scale_m))))) / t_1;
	elseif (y_45_scale_m <= 3.2e+151)
		tmp = 0.25 * ((b_m * (x_45_scale_m * ((y_45_scale_m * y_45_scale_m) * sqrt((8.0 * (((a ^ 4.0) * (sqrt(((t_5 ^ 4.0) / (y_45_scale_m ^ 4.0))) + ((t_5 ^ 2.0) / (y_45_scale_m * y_45_scale_m)))) / (y_45_scale_m * y_45_scale_m))))))) / (a * a));
	else
		tmp = 0.25 * ((b_m * ((x_45_scale_m * x_45_scale_m) * (y_45_scale_m * sqrt((8.0 * (((a ^ 4.0) * (sqrt(((t_3 ^ 4.0) / (x_45_scale_m ^ 4.0))) + ((t_3 ^ 2.0) / (x_45_scale_m * x_45_scale_m)))) / (x_45_scale_m * x_45_scale_m))))))) / (a * a));
	end
	tmp_2 = tmp;
end

Wolfram

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_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(N[(b$95$m * a), $MachinePrecision] * N[(b$95$m * (-a)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * t$95$0), $MachinePrecision] / N[Power[N[(x$45$scale$95$m * y$45$scale$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Cos[t$95$2], $MachinePrecision]}, Block[{t$95$4 = N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$5 = N[Sin[t$95$2], $MachinePrecision]}, If[LessEqual[y$45$scale$95$m, 8.8e-81], N[((-N[Sqrt[N[(N[(N[(2.0 * t$95$1), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(N[(N[(N[(N[Power[b$95$m, 2.0], $MachinePrecision] / x$45$scale$95$m), $MachinePrecision] / x$45$scale$95$m), $MachinePrecision] + N[(N[(N[(N[Power[N[(a * N[Cos[t$95$4], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b$95$m * N[Sin[t$95$4], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / y$45$scale$95$m), $MachinePrecision] / y$45$scale$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[Power[N[(N[Power[N[(a * t$95$3), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b$95$m * t$95$5), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] / N[(y$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], If[LessEqual[y$45$scale$95$m, 3.2e+151], N[(0.25 * N[(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[Power[a, 4.0], $MachinePrecision] * N[(N[Sqrt[N[(N[Power[t$95$5, 4.0], $MachinePrecision] / N[Power[y$45$scale$95$m, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[Power[t$95$5, 2.0], $MachinePrecision] / N[(y$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(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[Power[a, 4.0], $MachinePrecision] * N[(N[Sqrt[N[(N[Power[t$95$3, 4.0], $MachinePrecision] / N[Power[x$45$scale$95$m, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[Power[t$95$3, 2.0], $MachinePrecision] / N[(x$45$scale$95$m * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale$95$m * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y-scale < 8.7999999999999997e-81

   1. Initial program 2.9%

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

   2. Taylor expanded in y-scale around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 4.4%

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \color{blue}{\frac{\sqrt{{\left({\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right)}^{2}}}{y-scale \cdot y-scale}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   5. Taylor expanded in angle around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 4.4

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{b}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \frac{\sqrt{{\left({\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right)}^{2}}}{y-scale \cdot y-scale}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   7. Applied rewrites 4.4%

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\color{blue}{\frac{{b}^{2}}{x-scale}}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \frac{\sqrt{{\left({\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right)}^{2}}}{y-scale \cdot y-scale}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   if 8.7999999999999997e-81 < y-scale < 3.19999999999999994e151

   1. Initial program 2.9%

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

   2. Taylor expanded in b around -inf

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

   4. Applied rewrites 0.1%

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

   5. Taylor expanded in angle around 0

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

      1. lower-+.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 0.2

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

   7. Applied rewrites 0.2%

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

   8. Taylor expanded in x-scale around -inf

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

   10. Applied rewrites 2.5%

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

   if 3.19999999999999994e151 < y-scale

   1. Initial program 2.9%

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

   2. Taylor expanded in b around -inf

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

   4. Applied rewrites 0.1%

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

   5. Taylor expanded in angle around 0

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

      1. lower-+.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 0.2

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

   7. Applied rewrites 0.2%

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

   8. Taylor expanded in y-scale around -inf

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

   10. Applied rewrites 4.7%

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

Alternative 2: 7.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \sin t\_0\\ t_2 := \left(b\_m \cdot a\right) \cdot \left(b\_m \cdot \left(-a\right)\right)\\ t_3 := \frac{4 \cdot t\_2}{{\left(x-scale\_m \cdot y-scale\_m\right)}^{2}}\\ t_4 := \cos t\_0\\ t_5 := {\left(a \cdot t\_4\right)}^{2} + {\left(b\_m \cdot t\_1\right)}^{2}\\ \mathbf{if}\;y-scale\_m \leq 8.8 \cdot 10^{-81}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(2 \cdot t\_3\right) \cdot t\_2\right) \cdot \frac{\sqrt{{t\_5}^{2}} + t\_5}{y-scale\_m \cdot y-scale\_m}}}{t\_3}\\ \mathbf{elif}\;y-scale\_m \leq 3.2 \cdot 10^{+151}:\\ \;\;\;\;0.25 \cdot \frac{b\_m \cdot \left(x-scale\_m \cdot \left(\left(y-scale\_m \cdot y-scale\_m\right) \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\sqrt{\frac{{t\_1}^{4}}{{y-scale\_m}^{4}}} + \frac{{t\_1}^{2}}{y-scale\_m \cdot y-scale\_m}\right)}{y-scale\_m \cdot y-scale\_m}}\right)\right)}{a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{b\_m \cdot \left(\left(x-scale\_m \cdot x-scale\_m\right) \cdot \left(y-scale\_m \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\sqrt{\frac{{t\_4}^{4}}{{x-scale\_m}^{4}}} + \frac{{t\_4}^{2}}{x-scale\_m \cdot x-scale\_m}\right)}{x-scale\_m \cdot x-scale\_m}}\right)\right)}{a \cdot a}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b_m angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI)))
        (t_1 (sin t_0))
        (t_2 (* (* b_m a) (* b_m (- a))))
        (t_3 (/ (* 4.0 t_2) (pow (* x-scale_m y-scale_m) 2.0)))
        (t_4 (cos t_0))
        (t_5 (+ (pow (* a t_4) 2.0) (pow (* b_m t_1) 2.0))))
   (if (<= y-scale_m 8.8e-81)
     (/
      (-
       (sqrt
        (*
         (* (* 2.0 t_3) t_2)
         (/ (+ (sqrt (pow t_5 2.0)) t_5) (* y-scale_m y-scale_m)))))
      t_3)
     (if (<= y-scale_m 3.2e+151)
       (*
        0.25
        (/
         (*
          b_m
          (*
           x-scale_m
           (*
            (* y-scale_m y-scale_m)
            (sqrt
             (*
              8.0
              (/
               (*
                (pow a 4.0)
                (+
                 (sqrt (/ (pow t_1 4.0) (pow y-scale_m 4.0)))
                 (/ (pow t_1 2.0) (* y-scale_m y-scale_m))))
               (* y-scale_m y-scale_m)))))))
         (* a a)))
       (*
        0.25
        (/
         (*
          b_m
          (*
           (* x-scale_m x-scale_m)
           (*
            y-scale_m
            (sqrt
             (*
              8.0
              (/
               (*
                (pow a 4.0)
                (+
                 (sqrt (/ (pow t_4 4.0) (pow x-scale_m 4.0)))
                 (/ (pow t_4 2.0) (* x-scale_m x-scale_m))))
               (* x-scale_m x-scale_m)))))))
         (* a a)))))))

C

b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_1 = sin(t_0);
	double t_2 = (b_m * a) * (b_m * -a);
	double t_3 = (4.0 * t_2) / pow((x_45_scale_m * y_45_scale_m), 2.0);
	double t_4 = cos(t_0);
	double t_5 = pow((a * t_4), 2.0) + pow((b_m * t_1), 2.0);
	double tmp;
	if (y_45_scale_m <= 8.8e-81) {
		tmp = -sqrt((((2.0 * t_3) * t_2) * ((sqrt(pow(t_5, 2.0)) + t_5) / (y_45_scale_m * y_45_scale_m)))) / t_3;
	} else if (y_45_scale_m <= 3.2e+151) {
		tmp = 0.25 * ((b_m * (x_45_scale_m * ((y_45_scale_m * y_45_scale_m) * sqrt((8.0 * ((pow(a, 4.0) * (sqrt((pow(t_1, 4.0) / pow(y_45_scale_m, 4.0))) + (pow(t_1, 2.0) / (y_45_scale_m * y_45_scale_m)))) / (y_45_scale_m * y_45_scale_m))))))) / (a * a));
	} else {
		tmp = 0.25 * ((b_m * ((x_45_scale_m * x_45_scale_m) * (y_45_scale_m * sqrt((8.0 * ((pow(a, 4.0) * (sqrt((pow(t_4, 4.0) / pow(x_45_scale_m, 4.0))) + (pow(t_4, 2.0) / (x_45_scale_m * x_45_scale_m)))) / (x_45_scale_m * x_45_scale_m))))))) / (a * a));
	}
	return tmp;
}

Java

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, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = 0.005555555555555556 * (angle * Math.PI);
	double t_1 = Math.sin(t_0);
	double t_2 = (b_m * a) * (b_m * -a);
	double t_3 = (4.0 * t_2) / Math.pow((x_45_scale_m * y_45_scale_m), 2.0);
	double t_4 = Math.cos(t_0);
	double t_5 = Math.pow((a * t_4), 2.0) + Math.pow((b_m * t_1), 2.0);
	double tmp;
	if (y_45_scale_m <= 8.8e-81) {
		tmp = -Math.sqrt((((2.0 * t_3) * t_2) * ((Math.sqrt(Math.pow(t_5, 2.0)) + t_5) / (y_45_scale_m * y_45_scale_m)))) / t_3;
	} else if (y_45_scale_m <= 3.2e+151) {
		tmp = 0.25 * ((b_m * (x_45_scale_m * ((y_45_scale_m * y_45_scale_m) * Math.sqrt((8.0 * ((Math.pow(a, 4.0) * (Math.sqrt((Math.pow(t_1, 4.0) / Math.pow(y_45_scale_m, 4.0))) + (Math.pow(t_1, 2.0) / (y_45_scale_m * y_45_scale_m)))) / (y_45_scale_m * y_45_scale_m))))))) / (a * a));
	} else {
		tmp = 0.25 * ((b_m * ((x_45_scale_m * x_45_scale_m) * (y_45_scale_m * Math.sqrt((8.0 * ((Math.pow(a, 4.0) * (Math.sqrt((Math.pow(t_4, 4.0) / Math.pow(x_45_scale_m, 4.0))) + (Math.pow(t_4, 2.0) / (x_45_scale_m * x_45_scale_m)))) / (x_45_scale_m * x_45_scale_m))))))) / (a * a));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b_m, angle, x_45_scale_m, y_45_scale_m):
	t_0 = 0.005555555555555556 * (angle * math.pi)
	t_1 = math.sin(t_0)
	t_2 = (b_m * a) * (b_m * -a)
	t_3 = (4.0 * t_2) / math.pow((x_45_scale_m * y_45_scale_m), 2.0)
	t_4 = math.cos(t_0)
	t_5 = math.pow((a * t_4), 2.0) + math.pow((b_m * t_1), 2.0)
	tmp = 0
	if y_45_scale_m <= 8.8e-81:
		tmp = -math.sqrt((((2.0 * t_3) * t_2) * ((math.sqrt(math.pow(t_5, 2.0)) + t_5) / (y_45_scale_m * y_45_scale_m)))) / t_3
	elif y_45_scale_m <= 3.2e+151:
		tmp = 0.25 * ((b_m * (x_45_scale_m * ((y_45_scale_m * y_45_scale_m) * math.sqrt((8.0 * ((math.pow(a, 4.0) * (math.sqrt((math.pow(t_1, 4.0) / math.pow(y_45_scale_m, 4.0))) + (math.pow(t_1, 2.0) / (y_45_scale_m * y_45_scale_m)))) / (y_45_scale_m * y_45_scale_m))))))) / (a * a))
	else:
		tmp = 0.25 * ((b_m * ((x_45_scale_m * x_45_scale_m) * (y_45_scale_m * math.sqrt((8.0 * ((math.pow(a, 4.0) * (math.sqrt((math.pow(t_4, 4.0) / math.pow(x_45_scale_m, 4.0))) + (math.pow(t_4, 2.0) / (x_45_scale_m * x_45_scale_m)))) / (x_45_scale_m * x_45_scale_m))))))) / (a * a))
	return tmp

Julia

b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_1 = sin(t_0)
	t_2 = Float64(Float64(b_m * a) * Float64(b_m * Float64(-a)))
	t_3 = Float64(Float64(4.0 * t_2) / (Float64(x_45_scale_m * y_45_scale_m) ^ 2.0))
	t_4 = cos(t_0)
	t_5 = Float64((Float64(a * t_4) ^ 2.0) + (Float64(b_m * t_1) ^ 2.0))
	tmp = 0.0
	if (y_45_scale_m <= 8.8e-81)
		tmp = Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * t_3) * t_2) * Float64(Float64(sqrt((t_5 ^ 2.0)) + t_5) / Float64(y_45_scale_m * y_45_scale_m))))) / t_3);
	elseif (y_45_scale_m <= 3.2e+151)
		tmp = Float64(0.25 * Float64(Float64(b_m * Float64(x_45_scale_m * Float64(Float64(y_45_scale_m * y_45_scale_m) * sqrt(Float64(8.0 * Float64(Float64((a ^ 4.0) * Float64(sqrt(Float64((t_1 ^ 4.0) / (y_45_scale_m ^ 4.0))) + Float64((t_1 ^ 2.0) / Float64(y_45_scale_m * y_45_scale_m)))) / Float64(y_45_scale_m * y_45_scale_m))))))) / Float64(a * a)));
	else
		tmp = Float64(0.25 * Float64(Float64(b_m * Float64(Float64(x_45_scale_m * x_45_scale_m) * Float64(y_45_scale_m * sqrt(Float64(8.0 * Float64(Float64((a ^ 4.0) * Float64(sqrt(Float64((t_4 ^ 4.0) / (x_45_scale_m ^ 4.0))) + Float64((t_4 ^ 2.0) / Float64(x_45_scale_m * x_45_scale_m)))) / Float64(x_45_scale_m * x_45_scale_m))))))) / Float64(a * a)));
	end
	return tmp
end

MATLAB

b_m = abs(b);
x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = 0.005555555555555556 * (angle * pi);
	t_1 = sin(t_0);
	t_2 = (b_m * a) * (b_m * -a);
	t_3 = (4.0 * t_2) / ((x_45_scale_m * y_45_scale_m) ^ 2.0);
	t_4 = cos(t_0);
	t_5 = ((a * t_4) ^ 2.0) + ((b_m * t_1) ^ 2.0);
	tmp = 0.0;
	if (y_45_scale_m <= 8.8e-81)
		tmp = -sqrt((((2.0 * t_3) * t_2) * ((sqrt((t_5 ^ 2.0)) + t_5) / (y_45_scale_m * y_45_scale_m)))) / t_3;
	elseif (y_45_scale_m <= 3.2e+151)
		tmp = 0.25 * ((b_m * (x_45_scale_m * ((y_45_scale_m * y_45_scale_m) * sqrt((8.0 * (((a ^ 4.0) * (sqrt(((t_1 ^ 4.0) / (y_45_scale_m ^ 4.0))) + ((t_1 ^ 2.0) / (y_45_scale_m * y_45_scale_m)))) / (y_45_scale_m * y_45_scale_m))))))) / (a * a));
	else
		tmp = 0.25 * ((b_m * ((x_45_scale_m * x_45_scale_m) * (y_45_scale_m * sqrt((8.0 * (((a ^ 4.0) * (sqrt(((t_4 ^ 4.0) / (x_45_scale_m ^ 4.0))) + ((t_4 ^ 2.0) / (x_45_scale_m * x_45_scale_m)))) / (x_45_scale_m * x_45_scale_m))))))) / (a * a));
	end
	tmp_2 = tmp;
end

Wolfram

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_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[(N[(b$95$m * a), $MachinePrecision] * N[(b$95$m * (-a)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(4.0 * t$95$2), $MachinePrecision] / N[Power[N[(x$45$scale$95$m * y$45$scale$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$5 = N[(N[Power[N[(a * t$95$4), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b$95$m * t$95$1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$45$scale$95$m, 8.8e-81], N[((-N[Sqrt[N[(N[(N[(2.0 * t$95$3), $MachinePrecision] * t$95$2), $MachinePrecision] * N[(N[(N[Sqrt[N[Power[t$95$5, 2.0], $MachinePrecision]], $MachinePrecision] + t$95$5), $MachinePrecision] / N[(y$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$3), $MachinePrecision], If[LessEqual[y$45$scale$95$m, 3.2e+151], N[(0.25 * N[(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[Power[a, 4.0], $MachinePrecision] * N[(N[Sqrt[N[(N[Power[t$95$1, 4.0], $MachinePrecision] / N[Power[y$45$scale$95$m, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[Power[t$95$1, 2.0], $MachinePrecision] / N[(y$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(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[Power[a, 4.0], $MachinePrecision] * N[(N[Sqrt[N[(N[Power[t$95$4, 4.0], $MachinePrecision] / N[Power[x$45$scale$95$m, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[Power[t$95$4, 2.0], $MachinePrecision] / N[(x$45$scale$95$m * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale$95$m * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y-scale < 8.7999999999999997e-81

   1. Initial program 2.9%

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

   2. Taylor expanded in y-scale around 0

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

   4. Applied rewrites 3.9%

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

   if 8.7999999999999997e-81 < y-scale < 3.19999999999999994e151

   1. Initial program 2.9%

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

   2. Taylor expanded in b around -inf

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

   4. Applied rewrites 0.1%

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

   5. Taylor expanded in angle around 0

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

      1. lower-+.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 0.2

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

   7. Applied rewrites 0.2%

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

   8. Taylor expanded in x-scale around -inf

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

   10. Applied rewrites 2.5%

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

   if 3.19999999999999994e151 < y-scale

   1. Initial program 2.9%

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

   2. Taylor expanded in b around -inf

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

   4. Applied rewrites 0.1%

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

   5. Taylor expanded in angle around 0

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

      1. lower-+.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 0.2

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

   7. Applied rewrites 0.2%

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

   8. Taylor expanded in y-scale around -inf

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

   10. Applied rewrites 4.7%

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

Alternative 3: 7.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := \left(b\_m \cdot a\right) \cdot \left(b\_m \cdot \left(-a\right)\right)\\ t_1 := \frac{4 \cdot t\_0}{{\left(x-scale\_m \cdot y-scale\_m\right)}^{2}}\\ t_2 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_3 := \sin t\_2\\ t_4 := \cos t\_2\\ \mathbf{if}\;y-scale\_m \leq 8.8 \cdot 10^{-81}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(2 \cdot t\_1\right) \cdot t\_0\right) \cdot \left(\left(\frac{\frac{{b\_m}^{2}}{x-scale\_m}}{x-scale\_m} + \frac{\frac{a \cdot a}{y-scale\_m}}{y-scale\_m}\right) + \frac{\sqrt{{\left({\left(a \cdot t\_4\right)}^{2} + {\left(b\_m \cdot t\_3\right)}^{2}\right)}^{2}}}{y-scale\_m \cdot y-scale\_m}\right)}}{t\_1}\\ \mathbf{elif}\;y-scale\_m \leq 3.2 \cdot 10^{+151}:\\ \;\;\;\;0.25 \cdot \frac{b\_m \cdot \left(x-scale\_m \cdot \left(\left(y-scale\_m \cdot y-scale\_m\right) \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\sqrt{\frac{{t\_3}^{4}}{{y-scale\_m}^{4}}} + \frac{{t\_3}^{2}}{y-scale\_m \cdot y-scale\_m}\right)}{y-scale\_m \cdot y-scale\_m}}\right)\right)}{a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{b\_m \cdot \left(\left(x-scale\_m \cdot x-scale\_m\right) \cdot \left(y-scale\_m \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\sqrt{\frac{{t\_4}^{4}}{{x-scale\_m}^{4}}} + \frac{{t\_4}^{2}}{x-scale\_m \cdot x-scale\_m}\right)}{x-scale\_m \cdot x-scale\_m}}\right)\right)}{a \cdot a}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b_m angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (* (* b_m a) (* b_m (- a))))
        (t_1 (/ (* 4.0 t_0) (pow (* x-scale_m y-scale_m) 2.0)))
        (t_2 (* 0.005555555555555556 (* angle PI)))
        (t_3 (sin t_2))
        (t_4 (cos t_2)))
   (if (<= y-scale_m 8.8e-81)
     (/
      (-
       (sqrt
        (*
         (* (* 2.0 t_1) t_0)
         (+
          (+
           (/ (/ (pow b_m 2.0) x-scale_m) x-scale_m)
           (/ (/ (* a a) y-scale_m) y-scale_m))
          (/
           (sqrt (pow (+ (pow (* a t_4) 2.0) (pow (* b_m t_3) 2.0)) 2.0))
           (* y-scale_m y-scale_m))))))
      t_1)
     (if (<= y-scale_m 3.2e+151)
       (*
        0.25
        (/
         (*
          b_m
          (*
           x-scale_m
           (*
            (* y-scale_m y-scale_m)
            (sqrt
             (*
              8.0
              (/
               (*
                (pow a 4.0)
                (+
                 (sqrt (/ (pow t_3 4.0) (pow y-scale_m 4.0)))
                 (/ (pow t_3 2.0) (* y-scale_m y-scale_m))))
               (* y-scale_m y-scale_m)))))))
         (* a a)))
       (*
        0.25
        (/
         (*
          b_m
          (*
           (* x-scale_m x-scale_m)
           (*
            y-scale_m
            (sqrt
             (*
              8.0
              (/
               (*
                (pow a 4.0)
                (+
                 (sqrt (/ (pow t_4 4.0) (pow x-scale_m 4.0)))
                 (/ (pow t_4 2.0) (* x-scale_m x-scale_m))))
               (* x-scale_m x-scale_m)))))))
         (* a a)))))))

C

b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = (b_m * a) * (b_m * -a);
	double t_1 = (4.0 * t_0) / pow((x_45_scale_m * y_45_scale_m), 2.0);
	double t_2 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_3 = sin(t_2);
	double t_4 = cos(t_2);
	double tmp;
	if (y_45_scale_m <= 8.8e-81) {
		tmp = -sqrt((((2.0 * t_1) * t_0) * ((((pow(b_m, 2.0) / x_45_scale_m) / x_45_scale_m) + (((a * a) / y_45_scale_m) / y_45_scale_m)) + (sqrt(pow((pow((a * t_4), 2.0) + pow((b_m * t_3), 2.0)), 2.0)) / (y_45_scale_m * y_45_scale_m))))) / t_1;
	} else if (y_45_scale_m <= 3.2e+151) {
		tmp = 0.25 * ((b_m * (x_45_scale_m * ((y_45_scale_m * y_45_scale_m) * sqrt((8.0 * ((pow(a, 4.0) * (sqrt((pow(t_3, 4.0) / pow(y_45_scale_m, 4.0))) + (pow(t_3, 2.0) / (y_45_scale_m * y_45_scale_m)))) / (y_45_scale_m * y_45_scale_m))))))) / (a * a));
	} else {
		tmp = 0.25 * ((b_m * ((x_45_scale_m * x_45_scale_m) * (y_45_scale_m * sqrt((8.0 * ((pow(a, 4.0) * (sqrt((pow(t_4, 4.0) / pow(x_45_scale_m, 4.0))) + (pow(t_4, 2.0) / (x_45_scale_m * x_45_scale_m)))) / (x_45_scale_m * x_45_scale_m))))))) / (a * a));
	}
	return tmp;
}

Java

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, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = (b_m * a) * (b_m * -a);
	double t_1 = (4.0 * t_0) / Math.pow((x_45_scale_m * y_45_scale_m), 2.0);
	double t_2 = 0.005555555555555556 * (angle * Math.PI);
	double t_3 = Math.sin(t_2);
	double t_4 = Math.cos(t_2);
	double tmp;
	if (y_45_scale_m <= 8.8e-81) {
		tmp = -Math.sqrt((((2.0 * t_1) * t_0) * ((((Math.pow(b_m, 2.0) / x_45_scale_m) / x_45_scale_m) + (((a * a) / y_45_scale_m) / y_45_scale_m)) + (Math.sqrt(Math.pow((Math.pow((a * t_4), 2.0) + Math.pow((b_m * t_3), 2.0)), 2.0)) / (y_45_scale_m * y_45_scale_m))))) / t_1;
	} else if (y_45_scale_m <= 3.2e+151) {
		tmp = 0.25 * ((b_m * (x_45_scale_m * ((y_45_scale_m * y_45_scale_m) * Math.sqrt((8.0 * ((Math.pow(a, 4.0) * (Math.sqrt((Math.pow(t_3, 4.0) / Math.pow(y_45_scale_m, 4.0))) + (Math.pow(t_3, 2.0) / (y_45_scale_m * y_45_scale_m)))) / (y_45_scale_m * y_45_scale_m))))))) / (a * a));
	} else {
		tmp = 0.25 * ((b_m * ((x_45_scale_m * x_45_scale_m) * (y_45_scale_m * Math.sqrt((8.0 * ((Math.pow(a, 4.0) * (Math.sqrt((Math.pow(t_4, 4.0) / Math.pow(x_45_scale_m, 4.0))) + (Math.pow(t_4, 2.0) / (x_45_scale_m * x_45_scale_m)))) / (x_45_scale_m * x_45_scale_m))))))) / (a * a));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b_m, angle, x_45_scale_m, y_45_scale_m):
	t_0 = (b_m * a) * (b_m * -a)
	t_1 = (4.0 * t_0) / math.pow((x_45_scale_m * y_45_scale_m), 2.0)
	t_2 = 0.005555555555555556 * (angle * math.pi)
	t_3 = math.sin(t_2)
	t_4 = math.cos(t_2)
	tmp = 0
	if y_45_scale_m <= 8.8e-81:
		tmp = -math.sqrt((((2.0 * t_1) * t_0) * ((((math.pow(b_m, 2.0) / x_45_scale_m) / x_45_scale_m) + (((a * a) / y_45_scale_m) / y_45_scale_m)) + (math.sqrt(math.pow((math.pow((a * t_4), 2.0) + math.pow((b_m * t_3), 2.0)), 2.0)) / (y_45_scale_m * y_45_scale_m))))) / t_1
	elif y_45_scale_m <= 3.2e+151:
		tmp = 0.25 * ((b_m * (x_45_scale_m * ((y_45_scale_m * y_45_scale_m) * math.sqrt((8.0 * ((math.pow(a, 4.0) * (math.sqrt((math.pow(t_3, 4.0) / math.pow(y_45_scale_m, 4.0))) + (math.pow(t_3, 2.0) / (y_45_scale_m * y_45_scale_m)))) / (y_45_scale_m * y_45_scale_m))))))) / (a * a))
	else:
		tmp = 0.25 * ((b_m * ((x_45_scale_m * x_45_scale_m) * (y_45_scale_m * math.sqrt((8.0 * ((math.pow(a, 4.0) * (math.sqrt((math.pow(t_4, 4.0) / math.pow(x_45_scale_m, 4.0))) + (math.pow(t_4, 2.0) / (x_45_scale_m * x_45_scale_m)))) / (x_45_scale_m * x_45_scale_m))))))) / (a * a))
	return tmp

Julia

b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(Float64(b_m * a) * Float64(b_m * Float64(-a)))
	t_1 = Float64(Float64(4.0 * t_0) / (Float64(x_45_scale_m * y_45_scale_m) ^ 2.0))
	t_2 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_3 = sin(t_2)
	t_4 = cos(t_2)
	tmp = 0.0
	if (y_45_scale_m <= 8.8e-81)
		tmp = Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * t_1) * t_0) * Float64(Float64(Float64(Float64((b_m ^ 2.0) / x_45_scale_m) / x_45_scale_m) + Float64(Float64(Float64(a * a) / y_45_scale_m) / y_45_scale_m)) + Float64(sqrt((Float64((Float64(a * t_4) ^ 2.0) + (Float64(b_m * t_3) ^ 2.0)) ^ 2.0)) / Float64(y_45_scale_m * y_45_scale_m)))))) / t_1);
	elseif (y_45_scale_m <= 3.2e+151)
		tmp = Float64(0.25 * Float64(Float64(b_m * Float64(x_45_scale_m * Float64(Float64(y_45_scale_m * y_45_scale_m) * sqrt(Float64(8.0 * Float64(Float64((a ^ 4.0) * Float64(sqrt(Float64((t_3 ^ 4.0) / (y_45_scale_m ^ 4.0))) + Float64((t_3 ^ 2.0) / Float64(y_45_scale_m * y_45_scale_m)))) / Float64(y_45_scale_m * y_45_scale_m))))))) / Float64(a * a)));
	else
		tmp = Float64(0.25 * Float64(Float64(b_m * Float64(Float64(x_45_scale_m * x_45_scale_m) * Float64(y_45_scale_m * sqrt(Float64(8.0 * Float64(Float64((a ^ 4.0) * Float64(sqrt(Float64((t_4 ^ 4.0) / (x_45_scale_m ^ 4.0))) + Float64((t_4 ^ 2.0) / Float64(x_45_scale_m * x_45_scale_m)))) / Float64(x_45_scale_m * x_45_scale_m))))))) / Float64(a * a)));
	end
	return tmp
end

MATLAB

b_m = abs(b);
x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = (b_m * a) * (b_m * -a);
	t_1 = (4.0 * t_0) / ((x_45_scale_m * y_45_scale_m) ^ 2.0);
	t_2 = 0.005555555555555556 * (angle * pi);
	t_3 = sin(t_2);
	t_4 = cos(t_2);
	tmp = 0.0;
	if (y_45_scale_m <= 8.8e-81)
		tmp = -sqrt((((2.0 * t_1) * t_0) * (((((b_m ^ 2.0) / x_45_scale_m) / x_45_scale_m) + (((a * a) / y_45_scale_m) / y_45_scale_m)) + (sqrt(((((a * t_4) ^ 2.0) + ((b_m * t_3) ^ 2.0)) ^ 2.0)) / (y_45_scale_m * y_45_scale_m))))) / t_1;
	elseif (y_45_scale_m <= 3.2e+151)
		tmp = 0.25 * ((b_m * (x_45_scale_m * ((y_45_scale_m * y_45_scale_m) * sqrt((8.0 * (((a ^ 4.0) * (sqrt(((t_3 ^ 4.0) / (y_45_scale_m ^ 4.0))) + ((t_3 ^ 2.0) / (y_45_scale_m * y_45_scale_m)))) / (y_45_scale_m * y_45_scale_m))))))) / (a * a));
	else
		tmp = 0.25 * ((b_m * ((x_45_scale_m * x_45_scale_m) * (y_45_scale_m * sqrt((8.0 * (((a ^ 4.0) * (sqrt(((t_4 ^ 4.0) / (x_45_scale_m ^ 4.0))) + ((t_4 ^ 2.0) / (x_45_scale_m * x_45_scale_m)))) / (x_45_scale_m * x_45_scale_m))))))) / (a * a));
	end
	tmp_2 = tmp;
end

Wolfram

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_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(N[(b$95$m * a), $MachinePrecision] * N[(b$95$m * (-a)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * t$95$0), $MachinePrecision] / N[Power[N[(x$45$scale$95$m * y$45$scale$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sin[t$95$2], $MachinePrecision]}, Block[{t$95$4 = N[Cos[t$95$2], $MachinePrecision]}, If[LessEqual[y$45$scale$95$m, 8.8e-81], N[((-N[Sqrt[N[(N[(N[(2.0 * t$95$1), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(N[(N[(N[(N[Power[b$95$m, 2.0], $MachinePrecision] / x$45$scale$95$m), $MachinePrecision] / x$45$scale$95$m), $MachinePrecision] + N[(N[(N[(a * a), $MachinePrecision] / y$45$scale$95$m), $MachinePrecision] / y$45$scale$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[Power[N[(N[Power[N[(a * t$95$4), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b$95$m * t$95$3), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] / N[(y$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], If[LessEqual[y$45$scale$95$m, 3.2e+151], N[(0.25 * N[(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[Power[a, 4.0], $MachinePrecision] * N[(N[Sqrt[N[(N[Power[t$95$3, 4.0], $MachinePrecision] / N[Power[y$45$scale$95$m, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[Power[t$95$3, 2.0], $MachinePrecision] / N[(y$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(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[Power[a, 4.0], $MachinePrecision] * N[(N[Sqrt[N[(N[Power[t$95$4, 4.0], $MachinePrecision] / N[Power[x$45$scale$95$m, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[Power[t$95$4, 2.0], $MachinePrecision] / N[(x$45$scale$95$m * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale$95$m * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y-scale < 8.7999999999999997e-81

   1. Initial program 2.9%

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

   2. Taylor expanded in y-scale around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 4.4%

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \color{blue}{\frac{\sqrt{{\left({\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right)}^{2}}}{y-scale \cdot y-scale}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   5. Taylor expanded in angle around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 4.4

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{b}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \frac{\sqrt{{\left({\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right)}^{2}}}{y-scale \cdot y-scale}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   7. Applied rewrites 4.4%

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\color{blue}{\frac{{b}^{2}}{x-scale}}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \frac{\sqrt{{\left({\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right)}^{2}}}{y-scale \cdot y-scale}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   8. Taylor expanded in angle around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 4.4

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

   10. Applied rewrites 4.4%

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

   if 8.7999999999999997e-81 < y-scale < 3.19999999999999994e151

   1. Initial program 2.9%

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

   2. Taylor expanded in b around -inf

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

   4. Applied rewrites 0.1%

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

   5. Taylor expanded in angle around 0

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

      1. lower-+.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 0.2

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

   7. Applied rewrites 0.2%

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

   8. Taylor expanded in x-scale around -inf

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

   10. Applied rewrites 2.5%

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

   if 3.19999999999999994e151 < y-scale

   1. Initial program 2.9%

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

   2. Taylor expanded in b around -inf

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

   4. Applied rewrites 0.1%

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

   5. Taylor expanded in angle around 0

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

      1. lower-+.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 0.2

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

   7. Applied rewrites 0.2%

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

   8. Taylor expanded in y-scale around -inf

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

   10. Applied rewrites 4.7%

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

Alternative 4: 6.2% accurate, 3.8× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \sin t\_0\\ t_2 := \left(b\_m \cdot a\right) \cdot \left(b\_m \cdot \left(-a\right)\right)\\ t_3 := \frac{4 \cdot t\_2}{{\left(x-scale\_m \cdot y-scale\_m\right)}^{2}}\\ t_4 := \cos t\_0\\ \mathbf{if}\;y-scale\_m \leq 8.8 \cdot 10^{-81}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(2 \cdot t\_3\right) \cdot t\_2\right) \cdot \left(\left(\frac{\frac{{b\_m}^{2}}{x-scale\_m}}{x-scale\_m} + \frac{\frac{a \cdot a}{y-scale\_m}}{y-scale\_m}\right) + \frac{\sqrt{{\left(b\_m \cdot t\_1\right)}^{4}}}{y-scale\_m \cdot y-scale\_m}\right)}}{t\_3}\\ \mathbf{elif}\;y-scale\_m \leq 3.2 \cdot 10^{+151}:\\ \;\;\;\;0.25 \cdot \frac{b\_m \cdot \left(x-scale\_m \cdot \left(\left(y-scale\_m \cdot y-scale\_m\right) \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\sqrt{\frac{{t\_1}^{4}}{{y-scale\_m}^{4}}} + \frac{{t\_1}^{2}}{y-scale\_m \cdot y-scale\_m}\right)}{y-scale\_m \cdot y-scale\_m}}\right)\right)}{a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{b\_m \cdot \left(\left(x-scale\_m \cdot x-scale\_m\right) \cdot \left(y-scale\_m \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\sqrt{\frac{{t\_4}^{4}}{{x-scale\_m}^{4}}} + \frac{{t\_4}^{2}}{x-scale\_m \cdot x-scale\_m}\right)}{x-scale\_m \cdot x-scale\_m}}\right)\right)}{a \cdot a}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b_m angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI)))
        (t_1 (sin t_0))
        (t_2 (* (* b_m a) (* b_m (- a))))
        (t_3 (/ (* 4.0 t_2) (pow (* x-scale_m y-scale_m) 2.0)))
        (t_4 (cos t_0)))
   (if (<= y-scale_m 8.8e-81)
     (/
      (-
       (sqrt
        (*
         (* (* 2.0 t_3) t_2)
         (+
          (+
           (/ (/ (pow b_m 2.0) x-scale_m) x-scale_m)
           (/ (/ (* a a) y-scale_m) y-scale_m))
          (/ (sqrt (pow (* b_m t_1) 4.0)) (* y-scale_m y-scale_m))))))
      t_3)
     (if (<= y-scale_m 3.2e+151)
       (*
        0.25
        (/
         (*
          b_m
          (*
           x-scale_m
           (*
            (* y-scale_m y-scale_m)
            (sqrt
             (*
              8.0
              (/
               (*
                (pow a 4.0)
                (+
                 (sqrt (/ (pow t_1 4.0) (pow y-scale_m 4.0)))
                 (/ (pow t_1 2.0) (* y-scale_m y-scale_m))))
               (* y-scale_m y-scale_m)))))))
         (* a a)))
       (*
        0.25
        (/
         (*
          b_m
          (*
           (* x-scale_m x-scale_m)
           (*
            y-scale_m
            (sqrt
             (*
              8.0
              (/
               (*
                (pow a 4.0)
                (+
                 (sqrt (/ (pow t_4 4.0) (pow x-scale_m 4.0)))
                 (/ (pow t_4 2.0) (* x-scale_m x-scale_m))))
               (* x-scale_m x-scale_m)))))))
         (* a a)))))))

C

b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_1 = sin(t_0);
	double t_2 = (b_m * a) * (b_m * -a);
	double t_3 = (4.0 * t_2) / pow((x_45_scale_m * y_45_scale_m), 2.0);
	double t_4 = cos(t_0);
	double tmp;
	if (y_45_scale_m <= 8.8e-81) {
		tmp = -sqrt((((2.0 * t_3) * t_2) * ((((pow(b_m, 2.0) / x_45_scale_m) / x_45_scale_m) + (((a * a) / y_45_scale_m) / y_45_scale_m)) + (sqrt(pow((b_m * t_1), 4.0)) / (y_45_scale_m * y_45_scale_m))))) / t_3;
	} else if (y_45_scale_m <= 3.2e+151) {
		tmp = 0.25 * ((b_m * (x_45_scale_m * ((y_45_scale_m * y_45_scale_m) * sqrt((8.0 * ((pow(a, 4.0) * (sqrt((pow(t_1, 4.0) / pow(y_45_scale_m, 4.0))) + (pow(t_1, 2.0) / (y_45_scale_m * y_45_scale_m)))) / (y_45_scale_m * y_45_scale_m))))))) / (a * a));
	} else {
		tmp = 0.25 * ((b_m * ((x_45_scale_m * x_45_scale_m) * (y_45_scale_m * sqrt((8.0 * ((pow(a, 4.0) * (sqrt((pow(t_4, 4.0) / pow(x_45_scale_m, 4.0))) + (pow(t_4, 2.0) / (x_45_scale_m * x_45_scale_m)))) / (x_45_scale_m * x_45_scale_m))))))) / (a * a));
	}
	return tmp;
}

Java

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, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = 0.005555555555555556 * (angle * Math.PI);
	double t_1 = Math.sin(t_0);
	double t_2 = (b_m * a) * (b_m * -a);
	double t_3 = (4.0 * t_2) / Math.pow((x_45_scale_m * y_45_scale_m), 2.0);
	double t_4 = Math.cos(t_0);
	double tmp;
	if (y_45_scale_m <= 8.8e-81) {
		tmp = -Math.sqrt((((2.0 * t_3) * t_2) * ((((Math.pow(b_m, 2.0) / x_45_scale_m) / x_45_scale_m) + (((a * a) / y_45_scale_m) / y_45_scale_m)) + (Math.sqrt(Math.pow((b_m * t_1), 4.0)) / (y_45_scale_m * y_45_scale_m))))) / t_3;
	} else if (y_45_scale_m <= 3.2e+151) {
		tmp = 0.25 * ((b_m * (x_45_scale_m * ((y_45_scale_m * y_45_scale_m) * Math.sqrt((8.0 * ((Math.pow(a, 4.0) * (Math.sqrt((Math.pow(t_1, 4.0) / Math.pow(y_45_scale_m, 4.0))) + (Math.pow(t_1, 2.0) / (y_45_scale_m * y_45_scale_m)))) / (y_45_scale_m * y_45_scale_m))))))) / (a * a));
	} else {
		tmp = 0.25 * ((b_m * ((x_45_scale_m * x_45_scale_m) * (y_45_scale_m * Math.sqrt((8.0 * ((Math.pow(a, 4.0) * (Math.sqrt((Math.pow(t_4, 4.0) / Math.pow(x_45_scale_m, 4.0))) + (Math.pow(t_4, 2.0) / (x_45_scale_m * x_45_scale_m)))) / (x_45_scale_m * x_45_scale_m))))))) / (a * a));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b_m, angle, x_45_scale_m, y_45_scale_m):
	t_0 = 0.005555555555555556 * (angle * math.pi)
	t_1 = math.sin(t_0)
	t_2 = (b_m * a) * (b_m * -a)
	t_3 = (4.0 * t_2) / math.pow((x_45_scale_m * y_45_scale_m), 2.0)
	t_4 = math.cos(t_0)
	tmp = 0
	if y_45_scale_m <= 8.8e-81:
		tmp = -math.sqrt((((2.0 * t_3) * t_2) * ((((math.pow(b_m, 2.0) / x_45_scale_m) / x_45_scale_m) + (((a * a) / y_45_scale_m) / y_45_scale_m)) + (math.sqrt(math.pow((b_m * t_1), 4.0)) / (y_45_scale_m * y_45_scale_m))))) / t_3
	elif y_45_scale_m <= 3.2e+151:
		tmp = 0.25 * ((b_m * (x_45_scale_m * ((y_45_scale_m * y_45_scale_m) * math.sqrt((8.0 * ((math.pow(a, 4.0) * (math.sqrt((math.pow(t_1, 4.0) / math.pow(y_45_scale_m, 4.0))) + (math.pow(t_1, 2.0) / (y_45_scale_m * y_45_scale_m)))) / (y_45_scale_m * y_45_scale_m))))))) / (a * a))
	else:
		tmp = 0.25 * ((b_m * ((x_45_scale_m * x_45_scale_m) * (y_45_scale_m * math.sqrt((8.0 * ((math.pow(a, 4.0) * (math.sqrt((math.pow(t_4, 4.0) / math.pow(x_45_scale_m, 4.0))) + (math.pow(t_4, 2.0) / (x_45_scale_m * x_45_scale_m)))) / (x_45_scale_m * x_45_scale_m))))))) / (a * a))
	return tmp

Julia

b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_1 = sin(t_0)
	t_2 = Float64(Float64(b_m * a) * Float64(b_m * Float64(-a)))
	t_3 = Float64(Float64(4.0 * t_2) / (Float64(x_45_scale_m * y_45_scale_m) ^ 2.0))
	t_4 = cos(t_0)
	tmp = 0.0
	if (y_45_scale_m <= 8.8e-81)
		tmp = Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * t_3) * t_2) * Float64(Float64(Float64(Float64((b_m ^ 2.0) / x_45_scale_m) / x_45_scale_m) + Float64(Float64(Float64(a * a) / y_45_scale_m) / y_45_scale_m)) + Float64(sqrt((Float64(b_m * t_1) ^ 4.0)) / Float64(y_45_scale_m * y_45_scale_m)))))) / t_3);
	elseif (y_45_scale_m <= 3.2e+151)
		tmp = Float64(0.25 * Float64(Float64(b_m * Float64(x_45_scale_m * Float64(Float64(y_45_scale_m * y_45_scale_m) * sqrt(Float64(8.0 * Float64(Float64((a ^ 4.0) * Float64(sqrt(Float64((t_1 ^ 4.0) / (y_45_scale_m ^ 4.0))) + Float64((t_1 ^ 2.0) / Float64(y_45_scale_m * y_45_scale_m)))) / Float64(y_45_scale_m * y_45_scale_m))))))) / Float64(a * a)));
	else
		tmp = Float64(0.25 * Float64(Float64(b_m * Float64(Float64(x_45_scale_m * x_45_scale_m) * Float64(y_45_scale_m * sqrt(Float64(8.0 * Float64(Float64((a ^ 4.0) * Float64(sqrt(Float64((t_4 ^ 4.0) / (x_45_scale_m ^ 4.0))) + Float64((t_4 ^ 2.0) / Float64(x_45_scale_m * x_45_scale_m)))) / Float64(x_45_scale_m * x_45_scale_m))))))) / Float64(a * a)));
	end
	return tmp
end

MATLAB

b_m = abs(b);
x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = 0.005555555555555556 * (angle * pi);
	t_1 = sin(t_0);
	t_2 = (b_m * a) * (b_m * -a);
	t_3 = (4.0 * t_2) / ((x_45_scale_m * y_45_scale_m) ^ 2.0);
	t_4 = cos(t_0);
	tmp = 0.0;
	if (y_45_scale_m <= 8.8e-81)
		tmp = -sqrt((((2.0 * t_3) * t_2) * (((((b_m ^ 2.0) / x_45_scale_m) / x_45_scale_m) + (((a * a) / y_45_scale_m) / y_45_scale_m)) + (sqrt(((b_m * t_1) ^ 4.0)) / (y_45_scale_m * y_45_scale_m))))) / t_3;
	elseif (y_45_scale_m <= 3.2e+151)
		tmp = 0.25 * ((b_m * (x_45_scale_m * ((y_45_scale_m * y_45_scale_m) * sqrt((8.0 * (((a ^ 4.0) * (sqrt(((t_1 ^ 4.0) / (y_45_scale_m ^ 4.0))) + ((t_1 ^ 2.0) / (y_45_scale_m * y_45_scale_m)))) / (y_45_scale_m * y_45_scale_m))))))) / (a * a));
	else
		tmp = 0.25 * ((b_m * ((x_45_scale_m * x_45_scale_m) * (y_45_scale_m * sqrt((8.0 * (((a ^ 4.0) * (sqrt(((t_4 ^ 4.0) / (x_45_scale_m ^ 4.0))) + ((t_4 ^ 2.0) / (x_45_scale_m * x_45_scale_m)))) / (x_45_scale_m * x_45_scale_m))))))) / (a * a));
	end
	tmp_2 = tmp;
end

Wolfram

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_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[(N[(b$95$m * a), $MachinePrecision] * N[(b$95$m * (-a)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(4.0 * t$95$2), $MachinePrecision] / N[Power[N[(x$45$scale$95$m * y$45$scale$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Cos[t$95$0], $MachinePrecision]}, If[LessEqual[y$45$scale$95$m, 8.8e-81], N[((-N[Sqrt[N[(N[(N[(2.0 * t$95$3), $MachinePrecision] * t$95$2), $MachinePrecision] * N[(N[(N[(N[(N[Power[b$95$m, 2.0], $MachinePrecision] / x$45$scale$95$m), $MachinePrecision] / x$45$scale$95$m), $MachinePrecision] + N[(N[(N[(a * a), $MachinePrecision] / y$45$scale$95$m), $MachinePrecision] / y$45$scale$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[Power[N[(b$95$m * t$95$1), $MachinePrecision], 4.0], $MachinePrecision]], $MachinePrecision] / N[(y$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$3), $MachinePrecision], If[LessEqual[y$45$scale$95$m, 3.2e+151], N[(0.25 * N[(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[Power[a, 4.0], $MachinePrecision] * N[(N[Sqrt[N[(N[Power[t$95$1, 4.0], $MachinePrecision] / N[Power[y$45$scale$95$m, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[Power[t$95$1, 2.0], $MachinePrecision] / N[(y$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(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[Power[a, 4.0], $MachinePrecision] * N[(N[Sqrt[N[(N[Power[t$95$4, 4.0], $MachinePrecision] / N[Power[x$45$scale$95$m, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[Power[t$95$4, 2.0], $MachinePrecision] / N[(x$45$scale$95$m * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale$95$m * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y-scale < 8.7999999999999997e-81

   1. Initial program 2.9%

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

   2. Taylor expanded in y-scale around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 4.4%

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \color{blue}{\frac{\sqrt{{\left({\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right)}^{2}}}{y-scale \cdot y-scale}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   5. Taylor expanded in angle around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 4.4

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{b}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \frac{\sqrt{{\left({\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right)}^{2}}}{y-scale \cdot y-scale}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   7. Applied rewrites 4.4%

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\color{blue}{\frac{{b}^{2}}{x-scale}}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \frac{\sqrt{{\left({\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right)}^{2}}}{y-scale \cdot y-scale}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   8. Taylor expanded in a around 0

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. pow-prod-down N/A

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

      6. lift-*.f64 N/A

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

      7. lower-pow.f64 4.2

         \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{b}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \frac{\sqrt{{\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{4}}}{y-scale \cdot y-scale}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   10. Applied rewrites 4.2%

       \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{b}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \frac{\sqrt{{\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{4}}}{y-scale \cdot y-scale}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   11. Taylor expanded in angle around 0

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

       1. lower-/.f64 N/A

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

       2. pow2 N/A

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

       3. lift-*.f64 4.2

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

   13. Applied rewrites 4.2%

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

   if 8.7999999999999997e-81 < y-scale < 3.19999999999999994e151

   1. Initial program 2.9%

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

   2. Taylor expanded in b around -inf

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

   4. Applied rewrites 0.1%

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

   5. Taylor expanded in angle around 0

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

      1. lower-+.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 0.2

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

   7. Applied rewrites 0.2%

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

   8. Taylor expanded in x-scale around -inf

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

   10. Applied rewrites 2.5%

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

   if 3.19999999999999994e151 < y-scale

   1. Initial program 2.9%

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

   2. Taylor expanded in b around -inf

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

   4. Applied rewrites 0.1%

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

   5. Taylor expanded in angle around 0

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

      1. lower-+.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 0.2

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

   7. Applied rewrites 0.2%

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

   8. Taylor expanded in y-scale around -inf

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

   10. Applied rewrites 4.7%

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

Alternative 5: 6.1% accurate, 3.9× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \sin t\_0\\ t_2 := \cos t\_0\\ \mathbf{if}\;x-scale\_m \leq 8.2 \cdot 10^{+149}:\\ \;\;\;\;0.25 \cdot \frac{b\_m \cdot \left(\left(x-scale\_m \cdot x-scale\_m\right) \cdot \left(y-scale\_m \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\sqrt{\frac{{t\_2}^{4}}{{x-scale\_m}^{4}}} + \frac{{t\_2}^{2}}{x-scale\_m \cdot x-scale\_m}\right)}{x-scale\_m \cdot x-scale\_m}}\right)\right)}{a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{b\_m \cdot \left(x-scale\_m \cdot \left(\left(y-scale\_m \cdot y-scale\_m\right) \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\sqrt{\frac{{t\_1}^{4}}{{y-scale\_m}^{4}}} + \frac{{t\_1}^{2}}{y-scale\_m \cdot y-scale\_m}\right)}{y-scale\_m \cdot y-scale\_m}}\right)\right)}{a \cdot a}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b_m angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI)))
        (t_1 (sin t_0))
        (t_2 (cos t_0)))
   (if (<= x-scale_m 8.2e+149)
     (*
      0.25
      (/
       (*
        b_m
        (*
         (* x-scale_m x-scale_m)
         (*
          y-scale_m
          (sqrt
           (*
            8.0
            (/
             (*
              (pow a 4.0)
              (+
               (sqrt (/ (pow t_2 4.0) (pow x-scale_m 4.0)))
               (/ (pow t_2 2.0) (* x-scale_m x-scale_m))))
             (* x-scale_m x-scale_m)))))))
       (* a a)))
     (*
      0.25
      (/
       (*
        b_m
        (*
         x-scale_m
         (*
          (* y-scale_m y-scale_m)
          (sqrt
           (*
            8.0
            (/
             (*
              (pow a 4.0)
              (+
               (sqrt (/ (pow t_1 4.0) (pow y-scale_m 4.0)))
               (/ (pow t_1 2.0) (* y-scale_m y-scale_m))))
             (* y-scale_m y-scale_m)))))))
       (* a a))))))

C

b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_1 = sin(t_0);
	double t_2 = cos(t_0);
	double tmp;
	if (x_45_scale_m <= 8.2e+149) {
		tmp = 0.25 * ((b_m * ((x_45_scale_m * x_45_scale_m) * (y_45_scale_m * sqrt((8.0 * ((pow(a, 4.0) * (sqrt((pow(t_2, 4.0) / pow(x_45_scale_m, 4.0))) + (pow(t_2, 2.0) / (x_45_scale_m * x_45_scale_m)))) / (x_45_scale_m * x_45_scale_m))))))) / (a * a));
	} else {
		tmp = 0.25 * ((b_m * (x_45_scale_m * ((y_45_scale_m * y_45_scale_m) * sqrt((8.0 * ((pow(a, 4.0) * (sqrt((pow(t_1, 4.0) / pow(y_45_scale_m, 4.0))) + (pow(t_1, 2.0) / (y_45_scale_m * y_45_scale_m)))) / (y_45_scale_m * y_45_scale_m))))))) / (a * a));
	}
	return tmp;
}

Java

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, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = 0.005555555555555556 * (angle * Math.PI);
	double t_1 = Math.sin(t_0);
	double t_2 = Math.cos(t_0);
	double tmp;
	if (x_45_scale_m <= 8.2e+149) {
		tmp = 0.25 * ((b_m * ((x_45_scale_m * x_45_scale_m) * (y_45_scale_m * Math.sqrt((8.0 * ((Math.pow(a, 4.0) * (Math.sqrt((Math.pow(t_2, 4.0) / Math.pow(x_45_scale_m, 4.0))) + (Math.pow(t_2, 2.0) / (x_45_scale_m * x_45_scale_m)))) / (x_45_scale_m * x_45_scale_m))))))) / (a * a));
	} else {
		tmp = 0.25 * ((b_m * (x_45_scale_m * ((y_45_scale_m * y_45_scale_m) * Math.sqrt((8.0 * ((Math.pow(a, 4.0) * (Math.sqrt((Math.pow(t_1, 4.0) / Math.pow(y_45_scale_m, 4.0))) + (Math.pow(t_1, 2.0) / (y_45_scale_m * y_45_scale_m)))) / (y_45_scale_m * y_45_scale_m))))))) / (a * a));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b_m, angle, x_45_scale_m, y_45_scale_m):
	t_0 = 0.005555555555555556 * (angle * math.pi)
	t_1 = math.sin(t_0)
	t_2 = math.cos(t_0)
	tmp = 0
	if x_45_scale_m <= 8.2e+149:
		tmp = 0.25 * ((b_m * ((x_45_scale_m * x_45_scale_m) * (y_45_scale_m * math.sqrt((8.0 * ((math.pow(a, 4.0) * (math.sqrt((math.pow(t_2, 4.0) / math.pow(x_45_scale_m, 4.0))) + (math.pow(t_2, 2.0) / (x_45_scale_m * x_45_scale_m)))) / (x_45_scale_m * x_45_scale_m))))))) / (a * a))
	else:
		tmp = 0.25 * ((b_m * (x_45_scale_m * ((y_45_scale_m * y_45_scale_m) * math.sqrt((8.0 * ((math.pow(a, 4.0) * (math.sqrt((math.pow(t_1, 4.0) / math.pow(y_45_scale_m, 4.0))) + (math.pow(t_1, 2.0) / (y_45_scale_m * y_45_scale_m)))) / (y_45_scale_m * y_45_scale_m))))))) / (a * a))
	return tmp

Julia

b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_1 = sin(t_0)
	t_2 = cos(t_0)
	tmp = 0.0
	if (x_45_scale_m <= 8.2e+149)
		tmp = Float64(0.25 * Float64(Float64(b_m * Float64(Float64(x_45_scale_m * x_45_scale_m) * Float64(y_45_scale_m * sqrt(Float64(8.0 * Float64(Float64((a ^ 4.0) * Float64(sqrt(Float64((t_2 ^ 4.0) / (x_45_scale_m ^ 4.0))) + Float64((t_2 ^ 2.0) / Float64(x_45_scale_m * x_45_scale_m)))) / Float64(x_45_scale_m * x_45_scale_m))))))) / Float64(a * a)));
	else
		tmp = Float64(0.25 * Float64(Float64(b_m * Float64(x_45_scale_m * Float64(Float64(y_45_scale_m * y_45_scale_m) * sqrt(Float64(8.0 * Float64(Float64((a ^ 4.0) * Float64(sqrt(Float64((t_1 ^ 4.0) / (y_45_scale_m ^ 4.0))) + Float64((t_1 ^ 2.0) / Float64(y_45_scale_m * y_45_scale_m)))) / Float64(y_45_scale_m * y_45_scale_m))))))) / Float64(a * a)));
	end
	return tmp
end

MATLAB

b_m = abs(b);
x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = 0.005555555555555556 * (angle * pi);
	t_1 = sin(t_0);
	t_2 = cos(t_0);
	tmp = 0.0;
	if (x_45_scale_m <= 8.2e+149)
		tmp = 0.25 * ((b_m * ((x_45_scale_m * x_45_scale_m) * (y_45_scale_m * sqrt((8.0 * (((a ^ 4.0) * (sqrt(((t_2 ^ 4.0) / (x_45_scale_m ^ 4.0))) + ((t_2 ^ 2.0) / (x_45_scale_m * x_45_scale_m)))) / (x_45_scale_m * x_45_scale_m))))))) / (a * a));
	else
		tmp = 0.25 * ((b_m * (x_45_scale_m * ((y_45_scale_m * y_45_scale_m) * sqrt((8.0 * (((a ^ 4.0) * (sqrt(((t_1 ^ 4.0) / (y_45_scale_m ^ 4.0))) + ((t_1 ^ 2.0) / (y_45_scale_m * y_45_scale_m)))) / (y_45_scale_m * y_45_scale_m))))))) / (a * a));
	end
	tmp_2 = tmp;
end

Wolfram

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_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Cos[t$95$0], $MachinePrecision]}, If[LessEqual[x$45$scale$95$m, 8.2e+149], N[(0.25 * N[(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[Power[a, 4.0], $MachinePrecision] * N[(N[Sqrt[N[(N[Power[t$95$2, 4.0], $MachinePrecision] / N[Power[x$45$scale$95$m, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[Power[t$95$2, 2.0], $MachinePrecision] / N[(x$45$scale$95$m * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale$95$m * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(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[Power[a, 4.0], $MachinePrecision] * N[(N[Sqrt[N[(N[Power[t$95$1, 4.0], $MachinePrecision] / N[Power[y$45$scale$95$m, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[Power[t$95$1, 2.0], $MachinePrecision] / N[(y$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 8.1999999999999992e149

   1. Initial program 2.9%

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

   2. Taylor expanded in b around -inf

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

   4. Applied rewrites 0.1%

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

   5. Taylor expanded in angle around 0

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

      1. lower-+.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 0.2

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

   7. Applied rewrites 0.2%

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

   8. Taylor expanded in y-scale around -inf

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

   10. Applied rewrites 4.7%

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

   if 8.1999999999999992e149 < x-scale

   1. Initial program 2.9%

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

   2. Taylor expanded in b around -inf

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

   4. Applied rewrites 0.1%

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

   5. Taylor expanded in angle around 0

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

      1. lower-+.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 0.2

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

   7. Applied rewrites 0.2%

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

   8. Taylor expanded in x-scale around -inf

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

   10. Applied rewrites 2.5%

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

Alternative 6: 4.6% accurate, 6.3× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := \left(a \cdot a\right) \cdot \left(b\_m \cdot b\_m\right)\\ \mathbf{if}\;x-scale\_m \leq 2.8 \cdot 10^{+129}:\\ \;\;\;\;0.25 \cdot \frac{\left(x-scale\_m \cdot x-scale\_m\right) \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left({b\_m}^{4} \cdot \left(\sqrt{{a}^{4}} + a \cdot a\right)\right)}{x-scale\_m \cdot x-scale\_m}}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{\left(x-scale\_m \cdot x-scale\_m\right) \cdot \left(\left(y-scale\_m \cdot y-scale\_m\right) \cdot \frac{\left(b\_m \cdot b\_m\right) \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\sqrt{\frac{{a}^{4}}{{y-scale\_m}^{4}}} + \frac{a \cdot a}{y-scale\_m \cdot y-scale\_m}\right)}{y-scale\_m \cdot y-scale\_m}}}{x-scale\_m}\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b_m angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (* (* a a) (* b_m b_m))))
   (if (<= x-scale_m 2.8e+129)
     (*
      0.25
      (/
       (*
        (* x-scale_m x-scale_m)
        (sqrt
         (*
          8.0
          (/
           (* (pow a 4.0) (* (pow b_m 4.0) (+ (sqrt (pow a 4.0)) (* a a))))
           (* x-scale_m x-scale_m)))))
       t_0))
     (*
      0.25
      (/
       (*
        (* x-scale_m x-scale_m)
        (*
         (* y-scale_m y-scale_m)
         (/
          (*
           (* b_m b_m)
           (sqrt
            (*
             8.0
             (/
              (*
               (pow a 4.0)
               (+
                (sqrt (/ (pow a 4.0) (pow y-scale_m 4.0)))
                (/ (* a a) (* y-scale_m y-scale_m))))
              (* y-scale_m y-scale_m)))))
          x-scale_m)))
       t_0)))))

C

b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = (a * a) * (b_m * b_m);
	double tmp;
	if (x_45_scale_m <= 2.8e+129) {
		tmp = 0.25 * (((x_45_scale_m * x_45_scale_m) * sqrt((8.0 * ((pow(a, 4.0) * (pow(b_m, 4.0) * (sqrt(pow(a, 4.0)) + (a * a)))) / (x_45_scale_m * x_45_scale_m))))) / t_0);
	} else {
		tmp = 0.25 * (((x_45_scale_m * x_45_scale_m) * ((y_45_scale_m * y_45_scale_m) * (((b_m * b_m) * sqrt((8.0 * ((pow(a, 4.0) * (sqrt((pow(a, 4.0) / pow(y_45_scale_m, 4.0))) + ((a * a) / (y_45_scale_m * y_45_scale_m)))) / (y_45_scale_m * y_45_scale_m))))) / x_45_scale_m))) / t_0);
	}
	return tmp;
}

Fortran

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, b_m, angle, x_45scale_m, y_45scale_m)
use fmin_fmax_functions
    real(8), intent (in) :: a
    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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (a * a) * (b_m * b_m)
    if (x_45scale_m <= 2.8d+129) then
        tmp = 0.25d0 * (((x_45scale_m * x_45scale_m) * sqrt((8.0d0 * (((a ** 4.0d0) * ((b_m ** 4.0d0) * (sqrt((a ** 4.0d0)) + (a * a)))) / (x_45scale_m * x_45scale_m))))) / t_0)
    else
        tmp = 0.25d0 * (((x_45scale_m * x_45scale_m) * ((y_45scale_m * y_45scale_m) * (((b_m * b_m) * sqrt((8.0d0 * (((a ** 4.0d0) * (sqrt(((a ** 4.0d0) / (y_45scale_m ** 4.0d0))) + ((a * a) / (y_45scale_m * y_45scale_m)))) / (y_45scale_m * y_45scale_m))))) / x_45scale_m))) / t_0)
    end if
    code = tmp
end function

Java

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, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = (a * a) * (b_m * b_m);
	double tmp;
	if (x_45_scale_m <= 2.8e+129) {
		tmp = 0.25 * (((x_45_scale_m * x_45_scale_m) * Math.sqrt((8.0 * ((Math.pow(a, 4.0) * (Math.pow(b_m, 4.0) * (Math.sqrt(Math.pow(a, 4.0)) + (a * a)))) / (x_45_scale_m * x_45_scale_m))))) / t_0);
	} else {
		tmp = 0.25 * (((x_45_scale_m * x_45_scale_m) * ((y_45_scale_m * y_45_scale_m) * (((b_m * b_m) * Math.sqrt((8.0 * ((Math.pow(a, 4.0) * (Math.sqrt((Math.pow(a, 4.0) / Math.pow(y_45_scale_m, 4.0))) + ((a * a) / (y_45_scale_m * y_45_scale_m)))) / (y_45_scale_m * y_45_scale_m))))) / x_45_scale_m))) / t_0);
	}
	return tmp;
}

Python

b_m = math.fabs(b)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b_m, angle, x_45_scale_m, y_45_scale_m):
	t_0 = (a * a) * (b_m * b_m)
	tmp = 0
	if x_45_scale_m <= 2.8e+129:
		tmp = 0.25 * (((x_45_scale_m * x_45_scale_m) * math.sqrt((8.0 * ((math.pow(a, 4.0) * (math.pow(b_m, 4.0) * (math.sqrt(math.pow(a, 4.0)) + (a * a)))) / (x_45_scale_m * x_45_scale_m))))) / t_0)
	else:
		tmp = 0.25 * (((x_45_scale_m * x_45_scale_m) * ((y_45_scale_m * y_45_scale_m) * (((b_m * b_m) * math.sqrt((8.0 * ((math.pow(a, 4.0) * (math.sqrt((math.pow(a, 4.0) / math.pow(y_45_scale_m, 4.0))) + ((a * a) / (y_45_scale_m * y_45_scale_m)))) / (y_45_scale_m * y_45_scale_m))))) / x_45_scale_m))) / t_0)
	return tmp

Julia

b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(Float64(a * a) * Float64(b_m * b_m))
	tmp = 0.0
	if (x_45_scale_m <= 2.8e+129)
		tmp = Float64(0.25 * Float64(Float64(Float64(x_45_scale_m * x_45_scale_m) * sqrt(Float64(8.0 * Float64(Float64((a ^ 4.0) * Float64((b_m ^ 4.0) * Float64(sqrt((a ^ 4.0)) + Float64(a * a)))) / Float64(x_45_scale_m * x_45_scale_m))))) / t_0));
	else
		tmp = Float64(0.25 * Float64(Float64(Float64(x_45_scale_m * x_45_scale_m) * Float64(Float64(y_45_scale_m * y_45_scale_m) * Float64(Float64(Float64(b_m * b_m) * sqrt(Float64(8.0 * Float64(Float64((a ^ 4.0) * Float64(sqrt(Float64((a ^ 4.0) / (y_45_scale_m ^ 4.0))) + Float64(Float64(a * a) / Float64(y_45_scale_m * y_45_scale_m)))) / Float64(y_45_scale_m * y_45_scale_m))))) / x_45_scale_m))) / t_0));
	end
	return tmp
end

MATLAB

b_m = abs(b);
x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = (a * a) * (b_m * b_m);
	tmp = 0.0;
	if (x_45_scale_m <= 2.8e+129)
		tmp = 0.25 * (((x_45_scale_m * x_45_scale_m) * sqrt((8.0 * (((a ^ 4.0) * ((b_m ^ 4.0) * (sqrt((a ^ 4.0)) + (a * a)))) / (x_45_scale_m * x_45_scale_m))))) / t_0);
	else
		tmp = 0.25 * (((x_45_scale_m * x_45_scale_m) * ((y_45_scale_m * y_45_scale_m) * (((b_m * b_m) * sqrt((8.0 * (((a ^ 4.0) * (sqrt(((a ^ 4.0) / (y_45_scale_m ^ 4.0))) + ((a * a) / (y_45_scale_m * y_45_scale_m)))) / (y_45_scale_m * y_45_scale_m))))) / x_45_scale_m))) / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

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_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(N[(a * a), $MachinePrecision] * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$45$scale$95$m, 2.8e+129], N[(0.25 * N[(N[(N[(x$45$scale$95$m * x$45$scale$95$m), $MachinePrecision] * N[Sqrt[N[(8.0 * N[(N[(N[Power[a, 4.0], $MachinePrecision] * N[(N[Power[b$95$m, 4.0], $MachinePrecision] * N[(N[Sqrt[N[Power[a, 4.0], $MachinePrecision]], $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale$95$m * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(N[(x$45$scale$95$m * x$45$scale$95$m), $MachinePrecision] * N[(N[(y$45$scale$95$m * y$45$scale$95$m), $MachinePrecision] * N[(N[(N[(b$95$m * b$95$m), $MachinePrecision] * N[Sqrt[N[(8.0 * N[(N[(N[Power[a, 4.0], $MachinePrecision] * N[(N[Sqrt[N[(N[Power[a, 4.0], $MachinePrecision] / N[Power[y$45$scale$95$m, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[(a * a), $MachinePrecision] / N[(y$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 2.79999999999999975e129

   1. Initial program 2.9%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 0.3%

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

   5. Taylor expanded in b around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

   7. Applied rewrites 0.2%

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

   8. Taylor expanded in y-scale around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

   10. Applied rewrites 1.3%

       \[\leadsto 0.25 \cdot \frac{\left(x-scale \cdot x-scale\right) \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left({b}^{4} \cdot \left(\sqrt{{a}^{4}} + a \cdot a\right)\right)}{x-scale \cdot x-scale}}}{\left(a \cdot \color{blue}{a}\right) \cdot \left(b \cdot b\right)} \]

   if 2.79999999999999975e129 < x-scale

   1. Initial program 2.9%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 0.3%

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

   5. Taylor expanded in b around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

   7. Applied rewrites 0.2%

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

   8. Taylor expanded in x-scale around 0

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 0.9%

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

Alternative 7: 2.5% accurate, 4.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ 0.25 \cdot \frac{b\_m \cdot \left(x-scale\_m \cdot \left(\left(y-scale\_m \cdot y-scale\_m\right) \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\sqrt{\frac{{t\_0}^{4}}{{y-scale\_m}^{4}}} + \frac{{t\_0}^{2}}{y-scale\_m \cdot y-scale\_m}\right)}{y-scale\_m \cdot y-scale\_m}}\right)\right)}{a \cdot a} \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b_m angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (sin (* 0.005555555555555556 (* angle PI)))))
   (*
    0.25
    (/
     (*
      b_m
      (*
       x-scale_m
       (*
        (* y-scale_m y-scale_m)
        (sqrt
         (*
          8.0
          (/
           (*
            (pow a 4.0)
            (+
             (sqrt (/ (pow t_0 4.0) (pow y-scale_m 4.0)))
             (/ (pow t_0 2.0) (* y-scale_m y-scale_m))))
           (* y-scale_m y-scale_m)))))))
     (* a a)))))

C

b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = sin((0.005555555555555556 * (angle * ((double) M_PI))));
	return 0.25 * ((b_m * (x_45_scale_m * ((y_45_scale_m * y_45_scale_m) * sqrt((8.0 * ((pow(a, 4.0) * (sqrt((pow(t_0, 4.0) / pow(y_45_scale_m, 4.0))) + (pow(t_0, 2.0) / (y_45_scale_m * y_45_scale_m)))) / (y_45_scale_m * y_45_scale_m))))))) / (a * a));
}

Java

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, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = Math.sin((0.005555555555555556 * (angle * Math.PI)));
	return 0.25 * ((b_m * (x_45_scale_m * ((y_45_scale_m * y_45_scale_m) * Math.sqrt((8.0 * ((Math.pow(a, 4.0) * (Math.sqrt((Math.pow(t_0, 4.0) / Math.pow(y_45_scale_m, 4.0))) + (Math.pow(t_0, 2.0) / (y_45_scale_m * y_45_scale_m)))) / (y_45_scale_m * y_45_scale_m))))))) / (a * a));
}

Python

b_m = math.fabs(b)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b_m, angle, x_45_scale_m, y_45_scale_m):
	t_0 = math.sin((0.005555555555555556 * (angle * math.pi)))
	return 0.25 * ((b_m * (x_45_scale_m * ((y_45_scale_m * y_45_scale_m) * math.sqrt((8.0 * ((math.pow(a, 4.0) * (math.sqrt((math.pow(t_0, 4.0) / math.pow(y_45_scale_m, 4.0))) + (math.pow(t_0, 2.0) / (y_45_scale_m * y_45_scale_m)))) / (y_45_scale_m * y_45_scale_m))))))) / (a * a))

Julia

b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = sin(Float64(0.005555555555555556 * Float64(angle * pi)))
	return Float64(0.25 * Float64(Float64(b_m * Float64(x_45_scale_m * Float64(Float64(y_45_scale_m * y_45_scale_m) * sqrt(Float64(8.0 * Float64(Float64((a ^ 4.0) * Float64(sqrt(Float64((t_0 ^ 4.0) / (y_45_scale_m ^ 4.0))) + Float64((t_0 ^ 2.0) / Float64(y_45_scale_m * y_45_scale_m)))) / Float64(y_45_scale_m * y_45_scale_m))))))) / Float64(a * a)))
end

MATLAB

b_m = abs(b);
x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp = code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = sin((0.005555555555555556 * (angle * pi)));
	tmp = 0.25 * ((b_m * (x_45_scale_m * ((y_45_scale_m * y_45_scale_m) * sqrt((8.0 * (((a ^ 4.0) * (sqrt(((t_0 ^ 4.0) / (y_45_scale_m ^ 4.0))) + ((t_0 ^ 2.0) / (y_45_scale_m * y_45_scale_m)))) / (y_45_scale_m * y_45_scale_m))))))) / (a * a));
end

Wolfram

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_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(0.25 * N[(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[Power[a, 4.0], $MachinePrecision] * N[(N[Sqrt[N[(N[Power[t$95$0, 4.0], $MachinePrecision] / N[Power[y$45$scale$95$m, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[Power[t$95$0, 2.0], $MachinePrecision] / N[(y$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 2.9%

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

2. Taylor expanded in b around -inf

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

4. Applied rewrites 0.1%

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

5. Taylor expanded in angle around 0

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

   1. lower-+.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-pow.f64 0.2

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

7. Applied rewrites 0.2%

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

8. Taylor expanded in x-scale around -inf

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

10. Applied rewrites 2.5%

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

Alternative 8: 1.8% accurate, 7.8× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := \left(a \cdot a\right) \cdot \left(b\_m \cdot b\_m\right)\\ \mathbf{if}\;x-scale\_m \leq 5 \cdot 10^{+129}:\\ \;\;\;\;0.25 \cdot \frac{\left(x-scale\_m \cdot x-scale\_m\right) \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left({b\_m}^{4} \cdot \left(\sqrt{{a}^{4}} + a \cdot a\right)\right)}{x-scale\_m \cdot x-scale\_m}}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{\left(y-scale\_m \cdot y-scale\_m\right) \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left({b\_m}^{4} \cdot \left(\sqrt{{b\_m}^{4}} + b\_m \cdot b\_m\right)\right)}{y-scale\_m \cdot y-scale\_m}}}{t\_0}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b_m angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (* (* a a) (* b_m b_m))))
   (if (<= x-scale_m 5e+129)
     (*
      0.25
      (/
       (*
        (* x-scale_m x-scale_m)
        (sqrt
         (*
          8.0
          (/
           (* (pow a 4.0) (* (pow b_m 4.0) (+ (sqrt (pow a 4.0)) (* a a))))
           (* x-scale_m x-scale_m)))))
       t_0))
     (*
      0.25
      (/
       (*
        (* y-scale_m y-scale_m)
        (sqrt
         (*
          8.0
          (/
           (*
            (pow a 4.0)
            (* (pow b_m 4.0) (+ (sqrt (pow b_m 4.0)) (* b_m b_m))))
           (* y-scale_m y-scale_m)))))
       t_0)))))

C

b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = (a * a) * (b_m * b_m);
	double tmp;
	if (x_45_scale_m <= 5e+129) {
		tmp = 0.25 * (((x_45_scale_m * x_45_scale_m) * sqrt((8.0 * ((pow(a, 4.0) * (pow(b_m, 4.0) * (sqrt(pow(a, 4.0)) + (a * a)))) / (x_45_scale_m * x_45_scale_m))))) / t_0);
	} else {
		tmp = 0.25 * (((y_45_scale_m * y_45_scale_m) * sqrt((8.0 * ((pow(a, 4.0) * (pow(b_m, 4.0) * (sqrt(pow(b_m, 4.0)) + (b_m * b_m)))) / (y_45_scale_m * y_45_scale_m))))) / t_0);
	}
	return tmp;
}

Fortran

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, b_m, angle, x_45scale_m, y_45scale_m)
use fmin_fmax_functions
    real(8), intent (in) :: a
    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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (a * a) * (b_m * b_m)
    if (x_45scale_m <= 5d+129) then
        tmp = 0.25d0 * (((x_45scale_m * x_45scale_m) * sqrt((8.0d0 * (((a ** 4.0d0) * ((b_m ** 4.0d0) * (sqrt((a ** 4.0d0)) + (a * a)))) / (x_45scale_m * x_45scale_m))))) / t_0)
    else
        tmp = 0.25d0 * (((y_45scale_m * y_45scale_m) * sqrt((8.0d0 * (((a ** 4.0d0) * ((b_m ** 4.0d0) * (sqrt((b_m ** 4.0d0)) + (b_m * b_m)))) / (y_45scale_m * y_45scale_m))))) / t_0)
    end if
    code = tmp
end function

Java

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, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = (a * a) * (b_m * b_m);
	double tmp;
	if (x_45_scale_m <= 5e+129) {
		tmp = 0.25 * (((x_45_scale_m * x_45_scale_m) * Math.sqrt((8.0 * ((Math.pow(a, 4.0) * (Math.pow(b_m, 4.0) * (Math.sqrt(Math.pow(a, 4.0)) + (a * a)))) / (x_45_scale_m * x_45_scale_m))))) / t_0);
	} else {
		tmp = 0.25 * (((y_45_scale_m * y_45_scale_m) * Math.sqrt((8.0 * ((Math.pow(a, 4.0) * (Math.pow(b_m, 4.0) * (Math.sqrt(Math.pow(b_m, 4.0)) + (b_m * b_m)))) / (y_45_scale_m * y_45_scale_m))))) / t_0);
	}
	return tmp;
}

Python

b_m = math.fabs(b)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b_m, angle, x_45_scale_m, y_45_scale_m):
	t_0 = (a * a) * (b_m * b_m)
	tmp = 0
	if x_45_scale_m <= 5e+129:
		tmp = 0.25 * (((x_45_scale_m * x_45_scale_m) * math.sqrt((8.0 * ((math.pow(a, 4.0) * (math.pow(b_m, 4.0) * (math.sqrt(math.pow(a, 4.0)) + (a * a)))) / (x_45_scale_m * x_45_scale_m))))) / t_0)
	else:
		tmp = 0.25 * (((y_45_scale_m * y_45_scale_m) * math.sqrt((8.0 * ((math.pow(a, 4.0) * (math.pow(b_m, 4.0) * (math.sqrt(math.pow(b_m, 4.0)) + (b_m * b_m)))) / (y_45_scale_m * y_45_scale_m))))) / t_0)
	return tmp

Julia

b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(Float64(a * a) * Float64(b_m * b_m))
	tmp = 0.0
	if (x_45_scale_m <= 5e+129)
		tmp = Float64(0.25 * Float64(Float64(Float64(x_45_scale_m * x_45_scale_m) * sqrt(Float64(8.0 * Float64(Float64((a ^ 4.0) * Float64((b_m ^ 4.0) * Float64(sqrt((a ^ 4.0)) + Float64(a * a)))) / Float64(x_45_scale_m * x_45_scale_m))))) / t_0));
	else
		tmp = Float64(0.25 * Float64(Float64(Float64(y_45_scale_m * y_45_scale_m) * sqrt(Float64(8.0 * Float64(Float64((a ^ 4.0) * Float64((b_m ^ 4.0) * Float64(sqrt((b_m ^ 4.0)) + Float64(b_m * b_m)))) / Float64(y_45_scale_m * y_45_scale_m))))) / t_0));
	end
	return tmp
end

MATLAB

b_m = abs(b);
x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = (a * a) * (b_m * b_m);
	tmp = 0.0;
	if (x_45_scale_m <= 5e+129)
		tmp = 0.25 * (((x_45_scale_m * x_45_scale_m) * sqrt((8.0 * (((a ^ 4.0) * ((b_m ^ 4.0) * (sqrt((a ^ 4.0)) + (a * a)))) / (x_45_scale_m * x_45_scale_m))))) / t_0);
	else
		tmp = 0.25 * (((y_45_scale_m * y_45_scale_m) * sqrt((8.0 * (((a ^ 4.0) * ((b_m ^ 4.0) * (sqrt((b_m ^ 4.0)) + (b_m * b_m)))) / (y_45_scale_m * y_45_scale_m))))) / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

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_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(N[(a * a), $MachinePrecision] * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$45$scale$95$m, 5e+129], N[(0.25 * N[(N[(N[(x$45$scale$95$m * x$45$scale$95$m), $MachinePrecision] * N[Sqrt[N[(8.0 * N[(N[(N[Power[a, 4.0], $MachinePrecision] * N[(N[Power[b$95$m, 4.0], $MachinePrecision] * N[(N[Sqrt[N[Power[a, 4.0], $MachinePrecision]], $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale$95$m * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(N[(y$45$scale$95$m * y$45$scale$95$m), $MachinePrecision] * N[Sqrt[N[(8.0 * N[(N[(N[Power[a, 4.0], $MachinePrecision] * N[(N[Power[b$95$m, 4.0], $MachinePrecision] * N[(N[Sqrt[N[Power[b$95$m, 4.0], $MachinePrecision]], $MachinePrecision] + N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 5.0000000000000003e129

   1. Initial program 2.9%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 0.3%

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

   5. Taylor expanded in b around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

   7. Applied rewrites 0.2%

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

   8. Taylor expanded in y-scale around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

   10. Applied rewrites 1.3%

       \[\leadsto 0.25 \cdot \frac{\left(x-scale \cdot x-scale\right) \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left({b}^{4} \cdot \left(\sqrt{{a}^{4}} + a \cdot a\right)\right)}{x-scale \cdot x-scale}}}{\left(a \cdot \color{blue}{a}\right) \cdot \left(b \cdot b\right)} \]

   if 5.0000000000000003e129 < x-scale

   1. Initial program 2.9%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 0.3%

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

   5. Taylor expanded in b around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

   7. Applied rewrites 0.2%

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

   8. Taylor expanded in x-scale around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

   10. Applied rewrites 0.9%

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

Alternative 9: 1.5% accurate, 4.8× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := \frac{a \cdot a}{y-scale\_m \cdot y-scale\_m}\\ t_1 := \frac{b\_m \cdot b\_m}{x-scale\_m \cdot x-scale\_m}\\ t_2 := \left(b\_m \cdot a\right) \cdot \left(b\_m \cdot \left(-a\right)\right)\\ t_3 := \frac{4 \cdot t\_2}{{\left(x-scale\_m \cdot y-scale\_m\right)}^{2}}\\ \frac{-\sqrt{\left(\left(2 \cdot t\_3\right) \cdot t\_2\right) \cdot \left(\sqrt{{\left(t\_1 - t\_0\right)}^{2}} + \left(t\_0 + t\_1\right)\right)}}{t\_3} \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b_m angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (/ (* a a) (* y-scale_m y-scale_m)))
        (t_1 (/ (* b_m b_m) (* x-scale_m x-scale_m)))
        (t_2 (* (* b_m a) (* b_m (- a))))
        (t_3 (/ (* 4.0 t_2) (pow (* x-scale_m y-scale_m) 2.0))))
   (/
    (-
     (sqrt
      (* (* (* 2.0 t_3) t_2) (+ (sqrt (pow (- t_1 t_0) 2.0)) (+ t_0 t_1)))))
    t_3)))

C

b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = (a * a) / (y_45_scale_m * y_45_scale_m);
	double t_1 = (b_m * b_m) / (x_45_scale_m * x_45_scale_m);
	double t_2 = (b_m * a) * (b_m * -a);
	double t_3 = (4.0 * t_2) / pow((x_45_scale_m * y_45_scale_m), 2.0);
	return -sqrt((((2.0 * t_3) * t_2) * (sqrt(pow((t_1 - t_0), 2.0)) + (t_0 + t_1)))) / t_3;
}

Fortran

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, b_m, angle, x_45scale_m, y_45scale_m)
use fmin_fmax_functions
    real(8), intent (in) :: a
    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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    t_0 = (a * a) / (y_45scale_m * y_45scale_m)
    t_1 = (b_m * b_m) / (x_45scale_m * x_45scale_m)
    t_2 = (b_m * a) * (b_m * -a)
    t_3 = (4.0d0 * t_2) / ((x_45scale_m * y_45scale_m) ** 2.0d0)
    code = -sqrt((((2.0d0 * t_3) * t_2) * (sqrt(((t_1 - t_0) ** 2.0d0)) + (t_0 + t_1)))) / t_3
end function

Java

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, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = (a * a) / (y_45_scale_m * y_45_scale_m);
	double t_1 = (b_m * b_m) / (x_45_scale_m * x_45_scale_m);
	double t_2 = (b_m * a) * (b_m * -a);
	double t_3 = (4.0 * t_2) / Math.pow((x_45_scale_m * y_45_scale_m), 2.0);
	return -Math.sqrt((((2.0 * t_3) * t_2) * (Math.sqrt(Math.pow((t_1 - t_0), 2.0)) + (t_0 + t_1)))) / t_3;
}

Python

b_m = math.fabs(b)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b_m, angle, x_45_scale_m, y_45_scale_m):
	t_0 = (a * a) / (y_45_scale_m * y_45_scale_m)
	t_1 = (b_m * b_m) / (x_45_scale_m * x_45_scale_m)
	t_2 = (b_m * a) * (b_m * -a)
	t_3 = (4.0 * t_2) / math.pow((x_45_scale_m * y_45_scale_m), 2.0)
	return -math.sqrt((((2.0 * t_3) * t_2) * (math.sqrt(math.pow((t_1 - t_0), 2.0)) + (t_0 + t_1)))) / t_3

Julia

b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(Float64(a * a) / Float64(y_45_scale_m * y_45_scale_m))
	t_1 = Float64(Float64(b_m * b_m) / Float64(x_45_scale_m * x_45_scale_m))
	t_2 = Float64(Float64(b_m * a) * Float64(b_m * Float64(-a)))
	t_3 = Float64(Float64(4.0 * t_2) / (Float64(x_45_scale_m * y_45_scale_m) ^ 2.0))
	return Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * t_3) * t_2) * Float64(sqrt((Float64(t_1 - t_0) ^ 2.0)) + Float64(t_0 + t_1))))) / t_3)
end

MATLAB

b_m = abs(b);
x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp = code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = (a * a) / (y_45_scale_m * y_45_scale_m);
	t_1 = (b_m * b_m) / (x_45_scale_m * x_45_scale_m);
	t_2 = (b_m * a) * (b_m * -a);
	t_3 = (4.0 * t_2) / ((x_45_scale_m * y_45_scale_m) ^ 2.0);
	tmp = -sqrt((((2.0 * t_3) * t_2) * (sqrt(((t_1 - t_0) ^ 2.0)) + (t_0 + t_1)))) / t_3;
end

Wolfram

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_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(N[(a * a), $MachinePrecision] / N[(y$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b$95$m * b$95$m), $MachinePrecision] / N[(x$45$scale$95$m * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(b$95$m * a), $MachinePrecision] * N[(b$95$m * (-a)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(4.0 * t$95$2), $MachinePrecision] / N[Power[N[(x$45$scale$95$m * y$45$scale$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, N[((-N[Sqrt[N[(N[(N[(2.0 * t$95$3), $MachinePrecision] * t$95$2), $MachinePrecision] * N[(N[Sqrt[N[Power[N[(t$95$1 - t$95$0), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] + N[(t$95$0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$3), $MachinePrecision]]]]]

Derivation

1. Initial program 2.9%

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

2. Taylor expanded in angle around 0

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

4. Applied rewrites 4.6%

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

Alternative 10: 1.3% accurate, 8.1× speedup

Math

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

FPCore

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

C

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

Fortran

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, b_m, angle, x_45scale_m, y_45scale_m)
use fmin_fmax_functions
    real(8), intent (in) :: a
    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 * (((y_45scale_m * y_45scale_m) * sqrt((8.0d0 * (((a ** 4.0d0) * ((b_m ** 4.0d0) * (sqrt((b_m ** 4.0d0)) + (b_m * b_m)))) / (y_45scale_m * y_45scale_m))))) / ((a * a) * (b_m * b_m)))
end function

Java

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, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	return 0.25 * (((y_45_scale_m * y_45_scale_m) * Math.sqrt((8.0 * ((Math.pow(a, 4.0) * (Math.pow(b_m, 4.0) * (Math.sqrt(Math.pow(b_m, 4.0)) + (b_m * b_m)))) / (y_45_scale_m * y_45_scale_m))))) / ((a * a) * (b_m * b_m)));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := N[(0.25 * N[(N[(N[(y$45$scale$95$m * y$45$scale$95$m), $MachinePrecision] * N[Sqrt[N[(8.0 * N[(N[(N[Power[a, 4.0], $MachinePrecision] * N[(N[Power[b$95$m, 4.0], $MachinePrecision] * N[(N[Sqrt[N[Power[b$95$m, 4.0], $MachinePrecision]], $MachinePrecision] + N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(a * a), $MachinePrecision] * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 2.9%

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

2. Taylor expanded in angle around 0

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

4. Applied rewrites 0.3%

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

5. Taylor expanded in b around 0

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

   1. lower-*.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-*.f64 N/A

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

7. Applied rewrites 0.2%

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

8. Taylor expanded in x-scale around 0

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

   1. lower-*.f64 N/A

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

   2. pow2 N/A

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

   3. lift-*.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 N/A

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

10. Applied rewrites 0.9%

    \[\leadsto 0.25 \cdot \frac{\left(y-scale \cdot y-scale\right) \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left({b}^{4} \cdot \left(\sqrt{{b}^{4}} + b \cdot b\right)\right)}{y-scale \cdot y-scale}}}{\color{blue}{\left(a \cdot a\right)} \cdot \left(b \cdot b\right)} \]
11. Add Preprocessing

Alternative 11: 0.9% accurate, 8.7× speedup

Math

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

FPCore

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

C

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

Fortran

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, b_m, angle, x_45scale_m, y_45scale_m)
use fmin_fmax_functions
    real(8), intent (in) :: a
    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 * ((b_m * ((x_45scale_m * x_45scale_m) * ((y_45scale_m * y_45scale_m) * sqrt((8.0d0 * (((a ** 4.0d0) * (sqrt((x_45scale_m ** (-4.0d0))) + (1.0d0 / (x_45scale_m * x_45scale_m)))) / ((x_45scale_m * x_45scale_m) * (y_45scale_m * y_45scale_m)))))))) / (a * a))
end function

Java

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, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	return 0.25 * ((b_m * ((x_45_scale_m * x_45_scale_m) * ((y_45_scale_m * y_45_scale_m) * Math.sqrt((8.0 * ((Math.pow(a, 4.0) * (Math.sqrt(Math.pow(x_45_scale_m, -4.0)) + (1.0 / (x_45_scale_m * x_45_scale_m)))) / ((x_45_scale_m * x_45_scale_m) * (y_45_scale_m * y_45_scale_m)))))))) / (a * a));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := N[(0.25 * N[(N[(b$95$m * N[(N[(x$45$scale$95$m * x$45$scale$95$m), $MachinePrecision] * N[(N[(y$45$scale$95$m * y$45$scale$95$m), $MachinePrecision] * N[Sqrt[N[(8.0 * N[(N[(N[Power[a, 4.0], $MachinePrecision] * N[(N[Sqrt[N[Power[x$45$scale$95$m, -4.0], $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[(x$45$scale$95$m * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x$45$scale$95$m * x$45$scale$95$m), $MachinePrecision] * N[(y$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 2.9%

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

2. Taylor expanded in angle around 0

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

4. Applied rewrites 0.3%

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

5. Taylor expanded in b around 0

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

   1. lower-*.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-*.f64 N/A

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

7. Applied rewrites 0.2%

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

8. Taylor expanded in b around inf

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

10. Applied rewrites 1.3%

    \[\leadsto 0.25 \cdot \frac{b \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \left(\left(y-scale \cdot y-scale\right) \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\sqrt{{x-scale}^{-4}} + \frac{1}{x-scale \cdot x-scale}\right)}{\left(x-scale \cdot x-scale\right) \cdot \left(y-scale \cdot y-scale\right)}}\right)\right)}{\color{blue}{a \cdot a}} \]
11. Add Preprocessing

Alternative 12: 0.8% accurate, 8.7× speedup

Math

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

FPCore

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

C

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

Fortran

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, b_m, angle, x_45scale_m, y_45scale_m)
use fmin_fmax_functions
    real(8), intent (in) :: a
    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 * ((x_45scale_m * x_45scale_m) * ((y_45scale_m * y_45scale_m) * sqrt((8.0d0 * (((b_m ** 4.0d0) * (sqrt((y_45scale_m ** (-4.0d0))) + (1.0d0 / (y_45scale_m * y_45scale_m)))) / ((x_45scale_m * x_45scale_m) * (y_45scale_m * y_45scale_m)))))))) / (b_m * b_m))
end function

Java

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, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	return 0.25 * ((a * ((x_45_scale_m * x_45_scale_m) * ((y_45_scale_m * y_45_scale_m) * Math.sqrt((8.0 * ((Math.pow(b_m, 4.0) * (Math.sqrt(Math.pow(y_45_scale_m, -4.0)) + (1.0 / (y_45_scale_m * y_45_scale_m)))) / ((x_45_scale_m * x_45_scale_m) * (y_45_scale_m * y_45_scale_m)))))))) / (b_m * b_m));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := N[(0.25 * N[(N[(a * N[(N[(x$45$scale$95$m * x$45$scale$95$m), $MachinePrecision] * N[(N[(y$45$scale$95$m * y$45$scale$95$m), $MachinePrecision] * N[Sqrt[N[(8.0 * N[(N[(N[Power[b$95$m, 4.0], $MachinePrecision] * N[(N[Sqrt[N[Power[y$45$scale$95$m, -4.0], $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[(y$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x$45$scale$95$m * x$45$scale$95$m), $MachinePrecision] * N[(y$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 2.9%

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

2. Taylor expanded in angle around 0

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

4. Applied rewrites 0.3%

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

5. Taylor expanded in b around 0

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

   1. lower-*.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-*.f64 N/A

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

7. Applied rewrites 0.2%

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

8. Taylor expanded in a around inf

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

   1. lower-/.f64 N/A

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

10. Applied rewrites 0.8%

    \[\leadsto 0.25 \cdot \frac{a \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \left(\left(y-scale \cdot y-scale\right) \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\sqrt{{y-scale}^{-4}} + \frac{1}{y-scale \cdot y-scale}\right)}{\left(x-scale \cdot x-scale\right) \cdot \left(y-scale \cdot y-scale\right)}}\right)\right)}{\color{blue}{b \cdot b}} \]
11. Add Preprocessing

Alternative 13: 0.2% accurate, 11.3× speedup

Math

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

FPCore

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

C

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

Fortran

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, b_m, angle, x_45scale_m, y_45scale_m)
use fmin_fmax_functions
    real(8), intent (in) :: a
    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) * ((b_m * ((x_45scale_m * x_45scale_m) * ((y_45scale_m * y_45scale_m) * sqrt((8.0d0 * (((a ** 4.0d0) * (2.0d0 / (x_45scale_m * x_45scale_m))) / ((x_45scale_m * x_45scale_m) * (y_45scale_m * y_45scale_m)))))))) / (a * a))
end function

Java

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, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	return -0.25 * ((b_m * ((x_45_scale_m * x_45_scale_m) * ((y_45_scale_m * y_45_scale_m) * Math.sqrt((8.0 * ((Math.pow(a, 4.0) * (2.0 / (x_45_scale_m * x_45_scale_m))) / ((x_45_scale_m * x_45_scale_m) * (y_45_scale_m * y_45_scale_m)))))))) / (a * a));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := N[(-0.25 * N[(N[(b$95$m * N[(N[(x$45$scale$95$m * x$45$scale$95$m), $MachinePrecision] * N[(N[(y$45$scale$95$m * y$45$scale$95$m), $MachinePrecision] * N[Sqrt[N[(8.0 * N[(N[(N[Power[a, 4.0], $MachinePrecision] * N[(2.0 / N[(x$45$scale$95$m * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x$45$scale$95$m * x$45$scale$95$m), $MachinePrecision] * N[(y$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 2.9%

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

2. Taylor expanded in b around -inf

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

4. Applied rewrites 0.1%

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

5. Taylor expanded in angle around 0

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

   1. lower-+.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-pow.f64 0.2

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

7. Applied rewrites 0.2%

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

8. Taylor expanded in x-scale around 0

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

   1. lower-/.f64 N/A

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

   2. pow2 N/A

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

   3. lift-*.f64 0.2

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

10. Applied rewrites 0.2%

    \[\leadsto -0.25 \cdot \frac{b \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \left(\left(y-scale \cdot y-scale\right) \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \frac{2}{x-scale \cdot x-scale}}{\left(x-scale \cdot x-scale\right) \cdot \left(y-scale \cdot y-scale\right)}}\right)\right)}{a \cdot a} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025139 
(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))))