Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 24.4% → 52.1%

Time: 30.0s

Alternatives: 8

Speedup: 6.7×

• 34.1% 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: 24.4% 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: 52.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ t_1 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_2 := \sin t\_0\\ t_3 := \frac{\frac{-2 \cdot \left({a\_m}^{2} \cdot \left(\cos t\_1 \cdot \sin t\_1\right)\right)}{x-scale}}{y-scale}\\ t_4 := \cos t\_0\\ \mathbf{if}\;a\_m \leq 1.7 \cdot 10^{-146}:\\ \;\;\;\;t\_3 \cdot t\_3 - \left(4 \cdot \frac{\frac{{\left(a\_m \cdot t\_2\right)}^{2} + {\left(b \cdot t\_4\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a\_m \cdot t\_4\right)}^{2} + {\left(b \cdot t\_2\right)}^{2}}{y-scale}}{y-scale}\\ \mathbf{else}:\\ \;\;\;\;{a\_m}^{2} \cdot \left({b}^{2} \cdot \frac{-4}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (/ angle 180.0) PI))
        (t_1 (* 0.005555555555555556 (* angle PI)))
        (t_2 (sin t_0))
        (t_3
         (/
          (/ (* -2.0 (* (pow a_m 2.0) (* (cos t_1) (sin t_1)))) x-scale)
          y-scale))
        (t_4 (cos t_0)))
   (if (<= a_m 1.7e-146)
     (-
      (* t_3 t_3)
      (*
       (*
        4.0
        (/ (/ (+ (pow (* a_m t_2) 2.0) (pow (* b t_4) 2.0)) x-scale) x-scale))
       (/ (/ (+ (pow (* a_m t_4) 2.0) (pow (* b t_2) 2.0)) y-scale) y-scale)))
     (*
      (pow a_m 2.0)
      (* (pow b 2.0) (/ -4.0 (* (pow x-scale 2.0) (pow y-scale 2.0))))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * ((double) M_PI);
	double t_1 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_2 = sin(t_0);
	double t_3 = ((-2.0 * (pow(a_m, 2.0) * (cos(t_1) * sin(t_1)))) / x_45_scale) / y_45_scale;
	double t_4 = cos(t_0);
	double tmp;
	if (a_m <= 1.7e-146) {
		tmp = (t_3 * t_3) - ((4.0 * (((pow((a_m * t_2), 2.0) + pow((b * t_4), 2.0)) / x_45_scale) / x_45_scale)) * (((pow((a_m * t_4), 2.0) + pow((b * t_2), 2.0)) / y_45_scale) / y_45_scale));
	} else {
		tmp = pow(a_m, 2.0) * (pow(b, 2.0) * (-4.0 / (pow(x_45_scale, 2.0) * pow(y_45_scale, 2.0))));
	}
	return tmp;
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * Math.PI;
	double t_1 = 0.005555555555555556 * (angle * Math.PI);
	double t_2 = Math.sin(t_0);
	double t_3 = ((-2.0 * (Math.pow(a_m, 2.0) * (Math.cos(t_1) * Math.sin(t_1)))) / x_45_scale) / y_45_scale;
	double t_4 = Math.cos(t_0);
	double tmp;
	if (a_m <= 1.7e-146) {
		tmp = (t_3 * t_3) - ((4.0 * (((Math.pow((a_m * t_2), 2.0) + Math.pow((b * t_4), 2.0)) / x_45_scale) / x_45_scale)) * (((Math.pow((a_m * t_4), 2.0) + Math.pow((b * t_2), 2.0)) / y_45_scale) / y_45_scale));
	} else {
		tmp = Math.pow(a_m, 2.0) * (Math.pow(b, 2.0) * (-4.0 / (Math.pow(x_45_scale, 2.0) * Math.pow(y_45_scale, 2.0))));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle, x_45_scale, y_45_scale):
	t_0 = (angle / 180.0) * math.pi
	t_1 = 0.005555555555555556 * (angle * math.pi)
	t_2 = math.sin(t_0)
	t_3 = ((-2.0 * (math.pow(a_m, 2.0) * (math.cos(t_1) * math.sin(t_1)))) / x_45_scale) / y_45_scale
	t_4 = math.cos(t_0)
	tmp = 0
	if a_m <= 1.7e-146:
		tmp = (t_3 * t_3) - ((4.0 * (((math.pow((a_m * t_2), 2.0) + math.pow((b * t_4), 2.0)) / x_45_scale) / x_45_scale)) * (((math.pow((a_m * t_4), 2.0) + math.pow((b * t_2), 2.0)) / y_45_scale) / y_45_scale))
	else:
		tmp = math.pow(a_m, 2.0) * (math.pow(b, 2.0) * (-4.0 / (math.pow(x_45_scale, 2.0) * math.pow(y_45_scale, 2.0))))
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(angle / 180.0) * pi)
	t_1 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_2 = sin(t_0)
	t_3 = Float64(Float64(Float64(-2.0 * Float64((a_m ^ 2.0) * Float64(cos(t_1) * sin(t_1)))) / x_45_scale) / y_45_scale)
	t_4 = cos(t_0)
	tmp = 0.0
	if (a_m <= 1.7e-146)
		tmp = Float64(Float64(t_3 * t_3) - Float64(Float64(4.0 * Float64(Float64(Float64((Float64(a_m * t_2) ^ 2.0) + (Float64(b * t_4) ^ 2.0)) / x_45_scale) / x_45_scale)) * Float64(Float64(Float64((Float64(a_m * t_4) ^ 2.0) + (Float64(b * t_2) ^ 2.0)) / y_45_scale) / y_45_scale)));
	else
		tmp = Float64((a_m ^ 2.0) * Float64((b ^ 2.0) * Float64(-4.0 / Float64((x_45_scale ^ 2.0) * (y_45_scale ^ 2.0)))));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = (angle / 180.0) * pi;
	t_1 = 0.005555555555555556 * (angle * pi);
	t_2 = sin(t_0);
	t_3 = ((-2.0 * ((a_m ^ 2.0) * (cos(t_1) * sin(t_1)))) / x_45_scale) / y_45_scale;
	t_4 = cos(t_0);
	tmp = 0.0;
	if (a_m <= 1.7e-146)
		tmp = (t_3 * t_3) - ((4.0 * (((((a_m * t_2) ^ 2.0) + ((b * t_4) ^ 2.0)) / x_45_scale) / x_45_scale)) * (((((a_m * t_4) ^ 2.0) + ((b * t_2) ^ 2.0)) / y_45_scale) / y_45_scale));
	else
		tmp = (a_m ^ 2.0) * ((b ^ 2.0) * (-4.0 / ((x_45_scale ^ 2.0) * (y_45_scale ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(-2.0 * N[(N[Power[a$95$m, 2.0], $MachinePrecision] * N[(N[Cos[t$95$1], $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]}, Block[{t$95$4 = N[Cos[t$95$0], $MachinePrecision]}, If[LessEqual[a$95$m, 1.7e-146], N[(N[(t$95$3 * t$95$3), $MachinePrecision] - N[(N[(4.0 * N[(N[(N[(N[Power[N[(a$95$m * t$95$2), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$4), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Power[N[(a$95$m * t$95$4), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$2), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[a$95$m, 2.0], $MachinePrecision] * N[(N[Power[b, 2.0], $MachinePrecision] * N[(-4.0 / N[(N[Power[x$45$scale, 2.0], $MachinePrecision] * N[Power[y$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if a < 1.7e-146

   1. Initial program 43.0%

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

   2. Taylor expanded in a around inf

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

   4. Applied rewrites 45.0%

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

   5. Taylor expanded in a around inf

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

   7. Applied rewrites 54.5%

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

   if 1.7e-146 < a

   1. Initial program 17.9%

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

   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 40.4%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 42.2%

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

   8. Taylor expanded in angle around 0

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

   10. Applied rewrites 51.2%

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

Alternative 2: 52.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \frac{angle}{180} \cdot \pi\\ t_2 := \frac{\frac{-2 \cdot \left({a\_m}^{2} \cdot \left(\cos t\_0 \cdot \sin t\_0\right)\right)}{x-scale}}{y-scale}\\ \mathbf{if}\;a\_m \leq 1.7 \cdot 10^{-146}:\\ \;\;\;\;t\_2 \cdot t\_2 - \left(4 \cdot \frac{\frac{{b}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a\_m \cdot \cos t\_1\right)}^{2} + {\left(b \cdot \sin t\_1\right)}^{2}}{y-scale}}{y-scale}\\ \mathbf{else}:\\ \;\;\;\;{a\_m}^{2} \cdot \left({b}^{2} \cdot \frac{-4}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)\\ \end{array} \end{array} \]

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 (* (/ angle 180.0) PI))
        (t_2
         (/
          (/ (* -2.0 (* (pow a_m 2.0) (* (cos t_0) (sin t_0)))) x-scale)
          y-scale)))
   (if (<= a_m 1.7e-146)
     (-
      (* t_2 t_2)
      (*
       (* 4.0 (/ (/ (pow b 2.0) x-scale) x-scale))
       (/
        (/ (+ (pow (* a_m (cos t_1)) 2.0) (pow (* b (sin t_1)) 2.0)) y-scale)
        y-scale)))
     (*
      (pow a_m 2.0)
      (* (pow b 2.0) (/ -4.0 (* (pow x-scale 2.0) (pow y-scale 2.0))))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_1 = (angle / 180.0) * ((double) M_PI);
	double t_2 = ((-2.0 * (pow(a_m, 2.0) * (cos(t_0) * sin(t_0)))) / x_45_scale) / y_45_scale;
	double tmp;
	if (a_m <= 1.7e-146) {
		tmp = (t_2 * t_2) - ((4.0 * ((pow(b, 2.0) / x_45_scale) / x_45_scale)) * (((pow((a_m * cos(t_1)), 2.0) + pow((b * sin(t_1)), 2.0)) / y_45_scale) / y_45_scale));
	} else {
		tmp = pow(a_m, 2.0) * (pow(b, 2.0) * (-4.0 / (pow(x_45_scale, 2.0) * pow(y_45_scale, 2.0))));
	}
	return tmp;
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * Math.PI);
	double t_1 = (angle / 180.0) * Math.PI;
	double t_2 = ((-2.0 * (Math.pow(a_m, 2.0) * (Math.cos(t_0) * Math.sin(t_0)))) / x_45_scale) / y_45_scale;
	double tmp;
	if (a_m <= 1.7e-146) {
		tmp = (t_2 * t_2) - ((4.0 * ((Math.pow(b, 2.0) / x_45_scale) / x_45_scale)) * (((Math.pow((a_m * Math.cos(t_1)), 2.0) + Math.pow((b * Math.sin(t_1)), 2.0)) / y_45_scale) / y_45_scale));
	} else {
		tmp = Math.pow(a_m, 2.0) * (Math.pow(b, 2.0) * (-4.0 / (Math.pow(x_45_scale, 2.0) * Math.pow(y_45_scale, 2.0))));
	}
	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 = (angle / 180.0) * math.pi
	t_2 = ((-2.0 * (math.pow(a_m, 2.0) * (math.cos(t_0) * math.sin(t_0)))) / x_45_scale) / y_45_scale
	tmp = 0
	if a_m <= 1.7e-146:
		tmp = (t_2 * t_2) - ((4.0 * ((math.pow(b, 2.0) / x_45_scale) / x_45_scale)) * (((math.pow((a_m * math.cos(t_1)), 2.0) + math.pow((b * math.sin(t_1)), 2.0)) / y_45_scale) / y_45_scale))
	else:
		tmp = math.pow(a_m, 2.0) * (math.pow(b, 2.0) * (-4.0 / (math.pow(x_45_scale, 2.0) * math.pow(y_45_scale, 2.0))))
	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(angle / 180.0) * pi)
	t_2 = Float64(Float64(Float64(-2.0 * Float64((a_m ^ 2.0) * Float64(cos(t_0) * sin(t_0)))) / x_45_scale) / y_45_scale)
	tmp = 0.0
	if (a_m <= 1.7e-146)
		tmp = Float64(Float64(t_2 * t_2) - Float64(Float64(4.0 * Float64(Float64((b ^ 2.0) / x_45_scale) / x_45_scale)) * Float64(Float64(Float64((Float64(a_m * cos(t_1)) ^ 2.0) + (Float64(b * sin(t_1)) ^ 2.0)) / y_45_scale) / y_45_scale)));
	else
		tmp = Float64((a_m ^ 2.0) * Float64((b ^ 2.0) * Float64(-4.0 / Float64((x_45_scale ^ 2.0) * (y_45_scale ^ 2.0)))));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = 0.005555555555555556 * (angle * pi);
	t_1 = (angle / 180.0) * pi;
	t_2 = ((-2.0 * ((a_m ^ 2.0) * (cos(t_0) * sin(t_0)))) / x_45_scale) / y_45_scale;
	tmp = 0.0;
	if (a_m <= 1.7e-146)
		tmp = (t_2 * t_2) - ((4.0 * (((b ^ 2.0) / x_45_scale) / x_45_scale)) * (((((a_m * cos(t_1)) ^ 2.0) + ((b * sin(t_1)) ^ 2.0)) / y_45_scale) / y_45_scale));
	else
		tmp = (a_m ^ 2.0) * ((b ^ 2.0) * (-4.0 / ((x_45_scale ^ 2.0) * (y_45_scale ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(-2.0 * N[(N[Power[a$95$m, 2.0], $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, 1.7e-146], N[(N[(t$95$2 * t$95$2), $MachinePrecision] - N[(N[(4.0 * N[(N[(N[Power[b, 2.0], $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Power[N[(a$95$m * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[a$95$m, 2.0], $MachinePrecision] * N[(N[Power[b, 2.0], $MachinePrecision] * N[(-4.0 / N[(N[Power[x$45$scale, 2.0], $MachinePrecision] * N[Power[y$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if a < 1.7e-146

   1. Initial program 43.0%

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

   2. Taylor expanded in a around inf

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

   4. Applied rewrites 45.0%

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

   5. Taylor expanded in a around inf

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

   7. Applied rewrites 54.5%

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

   8. Taylor expanded in angle around 0

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

   10. Applied rewrites 54.5%

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

   if 1.7e-146 < a

   1. Initial program 17.9%

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

   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 40.4%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 42.2%

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

   8. Taylor expanded in angle around 0

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

   10. Applied rewrites 51.2%

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

Alternative 3: 51.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ t_1 := \cos t\_0\\ t_2 := \frac{\frac{-0.011111111111111112 \cdot \left({a\_m}^{2} \cdot \left(angle \cdot \pi\right)\right)}{x-scale}}{y-scale}\\ t_3 := \sin t\_0\\ \mathbf{if}\;a\_m \leq 6.8 \cdot 10^{-147}:\\ \;\;\;\;t\_2 \cdot t\_2 - \left(4 \cdot \frac{\frac{{\left(a\_m \cdot t\_3\right)}^{2} + {\left(b \cdot t\_1\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a\_m \cdot t\_1\right)}^{2} + {\left(b \cdot t\_3\right)}^{2}}{y-scale}}{y-scale}\\ \mathbf{else}:\\ \;\;\;\;{a\_m}^{2} \cdot \left({b}^{2} \cdot \frac{-4}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (/ angle 180.0) PI))
        (t_1 (cos t_0))
        (t_2
         (/
          (/ (* -0.011111111111111112 (* (pow a_m 2.0) (* angle PI))) x-scale)
          y-scale))
        (t_3 (sin t_0)))
   (if (<= a_m 6.8e-147)
     (-
      (* t_2 t_2)
      (*
       (*
        4.0
        (/ (/ (+ (pow (* a_m t_3) 2.0) (pow (* b t_1) 2.0)) x-scale) x-scale))
       (/ (/ (+ (pow (* a_m t_1) 2.0) (pow (* b t_3) 2.0)) y-scale) y-scale)))
     (*
      (pow a_m 2.0)
      (* (pow b 2.0) (/ -4.0 (* (pow x-scale 2.0) (pow y-scale 2.0))))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * ((double) M_PI);
	double t_1 = cos(t_0);
	double t_2 = ((-0.011111111111111112 * (pow(a_m, 2.0) * (angle * ((double) M_PI)))) / x_45_scale) / y_45_scale;
	double t_3 = sin(t_0);
	double tmp;
	if (a_m <= 6.8e-147) {
		tmp = (t_2 * t_2) - ((4.0 * (((pow((a_m * t_3), 2.0) + pow((b * t_1), 2.0)) / x_45_scale) / x_45_scale)) * (((pow((a_m * t_1), 2.0) + pow((b * t_3), 2.0)) / y_45_scale) / y_45_scale));
	} else {
		tmp = pow(a_m, 2.0) * (pow(b, 2.0) * (-4.0 / (pow(x_45_scale, 2.0) * pow(y_45_scale, 2.0))));
	}
	return tmp;
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * Math.PI;
	double t_1 = Math.cos(t_0);
	double t_2 = ((-0.011111111111111112 * (Math.pow(a_m, 2.0) * (angle * Math.PI))) / x_45_scale) / y_45_scale;
	double t_3 = Math.sin(t_0);
	double tmp;
	if (a_m <= 6.8e-147) {
		tmp = (t_2 * t_2) - ((4.0 * (((Math.pow((a_m * t_3), 2.0) + Math.pow((b * t_1), 2.0)) / x_45_scale) / x_45_scale)) * (((Math.pow((a_m * t_1), 2.0) + Math.pow((b * t_3), 2.0)) / y_45_scale) / y_45_scale));
	} else {
		tmp = Math.pow(a_m, 2.0) * (Math.pow(b, 2.0) * (-4.0 / (Math.pow(x_45_scale, 2.0) * Math.pow(y_45_scale, 2.0))));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle, x_45_scale, y_45_scale):
	t_0 = (angle / 180.0) * math.pi
	t_1 = math.cos(t_0)
	t_2 = ((-0.011111111111111112 * (math.pow(a_m, 2.0) * (angle * math.pi))) / x_45_scale) / y_45_scale
	t_3 = math.sin(t_0)
	tmp = 0
	if a_m <= 6.8e-147:
		tmp = (t_2 * t_2) - ((4.0 * (((math.pow((a_m * t_3), 2.0) + math.pow((b * t_1), 2.0)) / x_45_scale) / x_45_scale)) * (((math.pow((a_m * t_1), 2.0) + math.pow((b * t_3), 2.0)) / y_45_scale) / y_45_scale))
	else:
		tmp = math.pow(a_m, 2.0) * (math.pow(b, 2.0) * (-4.0 / (math.pow(x_45_scale, 2.0) * math.pow(y_45_scale, 2.0))))
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(angle / 180.0) * pi)
	t_1 = cos(t_0)
	t_2 = Float64(Float64(Float64(-0.011111111111111112 * Float64((a_m ^ 2.0) * Float64(angle * pi))) / x_45_scale) / y_45_scale)
	t_3 = sin(t_0)
	tmp = 0.0
	if (a_m <= 6.8e-147)
		tmp = Float64(Float64(t_2 * t_2) - Float64(Float64(4.0 * Float64(Float64(Float64((Float64(a_m * t_3) ^ 2.0) + (Float64(b * t_1) ^ 2.0)) / x_45_scale) / x_45_scale)) * Float64(Float64(Float64((Float64(a_m * t_1) ^ 2.0) + (Float64(b * t_3) ^ 2.0)) / y_45_scale) / y_45_scale)));
	else
		tmp = Float64((a_m ^ 2.0) * Float64((b ^ 2.0) * Float64(-4.0 / Float64((x_45_scale ^ 2.0) * (y_45_scale ^ 2.0)))));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = (angle / 180.0) * pi;
	t_1 = cos(t_0);
	t_2 = ((-0.011111111111111112 * ((a_m ^ 2.0) * (angle * pi))) / x_45_scale) / y_45_scale;
	t_3 = sin(t_0);
	tmp = 0.0;
	if (a_m <= 6.8e-147)
		tmp = (t_2 * t_2) - ((4.0 * (((((a_m * t_3) ^ 2.0) + ((b * t_1) ^ 2.0)) / x_45_scale) / x_45_scale)) * (((((a_m * t_1) ^ 2.0) + ((b * t_3) ^ 2.0)) / y_45_scale) / y_45_scale));
	else
		tmp = (a_m ^ 2.0) * ((b ^ 2.0) * (-4.0 / ((x_45_scale ^ 2.0) * (y_45_scale ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(-0.011111111111111112 * N[(N[Power[a$95$m, 2.0], $MachinePrecision] * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]}, Block[{t$95$3 = N[Sin[t$95$0], $MachinePrecision]}, If[LessEqual[a$95$m, 6.8e-147], N[(N[(t$95$2 * t$95$2), $MachinePrecision] - N[(N[(4.0 * N[(N[(N[(N[Power[N[(a$95$m * t$95$3), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Power[N[(a$95$m * t$95$1), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$3), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[a$95$m, 2.0], $MachinePrecision] * N[(N[Power[b, 2.0], $MachinePrecision] * N[(-4.0 / N[(N[Power[x$45$scale, 2.0], $MachinePrecision] * N[Power[y$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if a < 6.79999999999999991e-147

   1. Initial program 43.1%

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

   2. Taylor expanded in a around inf

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

   4. Applied rewrites 45.0%

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

   5. Taylor expanded in a around inf

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

   7. Applied rewrites 54.5%

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

   8. Taylor expanded in angle around 0

      \[\leadsto \frac{\frac{\frac{-1}{90} \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale}}{y-scale} \cdot \frac{\frac{-2 \cdot \left({a}^{2} \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]

   10. Applied rewrites 54.5%

       \[\leadsto \frac{\frac{-0.011111111111111112 \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)}}{x-scale}}{y-scale} \cdot \frac{\frac{-2 \cdot \left({a}^{2} \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]

   11. Taylor expanded in angle around 0

       \[\leadsto \frac{\frac{\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\frac{-1}{90} \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]

   13. Applied rewrites 53.8%

       \[\leadsto \frac{\frac{-0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)}{x-scale}}{y-scale} \cdot \frac{\frac{-0.011111111111111112 \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)}}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]

   if 6.79999999999999991e-147 < a

   1. Initial program 17.9%

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

   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 40.4%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 42.2%

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

   8. Taylor expanded in angle around 0

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

   10. Applied rewrites 51.3%

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

Alternative 4: 51.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := \frac{\frac{-0.011111111111111112 \cdot \left({a\_m}^{2} \cdot \left(angle \cdot \pi\right)\right)}{x-scale}}{y-scale}\\ t_1 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_2 := \sin t\_1\\ \mathbf{if}\;a\_m \leq 7 \cdot 10^{-147}:\\ \;\;\;\;t\_0 \cdot t\_0 - \left(4 \cdot \frac{\frac{{a\_m}^{2} \cdot {t\_2}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a\_m \cdot \cos t\_1\right)}^{2} + {\left(b \cdot t\_2\right)}^{2}}{y-scale}}{y-scale}\\ \mathbf{else}:\\ \;\;\;\;{a\_m}^{2} \cdot \left({b}^{2} \cdot \frac{-4}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0
         (/
          (/ (* -0.011111111111111112 (* (pow a_m 2.0) (* angle PI))) x-scale)
          y-scale))
        (t_1 (* 0.005555555555555556 (* angle PI)))
        (t_2 (sin t_1)))
   (if (<= a_m 7e-147)
     (-
      (* t_0 t_0)
      (*
       (* 4.0 (/ (/ (* (pow a_m 2.0) (pow t_2 2.0)) x-scale) x-scale))
       (/
        (/ (+ (pow (* a_m (cos t_1)) 2.0) (pow (* b t_2) 2.0)) y-scale)
        y-scale)))
     (*
      (pow a_m 2.0)
      (* (pow b 2.0) (/ -4.0 (* (pow x-scale 2.0) (pow y-scale 2.0))))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = ((-0.011111111111111112 * (pow(a_m, 2.0) * (angle * ((double) M_PI)))) / x_45_scale) / y_45_scale;
	double t_1 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_2 = sin(t_1);
	double tmp;
	if (a_m <= 7e-147) {
		tmp = (t_0 * t_0) - ((4.0 * (((pow(a_m, 2.0) * pow(t_2, 2.0)) / x_45_scale) / x_45_scale)) * (((pow((a_m * cos(t_1)), 2.0) + pow((b * t_2), 2.0)) / y_45_scale) / y_45_scale));
	} else {
		tmp = pow(a_m, 2.0) * (pow(b, 2.0) * (-4.0 / (pow(x_45_scale, 2.0) * pow(y_45_scale, 2.0))));
	}
	return tmp;
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = ((-0.011111111111111112 * (Math.pow(a_m, 2.0) * (angle * Math.PI))) / x_45_scale) / y_45_scale;
	double t_1 = 0.005555555555555556 * (angle * Math.PI);
	double t_2 = Math.sin(t_1);
	double tmp;
	if (a_m <= 7e-147) {
		tmp = (t_0 * t_0) - ((4.0 * (((Math.pow(a_m, 2.0) * Math.pow(t_2, 2.0)) / x_45_scale) / x_45_scale)) * (((Math.pow((a_m * Math.cos(t_1)), 2.0) + Math.pow((b * t_2), 2.0)) / y_45_scale) / y_45_scale));
	} else {
		tmp = Math.pow(a_m, 2.0) * (Math.pow(b, 2.0) * (-4.0 / (Math.pow(x_45_scale, 2.0) * Math.pow(y_45_scale, 2.0))));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle, x_45_scale, y_45_scale):
	t_0 = ((-0.011111111111111112 * (math.pow(a_m, 2.0) * (angle * math.pi))) / x_45_scale) / y_45_scale
	t_1 = 0.005555555555555556 * (angle * math.pi)
	t_2 = math.sin(t_1)
	tmp = 0
	if a_m <= 7e-147:
		tmp = (t_0 * t_0) - ((4.0 * (((math.pow(a_m, 2.0) * math.pow(t_2, 2.0)) / x_45_scale) / x_45_scale)) * (((math.pow((a_m * math.cos(t_1)), 2.0) + math.pow((b * t_2), 2.0)) / y_45_scale) / y_45_scale))
	else:
		tmp = math.pow(a_m, 2.0) * (math.pow(b, 2.0) * (-4.0 / (math.pow(x_45_scale, 2.0) * math.pow(y_45_scale, 2.0))))
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(Float64(-0.011111111111111112 * Float64((a_m ^ 2.0) * Float64(angle * pi))) / x_45_scale) / y_45_scale)
	t_1 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_2 = sin(t_1)
	tmp = 0.0
	if (a_m <= 7e-147)
		tmp = Float64(Float64(t_0 * t_0) - Float64(Float64(4.0 * Float64(Float64(Float64((a_m ^ 2.0) * (t_2 ^ 2.0)) / x_45_scale) / x_45_scale)) * Float64(Float64(Float64((Float64(a_m * cos(t_1)) ^ 2.0) + (Float64(b * t_2) ^ 2.0)) / y_45_scale) / y_45_scale)));
	else
		tmp = Float64((a_m ^ 2.0) * Float64((b ^ 2.0) * Float64(-4.0 / Float64((x_45_scale ^ 2.0) * (y_45_scale ^ 2.0)))));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = ((-0.011111111111111112 * ((a_m ^ 2.0) * (angle * pi))) / x_45_scale) / y_45_scale;
	t_1 = 0.005555555555555556 * (angle * pi);
	t_2 = sin(t_1);
	tmp = 0.0;
	if (a_m <= 7e-147)
		tmp = (t_0 * t_0) - ((4.0 * ((((a_m ^ 2.0) * (t_2 ^ 2.0)) / x_45_scale) / x_45_scale)) * (((((a_m * cos(t_1)) ^ 2.0) + ((b * t_2) ^ 2.0)) / y_45_scale) / y_45_scale));
	else
		tmp = (a_m ^ 2.0) * ((b ^ 2.0) * (-4.0 / ((x_45_scale ^ 2.0) * (y_45_scale ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(N[(-0.011111111111111112 * N[(N[Power[a$95$m, 2.0], $MachinePrecision] * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]}, Block[{t$95$1 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[t$95$1], $MachinePrecision]}, If[LessEqual[a$95$m, 7e-147], N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(N[(4.0 * N[(N[(N[(N[Power[a$95$m, 2.0], $MachinePrecision] * N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Power[N[(a$95$m * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$2), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[a$95$m, 2.0], $MachinePrecision] * N[(N[Power[b, 2.0], $MachinePrecision] * N[(-4.0 / N[(N[Power[x$45$scale, 2.0], $MachinePrecision] * N[Power[y$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if a < 7.00000000000000007e-147

   1. Initial program 43.1%

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

   2. Taylor expanded in a around inf

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

   4. Applied rewrites 45.0%

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

   5. Taylor expanded in a around inf

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

   7. Applied rewrites 54.5%

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

   8. Taylor expanded in a around inf

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

   10. Applied rewrites 54.1%

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

   11. Taylor expanded in angle around 0

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

   13. Applied rewrites 54.1%

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

   14. Taylor expanded in angle around 0

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

   16. Applied rewrites 54.1%

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

   17. Taylor expanded in angle around 0

       \[\leadsto \frac{\frac{\frac{-1}{90} \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale}}{y-scale} \cdot \frac{\frac{-2 \cdot \left({a}^{2} \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}{y-scale}}{y-scale} \]

   19. Applied rewrites 54.1%

       \[\leadsto \frac{\frac{-0.011111111111111112 \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)}}{x-scale}}{y-scale} \cdot \frac{\frac{-2 \cdot \left({a}^{2} \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}{y-scale}}{y-scale} \]

   20. Taylor expanded in angle around 0

       \[\leadsto \frac{\frac{\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\frac{-1}{90} \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}{y-scale}}{y-scale} \]

   22. Applied rewrites 53.4%

       \[\leadsto \frac{\frac{-0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)}{x-scale}}{y-scale} \cdot \frac{\frac{-0.011111111111111112 \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)}}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}{y-scale}}{y-scale} \]

   if 7.00000000000000007e-147 < a

   1. Initial program 17.9%

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

   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 40.4%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 42.2%

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

   8. Taylor expanded in angle around 0

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

   10. Applied rewrites 51.3%

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

Alternative 5: 50.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ t_1 := 0.011111111111111112 \cdot \frac{angle \cdot \left(\pi \cdot \left(-1 \cdot {a\_m}^{2}\right)\right)}{x-scale \cdot y-scale}\\ \mathbf{if}\;a\_m \leq 6.8 \cdot 10^{-147}:\\ \;\;\;\;t\_1 \cdot t\_1 - \left(4 \cdot \frac{\frac{{b}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a\_m \cdot \cos t\_0\right)}^{2} + {\left(b \cdot \sin t\_0\right)}^{2}}{y-scale}}{y-scale}\\ \mathbf{else}:\\ \;\;\;\;{a\_m}^{2} \cdot \left({b}^{2} \cdot \frac{-4}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (/ angle 180.0) PI))
        (t_1
         (*
          0.011111111111111112
          (/ (* angle (* PI (* -1.0 (pow a_m 2.0)))) (* x-scale y-scale)))))
   (if (<= a_m 6.8e-147)
     (-
      (* t_1 t_1)
      (*
       (* 4.0 (/ (/ (pow b 2.0) x-scale) x-scale))
       (/
        (/ (+ (pow (* a_m (cos t_0)) 2.0) (pow (* b (sin t_0)) 2.0)) y-scale)
        y-scale)))
     (*
      (pow a_m 2.0)
      (* (pow b 2.0) (/ -4.0 (* (pow x-scale 2.0) (pow y-scale 2.0))))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * ((double) M_PI);
	double t_1 = 0.011111111111111112 * ((angle * (((double) M_PI) * (-1.0 * pow(a_m, 2.0)))) / (x_45_scale * y_45_scale));
	double tmp;
	if (a_m <= 6.8e-147) {
		tmp = (t_1 * t_1) - ((4.0 * ((pow(b, 2.0) / x_45_scale) / x_45_scale)) * (((pow((a_m * cos(t_0)), 2.0) + pow((b * sin(t_0)), 2.0)) / y_45_scale) / y_45_scale));
	} else {
		tmp = pow(a_m, 2.0) * (pow(b, 2.0) * (-4.0 / (pow(x_45_scale, 2.0) * pow(y_45_scale, 2.0))));
	}
	return tmp;
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * Math.PI;
	double t_1 = 0.011111111111111112 * ((angle * (Math.PI * (-1.0 * Math.pow(a_m, 2.0)))) / (x_45_scale * y_45_scale));
	double tmp;
	if (a_m <= 6.8e-147) {
		tmp = (t_1 * t_1) - ((4.0 * ((Math.pow(b, 2.0) / x_45_scale) / x_45_scale)) * (((Math.pow((a_m * Math.cos(t_0)), 2.0) + Math.pow((b * Math.sin(t_0)), 2.0)) / y_45_scale) / y_45_scale));
	} else {
		tmp = Math.pow(a_m, 2.0) * (Math.pow(b, 2.0) * (-4.0 / (Math.pow(x_45_scale, 2.0) * Math.pow(y_45_scale, 2.0))));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle, x_45_scale, y_45_scale):
	t_0 = (angle / 180.0) * math.pi
	t_1 = 0.011111111111111112 * ((angle * (math.pi * (-1.0 * math.pow(a_m, 2.0)))) / (x_45_scale * y_45_scale))
	tmp = 0
	if a_m <= 6.8e-147:
		tmp = (t_1 * t_1) - ((4.0 * ((math.pow(b, 2.0) / x_45_scale) / x_45_scale)) * (((math.pow((a_m * math.cos(t_0)), 2.0) + math.pow((b * math.sin(t_0)), 2.0)) / y_45_scale) / y_45_scale))
	else:
		tmp = math.pow(a_m, 2.0) * (math.pow(b, 2.0) * (-4.0 / (math.pow(x_45_scale, 2.0) * math.pow(y_45_scale, 2.0))))
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(angle / 180.0) * pi)
	t_1 = Float64(0.011111111111111112 * Float64(Float64(angle * Float64(pi * Float64(-1.0 * (a_m ^ 2.0)))) / Float64(x_45_scale * y_45_scale)))
	tmp = 0.0
	if (a_m <= 6.8e-147)
		tmp = Float64(Float64(t_1 * t_1) - Float64(Float64(4.0 * Float64(Float64((b ^ 2.0) / x_45_scale) / x_45_scale)) * Float64(Float64(Float64((Float64(a_m * cos(t_0)) ^ 2.0) + (Float64(b * sin(t_0)) ^ 2.0)) / y_45_scale) / y_45_scale)));
	else
		tmp = Float64((a_m ^ 2.0) * Float64((b ^ 2.0) * Float64(-4.0 / Float64((x_45_scale ^ 2.0) * (y_45_scale ^ 2.0)))));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = (angle / 180.0) * pi;
	t_1 = 0.011111111111111112 * ((angle * (pi * (-1.0 * (a_m ^ 2.0)))) / (x_45_scale * y_45_scale));
	tmp = 0.0;
	if (a_m <= 6.8e-147)
		tmp = (t_1 * t_1) - ((4.0 * (((b ^ 2.0) / x_45_scale) / x_45_scale)) * (((((a_m * cos(t_0)) ^ 2.0) + ((b * sin(t_0)) ^ 2.0)) / y_45_scale) / y_45_scale));
	else
		tmp = (a_m ^ 2.0) * ((b ^ 2.0) * (-4.0 / ((x_45_scale ^ 2.0) * (y_45_scale ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[(0.011111111111111112 * N[(N[(angle * N[(Pi * N[(-1.0 * N[Power[a$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a$95$m, 6.8e-147], N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(N[(4.0 * N[(N[(N[Power[b, 2.0], $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Power[N[(a$95$m * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[a$95$m, 2.0], $MachinePrecision] * N[(N[Power[b, 2.0], $MachinePrecision] * N[(-4.0 / N[(N[Power[x$45$scale, 2.0], $MachinePrecision] * N[Power[y$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if a < 6.79999999999999991e-147

   1. Initial program 43.1%

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

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \frac{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}{x-scale \cdot y-scale}\right)} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]

   4. Applied rewrites 43.0%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \frac{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}{x-scale \cdot y-scale}\right)} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]

   5. Taylor expanded in angle around 0

      \[\leadsto \left(\frac{1}{90} \cdot \frac{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}{x-scale \cdot y-scale}\right) \cdot \color{blue}{\left(\frac{1}{90} \cdot \frac{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}{x-scale \cdot y-scale}\right)} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]

   7. Applied rewrites 37.9%

      \[\leadsto \left(0.011111111111111112 \cdot \frac{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}{x-scale \cdot y-scale}\right) \cdot \color{blue}{\left(0.011111111111111112 \cdot \frac{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}{x-scale \cdot y-scale}\right)} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]

   8. Taylor expanded in angle around 0

      \[\leadsto \left(\frac{1}{90} \cdot \frac{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}{x-scale \cdot y-scale}\right) \cdot \left(\frac{1}{90} \cdot \frac{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}{x-scale \cdot y-scale}\right) - \left(4 \cdot \frac{\frac{\color{blue}{{b}^{2}}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]

   10. Applied rewrites 37.9%

       \[\leadsto \left(0.011111111111111112 \cdot \frac{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}{x-scale \cdot y-scale}\right) \cdot \left(0.011111111111111112 \cdot \frac{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}{x-scale \cdot y-scale}\right) - \left(4 \cdot \frac{\frac{\color{blue}{{b}^{2}}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]

   11. Taylor expanded in a around inf

       \[\leadsto \left(\frac{1}{90} \cdot \frac{angle \cdot \left(\pi \cdot \left(-1 \cdot {a}^{2}\right)\right)}{x-scale \cdot y-scale}\right) \cdot \left(\frac{1}{90} \cdot \frac{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}{x-scale \cdot y-scale}\right) - \left(4 \cdot \frac{\frac{{b}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]

   13. Applied rewrites 43.3%

       \[\leadsto \left(0.011111111111111112 \cdot \frac{angle \cdot \left(\pi \cdot \left(-1 \cdot {a}^{2}\right)\right)}{x-scale \cdot y-scale}\right) \cdot \left(0.011111111111111112 \cdot \frac{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}{x-scale \cdot y-scale}\right) - \left(4 \cdot \frac{\frac{{b}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]

   14. Taylor expanded in a around inf

       \[\leadsto \left(\frac{1}{90} \cdot \frac{angle \cdot \left(\pi \cdot \left(-1 \cdot {a}^{2}\right)\right)}{x-scale \cdot y-scale}\right) \cdot \left(\frac{1}{90} \cdot \frac{angle \cdot \left(\pi \cdot \left(-1 \cdot {a}^{2}\right)\right)}{x-scale \cdot y-scale}\right) - \left(4 \cdot \frac{\frac{{b}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]

   16. Applied rewrites 48.5%

       \[\leadsto \left(0.011111111111111112 \cdot \frac{angle \cdot \left(\pi \cdot \left(-1 \cdot {a}^{2}\right)\right)}{x-scale \cdot y-scale}\right) \cdot \left(0.011111111111111112 \cdot \frac{angle \cdot \left(\pi \cdot \left(-1 \cdot {a}^{2}\right)\right)}{x-scale \cdot y-scale}\right) - \left(4 \cdot \frac{\frac{{b}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]

   if 6.79999999999999991e-147 < a

   1. Initial program 17.9%

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

   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 40.4%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 42.2%

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

   8. Taylor expanded in angle around 0

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

   10. Applied rewrites 51.3%

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

Alternative 6: 48.2% accurate, 4.1× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := 0.011111111111111112 \cdot \frac{angle \cdot \left({b}^{2} \cdot \pi\right)}{x-scale \cdot y-scale}\\ \mathbf{if}\;b \leq 1.35 \cdot 10^{-185}:\\ \;\;\;\;t\_0 \cdot t\_0 - \left(4 \cdot \frac{\frac{{b}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{a\_m}^{2}}{y-scale}}{y-scale}\\ \mathbf{else}:\\ \;\;\;\;{a\_m}^{2} \cdot \left({b}^{2} \cdot \frac{-4}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0
         (*
          0.011111111111111112
          (/ (* angle (* (pow b 2.0) PI)) (* x-scale y-scale)))))
   (if (<= b 1.35e-185)
     (-
      (* t_0 t_0)
      (*
       (* 4.0 (/ (/ (pow b 2.0) x-scale) x-scale))
       (/ (/ (pow a_m 2.0) y-scale) y-scale)))
     (*
      (pow a_m 2.0)
      (* (pow b 2.0) (/ -4.0 (* (pow x-scale 2.0) (pow y-scale 2.0))))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.011111111111111112 * ((angle * (pow(b, 2.0) * ((double) M_PI))) / (x_45_scale * y_45_scale));
	double tmp;
	if (b <= 1.35e-185) {
		tmp = (t_0 * t_0) - ((4.0 * ((pow(b, 2.0) / x_45_scale) / x_45_scale)) * ((pow(a_m, 2.0) / y_45_scale) / y_45_scale));
	} else {
		tmp = pow(a_m, 2.0) * (pow(b, 2.0) * (-4.0 / (pow(x_45_scale, 2.0) * pow(y_45_scale, 2.0))));
	}
	return tmp;
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.011111111111111112 * ((angle * (Math.pow(b, 2.0) * Math.PI)) / (x_45_scale * y_45_scale));
	double tmp;
	if (b <= 1.35e-185) {
		tmp = (t_0 * t_0) - ((4.0 * ((Math.pow(b, 2.0) / x_45_scale) / x_45_scale)) * ((Math.pow(a_m, 2.0) / y_45_scale) / y_45_scale));
	} else {
		tmp = Math.pow(a_m, 2.0) * (Math.pow(b, 2.0) * (-4.0 / (Math.pow(x_45_scale, 2.0) * Math.pow(y_45_scale, 2.0))));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle, x_45_scale, y_45_scale):
	t_0 = 0.011111111111111112 * ((angle * (math.pow(b, 2.0) * math.pi)) / (x_45_scale * y_45_scale))
	tmp = 0
	if b <= 1.35e-185:
		tmp = (t_0 * t_0) - ((4.0 * ((math.pow(b, 2.0) / x_45_scale) / x_45_scale)) * ((math.pow(a_m, 2.0) / y_45_scale) / y_45_scale))
	else:
		tmp = math.pow(a_m, 2.0) * (math.pow(b, 2.0) * (-4.0 / (math.pow(x_45_scale, 2.0) * math.pow(y_45_scale, 2.0))))
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.011111111111111112 * Float64(Float64(angle * Float64((b ^ 2.0) * pi)) / Float64(x_45_scale * y_45_scale)))
	tmp = 0.0
	if (b <= 1.35e-185)
		tmp = Float64(Float64(t_0 * t_0) - Float64(Float64(4.0 * Float64(Float64((b ^ 2.0) / x_45_scale) / x_45_scale)) * Float64(Float64((a_m ^ 2.0) / y_45_scale) / y_45_scale)));
	else
		tmp = Float64((a_m ^ 2.0) * Float64((b ^ 2.0) * Float64(-4.0 / Float64((x_45_scale ^ 2.0) * (y_45_scale ^ 2.0)))));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = 0.011111111111111112 * ((angle * ((b ^ 2.0) * pi)) / (x_45_scale * y_45_scale));
	tmp = 0.0;
	if (b <= 1.35e-185)
		tmp = (t_0 * t_0) - ((4.0 * (((b ^ 2.0) / x_45_scale) / x_45_scale)) * (((a_m ^ 2.0) / y_45_scale) / y_45_scale));
	else
		tmp = (a_m ^ 2.0) * ((b ^ 2.0) * (-4.0 / ((x_45_scale ^ 2.0) * (y_45_scale ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.011111111111111112 * N[(N[(angle * N[(N[Power[b, 2.0], $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1.35e-185], N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(N[(4.0 * N[(N[(N[Power[b, 2.0], $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[a$95$m, 2.0], $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[a$95$m, 2.0], $MachinePrecision] * N[(N[Power[b, 2.0], $MachinePrecision] * N[(-4.0 / N[(N[Power[x$45$scale, 2.0], $MachinePrecision] * N[Power[y$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.34999999999999994e-185

   1. Initial program 27.6%

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

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \frac{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}{x-scale \cdot y-scale}\right)} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]

   4. Applied rewrites 31.3%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \frac{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}{x-scale \cdot y-scale}\right)} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]

   5. Taylor expanded in angle around 0

      \[\leadsto \left(\frac{1}{90} \cdot \frac{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}{x-scale \cdot y-scale}\right) \cdot \color{blue}{\left(\frac{1}{90} \cdot \frac{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}{x-scale \cdot y-scale}\right)} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]

   7. Applied rewrites 23.6%

      \[\leadsto \left(0.011111111111111112 \cdot \frac{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}{x-scale \cdot y-scale}\right) \cdot \color{blue}{\left(0.011111111111111112 \cdot \frac{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}{x-scale \cdot y-scale}\right)} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]

   8. Taylor expanded in angle around 0

      \[\leadsto \left(\frac{1}{90} \cdot \frac{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}{x-scale \cdot y-scale}\right) \cdot \left(\frac{1}{90} \cdot \frac{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}{x-scale \cdot y-scale}\right) - \left(4 \cdot \frac{\frac{\color{blue}{{b}^{2}}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]

   10. Applied rewrites 23.5%

       \[\leadsto \left(0.011111111111111112 \cdot \frac{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}{x-scale \cdot y-scale}\right) \cdot \left(0.011111111111111112 \cdot \frac{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}{x-scale \cdot y-scale}\right) - \left(4 \cdot \frac{\frac{\color{blue}{{b}^{2}}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]

   11. Taylor expanded in angle around 0

       \[\leadsto \left(\frac{1}{90} \cdot \frac{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}{x-scale \cdot y-scale}\right) \cdot \left(\frac{1}{90} \cdot \frac{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}{x-scale \cdot y-scale}\right) - \left(4 \cdot \frac{\frac{{b}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{\color{blue}{{a}^{2}}}{y-scale}}{y-scale} \]

   13. Applied rewrites 23.6%

       \[\leadsto \left(0.011111111111111112 \cdot \frac{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}{x-scale \cdot y-scale}\right) \cdot \left(0.011111111111111112 \cdot \frac{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}{x-scale \cdot y-scale}\right) - \left(4 \cdot \frac{\frac{{b}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{\color{blue}{{a}^{2}}}{y-scale}}{y-scale} \]

   14. Taylor expanded in a around 0

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

   16. Applied rewrites 35.6%

       \[\leadsto \left(0.011111111111111112 \cdot \frac{angle \cdot \left({b}^{2} \cdot \pi\right)}{x-scale \cdot y-scale}\right) \cdot \left(0.011111111111111112 \cdot \frac{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}{x-scale \cdot y-scale}\right) - \left(4 \cdot \frac{\frac{{b}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{a}^{2}}{y-scale}}{y-scale} \]

   17. Taylor expanded in a around 0

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

   19. Applied rewrites 32.3%

       \[\leadsto \left(0.011111111111111112 \cdot \frac{angle \cdot \left({b}^{2} \cdot \pi\right)}{x-scale \cdot y-scale}\right) \cdot \left(0.011111111111111112 \cdot \frac{angle \cdot \left({b}^{2} \cdot \pi\right)}{x-scale \cdot y-scale}\right) - \left(4 \cdot \frac{\frac{{b}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{a}^{2}}{y-scale}}{y-scale} \]

   if 1.34999999999999994e-185 < b

   1. Initial program 19.7%

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

   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 39.5%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 41.6%

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

   8. Taylor expanded in angle around 0

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

   10. Applied rewrites 50.1%

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

Alternative 7: 48.2% accurate, 6.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 24.4%

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

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 40.0%

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

5. Taylor expanded in a around 0

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

7. Applied rewrites 41.3%

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

8. Taylor expanded in y-scale around 0

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

10. Applied rewrites 44.0%

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

11. Taylor expanded in angle around 0

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

13. Applied rewrites 48.2%

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

Alternative 8: 39.4% accurate, 6.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 24.4%

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

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 40.0%

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

5. Taylor expanded in a around 0

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

7. Applied rewrites 41.3%

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

8. Taylor expanded in angle around 0

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

10. Applied rewrites 48.2%

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

Reproduce

herbie shell --seed 2025121 
(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))))