b from scale-rotated-ellipse

Percentage Accurate: 0.1% → 14.9%

Time: 39.9s

Alternatives: 4

Speedup: 31.8×

• could not determine a ground truth

  1. a = -3.82982505763553e+214
  2. b = 3.054642616747308e-301
  3. angle = -3.315596310044344e+222
  4. x-scale = -6.168598832121298e+298
  5. y-scale = -2.0787219940878388e-287

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: 14.9% accurate, 5.1× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{if}\;a\_m \leq 1.4 \cdot 10^{+78}:\\ \;\;\;\;0.25 \cdot \frac{\left(x-scale \cdot x-scale\right) \cdot \frac{\left(\left(a\_m \cdot a\_m\right) \cdot a\_m\right) \cdot \sqrt{8 \cdot \left({t\_0}^{2} - \sqrt{{t\_0}^{4}}\right)}}{x-scale}}{a\_m \cdot a\_m}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{\left(x-scale \cdot x-scale\right) \cdot \sqrt{0}}{{\left(a\_m \cdot b\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (cos (* 0.005555555555555556 (* angle PI)))))
   (if (<= a_m 1.4e+78)
     (*
      0.25
      (/
       (*
        (* x-scale x-scale)
        (/
         (*
          (* (* a_m a_m) a_m)
          (sqrt (* 8.0 (- (pow t_0 2.0) (sqrt (pow t_0 4.0))))))
         x-scale))
       (* a_m a_m)))
     (* 0.25 (/ (* (* x-scale x-scale) (sqrt 0.0)) (pow (* a_m b) 2.0))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = cos((0.005555555555555556 * (angle * ((double) M_PI))));
	double tmp;
	if (a_m <= 1.4e+78) {
		tmp = 0.25 * (((x_45_scale * x_45_scale) * ((((a_m * a_m) * a_m) * sqrt((8.0 * (pow(t_0, 2.0) - sqrt(pow(t_0, 4.0)))))) / x_45_scale)) / (a_m * a_m));
	} else {
		tmp = 0.25 * (((x_45_scale * x_45_scale) * sqrt(0.0)) / pow((a_m * b), 2.0));
	}
	return tmp;
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = Math.cos((0.005555555555555556 * (angle * Math.PI)));
	double tmp;
	if (a_m <= 1.4e+78) {
		tmp = 0.25 * (((x_45_scale * x_45_scale) * ((((a_m * a_m) * a_m) * Math.sqrt((8.0 * (Math.pow(t_0, 2.0) - Math.sqrt(Math.pow(t_0, 4.0)))))) / x_45_scale)) / (a_m * a_m));
	} else {
		tmp = 0.25 * (((x_45_scale * x_45_scale) * Math.sqrt(0.0)) / Math.pow((a_m * b), 2.0));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle, x_45_scale, y_45_scale):
	t_0 = math.cos((0.005555555555555556 * (angle * math.pi)))
	tmp = 0
	if a_m <= 1.4e+78:
		tmp = 0.25 * (((x_45_scale * x_45_scale) * ((((a_m * a_m) * a_m) * math.sqrt((8.0 * (math.pow(t_0, 2.0) - math.sqrt(math.pow(t_0, 4.0)))))) / x_45_scale)) / (a_m * a_m))
	else:
		tmp = 0.25 * (((x_45_scale * x_45_scale) * math.sqrt(0.0)) / math.pow((a_m * b), 2.0))
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = cos(Float64(0.005555555555555556 * Float64(angle * pi)))
	tmp = 0.0
	if (a_m <= 1.4e+78)
		tmp = Float64(0.25 * Float64(Float64(Float64(x_45_scale * x_45_scale) * Float64(Float64(Float64(Float64(a_m * a_m) * a_m) * sqrt(Float64(8.0 * Float64((t_0 ^ 2.0) - sqrt((t_0 ^ 4.0)))))) / x_45_scale)) / Float64(a_m * a_m)));
	else
		tmp = Float64(0.25 * Float64(Float64(Float64(x_45_scale * x_45_scale) * sqrt(0.0)) / (Float64(a_m * b) ^ 2.0)));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = cos((0.005555555555555556 * (angle * pi)));
	tmp = 0.0;
	if (a_m <= 1.4e+78)
		tmp = 0.25 * (((x_45_scale * x_45_scale) * ((((a_m * a_m) * a_m) * sqrt((8.0 * ((t_0 ^ 2.0) - sqrt((t_0 ^ 4.0)))))) / x_45_scale)) / (a_m * a_m));
	else
		tmp = 0.25 * (((x_45_scale * x_45_scale) * sqrt(0.0)) / ((a_m * b) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[Cos[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[a$95$m, 1.4e+78], N[(0.25 * N[(N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * N[(N[(N[(N[(a$95$m * a$95$m), $MachinePrecision] * a$95$m), $MachinePrecision] * N[Sqrt[N[(8.0 * N[(N[Power[t$95$0, 2.0], $MachinePrecision] - N[Sqrt[N[Power[t$95$0, 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] / N[(a$95$m * a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * N[Sqrt[0.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[(a$95$m * b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 1.4000000000000001e78

   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 0.8%

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

   5. Taylor expanded in y-scale around 0

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

      1. lower-sqrt.f64 N/A

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

   7. Applied rewrites 3.9%

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

   8. Taylor expanded in x-scale around 0

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

   10. Applied rewrites 8.9%

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

   11. Taylor expanded in a around 0

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

       1. lower-*.f64 N/A

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

       2. unpow3 N/A

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

       3. pow2 N/A

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

       4. lower-*.f64 N/A

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

       5. pow2 N/A

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

       6. lift-*.f64 N/A

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

       7. lower-sqrt.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. lower--.f64 N/A

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

   13. Applied rewrites 9.6%

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

   if 1.4000000000000001e78 < a

   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 angle around 0

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

   4. Applied rewrites 0.1%

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

   5. Taylor expanded in y-scale around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

   7. Applied rewrites 0.8%

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

   8. Taylor expanded in a around 0

      \[\leadsto \frac{1}{4} \cdot \frac{\left(x-scale \cdot x-scale\right) \cdot \sqrt{0}}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 6.2%

       \[\leadsto 0.25 \cdot \frac{\left(x-scale \cdot x-scale\right) \cdot \sqrt{0}}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{1}{4} \cdot \frac{\left(x-scale \cdot x-scale\right) \cdot \sqrt{0}}{\left(a \cdot a\right) \cdot \color{blue}{\left(b \cdot b\right)}} \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{1}{4} \cdot \frac{\left(x-scale \cdot x-scale\right) \cdot \sqrt{0}}{\left(a \cdot a\right) \cdot \left(\color{blue}{b} \cdot b\right)} \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{1}{4} \cdot \frac{\left(x-scale \cdot x-scale\right) \cdot \sqrt{0}}{\left(a \cdot a\right) \cdot \left(b \cdot \color{blue}{b}\right)} \]

       4. pow2 N/A

          \[\leadsto \frac{1}{4} \cdot \frac{\left(x-scale \cdot x-scale\right) \cdot \sqrt{0}}{{a}^{2} \cdot \left(\color{blue}{b} \cdot b\right)} \]

       5. pow2 N/A

          \[\leadsto \frac{1}{4} \cdot \frac{\left(x-scale \cdot x-scale\right) \cdot \sqrt{0}}{{a}^{2} \cdot {b}^{\color{blue}{2}}} \]

       6. pow-prod-down N/A

          \[\leadsto \frac{1}{4} \cdot \frac{\left(x-scale \cdot x-scale\right) \cdot \sqrt{0}}{{\left(a \cdot b\right)}^{\color{blue}{2}}} \]

       7. lower-pow.f64 N/A

          \[\leadsto \frac{1}{4} \cdot \frac{\left(x-scale \cdot x-scale\right) \cdot \sqrt{0}}{{\left(a \cdot b\right)}^{\color{blue}{2}}} \]

       8. lower-*.f64 14.1

          \[\leadsto 0.25 \cdot \frac{\left(x-scale \cdot x-scale\right) \cdot \sqrt{0}}{{\left(a \cdot b\right)}^{2}} \]

   12. Applied rewrites 14.1%

       \[\leadsto 0.25 \cdot \frac{\left(x-scale \cdot x-scale\right) \cdot \sqrt{0}}{{\left(a \cdot b\right)}^{\color{blue}{2}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 14.8% accurate, 10.8× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;a\_m \leq 2.75 \cdot 10^{+72}:\\ \;\;\;\;0.25 \cdot \frac{\left(x-scale \cdot x-scale\right) \cdot \frac{\sqrt{8 \cdot \left({a\_m}^{4} \cdot \left(a\_m \cdot a\_m - \sqrt{{a\_m}^{4}}\right)\right)}}{x-scale}}{a\_m \cdot a\_m}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{\left(x-scale \cdot x-scale\right) \cdot \sqrt{0}}{{\left(a\_m \cdot b\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (if (<= a_m 2.75e+72)
   (*
    0.25
    (/
     (*
      (* x-scale x-scale)
      (/
       (sqrt (* 8.0 (* (pow a_m 4.0) (- (* a_m a_m) (sqrt (pow a_m 4.0))))))
       x-scale))
     (* a_m a_m)))
   (* 0.25 (/ (* (* x-scale x-scale) (sqrt 0.0)) (pow (* a_m b) 2.0)))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (a_m <= 2.75e+72) {
		tmp = 0.25 * (((x_45_scale * x_45_scale) * (sqrt((8.0 * (pow(a_m, 4.0) * ((a_m * a_m) - sqrt(pow(a_m, 4.0)))))) / x_45_scale)) / (a_m * a_m));
	} else {
		tmp = 0.25 * (((x_45_scale * x_45_scale) * sqrt(0.0)) / pow((a_m * b), 2.0));
	}
	return tmp;
}

Fortran

a_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a_m, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a_m
    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 (a_m <= 2.75d+72) then
        tmp = 0.25d0 * (((x_45scale * x_45scale) * (sqrt((8.0d0 * ((a_m ** 4.0d0) * ((a_m * a_m) - sqrt((a_m ** 4.0d0)))))) / x_45scale)) / (a_m * a_m))
    else
        tmp = 0.25d0 * (((x_45scale * x_45scale) * sqrt(0.0d0)) / ((a_m * b) ** 2.0d0))
    end if
    code = tmp
end function

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (a_m <= 2.75e+72) {
		tmp = 0.25 * (((x_45_scale * x_45_scale) * (Math.sqrt((8.0 * (Math.pow(a_m, 4.0) * ((a_m * a_m) - Math.sqrt(Math.pow(a_m, 4.0)))))) / x_45_scale)) / (a_m * a_m));
	} else {
		tmp = 0.25 * (((x_45_scale * x_45_scale) * Math.sqrt(0.0)) / Math.pow((a_m * b), 2.0));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if a_m <= 2.75e+72:
		tmp = 0.25 * (((x_45_scale * x_45_scale) * (math.sqrt((8.0 * (math.pow(a_m, 4.0) * ((a_m * a_m) - math.sqrt(math.pow(a_m, 4.0)))))) / x_45_scale)) / (a_m * a_m))
	else:
		tmp = 0.25 * (((x_45_scale * x_45_scale) * math.sqrt(0.0)) / math.pow((a_m * b), 2.0))
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (a_m <= 2.75e+72)
		tmp = Float64(0.25 * Float64(Float64(Float64(x_45_scale * x_45_scale) * Float64(sqrt(Float64(8.0 * Float64((a_m ^ 4.0) * Float64(Float64(a_m * a_m) - sqrt((a_m ^ 4.0)))))) / x_45_scale)) / Float64(a_m * a_m)));
	else
		tmp = Float64(0.25 * Float64(Float64(Float64(x_45_scale * x_45_scale) * sqrt(0.0)) / (Float64(a_m * b) ^ 2.0)));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (a_m <= 2.75e+72)
		tmp = 0.25 * (((x_45_scale * x_45_scale) * (sqrt((8.0 * ((a_m ^ 4.0) * ((a_m * a_m) - sqrt((a_m ^ 4.0)))))) / x_45_scale)) / (a_m * a_m));
	else
		tmp = 0.25 * (((x_45_scale * x_45_scale) * sqrt(0.0)) / ((a_m * b) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[a$95$m, 2.75e+72], N[(0.25 * N[(N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * N[(N[Sqrt[N[(8.0 * N[(N[Power[a$95$m, 4.0], $MachinePrecision] * N[(N[(a$95$m * a$95$m), $MachinePrecision] - N[Sqrt[N[Power[a$95$m, 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] / N[(a$95$m * a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * N[Sqrt[0.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[(a$95$m * b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.75e72

   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 0.8%

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

   5. Taylor expanded in y-scale around 0

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

      1. lower-sqrt.f64 N/A

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

   7. Applied rewrites 3.9%

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

   8. Taylor expanded in x-scale around 0

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

   10. Applied rewrites 8.9%

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

   11. Taylor expanded in angle around 0

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

       1. lower--.f64 N/A

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

       2. pow2 N/A

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

       3. lift-*.f64 N/A

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

       4. lower-sqrt.f64 N/A

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

       5. lift-pow.f64 9.0

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

   13. Applied rewrites 9.0%

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

   if 2.75e72 < a

   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 angle around 0

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

   4. Applied rewrites 0.1%

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

   5. Taylor expanded in y-scale around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

   7. Applied rewrites 0.8%

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

   8. Taylor expanded in a around 0

      \[\leadsto \frac{1}{4} \cdot \frac{\left(x-scale \cdot x-scale\right) \cdot \sqrt{0}}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 6.2%

       \[\leadsto 0.25 \cdot \frac{\left(x-scale \cdot x-scale\right) \cdot \sqrt{0}}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{1}{4} \cdot \frac{\left(x-scale \cdot x-scale\right) \cdot \sqrt{0}}{\left(a \cdot a\right) \cdot \color{blue}{\left(b \cdot b\right)}} \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{1}{4} \cdot \frac{\left(x-scale \cdot x-scale\right) \cdot \sqrt{0}}{\left(a \cdot a\right) \cdot \left(\color{blue}{b} \cdot b\right)} \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{1}{4} \cdot \frac{\left(x-scale \cdot x-scale\right) \cdot \sqrt{0}}{\left(a \cdot a\right) \cdot \left(b \cdot \color{blue}{b}\right)} \]

       4. pow2 N/A

          \[\leadsto \frac{1}{4} \cdot \frac{\left(x-scale \cdot x-scale\right) \cdot \sqrt{0}}{{a}^{2} \cdot \left(\color{blue}{b} \cdot b\right)} \]

       5. pow2 N/A

          \[\leadsto \frac{1}{4} \cdot \frac{\left(x-scale \cdot x-scale\right) \cdot \sqrt{0}}{{a}^{2} \cdot {b}^{\color{blue}{2}}} \]

       6. pow-prod-down N/A

          \[\leadsto \frac{1}{4} \cdot \frac{\left(x-scale \cdot x-scale\right) \cdot \sqrt{0}}{{\left(a \cdot b\right)}^{\color{blue}{2}}} \]

       7. lower-pow.f64 N/A

          \[\leadsto \frac{1}{4} \cdot \frac{\left(x-scale \cdot x-scale\right) \cdot \sqrt{0}}{{\left(a \cdot b\right)}^{\color{blue}{2}}} \]

       8. lower-*.f64 14.1

          \[\leadsto 0.25 \cdot \frac{\left(x-scale \cdot x-scale\right) \cdot \sqrt{0}}{{\left(a \cdot b\right)}^{2}} \]

   12. Applied rewrites 14.1%

       \[\leadsto 0.25 \cdot \frac{\left(x-scale \cdot x-scale\right) \cdot \sqrt{0}}{{\left(a \cdot b\right)}^{\color{blue}{2}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 14.1% accurate, 22.5× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ 0.25 \cdot \frac{\left(x-scale \cdot x-scale\right) \cdot \sqrt{0}}{{\left(a\_m \cdot b\right)}^{2}} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (* 0.25 (/ (* (* x-scale x-scale) (sqrt 0.0)) (pow (* a_m b) 2.0))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	return 0.25 * (((x_45_scale * x_45_scale) * sqrt(0.0)) / pow((a_m * b), 2.0));
}

Fortran

a_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a_m, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a_m
    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(0.0d0)) / ((a_m * b) ** 2.0d0))
end function

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	return 0.25 * (((x_45_scale * x_45_scale) * Math.sqrt(0.0)) / Math.pow((a_m * b), 2.0));
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle, x_45_scale, y_45_scale):
	return 0.25 * (((x_45_scale * x_45_scale) * math.sqrt(0.0)) / math.pow((a_m * b), 2.0))

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	return Float64(0.25 * Float64(Float64(Float64(x_45_scale * x_45_scale) * sqrt(0.0)) / (Float64(a_m * b) ^ 2.0)))
end

MATLAB

a_m = abs(a);
function tmp = code(a_m, b, angle, x_45_scale, y_45_scale)
	tmp = 0.25 * (((x_45_scale * x_45_scale) * sqrt(0.0)) / ((a_m * b) ^ 2.0));
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := N[(0.25 * N[(N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * N[Sqrt[0.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[(a$95$m * b), $MachinePrecision], 2.0], $MachinePrecision]), $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 angle around 0

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

4. Applied rewrites 0.1%

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

5. Taylor expanded in y-scale around 0

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

   1. lower-sqrt.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

7. Applied rewrites 0.8%

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

8. Taylor expanded in a around 0

   \[\leadsto \frac{1}{4} \cdot \frac{\left(x-scale \cdot x-scale\right) \cdot \sqrt{0}}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]
9. Step-by-step derivation

10. Applied rewrites 6.2%

    \[\leadsto 0.25 \cdot \frac{\left(x-scale \cdot x-scale\right) \cdot \sqrt{0}}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]
11. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \frac{1}{4} \cdot \frac{\left(x-scale \cdot x-scale\right) \cdot \sqrt{0}}{\left(a \cdot a\right) \cdot \color{blue}{\left(b \cdot b\right)}} \]

    2. lift-*.f64 N/A

       \[\leadsto \frac{1}{4} \cdot \frac{\left(x-scale \cdot x-scale\right) \cdot \sqrt{0}}{\left(a \cdot a\right) \cdot \left(\color{blue}{b} \cdot b\right)} \]

    3. lift-*.f64 N/A

       \[\leadsto \frac{1}{4} \cdot \frac{\left(x-scale \cdot x-scale\right) \cdot \sqrt{0}}{\left(a \cdot a\right) \cdot \left(b \cdot \color{blue}{b}\right)} \]

    4. pow2 N/A

       \[\leadsto \frac{1}{4} \cdot \frac{\left(x-scale \cdot x-scale\right) \cdot \sqrt{0}}{{a}^{2} \cdot \left(\color{blue}{b} \cdot b\right)} \]

    5. pow2 N/A

       \[\leadsto \frac{1}{4} \cdot \frac{\left(x-scale \cdot x-scale\right) \cdot \sqrt{0}}{{a}^{2} \cdot {b}^{\color{blue}{2}}} \]

    6. pow-prod-down N/A

       \[\leadsto \frac{1}{4} \cdot \frac{\left(x-scale \cdot x-scale\right) \cdot \sqrt{0}}{{\left(a \cdot b\right)}^{\color{blue}{2}}} \]

    7. lower-pow.f64 N/A

       \[\leadsto \frac{1}{4} \cdot \frac{\left(x-scale \cdot x-scale\right) \cdot \sqrt{0}}{{\left(a \cdot b\right)}^{\color{blue}{2}}} \]

    8. lower-*.f64 14.1

       \[\leadsto 0.25 \cdot \frac{\left(x-scale \cdot x-scale\right) \cdot \sqrt{0}}{{\left(a \cdot b\right)}^{2}} \]

12. Applied rewrites 14.1%

    \[\leadsto 0.25 \cdot \frac{\left(x-scale \cdot x-scale\right) \cdot \sqrt{0}}{{\left(a \cdot b\right)}^{\color{blue}{2}}} \]
13. Add Preprocessing

Alternative 4: 6.2% accurate, 31.8× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ 0.25 \cdot \frac{\left(x-scale \cdot x-scale\right) \cdot \sqrt{0}}{\left(a\_m \cdot a\_m\right) \cdot \left(b \cdot b\right)} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (* 0.25 (/ (* (* x-scale x-scale) (sqrt 0.0)) (* (* a_m a_m) (* b b)))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	return 0.25 * (((x_45_scale * x_45_scale) * sqrt(0.0)) / ((a_m * a_m) * (b * b)));
}

Fortran

a_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a_m, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a_m
    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(0.0d0)) / ((a_m * a_m) * (b * b)))
end function

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	return 0.25 * (((x_45_scale * x_45_scale) * Math.sqrt(0.0)) / ((a_m * a_m) * (b * b)));
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle, x_45_scale, y_45_scale):
	return 0.25 * (((x_45_scale * x_45_scale) * math.sqrt(0.0)) / ((a_m * a_m) * (b * b)))

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	return Float64(0.25 * Float64(Float64(Float64(x_45_scale * x_45_scale) * sqrt(0.0)) / Float64(Float64(a_m * a_m) * Float64(b * b))))
end

MATLAB

a_m = abs(a);
function tmp = code(a_m, b, angle, x_45_scale, y_45_scale)
	tmp = 0.25 * (((x_45_scale * x_45_scale) * sqrt(0.0)) / ((a_m * a_m) * (b * b)));
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := N[(0.25 * N[(N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * N[Sqrt[0.0], $MachinePrecision]), $MachinePrecision] / N[(N[(a$95$m * a$95$m), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $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 angle around 0

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

4. Applied rewrites 0.1%

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

5. Taylor expanded in y-scale around 0

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

   1. lower-sqrt.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

7. Applied rewrites 0.8%

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

8. Taylor expanded in a around 0

   \[\leadsto \frac{1}{4} \cdot \frac{\left(x-scale \cdot x-scale\right) \cdot \sqrt{0}}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]
9. Step-by-step derivation

10. Applied rewrites 6.2%

    \[\leadsto 0.25 \cdot \frac{\left(x-scale \cdot x-scale\right) \cdot \sqrt{0}}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025142 
(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))))