Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 25.7% → 77.5%

Time: 12.7s

Alternatives: 7

Speedup: 20.4×

• 33.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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
         (/
          (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) t_1) t_2) x-scale)
          y-scale)))
   (-
    (* t_3 t_3)
    (*
     (*
      4.0
      (/ (/ (+ (pow (* a t_1) 2.0) (pow (* b t_2) 2.0)) x-scale) x-scale))
     (/ (/ (+ (pow (* a t_2) 2.0) (pow (* b t_1) 2.0)) y-scale) y-scale)))))

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 = ((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale;
	return (t_3 * t_3) - ((4.0 * (((pow((a * t_1), 2.0) + pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale)) * (((pow((a * t_2), 2.0) + pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale));
}

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 = ((((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale;
	return (t_3 * t_3) - ((4.0 * (((Math.pow((a * t_1), 2.0) + Math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale)) * (((Math.pow((a * t_2), 2.0) + Math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale));
}

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 = ((((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale
	return (t_3 * t_3) - ((4.0 * (((math.pow((a * t_1), 2.0) + math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale)) * (((math.pow((a * t_2), 2.0) + math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale))

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(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale)
	return Float64(Float64(t_3 * t_3) - Float64(Float64(4.0 * Float64(Float64(Float64((Float64(a * t_1) ^ 2.0) + (Float64(b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale)) * Float64(Float64(Float64((Float64(a * t_2) ^ 2.0) + (Float64(b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale)))
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 = ((((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale;
	tmp = (t_3 * t_3) - ((4.0 * (((((a * t_1) ^ 2.0) + ((b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale)) * (((((a * t_2) ^ 2.0) + ((b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale));
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[(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]}, N[(N[(t$95$3 * t$95$3), $MachinePrecision] - N[(N[(4.0 * 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]), $MachinePrecision] * 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]), $MachinePrecision]), $MachinePrecision]]]]]

Initial Program: 25.7% accurate, 1.0× speedup

Math

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

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
         (/
          (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) t_1) t_2) x-scale)
          y-scale)))
   (-
    (* t_3 t_3)
    (*
     (*
      4.0
      (/ (/ (+ (pow (* a t_1) 2.0) (pow (* b t_2) 2.0)) x-scale) x-scale))
     (/ (/ (+ (pow (* a t_2) 2.0) (pow (* b t_1) 2.0)) y-scale) y-scale)))))

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 = ((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale;
	return (t_3 * t_3) - ((4.0 * (((pow((a * t_1), 2.0) + pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale)) * (((pow((a * t_2), 2.0) + pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale));
}

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 = ((((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale;
	return (t_3 * t_3) - ((4.0 * (((Math.pow((a * t_1), 2.0) + Math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale)) * (((Math.pow((a * t_2), 2.0) + Math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale));
}

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 = ((((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale
	return (t_3 * t_3) - ((4.0 * (((math.pow((a * t_1), 2.0) + math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale)) * (((math.pow((a * t_2), 2.0) + math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale))

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(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale)
	return Float64(Float64(t_3 * t_3) - Float64(Float64(4.0 * Float64(Float64(Float64((Float64(a * t_1) ^ 2.0) + (Float64(b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale)) * Float64(Float64(Float64((Float64(a * t_2) ^ 2.0) + (Float64(b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale)))
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 = ((((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale;
	tmp = (t_3 * t_3) - ((4.0 * (((((a * t_1) ^ 2.0) + ((b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale)) * (((((a * t_2) ^ 2.0) + ((b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale));
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[(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]}, N[(N[(t$95$3 * t$95$3), $MachinePrecision] - N[(N[(4.0 * 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]), $MachinePrecision] * 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]), $MachinePrecision]), $MachinePrecision]]]]]

Alternative 1: 77.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \frac{\frac{-2 \cdot \left(\left(a\_m \cdot a\_m\right) \cdot \left(\cos t\_0 \cdot \sin t\_0\right)\right)}{x-scale}}{y-scale}\\ \mathbf{if}\;a\_m \leq 2.15 \cdot 10^{-260}:\\ \;\;\;\;t\_1 \cdot t\_1 - \left(4 \cdot \frac{\frac{b \cdot b}{x-scale}}{x-scale}\right) \cdot \frac{\frac{a\_m \cdot a\_m}{y-scale}}{y-scale}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a\_m \cdot b\right) \cdot \left(a\_m \cdot b\right)}{{\left(x-scale \cdot y-scale\right)}^{2}} \cdot -4\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI)))
        (t_1
         (/
          (/ (* -2.0 (* (* a_m a_m) (* (cos t_0) (sin t_0)))) x-scale)
          y-scale)))
   (if (<= a_m 2.15e-260)
     (-
      (* t_1 t_1)
      (*
       (* 4.0 (/ (/ (* b b) x-scale) x-scale))
       (/ (/ (* a_m a_m) y-scale) y-scale)))
     (* (/ (* (* a_m b) (* a_m b)) (pow (* x-scale y-scale) 2.0)) -4.0))))

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 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_1 = ((-2.0 * ((a_m * a_m) * (cos(t_0) * sin(t_0)))) / x_45_scale) / y_45_scale;
	double tmp;
	if (a_m <= 2.15e-260) {
		tmp = (t_1 * t_1) - ((4.0 * (((b * b) / x_45_scale) / x_45_scale)) * (((a_m * a_m) / y_45_scale) / y_45_scale));
	} else {
		tmp = (((a_m * b) * (a_m * b)) / pow((x_45_scale * y_45_scale), 2.0)) * -4.0;
	}
	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 = 0.005555555555555556 * (angle * Math.PI);
	double t_1 = ((-2.0 * ((a_m * a_m) * (Math.cos(t_0) * Math.sin(t_0)))) / x_45_scale) / y_45_scale;
	double tmp;
	if (a_m <= 2.15e-260) {
		tmp = (t_1 * t_1) - ((4.0 * (((b * b) / x_45_scale) / x_45_scale)) * (((a_m * a_m) / y_45_scale) / y_45_scale));
	} else {
		tmp = (((a_m * b) * (a_m * b)) / Math.pow((x_45_scale * y_45_scale), 2.0)) * -4.0;
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle, x_45_scale, y_45_scale):
	t_0 = 0.005555555555555556 * (angle * math.pi)
	t_1 = ((-2.0 * ((a_m * a_m) * (math.cos(t_0) * math.sin(t_0)))) / x_45_scale) / y_45_scale
	tmp = 0
	if a_m <= 2.15e-260:
		tmp = (t_1 * t_1) - ((4.0 * (((b * b) / x_45_scale) / x_45_scale)) * (((a_m * a_m) / y_45_scale) / y_45_scale))
	else:
		tmp = (((a_m * b) * (a_m * b)) / math.pow((x_45_scale * y_45_scale), 2.0)) * -4.0
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_1 = Float64(Float64(Float64(-2.0 * Float64(Float64(a_m * a_m) * Float64(cos(t_0) * sin(t_0)))) / x_45_scale) / y_45_scale)
	tmp = 0.0
	if (a_m <= 2.15e-260)
		tmp = Float64(Float64(t_1 * t_1) - Float64(Float64(4.0 * Float64(Float64(Float64(b * b) / x_45_scale) / x_45_scale)) * Float64(Float64(Float64(a_m * a_m) / y_45_scale) / y_45_scale)));
	else
		tmp = Float64(Float64(Float64(Float64(a_m * b) * Float64(a_m * b)) / (Float64(x_45_scale * y_45_scale) ^ 2.0)) * -4.0);
	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 = 0.005555555555555556 * (angle * pi);
	t_1 = ((-2.0 * ((a_m * a_m) * (cos(t_0) * sin(t_0)))) / x_45_scale) / y_45_scale;
	tmp = 0.0;
	if (a_m <= 2.15e-260)
		tmp = (t_1 * t_1) - ((4.0 * (((b * b) / x_45_scale) / x_45_scale)) * (((a_m * a_m) / y_45_scale) / y_45_scale));
	else
		tmp = (((a_m * b) * (a_m * b)) / ((x_45_scale * y_45_scale) ^ 2.0)) * -4.0;
	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[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * N[(N[(a$95$m * a$95$m), $MachinePrecision] * N[(N[Cos[t$95$0], $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]}, If[LessEqual[a$95$m, 2.15e-260], N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(N[(4.0 * N[(N[(N[(b * b), $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(a$95$m * a$95$m), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(a$95$m * b), $MachinePrecision] * N[(a$95$m * b), $MachinePrecision]), $MachinePrecision] / N[Power[N[(x$45$scale * y$45$scale), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if a < 2.15000000000000011e-260

   1. Initial program 44.3%

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

   2. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 43.9

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

   4. Applied rewrites 43.9%

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

   5. Taylor expanded in angle around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 43.9

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

   7. Applied rewrites 43.9%

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

   8. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-PI.f64 52.1

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

   10. Applied rewrites 52.1%

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

   11. Taylor expanded in a around inf

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-cos.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. lift-PI.f64 N/A

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

       10. lower-sin.f64 N/A

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

       11. lower-*.f64 N/A

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

       12. lower-*.f64 N/A

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

       13. lift-PI.f64 52.2

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

   13. Applied rewrites 52.2%

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

   if 2.15000000000000011e-260 < a

   1. Initial program 24.0%

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

   2. Taylor expanded in angle around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \cdot \color{blue}{-4} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \cdot \color{blue}{-4} \]

   4. Applied rewrites 49.1%

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

   5. Taylor expanded in a around 0

      \[\leadsto \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \cdot -4 \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \cdot -4 \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \cdot -4 \]

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 48.5

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

   7. Applied rewrites 48.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-*.f64 61.4

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

   9. Applied rewrites 61.4%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. unswap-sqr N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 77.8

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

   11. Applied rewrites 77.8%

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

Alternative 2: 75.7% accurate, 14.6× speedup

Math

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

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (* (/ (* (* a_m b) (* a_m b)) (pow (* x-scale y-scale) 2.0)) -4.0))

C

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

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 = (((a_m * b) * (a_m * b)) / ((x_45scale * y_45scale) ** 2.0d0)) * (-4.0d0)
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 (((a_m * b) * (a_m * b)) / Math.pow((x_45_scale * y_45_scale), 2.0)) * -4.0;
}

Python

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

Julia

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

MATLAB

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

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := N[(N[(N[(N[(a$95$m * b), $MachinePrecision] * N[(a$95$m * b), $MachinePrecision]), $MachinePrecision] / N[Power[N[(x$45$scale * y$45$scale), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]

Derivation

1. Initial program 25.7%

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

2. Taylor expanded in angle around 0

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

   1. *-commutative N/A

      \[\leadsto \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \cdot \color{blue}{-4} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \cdot \color{blue}{-4} \]

4. Applied rewrites 48.0%

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

5. Taylor expanded in a around 0

   \[\leadsto \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \cdot -4 \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \cdot -4 \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \cdot -4 \]

   3. pow2 N/A

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

   4. lift-*.f64 N/A

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

   5. pow2 N/A

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

   6. lift-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. pow2 N/A

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

   9. lift-*.f64 N/A

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

   10. pow2 N/A

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

   11. lift-*.f64 47.4

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

7. Applied rewrites 47.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. pow2 N/A

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

   5. pow2 N/A

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

   6. pow-prod-down N/A

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

   7. lower-pow.f64 N/A

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

   8. lift-*.f64 60.3

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

9. Applied rewrites 60.3%

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

    1. lift-*.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. unswap-sqr N/A

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

    5. lower-*.f64 N/A

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

    6. lower-*.f64 N/A

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

    7. lower-*.f64 77.5

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

11. Applied rewrites 77.5%

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

Alternative 3: 67.2% accurate, 13.2× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;a\_m \leq 8.2 \cdot 10^{+50}:\\ \;\;\;\;\frac{a\_m \cdot \left(a\_m \cdot \left(b \cdot b\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-4 \cdot \left(a\_m \cdot \frac{a\_m}{\left(\left(x-scale \cdot x-scale\right) \cdot y-scale\right) \cdot y-scale}\right)\right) \cdot b\right) \cdot b\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (if (<= a_m 8.2e+50)
   (* (/ (* a_m (* a_m (* b b))) (pow (* x-scale y-scale) 2.0)) -4.0)
   (*
    (* (* -4.0 (* a_m (/ a_m (* (* (* x-scale x-scale) y-scale) y-scale)))) b)
    b)))

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 <= 8.2e+50) {
		tmp = ((a_m * (a_m * (b * b))) / pow((x_45_scale * y_45_scale), 2.0)) * -4.0;
	} else {
		tmp = ((-4.0 * (a_m * (a_m / (((x_45_scale * x_45_scale) * y_45_scale) * y_45_scale)))) * b) * b;
	}
	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 <= 8.2d+50) then
        tmp = ((a_m * (a_m * (b * b))) / ((x_45scale * y_45scale) ** 2.0d0)) * (-4.0d0)
    else
        tmp = (((-4.0d0) * (a_m * (a_m / (((x_45scale * x_45scale) * y_45scale) * y_45scale)))) * b) * b
    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 <= 8.2e+50) {
		tmp = ((a_m * (a_m * (b * b))) / Math.pow((x_45_scale * y_45_scale), 2.0)) * -4.0;
	} else {
		tmp = ((-4.0 * (a_m * (a_m / (((x_45_scale * x_45_scale) * y_45_scale) * y_45_scale)))) * b) * b;
	}
	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 <= 8.2e+50:
		tmp = ((a_m * (a_m * (b * b))) / math.pow((x_45_scale * y_45_scale), 2.0)) * -4.0
	else:
		tmp = ((-4.0 * (a_m * (a_m / (((x_45_scale * x_45_scale) * y_45_scale) * y_45_scale)))) * b) * b
	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 <= 8.2e+50)
		tmp = Float64(Float64(Float64(a_m * Float64(a_m * Float64(b * b))) / (Float64(x_45_scale * y_45_scale) ^ 2.0)) * -4.0);
	else
		tmp = Float64(Float64(Float64(-4.0 * Float64(a_m * Float64(a_m / Float64(Float64(Float64(x_45_scale * x_45_scale) * y_45_scale) * y_45_scale)))) * b) * b);
	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 <= 8.2e+50)
		tmp = ((a_m * (a_m * (b * b))) / ((x_45_scale * y_45_scale) ^ 2.0)) * -4.0;
	else
		tmp = ((-4.0 * (a_m * (a_m / (((x_45_scale * x_45_scale) * y_45_scale) * y_45_scale)))) * b) * b;
	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, 8.2e+50], N[(N[(N[(a$95$m * N[(a$95$m * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[(x$45$scale * y$45$scale), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision], N[(N[(N[(-4.0 * N[(a$95$m * N[(a$95$m / N[(N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 8.2000000000000002e50

   1. Initial program 38.6%

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

   2. Taylor expanded in angle around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \cdot \color{blue}{-4} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \cdot \color{blue}{-4} \]

   4. Applied rewrites 50.0%

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

   5. Taylor expanded in a around 0

      \[\leadsto \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \cdot -4 \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \cdot -4 \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \cdot -4 \]

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 49.5

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

   7. Applied rewrites 49.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-*.f64 62.7

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

   9. Applied rewrites 62.7%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. pow2 N/A

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

       5. associate-*l* N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 68.1

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

   11. Applied rewrites 68.1%

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

   if 8.2000000000000002e50 < a

   1. Initial program 6.2%

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

   2. Taylor expanded in b around 0

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

   4. Applied rewrites 34.0%

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

   5. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 45.0

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

   7. Applied rewrites 45.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

   9. Applied rewrites 65.7%

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

Alternative 4: 66.2% accurate, 17.7× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;y-scale \leq 1.75 \cdot 10^{+121}:\\ \;\;\;\;\left(\left(-4 \cdot \left(a\_m \cdot \frac{a\_m}{\left(\left(x-scale \cdot x-scale\right) \cdot y-scale\right) \cdot y-scale}\right)\right) \cdot b\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a\_m \cdot a\_m\right) \cdot \left(b \cdot b\right)}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)} \cdot -4\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (if (<= y-scale 1.75e+121)
   (*
    (* (* -4.0 (* a_m (/ a_m (* (* (* x-scale x-scale) y-scale) y-scale)))) b)
    b)
   (*
    (/ (* (* a_m a_m) (* b b)) (* (* x-scale y-scale) (* x-scale y-scale)))
    -4.0)))

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 (y_45_scale <= 1.75e+121) {
		tmp = ((-4.0 * (a_m * (a_m / (((x_45_scale * x_45_scale) * y_45_scale) * y_45_scale)))) * b) * b;
	} else {
		tmp = (((a_m * a_m) * (b * b)) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale))) * -4.0;
	}
	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 (y_45scale <= 1.75d+121) then
        tmp = (((-4.0d0) * (a_m * (a_m / (((x_45scale * x_45scale) * y_45scale) * y_45scale)))) * b) * b
    else
        tmp = (((a_m * a_m) * (b * b)) / ((x_45scale * y_45scale) * (x_45scale * y_45scale))) * (-4.0d0)
    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 (y_45_scale <= 1.75e+121) {
		tmp = ((-4.0 * (a_m * (a_m / (((x_45_scale * x_45_scale) * y_45_scale) * y_45_scale)))) * b) * b;
	} else {
		tmp = (((a_m * a_m) * (b * b)) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale))) * -4.0;
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if y_45_scale <= 1.75e+121:
		tmp = ((-4.0 * (a_m * (a_m / (((x_45_scale * x_45_scale) * y_45_scale) * y_45_scale)))) * b) * b
	else:
		tmp = (((a_m * a_m) * (b * b)) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale))) * -4.0
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (y_45_scale <= 1.75e+121)
		tmp = Float64(Float64(Float64(-4.0 * Float64(a_m * Float64(a_m / Float64(Float64(Float64(x_45_scale * x_45_scale) * y_45_scale) * y_45_scale)))) * b) * b);
	else
		tmp = Float64(Float64(Float64(Float64(a_m * a_m) * Float64(b * b)) / Float64(Float64(x_45_scale * y_45_scale) * Float64(x_45_scale * y_45_scale))) * -4.0);
	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 (y_45_scale <= 1.75e+121)
		tmp = ((-4.0 * (a_m * (a_m / (((x_45_scale * x_45_scale) * y_45_scale) * y_45_scale)))) * b) * b;
	else
		tmp = (((a_m * a_m) * (b * b)) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale))) * -4.0;
	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[y$45$scale, 1.75e+121], N[(N[(N[(-4.0 * N[(a$95$m * N[(a$95$m / N[(N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision], N[(N[(N[(N[(a$95$m * a$95$m), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[(x$45$scale * y$45$scale), $MachinePrecision] * N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 1.75e121

   1. Initial program 23.7%

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

   2. Taylor expanded in b around 0

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

   4. Applied rewrites 41.3%

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

   5. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 49.2

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

   7. Applied rewrites 49.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

   9. Applied rewrites 67.6%

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

   if 1.75e121 < y-scale

   1. Initial program 37.3%

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

   2. Taylor expanded in angle around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \cdot \color{blue}{-4} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \cdot \color{blue}{-4} \]

   4. Applied rewrites 41.6%

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

   5. Taylor expanded in a around 0

      \[\leadsto \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \cdot -4 \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \cdot -4 \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \cdot -4 \]

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 41.0

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

   7. Applied rewrites 41.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-*.f64 57.9

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

   9. Applied rewrites 57.9%

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

       1. lift-*.f64 N/A

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

       2. lift-pow.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 57.9

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

   11. Applied rewrites 57.9%

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

Alternative 5: 60.3% accurate, 17.7× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;a\_m \leq 9 \cdot 10^{+50}:\\ \;\;\;\;\frac{\left(a\_m \cdot a\_m\right) \cdot \left(b \cdot b\right)}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(a\_m \cdot a\_m\right) \cdot b\right) \cdot b}{\left(\left(x-scale \cdot x-scale\right) \cdot y-scale\right) \cdot y-scale} \cdot -4\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (if (<= a_m 9e+50)
   (*
    (/ (* (* a_m a_m) (* b b)) (* (* x-scale y-scale) (* x-scale y-scale)))
    -4.0)
   (*
    (/ (* (* (* a_m a_m) b) b) (* (* (* x-scale x-scale) y-scale) y-scale))
    -4.0)))

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 <= 9e+50) {
		tmp = (((a_m * a_m) * (b * b)) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale))) * -4.0;
	} else {
		tmp = ((((a_m * a_m) * b) * b) / (((x_45_scale * x_45_scale) * y_45_scale) * y_45_scale)) * -4.0;
	}
	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 <= 9d+50) then
        tmp = (((a_m * a_m) * (b * b)) / ((x_45scale * y_45scale) * (x_45scale * y_45scale))) * (-4.0d0)
    else
        tmp = ((((a_m * a_m) * b) * b) / (((x_45scale * x_45scale) * y_45scale) * y_45scale)) * (-4.0d0)
    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 <= 9e+50) {
		tmp = (((a_m * a_m) * (b * b)) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale))) * -4.0;
	} else {
		tmp = ((((a_m * a_m) * b) * b) / (((x_45_scale * x_45_scale) * y_45_scale) * y_45_scale)) * -4.0;
	}
	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 <= 9e+50:
		tmp = (((a_m * a_m) * (b * b)) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale))) * -4.0
	else:
		tmp = ((((a_m * a_m) * b) * b) / (((x_45_scale * x_45_scale) * y_45_scale) * y_45_scale)) * -4.0
	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 <= 9e+50)
		tmp = Float64(Float64(Float64(Float64(a_m * a_m) * Float64(b * b)) / Float64(Float64(x_45_scale * y_45_scale) * Float64(x_45_scale * y_45_scale))) * -4.0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(a_m * a_m) * b) * b) / Float64(Float64(Float64(x_45_scale * x_45_scale) * y_45_scale) * y_45_scale)) * -4.0);
	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 <= 9e+50)
		tmp = (((a_m * a_m) * (b * b)) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale))) * -4.0;
	else
		tmp = ((((a_m * a_m) * b) * b) / (((x_45_scale * x_45_scale) * y_45_scale) * y_45_scale)) * -4.0;
	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, 9e+50], N[(N[(N[(N[(a$95$m * a$95$m), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[(x$45$scale * y$45$scale), $MachinePrecision] * N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision], N[(N[(N[(N[(N[(a$95$m * a$95$m), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] / N[(N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 9.00000000000000027e50

   1. Initial program 38.6%

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

   2. Taylor expanded in angle around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \cdot \color{blue}{-4} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \cdot \color{blue}{-4} \]

   4. Applied rewrites 50.0%

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

   5. Taylor expanded in a around 0

      \[\leadsto \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \cdot -4 \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \cdot -4 \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \cdot -4 \]

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 49.5

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

   7. Applied rewrites 49.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-*.f64 62.7

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

   9. Applied rewrites 62.7%

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

       1. lift-*.f64 N/A

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

       2. lift-pow.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 62.7

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

   11. Applied rewrites 62.7%

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

   if 9.00000000000000027e50 < a

   1. Initial program 6.2%

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

   2. Taylor expanded in angle around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \cdot \color{blue}{-4} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \cdot \color{blue}{-4} \]

   4. Applied rewrites 45.0%

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

   5. Taylor expanded in a around 0

      \[\leadsto \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \cdot -4 \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \cdot -4 \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \cdot -4 \]

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 44.3

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

   7. Applied rewrites 44.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-*.f64 56.7

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

   9. Applied rewrites 56.7%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. pow2 N/A

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

       5. associate-*r* N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 64.4

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

       10. lift-*.f64 N/A

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

       11. lift-pow.f64 N/A

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

       12. unpow-prod-down N/A

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

       13. pow2 N/A

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

       14. associate-*r* N/A

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

       15. lower-*.f64 N/A

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

       16. lower-*.f64 N/A

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

       17. pow2 N/A

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

       18. lift-*.f64 55.9

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

   11. Applied rewrites 55.9%

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

Alternative 6: 60.0% accurate, 20.4× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \frac{\left(a\_m \cdot a\_m\right) \cdot \left(b \cdot b\right)}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)} \cdot -4 \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (*
  (/ (* (* a_m a_m) (* b b)) (* (* x-scale y-scale) (* x-scale y-scale)))
  -4.0))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	return (((a_m * a_m) * (b * b)) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale))) * -4.0;
}

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 = (((a_m * a_m) * (b * b)) / ((x_45scale * y_45scale) * (x_45scale * y_45scale))) * (-4.0d0)
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 (((a_m * a_m) * (b * b)) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale))) * -4.0;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle, x_45_scale, y_45_scale):
	return (((a_m * a_m) * (b * b)) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale))) * -4.0

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	return Float64(Float64(Float64(Float64(a_m * a_m) * Float64(b * b)) / Float64(Float64(x_45_scale * y_45_scale) * Float64(x_45_scale * y_45_scale))) * -4.0)
end

MATLAB

a_m = abs(a);
function tmp = code(a_m, b, angle, x_45_scale, y_45_scale)
	tmp = (((a_m * a_m) * (b * b)) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale))) * -4.0;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := N[(N[(N[(N[(a$95$m * a$95$m), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[(x$45$scale * y$45$scale), $MachinePrecision] * N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]

Derivation

1. Initial program 25.7%

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

2. Taylor expanded in angle around 0

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

   1. *-commutative N/A

      \[\leadsto \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \cdot \color{blue}{-4} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \cdot \color{blue}{-4} \]

4. Applied rewrites 48.0%

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

5. Taylor expanded in a around 0

   \[\leadsto \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \cdot -4 \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \cdot -4 \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \cdot -4 \]

   3. pow2 N/A

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

   4. lift-*.f64 N/A

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

   5. pow2 N/A

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

   6. lift-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. pow2 N/A

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

   9. lift-*.f64 N/A

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

   10. pow2 N/A

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

   11. lift-*.f64 47.4

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

7. Applied rewrites 47.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. pow2 N/A

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

   5. pow2 N/A

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

   6. pow-prod-down N/A

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

   7. lower-pow.f64 N/A

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

   8. lift-*.f64 60.3

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

9. Applied rewrites 60.3%

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

    1. lift-*.f64 N/A

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

    2. lift-pow.f64 N/A

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

    3. unpow2 N/A

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

    4. lower-*.f64 N/A

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

    5. lift-*.f64 N/A

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

    6. lift-*.f64 60.3

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

11. Applied rewrites 60.3%

    \[\leadsto \frac{\left(a \cdot a\right) \cdot \left(b \cdot b\right)}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)} \cdot -4 \]
12. Add Preprocessing

Alternative 7: 53.4% accurate, 20.4× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \frac{\left(a\_m \cdot a\_m\right) \cdot \left(b \cdot b\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)} \cdot -4 \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (*
  (/ (* (* a_m a_m) (* b b)) (* x-scale (* x-scale (* y-scale y-scale))))
  -4.0))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	return (((a_m * a_m) * (b * b)) / (x_45_scale * (x_45_scale * (y_45_scale * y_45_scale)))) * -4.0;
}

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 = (((a_m * a_m) * (b * b)) / (x_45scale * (x_45scale * (y_45scale * y_45scale)))) * (-4.0d0)
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 (((a_m * a_m) * (b * b)) / (x_45_scale * (x_45_scale * (y_45_scale * y_45_scale)))) * -4.0;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle, x_45_scale, y_45_scale):
	return (((a_m * a_m) * (b * b)) / (x_45_scale * (x_45_scale * (y_45_scale * y_45_scale)))) * -4.0

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	return Float64(Float64(Float64(Float64(a_m * a_m) * Float64(b * b)) / Float64(x_45_scale * Float64(x_45_scale * Float64(y_45_scale * y_45_scale)))) * -4.0)
end

MATLAB

a_m = abs(a);
function tmp = code(a_m, b, angle, x_45_scale, y_45_scale)
	tmp = (((a_m * a_m) * (b * b)) / (x_45_scale * (x_45_scale * (y_45_scale * y_45_scale)))) * -4.0;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := N[(N[(N[(N[(a$95$m * a$95$m), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * N[(x$45$scale * N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]

Derivation

1. Initial program 25.7%

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

2. Taylor expanded in angle around 0

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

   1. *-commutative N/A

      \[\leadsto \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \cdot \color{blue}{-4} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \cdot \color{blue}{-4} \]

4. Applied rewrites 48.0%

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

5. Taylor expanded in a around 0

   \[\leadsto \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \cdot -4 \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \cdot -4 \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \cdot -4 \]

   3. pow2 N/A

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

   4. lift-*.f64 N/A

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

   5. pow2 N/A

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

   6. lift-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. pow2 N/A

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

   9. lift-*.f64 N/A

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

   10. pow2 N/A

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

   11. lift-*.f64 47.4

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

7. Applied rewrites 47.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. pow2 N/A

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

   5. pow2 N/A

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

   6. pow-prod-down N/A

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

   7. lower-pow.f64 N/A

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

   8. lift-*.f64 60.3

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

9. Applied rewrites 60.3%

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

    1. lift-*.f64 N/A

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

    2. lift-pow.f64 N/A

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

    3. unpow-prod-down N/A

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

    4. pow2 N/A

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

    5. associate-*l* N/A

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

    6. lower-*.f64 N/A

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

    7. lower-*.f64 N/A

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

    8. pow2 N/A

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

    9. lift-*.f64 53.4

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

11. Applied rewrites 53.4%

    \[\leadsto \frac{\left(a \cdot a\right) \cdot \left(b \cdot b\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)} \cdot -4 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025120 
(FPCore (a b angle x-scale y-scale)
  :name "Simplification of discriminant from scale-rotated-ellipse"
  :precision binary64
  (- (* (/ (/ (* (* (* 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 (- (pow b 2.0) (pow a 2.0))) (sin (* (/ angle 180.0) PI))) (cos (* (/ angle 180.0) PI))) x-scale) y-scale)) (* (* 4.0 (/ (/ (+ (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))))