b from scale-rotated-ellipse

Percentage Accurate: 0.1% → 24.0%

Time: 37.0s

Alternatives: 7

Speedup: 8.1×

• could not determine a ground truth

  1. a = 9.352828484200807e+284
  2. b = -2.815985373355598e-283
  3. angle = -1.4980788717638468e-295
  4. x-scale = 6.957013205961828e+304
  5. y-scale = -3.0236101855690335e-164

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: 0.1% 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: 24.0% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\cos \left(-0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right), 0.5, 0.5\right)\\ \frac{\frac{0.25 \cdot \left(x-scale \cdot \left(x-scale \cdot \sqrt{\frac{8}{x-scale} \cdot \frac{\left(\left(a \cdot a\right) \cdot t\_0 - \sqrt{{t\_0}^{2} \cdot {a}^{4}}\right) \cdot {a}^{4}}{x-scale}}\right)\right)}{a}}{a} \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (fma (cos (* -0.011111111111111112 (* PI angle))) 0.5 0.5)))
   (/
    (/
     (*
      0.25
      (*
       x-scale
       (*
        x-scale
        (sqrt
         (*
          (/ 8.0 x-scale)
          (/
           (*
            (- (* (* a a) t_0) (sqrt (* (pow t_0 2.0) (pow a 4.0))))
            (pow a 4.0))
           x-scale))))))
     a)
    a)))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = fma(cos((-0.011111111111111112 * (((double) M_PI) * angle))), 0.5, 0.5);
	return ((0.25 * (x_45_scale * (x_45_scale * sqrt(((8.0 / x_45_scale) * (((((a * a) * t_0) - sqrt((pow(t_0, 2.0) * pow(a, 4.0)))) * pow(a, 4.0)) / x_45_scale)))))) / a) / a;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = fma(cos(Float64(-0.011111111111111112 * Float64(pi * angle))), 0.5, 0.5)
	return Float64(Float64(Float64(0.25 * Float64(x_45_scale * Float64(x_45_scale * sqrt(Float64(Float64(8.0 / x_45_scale) * Float64(Float64(Float64(Float64(Float64(a * a) * t_0) - sqrt(Float64((t_0 ^ 2.0) * (a ^ 4.0)))) * (a ^ 4.0)) / x_45_scale)))))) / a) / a)
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[Cos[N[(-0.011111111111111112 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]}, N[(N[(N[(0.25 * N[(x$45$scale * N[(x$45$scale * N[Sqrt[N[(N[(8.0 / x$45$scale), $MachinePrecision] * N[(N[(N[(N[(N[(a * a), $MachinePrecision] * t$95$0), $MachinePrecision] - N[Sqrt[N[(N[Power[t$95$0, 2.0], $MachinePrecision] * N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Initial program 0.1%

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

2. Taylor expanded in b around 0

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

4. Applied rewrites 0.6%

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

6. Applied rewrites 4.8%

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

7. Taylor expanded in y-scale around 0

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

9. Applied rewrites 13.8%

   \[\leadsto \frac{\frac{0.25 \cdot \left(x-scale \cdot \left(x-scale \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left({a}^{2} \cdot \left(0.5 + 0.5 \cdot \cos \left(-0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right) - \sqrt{{a}^{4} \cdot {\left(0.5 + 0.5 \cdot \cos \left(-0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}\right)}{{x-scale}^{2}}}\right)\right)}{a}}{a} \]
10. Step-by-step derivation

    1. lift-*.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. associate-*r/ N/A

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

    4. lift-pow.f64 N/A

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

    5. pow2 N/A

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

    6. times-frac N/A

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

11. Applied rewrites 24.0%

    \[\leadsto \frac{\frac{0.25 \cdot \left(x-scale \cdot \left(x-scale \cdot \sqrt{\frac{8}{x-scale} \cdot \frac{\left(\left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(-0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right), 0.5, 0.5\right) - \sqrt{{\left(\mathsf{fma}\left(\cos \left(-0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right), 0.5, 0.5\right)\right)}^{2} \cdot {a}^{4}}\right) \cdot {a}^{4}}{x-scale}}\right)\right)}{a}}{a} \]
12. Add Preprocessing

Alternative 2: 15.5% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{{y-scale}^{2}}\\ \mathbf{if}\;x-scale \leq 2 \cdot 10^{-166}:\\ \;\;\;\;\frac{\frac{0.25 \cdot \left(x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot \left(y-scale \cdot \frac{\sqrt{8 \cdot \frac{{a}^{4} \cdot \left({a}^{2} \cdot t\_0 - \sqrt{{a}^{4} \cdot {t\_0}^{2}}\right)}{{y-scale}^{2}}}}{x-scale}\right)\right)\right)\right)}{a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.25 \cdot \left(x-scale \cdot \left(x-scale \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left({a}^{2} - \sqrt{{a}^{4}}\right)}{{x-scale}^{2}}}\right)\right)}{a}}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ 1.0 (pow y-scale 2.0))))
   (if (<= x-scale 2e-166)
     (/
      (/
       (*
        0.25
        (*
         x-scale
         (*
          x-scale
          (*
           y-scale
           (*
            y-scale
            (/
             (sqrt
              (*
               8.0
               (/
                (*
                 (pow a 4.0)
                 (- (* (pow a 2.0) t_0) (sqrt (* (pow a 4.0) (pow t_0 2.0)))))
                (pow y-scale 2.0))))
             x-scale))))))
       a)
      a)
     (/
      (/
       (*
        0.25
        (*
         x-scale
         (*
          x-scale
          (sqrt
           (*
            8.0
            (/
             (* (pow a 4.0) (- (pow a 2.0) (sqrt (pow a 4.0))))
             (pow x-scale 2.0)))))))
       a)
      a))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 1.0 / pow(y_45_scale, 2.0);
	double tmp;
	if (x_45_scale <= 2e-166) {
		tmp = ((0.25 * (x_45_scale * (x_45_scale * (y_45_scale * (y_45_scale * (sqrt((8.0 * ((pow(a, 4.0) * ((pow(a, 2.0) * t_0) - sqrt((pow(a, 4.0) * pow(t_0, 2.0))))) / pow(y_45_scale, 2.0)))) / x_45_scale)))))) / a) / a;
	} else {
		tmp = ((0.25 * (x_45_scale * (x_45_scale * sqrt((8.0 * ((pow(a, 4.0) * (pow(a, 2.0) - sqrt(pow(a, 4.0)))) / pow(x_45_scale, 2.0))))))) / a) / a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 / (y_45scale ** 2.0d0)
    if (x_45scale <= 2d-166) then
        tmp = ((0.25d0 * (x_45scale * (x_45scale * (y_45scale * (y_45scale * (sqrt((8.0d0 * (((a ** 4.0d0) * (((a ** 2.0d0) * t_0) - sqrt(((a ** 4.0d0) * (t_0 ** 2.0d0))))) / (y_45scale ** 2.0d0)))) / x_45scale)))))) / a) / a
    else
        tmp = ((0.25d0 * (x_45scale * (x_45scale * sqrt((8.0d0 * (((a ** 4.0d0) * ((a ** 2.0d0) - sqrt((a ** 4.0d0)))) / (x_45scale ** 2.0d0))))))) / a) / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 1.0 / Math.pow(y_45_scale, 2.0);
	double tmp;
	if (x_45_scale <= 2e-166) {
		tmp = ((0.25 * (x_45_scale * (x_45_scale * (y_45_scale * (y_45_scale * (Math.sqrt((8.0 * ((Math.pow(a, 4.0) * ((Math.pow(a, 2.0) * t_0) - Math.sqrt((Math.pow(a, 4.0) * Math.pow(t_0, 2.0))))) / Math.pow(y_45_scale, 2.0)))) / x_45_scale)))))) / a) / a;
	} else {
		tmp = ((0.25 * (x_45_scale * (x_45_scale * Math.sqrt((8.0 * ((Math.pow(a, 4.0) * (Math.pow(a, 2.0) - Math.sqrt(Math.pow(a, 4.0)))) / Math.pow(x_45_scale, 2.0))))))) / a) / a;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = 1.0 / math.pow(y_45_scale, 2.0)
	tmp = 0
	if x_45_scale <= 2e-166:
		tmp = ((0.25 * (x_45_scale * (x_45_scale * (y_45_scale * (y_45_scale * (math.sqrt((8.0 * ((math.pow(a, 4.0) * ((math.pow(a, 2.0) * t_0) - math.sqrt((math.pow(a, 4.0) * math.pow(t_0, 2.0))))) / math.pow(y_45_scale, 2.0)))) / x_45_scale)))))) / a) / a
	else:
		tmp = ((0.25 * (x_45_scale * (x_45_scale * math.sqrt((8.0 * ((math.pow(a, 4.0) * (math.pow(a, 2.0) - math.sqrt(math.pow(a, 4.0)))) / math.pow(x_45_scale, 2.0))))))) / a) / a
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(1.0 / (y_45_scale ^ 2.0))
	tmp = 0.0
	if (x_45_scale <= 2e-166)
		tmp = Float64(Float64(Float64(0.25 * Float64(x_45_scale * Float64(x_45_scale * Float64(y_45_scale * Float64(y_45_scale * Float64(sqrt(Float64(8.0 * Float64(Float64((a ^ 4.0) * Float64(Float64((a ^ 2.0) * t_0) - sqrt(Float64((a ^ 4.0) * (t_0 ^ 2.0))))) / (y_45_scale ^ 2.0)))) / x_45_scale)))))) / a) / a);
	else
		tmp = Float64(Float64(Float64(0.25 * Float64(x_45_scale * Float64(x_45_scale * sqrt(Float64(8.0 * Float64(Float64((a ^ 4.0) * Float64((a ^ 2.0) - sqrt((a ^ 4.0)))) / (x_45_scale ^ 2.0))))))) / a) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = 1.0 / (y_45_scale ^ 2.0);
	tmp = 0.0;
	if (x_45_scale <= 2e-166)
		tmp = ((0.25 * (x_45_scale * (x_45_scale * (y_45_scale * (y_45_scale * (sqrt((8.0 * (((a ^ 4.0) * (((a ^ 2.0) * t_0) - sqrt(((a ^ 4.0) * (t_0 ^ 2.0))))) / (y_45_scale ^ 2.0)))) / x_45_scale)))))) / a) / a;
	else
		tmp = ((0.25 * (x_45_scale * (x_45_scale * sqrt((8.0 * (((a ^ 4.0) * ((a ^ 2.0) - sqrt((a ^ 4.0)))) / (x_45_scale ^ 2.0))))))) / a) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(1.0 / N[Power[y$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$45$scale, 2e-166], N[(N[(N[(0.25 * N[(x$45$scale * N[(x$45$scale * N[(y$45$scale * N[(y$45$scale * N[(N[Sqrt[N[(8.0 * N[(N[(N[Power[a, 4.0], $MachinePrecision] * N[(N[(N[Power[a, 2.0], $MachinePrecision] * t$95$0), $MachinePrecision] - N[Sqrt[N[(N[Power[a, 4.0], $MachinePrecision] * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[y$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / a), $MachinePrecision], N[(N[(N[(0.25 * N[(x$45$scale * N[(x$45$scale * N[Sqrt[N[(8.0 * N[(N[(N[Power[a, 4.0], $MachinePrecision] * N[(N[Power[a, 2.0], $MachinePrecision] - N[Sqrt[N[Power[a, 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 2.00000000000000008e-166

   1. Initial program 0.1%

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

   2. Taylor expanded in b around 0

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

   4. Applied rewrites 0.6%

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

   6. Applied rewrites 4.8%

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

   7. Taylor expanded in x-scale around inf

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

   9. Applied rewrites 10.1%

      \[\leadsto \frac{\frac{0.25 \cdot \left(x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot \left(y-scale \cdot \frac{\sqrt{8 \cdot \frac{{a}^{4} \cdot \left({a}^{2} \cdot \mathsf{fma}\left(0.5, \frac{\cos \left(-0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}, 0.5 \cdot \frac{1}{{y-scale}^{2}}\right) - \sqrt{{a}^{4} \cdot {\left(\mathsf{fma}\left(0.5, \frac{\cos \left(-0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}, 0.5 \cdot \frac{1}{{y-scale}^{2}}\right)\right)}^{2}}\right)}{{y-scale}^{2}}}}{x-scale}\right)\right)\right)\right)}{a}}{a} \]

   10. Taylor expanded in angle around 0

       \[\leadsto \frac{\frac{\frac{1}{4} \cdot \left(x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot \left(y-scale \cdot \frac{\sqrt{8 \cdot \frac{{a}^{4} \cdot \left({a}^{2} \cdot \frac{1}{{y-scale}^{2}} - \sqrt{{a}^{4} \cdot {\left(\mathsf{fma}\left(\frac{1}{2}, \frac{\cos \left(\frac{-1}{90} \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}, \frac{1}{2} \cdot \frac{1}{{y-scale}^{2}}\right)\right)}^{2}}\right)}{{y-scale}^{2}}}}{x-scale}\right)\right)\right)\right)}{a}}{a} \]
   11. Step-by-step derivation

       1. lower-/.f64 N/A

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

       2. lower-pow.f64 9.9

          \[\leadsto \frac{\frac{0.25 \cdot \left(x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot \left(y-scale \cdot \frac{\sqrt{8 \cdot \frac{{a}^{4} \cdot \left({a}^{2} \cdot \frac{1}{{y-scale}^{2}} - \sqrt{{a}^{4} \cdot {\left(\mathsf{fma}\left(0.5, \frac{\cos \left(-0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}, 0.5 \cdot \frac{1}{{y-scale}^{2}}\right)\right)}^{2}}\right)}{{y-scale}^{2}}}}{x-scale}\right)\right)\right)\right)}{a}}{a} \]

   12. Applied rewrites 9.9%

       \[\leadsto \frac{\frac{0.25 \cdot \left(x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot \left(y-scale \cdot \frac{\sqrt{8 \cdot \frac{{a}^{4} \cdot \left({a}^{2} \cdot \frac{1}{{y-scale}^{2}} - \sqrt{{a}^{4} \cdot {\left(\mathsf{fma}\left(0.5, \frac{\cos \left(-0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}, 0.5 \cdot \frac{1}{{y-scale}^{2}}\right)\right)}^{2}}\right)}{{y-scale}^{2}}}}{x-scale}\right)\right)\right)\right)}{a}}{a} \]

   13. Taylor expanded in angle around 0

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

       1. lower-/.f64 N/A

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

       2. lower-pow.f64 10.0

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

   15. Applied rewrites 10.0%

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

   if 2.00000000000000008e-166 < x-scale

   1. Initial program 0.1%

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

   2. Taylor expanded in b around 0

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

   4. Applied rewrites 0.6%

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

   6. Applied rewrites 4.8%

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

   7. Taylor expanded in y-scale around 0

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

   9. Applied rewrites 13.8%

      \[\leadsto \frac{\frac{0.25 \cdot \left(x-scale \cdot \left(x-scale \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left({a}^{2} \cdot \left(0.5 + 0.5 \cdot \cos \left(-0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right) - \sqrt{{a}^{4} \cdot {\left(0.5 + 0.5 \cdot \cos \left(-0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}\right)}{{x-scale}^{2}}}\right)\right)}{a}}{a} \]

   10. Taylor expanded in angle around 0

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

       1. lower--.f64 N/A

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

       2. lower-pow.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-pow.f64 14.1

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

   12. Applied rewrites 14.1%

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

Alternative 3: 15.4% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x-scale \leq 2 \cdot 10^{-166}:\\ \;\;\;\;\frac{\frac{0.25 \cdot \left(x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot \left(y-scale \cdot \frac{\sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\frac{{a}^{2}}{{y-scale}^{2}} - \sqrt{\frac{{a}^{4}}{{y-scale}^{4}}}\right)}{{y-scale}^{2}}}}{x-scale}\right)\right)\right)\right)}{a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.25 \cdot \left(x-scale \cdot \left(x-scale \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left({a}^{2} - \sqrt{{a}^{4}}\right)}{{x-scale}^{2}}}\right)\right)}{a}}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= x-scale 2e-166)
   (/
    (/
     (*
      0.25
      (*
       x-scale
       (*
        x-scale
        (*
         y-scale
         (*
          y-scale
          (/
           (sqrt
            (*
             8.0
             (/
              (*
               (pow a 4.0)
               (-
                (/ (pow a 2.0) (pow y-scale 2.0))
                (sqrt (/ (pow a 4.0) (pow y-scale 4.0)))))
              (pow y-scale 2.0))))
           x-scale))))))
     a)
    a)
   (/
    (/
     (*
      0.25
      (*
       x-scale
       (*
        x-scale
        (sqrt
         (*
          8.0
          (/
           (* (pow a 4.0) (- (pow a 2.0) (sqrt (pow a 4.0))))
           (pow x-scale 2.0)))))))
     a)
    a)))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (x_45_scale <= 2e-166) {
		tmp = ((0.25 * (x_45_scale * (x_45_scale * (y_45_scale * (y_45_scale * (sqrt((8.0 * ((pow(a, 4.0) * ((pow(a, 2.0) / pow(y_45_scale, 2.0)) - sqrt((pow(a, 4.0) / pow(y_45_scale, 4.0))))) / pow(y_45_scale, 2.0)))) / x_45_scale)))))) / a) / a;
	} else {
		tmp = ((0.25 * (x_45_scale * (x_45_scale * sqrt((8.0 * ((pow(a, 4.0) * (pow(a, 2.0) - sqrt(pow(a, 4.0)))) / pow(x_45_scale, 2.0))))))) / a) / a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: tmp
    if (x_45scale <= 2d-166) then
        tmp = ((0.25d0 * (x_45scale * (x_45scale * (y_45scale * (y_45scale * (sqrt((8.0d0 * (((a ** 4.0d0) * (((a ** 2.0d0) / (y_45scale ** 2.0d0)) - sqrt(((a ** 4.0d0) / (y_45scale ** 4.0d0))))) / (y_45scale ** 2.0d0)))) / x_45scale)))))) / a) / a
    else
        tmp = ((0.25d0 * (x_45scale * (x_45scale * sqrt((8.0d0 * (((a ** 4.0d0) * ((a ** 2.0d0) - sqrt((a ** 4.0d0)))) / (x_45scale ** 2.0d0))))))) / a) / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (x_45_scale <= 2e-166) {
		tmp = ((0.25 * (x_45_scale * (x_45_scale * (y_45_scale * (y_45_scale * (Math.sqrt((8.0 * ((Math.pow(a, 4.0) * ((Math.pow(a, 2.0) / Math.pow(y_45_scale, 2.0)) - Math.sqrt((Math.pow(a, 4.0) / Math.pow(y_45_scale, 4.0))))) / Math.pow(y_45_scale, 2.0)))) / x_45_scale)))))) / a) / a;
	} else {
		tmp = ((0.25 * (x_45_scale * (x_45_scale * Math.sqrt((8.0 * ((Math.pow(a, 4.0) * (Math.pow(a, 2.0) - Math.sqrt(Math.pow(a, 4.0)))) / Math.pow(x_45_scale, 2.0))))))) / a) / a;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if x_45_scale <= 2e-166:
		tmp = ((0.25 * (x_45_scale * (x_45_scale * (y_45_scale * (y_45_scale * (math.sqrt((8.0 * ((math.pow(a, 4.0) * ((math.pow(a, 2.0) / math.pow(y_45_scale, 2.0)) - math.sqrt((math.pow(a, 4.0) / math.pow(y_45_scale, 4.0))))) / math.pow(y_45_scale, 2.0)))) / x_45_scale)))))) / a) / a
	else:
		tmp = ((0.25 * (x_45_scale * (x_45_scale * math.sqrt((8.0 * ((math.pow(a, 4.0) * (math.pow(a, 2.0) - math.sqrt(math.pow(a, 4.0)))) / math.pow(x_45_scale, 2.0))))))) / a) / a
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (x_45_scale <= 2e-166)
		tmp = Float64(Float64(Float64(0.25 * Float64(x_45_scale * Float64(x_45_scale * Float64(y_45_scale * Float64(y_45_scale * Float64(sqrt(Float64(8.0 * Float64(Float64((a ^ 4.0) * Float64(Float64((a ^ 2.0) / (y_45_scale ^ 2.0)) - sqrt(Float64((a ^ 4.0) / (y_45_scale ^ 4.0))))) / (y_45_scale ^ 2.0)))) / x_45_scale)))))) / a) / a);
	else
		tmp = Float64(Float64(Float64(0.25 * Float64(x_45_scale * Float64(x_45_scale * sqrt(Float64(8.0 * Float64(Float64((a ^ 4.0) * Float64((a ^ 2.0) - sqrt((a ^ 4.0)))) / (x_45_scale ^ 2.0))))))) / a) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (x_45_scale <= 2e-166)
		tmp = ((0.25 * (x_45_scale * (x_45_scale * (y_45_scale * (y_45_scale * (sqrt((8.0 * (((a ^ 4.0) * (((a ^ 2.0) / (y_45_scale ^ 2.0)) - sqrt(((a ^ 4.0) / (y_45_scale ^ 4.0))))) / (y_45_scale ^ 2.0)))) / x_45_scale)))))) / a) / a;
	else
		tmp = ((0.25 * (x_45_scale * (x_45_scale * sqrt((8.0 * (((a ^ 4.0) * ((a ^ 2.0) - sqrt((a ^ 4.0)))) / (x_45_scale ^ 2.0))))))) / a) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[x$45$scale, 2e-166], N[(N[(N[(0.25 * N[(x$45$scale * N[(x$45$scale * N[(y$45$scale * N[(y$45$scale * N[(N[Sqrt[N[(8.0 * N[(N[(N[Power[a, 4.0], $MachinePrecision] * N[(N[(N[Power[a, 2.0], $MachinePrecision] / N[Power[y$45$scale, 2.0], $MachinePrecision]), $MachinePrecision] - N[Sqrt[N[(N[Power[a, 4.0], $MachinePrecision] / N[Power[y$45$scale, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[y$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / a), $MachinePrecision], N[(N[(N[(0.25 * N[(x$45$scale * N[(x$45$scale * N[Sqrt[N[(8.0 * N[(N[(N[Power[a, 4.0], $MachinePrecision] * N[(N[Power[a, 2.0], $MachinePrecision] - N[Sqrt[N[Power[a, 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 2.00000000000000008e-166

   1. Initial program 0.1%

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

   2. Taylor expanded in b around 0

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

   4. Applied rewrites 0.6%

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

   6. Applied rewrites 4.8%

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

   7. Taylor expanded in x-scale around inf

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

   9. Applied rewrites 10.1%

      \[\leadsto \frac{\frac{0.25 \cdot \left(x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot \left(y-scale \cdot \frac{\sqrt{8 \cdot \frac{{a}^{4} \cdot \left({a}^{2} \cdot \mathsf{fma}\left(0.5, \frac{\cos \left(-0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}, 0.5 \cdot \frac{1}{{y-scale}^{2}}\right) - \sqrt{{a}^{4} \cdot {\left(\mathsf{fma}\left(0.5, \frac{\cos \left(-0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}, 0.5 \cdot \frac{1}{{y-scale}^{2}}\right)\right)}^{2}}\right)}{{y-scale}^{2}}}}{x-scale}\right)\right)\right)\right)}{a}}{a} \]

   10. Taylor expanded in angle around 0

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

       1. lower--.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower-pow.f64 N/A

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

       4. lower-pow.f64 N/A

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

       5. lower-sqrt.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. lower-pow.f64 N/A

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

       8. lower-pow.f64 10.2

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

   12. Applied rewrites 10.2%

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

   if 2.00000000000000008e-166 < x-scale

   1. Initial program 0.1%

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

   2. Taylor expanded in b around 0

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

   4. Applied rewrites 0.6%

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

   6. Applied rewrites 4.8%

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

   7. Taylor expanded in y-scale around 0

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

   9. Applied rewrites 13.8%

      \[\leadsto \frac{\frac{0.25 \cdot \left(x-scale \cdot \left(x-scale \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left({a}^{2} \cdot \left(0.5 + 0.5 \cdot \cos \left(-0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right) - \sqrt{{a}^{4} \cdot {\left(0.5 + 0.5 \cdot \cos \left(-0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}\right)}{{x-scale}^{2}}}\right)\right)}{a}}{a} \]

   10. Taylor expanded in angle around 0

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

       1. lower--.f64 N/A

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

       2. lower-pow.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-pow.f64 14.1

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

   12. Applied rewrites 14.1%

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

Alternative 4: 14.1% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{{a}^{4}}\\ \mathbf{if}\;x-scale \leq 3.5 \cdot 10^{-178}:\\ \;\;\;\;0.25 \cdot \frac{{x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{\frac{8}{y-scale \cdot x-scale} \cdot \frac{\left(a \cdot \frac{a}{y-scale \cdot y-scale} - \frac{t\_0}{y-scale \cdot y-scale}\right) \cdot {a}^{4}}{y-scale \cdot x-scale}}\right)}{{a}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.25 \cdot \left(x-scale \cdot \left(x-scale \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left({a}^{2} - t\_0\right)}{{x-scale}^{2}}}\right)\right)}{a}}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (sqrt (pow a 4.0))))
   (if (<= x-scale 3.5e-178)
     (*
      0.25
      (/
       (*
        (pow x-scale 2.0)
        (*
         (pow y-scale 2.0)
         (sqrt
          (*
           (/ 8.0 (* y-scale x-scale))
           (/
            (*
             (- (* a (/ a (* y-scale y-scale))) (/ t_0 (* y-scale y-scale)))
             (pow a 4.0))
            (* y-scale x-scale))))))
       (pow a 2.0)))
     (/
      (/
       (*
        0.25
        (*
         x-scale
         (*
          x-scale
          (sqrt
           (*
            8.0
            (/ (* (pow a 4.0) (- (pow a 2.0) t_0)) (pow x-scale 2.0)))))))
       a)
      a))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = sqrt(pow(a, 4.0));
	double tmp;
	if (x_45_scale <= 3.5e-178) {
		tmp = 0.25 * ((pow(x_45_scale, 2.0) * (pow(y_45_scale, 2.0) * sqrt(((8.0 / (y_45_scale * x_45_scale)) * ((((a * (a / (y_45_scale * y_45_scale))) - (t_0 / (y_45_scale * y_45_scale))) * pow(a, 4.0)) / (y_45_scale * x_45_scale)))))) / pow(a, 2.0));
	} else {
		tmp = ((0.25 * (x_45_scale * (x_45_scale * sqrt((8.0 * ((pow(a, 4.0) * (pow(a, 2.0) - t_0)) / pow(x_45_scale, 2.0))))))) / a) / a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((a ** 4.0d0))
    if (x_45scale <= 3.5d-178) then
        tmp = 0.25d0 * (((x_45scale ** 2.0d0) * ((y_45scale ** 2.0d0) * sqrt(((8.0d0 / (y_45scale * x_45scale)) * ((((a * (a / (y_45scale * y_45scale))) - (t_0 / (y_45scale * y_45scale))) * (a ** 4.0d0)) / (y_45scale * x_45scale)))))) / (a ** 2.0d0))
    else
        tmp = ((0.25d0 * (x_45scale * (x_45scale * sqrt((8.0d0 * (((a ** 4.0d0) * ((a ** 2.0d0) - t_0)) / (x_45scale ** 2.0d0))))))) / a) / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = Math.sqrt(Math.pow(a, 4.0));
	double tmp;
	if (x_45_scale <= 3.5e-178) {
		tmp = 0.25 * ((Math.pow(x_45_scale, 2.0) * (Math.pow(y_45_scale, 2.0) * Math.sqrt(((8.0 / (y_45_scale * x_45_scale)) * ((((a * (a / (y_45_scale * y_45_scale))) - (t_0 / (y_45_scale * y_45_scale))) * Math.pow(a, 4.0)) / (y_45_scale * x_45_scale)))))) / Math.pow(a, 2.0));
	} else {
		tmp = ((0.25 * (x_45_scale * (x_45_scale * Math.sqrt((8.0 * ((Math.pow(a, 4.0) * (Math.pow(a, 2.0) - t_0)) / Math.pow(x_45_scale, 2.0))))))) / a) / a;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = math.sqrt(math.pow(a, 4.0))
	tmp = 0
	if x_45_scale <= 3.5e-178:
		tmp = 0.25 * ((math.pow(x_45_scale, 2.0) * (math.pow(y_45_scale, 2.0) * math.sqrt(((8.0 / (y_45_scale * x_45_scale)) * ((((a * (a / (y_45_scale * y_45_scale))) - (t_0 / (y_45_scale * y_45_scale))) * math.pow(a, 4.0)) / (y_45_scale * x_45_scale)))))) / math.pow(a, 2.0))
	else:
		tmp = ((0.25 * (x_45_scale * (x_45_scale * math.sqrt((8.0 * ((math.pow(a, 4.0) * (math.pow(a, 2.0) - t_0)) / math.pow(x_45_scale, 2.0))))))) / a) / a
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = sqrt((a ^ 4.0))
	tmp = 0.0
	if (x_45_scale <= 3.5e-178)
		tmp = Float64(0.25 * Float64(Float64((x_45_scale ^ 2.0) * Float64((y_45_scale ^ 2.0) * sqrt(Float64(Float64(8.0 / Float64(y_45_scale * x_45_scale)) * Float64(Float64(Float64(Float64(a * Float64(a / Float64(y_45_scale * y_45_scale))) - Float64(t_0 / Float64(y_45_scale * y_45_scale))) * (a ^ 4.0)) / Float64(y_45_scale * x_45_scale)))))) / (a ^ 2.0)));
	else
		tmp = Float64(Float64(Float64(0.25 * Float64(x_45_scale * Float64(x_45_scale * sqrt(Float64(8.0 * Float64(Float64((a ^ 4.0) * Float64((a ^ 2.0) - t_0)) / (x_45_scale ^ 2.0))))))) / a) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = sqrt((a ^ 4.0));
	tmp = 0.0;
	if (x_45_scale <= 3.5e-178)
		tmp = 0.25 * (((x_45_scale ^ 2.0) * ((y_45_scale ^ 2.0) * sqrt(((8.0 / (y_45_scale * x_45_scale)) * ((((a * (a / (y_45_scale * y_45_scale))) - (t_0 / (y_45_scale * y_45_scale))) * (a ^ 4.0)) / (y_45_scale * x_45_scale)))))) / (a ^ 2.0));
	else
		tmp = ((0.25 * (x_45_scale * (x_45_scale * sqrt((8.0 * (((a ^ 4.0) * ((a ^ 2.0) - t_0)) / (x_45_scale ^ 2.0))))))) / a) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[Sqrt[N[Power[a, 4.0], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$45$scale, 3.5e-178], N[(0.25 * N[(N[(N[Power[x$45$scale, 2.0], $MachinePrecision] * N[(N[Power[y$45$scale, 2.0], $MachinePrecision] * N[Sqrt[N[(N[(8.0 / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(a * N[(a / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t$95$0 / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.25 * N[(x$45$scale * N[(x$45$scale * N[Sqrt[N[(8.0 * N[(N[(N[Power[a, 4.0], $MachinePrecision] * N[(N[Power[a, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision] / N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 3.49999999999999983e-178

   1. Initial program 0.1%

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

   2. Taylor expanded in b around 0

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

   4. Applied rewrites 0.6%

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

   5. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower--.f64 N/A

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

   7. Applied rewrites 0.5%

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

   8. Taylor expanded in y-scale around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-pow.f64 0.8

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

   10. Applied rewrites 0.8%

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

   12. Applied rewrites 2.4%

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

   if 3.49999999999999983e-178 < x-scale

   1. Initial program 0.1%

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

   2. Taylor expanded in b around 0

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

   4. Applied rewrites 0.6%

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

   6. Applied rewrites 4.8%

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

   7. Taylor expanded in y-scale around 0

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

   9. Applied rewrites 13.8%

      \[\leadsto \frac{\frac{0.25 \cdot \left(x-scale \cdot \left(x-scale \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left({a}^{2} \cdot \left(0.5 + 0.5 \cdot \cos \left(-0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right) - \sqrt{{a}^{4} \cdot {\left(0.5 + 0.5 \cdot \cos \left(-0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}\right)}{{x-scale}^{2}}}\right)\right)}{a}}{a} \]

   10. Taylor expanded in angle around 0

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

       1. lower--.f64 N/A

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

       2. lower-pow.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-pow.f64 14.1

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

   12. Applied rewrites 14.1%

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

Alternative 5: 9.0% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.25 \cdot \left(x-scale \cdot \left(x-scale \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left({a}^{2} - \sqrt{{a}^{4}}\right)}{{x-scale}^{2}}}\right)\right)}{a}}{a} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (/
  (/
   (*
    0.25
    (*
     x-scale
     (*
      x-scale
      (sqrt
       (*
        8.0
        (/
         (* (pow a 4.0) (- (pow a 2.0) (sqrt (pow a 4.0))))
         (pow x-scale 2.0)))))))
   a)
  a))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return ((0.25 * (x_45_scale * (x_45_scale * sqrt((8.0 * ((pow(a, 4.0) * (pow(a, 2.0) - sqrt(pow(a, 4.0)))) / pow(x_45_scale, 2.0))))))) / a) / a;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    code = ((0.25d0 * (x_45scale * (x_45scale * sqrt((8.0d0 * (((a ** 4.0d0) * ((a ** 2.0d0) - sqrt((a ** 4.0d0)))) / (x_45scale ** 2.0d0))))))) / a) / a
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return ((0.25 * (x_45_scale * (x_45_scale * Math.sqrt((8.0 * ((Math.pow(a, 4.0) * (Math.pow(a, 2.0) - Math.sqrt(Math.pow(a, 4.0)))) / Math.pow(x_45_scale, 2.0))))))) / a) / a;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	return ((0.25 * (x_45_scale * (x_45_scale * math.sqrt((8.0 * ((math.pow(a, 4.0) * (math.pow(a, 2.0) - math.sqrt(math.pow(a, 4.0)))) / math.pow(x_45_scale, 2.0))))))) / a) / a

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return Float64(Float64(Float64(0.25 * Float64(x_45_scale * Float64(x_45_scale * sqrt(Float64(8.0 * Float64(Float64((a ^ 4.0) * Float64((a ^ 2.0) - sqrt((a ^ 4.0)))) / (x_45_scale ^ 2.0))))))) / a) / a)
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = ((0.25 * (x_45_scale * (x_45_scale * sqrt((8.0 * (((a ^ 4.0) * ((a ^ 2.0) - sqrt((a ^ 4.0)))) / (x_45_scale ^ 2.0))))))) / a) / a;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[(N[(N[(0.25 * N[(x$45$scale * N[(x$45$scale * N[Sqrt[N[(8.0 * N[(N[(N[Power[a, 4.0], $MachinePrecision] * N[(N[Power[a, 2.0], $MachinePrecision] - N[Sqrt[N[Power[a, 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / a), $MachinePrecision]

Derivation

1. Initial program 0.1%

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

2. Taylor expanded in b around 0

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

4. Applied rewrites 0.6%

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

6. Applied rewrites 4.8%

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

7. Taylor expanded in y-scale around 0

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

9. Applied rewrites 13.8%

   \[\leadsto \frac{\frac{0.25 \cdot \left(x-scale \cdot \left(x-scale \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left({a}^{2} \cdot \left(0.5 + 0.5 \cdot \cos \left(-0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right) - \sqrt{{a}^{4} \cdot {\left(0.5 + 0.5 \cdot \cos \left(-0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}\right)}{{x-scale}^{2}}}\right)\right)}{a}}{a} \]

10. Taylor expanded in angle around 0

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

    1. lower--.f64 N/A

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

    2. lower-pow.f64 N/A

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

    3. lower-sqrt.f64 N/A

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

    4. lower-pow.f64 14.1

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

12. Applied rewrites 14.1%

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

Alternative 6: 2.8% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \frac{0.25}{a} \cdot \frac{\left(\sqrt{\frac{\left(a \cdot \frac{a}{y-scale \cdot y-scale} - \frac{\sqrt{{a}^{4}}}{y-scale \cdot y-scale}\right) \cdot {a}^{4}}{\left(\left(x-scale \cdot x-scale\right) \cdot y-scale\right) \cdot y-scale} \cdot 8} \cdot \left(y-scale \cdot y-scale\right)\right) \cdot \left(x-scale \cdot x-scale\right)}{a} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (*
  (/ 0.25 a)
  (/
   (*
    (*
     (sqrt
      (*
       (/
        (*
         (-
          (* a (/ a (* y-scale y-scale)))
          (/ (sqrt (pow a 4.0)) (* y-scale y-scale)))
         (pow a 4.0))
        (* (* (* x-scale x-scale) y-scale) y-scale))
       8.0))
     (* y-scale y-scale))
    (* x-scale x-scale))
   a)))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return (0.25 / a) * (((sqrt((((((a * (a / (y_45_scale * y_45_scale))) - (sqrt(pow(a, 4.0)) / (y_45_scale * y_45_scale))) * pow(a, 4.0)) / (((x_45_scale * x_45_scale) * y_45_scale) * y_45_scale)) * 8.0)) * (y_45_scale * y_45_scale)) * (x_45_scale * x_45_scale)) / a);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    code = (0.25d0 / a) * (((sqrt((((((a * (a / (y_45scale * y_45scale))) - (sqrt((a ** 4.0d0)) / (y_45scale * y_45scale))) * (a ** 4.0d0)) / (((x_45scale * x_45scale) * y_45scale) * y_45scale)) * 8.0d0)) * (y_45scale * y_45scale)) * (x_45scale * x_45scale)) / a)
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return (0.25 / a) * (((Math.sqrt((((((a * (a / (y_45_scale * y_45_scale))) - (Math.sqrt(Math.pow(a, 4.0)) / (y_45_scale * y_45_scale))) * Math.pow(a, 4.0)) / (((x_45_scale * x_45_scale) * y_45_scale) * y_45_scale)) * 8.0)) * (y_45_scale * y_45_scale)) * (x_45_scale * x_45_scale)) / a);
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	return (0.25 / a) * (((math.sqrt((((((a * (a / (y_45_scale * y_45_scale))) - (math.sqrt(math.pow(a, 4.0)) / (y_45_scale * y_45_scale))) * math.pow(a, 4.0)) / (((x_45_scale * x_45_scale) * y_45_scale) * y_45_scale)) * 8.0)) * (y_45_scale * y_45_scale)) * (x_45_scale * x_45_scale)) / a)

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return Float64(Float64(0.25 / a) * Float64(Float64(Float64(sqrt(Float64(Float64(Float64(Float64(Float64(a * Float64(a / Float64(y_45_scale * y_45_scale))) - Float64(sqrt((a ^ 4.0)) / Float64(y_45_scale * y_45_scale))) * (a ^ 4.0)) / Float64(Float64(Float64(x_45_scale * x_45_scale) * y_45_scale) * y_45_scale)) * 8.0)) * Float64(y_45_scale * y_45_scale)) * Float64(x_45_scale * x_45_scale)) / a))
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = (0.25 / a) * (((sqrt((((((a * (a / (y_45_scale * y_45_scale))) - (sqrt((a ^ 4.0)) / (y_45_scale * y_45_scale))) * (a ^ 4.0)) / (((x_45_scale * x_45_scale) * y_45_scale) * y_45_scale)) * 8.0)) * (y_45_scale * y_45_scale)) * (x_45_scale * x_45_scale)) / a);
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[(N[(0.25 / a), $MachinePrecision] * N[(N[(N[(N[Sqrt[N[(N[(N[(N[(N[(a * N[(a / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[N[Power[a, 4.0], $MachinePrecision]], $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision]), $MachinePrecision] * 8.0), $MachinePrecision]], $MachinePrecision] * N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] * N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.1%

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

2. Taylor expanded in b around 0

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

4. Applied rewrites 0.6%

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

5. Taylor expanded in angle around 0

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

   1. lower-*.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower--.f64 N/A

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

7. Applied rewrites 0.5%

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

8. Taylor expanded in y-scale around 0

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

   1. lower-/.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-pow.f64 0.8

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

10. Applied rewrites 0.8%

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

12. Applied rewrites 2.8%

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

Alternative 7: 0.8% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \frac{0.25 \cdot \left(\left(\sqrt{\frac{\left(a \cdot \frac{a}{y-scale \cdot y-scale} - \frac{\sqrt{{a}^{4}}}{y-scale \cdot y-scale}\right) \cdot {a}^{4}}{\left(\left(x-scale \cdot x-scale\right) \cdot y-scale\right) \cdot y-scale} \cdot 8} \cdot \left(y-scale \cdot y-scale\right)\right) \cdot \left(x-scale \cdot x-scale\right)\right)}{a \cdot a} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (/
  (*
   0.25
   (*
    (*
     (sqrt
      (*
       (/
        (*
         (-
          (* a (/ a (* y-scale y-scale)))
          (/ (sqrt (pow a 4.0)) (* y-scale y-scale)))
         (pow a 4.0))
        (* (* (* x-scale x-scale) y-scale) y-scale))
       8.0))
     (* y-scale y-scale))
    (* x-scale x-scale)))
  (* a a)))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return (0.25 * ((sqrt((((((a * (a / (y_45_scale * y_45_scale))) - (sqrt(pow(a, 4.0)) / (y_45_scale * y_45_scale))) * pow(a, 4.0)) / (((x_45_scale * x_45_scale) * y_45_scale) * y_45_scale)) * 8.0)) * (y_45_scale * y_45_scale)) * (x_45_scale * x_45_scale))) / (a * a);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    code = (0.25d0 * ((sqrt((((((a * (a / (y_45scale * y_45scale))) - (sqrt((a ** 4.0d0)) / (y_45scale * y_45scale))) * (a ** 4.0d0)) / (((x_45scale * x_45scale) * y_45scale) * y_45scale)) * 8.0d0)) * (y_45scale * y_45scale)) * (x_45scale * x_45scale))) / (a * a)
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return (0.25 * ((Math.sqrt((((((a * (a / (y_45_scale * y_45_scale))) - (Math.sqrt(Math.pow(a, 4.0)) / (y_45_scale * y_45_scale))) * Math.pow(a, 4.0)) / (((x_45_scale * x_45_scale) * y_45_scale) * y_45_scale)) * 8.0)) * (y_45_scale * y_45_scale)) * (x_45_scale * x_45_scale))) / (a * a);
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	return (0.25 * ((math.sqrt((((((a * (a / (y_45_scale * y_45_scale))) - (math.sqrt(math.pow(a, 4.0)) / (y_45_scale * y_45_scale))) * math.pow(a, 4.0)) / (((x_45_scale * x_45_scale) * y_45_scale) * y_45_scale)) * 8.0)) * (y_45_scale * y_45_scale)) * (x_45_scale * x_45_scale))) / (a * a)

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return Float64(Float64(0.25 * Float64(Float64(sqrt(Float64(Float64(Float64(Float64(Float64(a * Float64(a / Float64(y_45_scale * y_45_scale))) - Float64(sqrt((a ^ 4.0)) / Float64(y_45_scale * y_45_scale))) * (a ^ 4.0)) / Float64(Float64(Float64(x_45_scale * x_45_scale) * y_45_scale) * y_45_scale)) * 8.0)) * Float64(y_45_scale * y_45_scale)) * Float64(x_45_scale * x_45_scale))) / Float64(a * a))
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = (0.25 * ((sqrt((((((a * (a / (y_45_scale * y_45_scale))) - (sqrt((a ^ 4.0)) / (y_45_scale * y_45_scale))) * (a ^ 4.0)) / (((x_45_scale * x_45_scale) * y_45_scale) * y_45_scale)) * 8.0)) * (y_45_scale * y_45_scale)) * (x_45_scale * x_45_scale))) / (a * a);
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[(N[(0.25 * N[(N[(N[Sqrt[N[(N[(N[(N[(N[(a * N[(a / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[N[Power[a, 4.0], $MachinePrecision]], $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision]), $MachinePrecision] * 8.0), $MachinePrecision]], $MachinePrecision] * N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] * N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.1%

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

2. Taylor expanded in b around 0

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

4. Applied rewrites 0.6%

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

5. Taylor expanded in angle around 0

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

   1. lower-*.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower--.f64 N/A

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

7. Applied rewrites 0.5%

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

8. Taylor expanded in y-scale around 0

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

   1. lower-/.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-pow.f64 0.8

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

10. Applied rewrites 0.8%

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

12. Applied rewrites 0.8%

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

Reproduce

herbie shell --seed 2025156 
(FPCore (a b angle x-scale y-scale)
  :name "b 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))))