Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 25.5% → 95.0%

Time: 2.2min

Alternatives: 5

Speedup: 2485.0×

• 33.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (/ angle 180.0) PI))
        (t_1 (sin t_0))
        (t_2 (cos t_0))
        (t_3
         (/
          (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) t_1) t_2) x-scale)
          y-scale)))
   (-
    (* t_3 t_3)
    (*
     (*
      4.0
      (/ (/ (+ (pow (* a t_1) 2.0) (pow (* b t_2) 2.0)) x-scale) x-scale))
     (/ (/ (+ (pow (* a t_2) 2.0) (pow (* b t_1) 2.0)) y-scale) y-scale)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * ((double) M_PI);
	double t_1 = sin(t_0);
	double t_2 = cos(t_0);
	double t_3 = ((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale;
	return (t_3 * t_3) - ((4.0 * (((pow((a * t_1), 2.0) + pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale)) * (((pow((a * t_2), 2.0) + pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale));
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * Math.PI;
	double t_1 = Math.sin(t_0);
	double t_2 = Math.cos(t_0);
	double t_3 = ((((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale;
	return (t_3 * t_3) - ((4.0 * (((Math.pow((a * t_1), 2.0) + Math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale)) * (((Math.pow((a * t_2), 2.0) + Math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale));
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (angle / 180.0) * math.pi
	t_1 = math.sin(t_0)
	t_2 = math.cos(t_0)
	t_3 = ((((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale
	return (t_3 * t_3) - ((4.0 * (((math.pow((a * t_1), 2.0) + math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale)) * (((math.pow((a * t_2), 2.0) + math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale))

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(angle / 180.0) * pi)
	t_1 = sin(t_0)
	t_2 = cos(t_0)
	t_3 = Float64(Float64(Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale)
	return Float64(Float64(t_3 * t_3) - Float64(Float64(4.0 * Float64(Float64(Float64((Float64(a * t_1) ^ 2.0) + (Float64(b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale)) * Float64(Float64(Float64((Float64(a * t_2) ^ 2.0) + (Float64(b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale)))
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (angle / 180.0) * pi;
	t_1 = sin(t_0);
	t_2 = cos(t_0);
	t_3 = ((((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale;
	tmp = (t_3 * t_3) - ((4.0 * (((((a * t_1) ^ 2.0) + ((b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale)) * (((((a * t_2) ^ 2.0) + ((b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale));
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision] / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]}, N[(N[(t$95$3 * t$95$3), $MachinePrecision] - N[(N[(4.0 * N[(N[(N[(N[Power[N[(a * t$95$1), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$2), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Power[N[(a * t$95$2), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Initial Program: 25.5% 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: 95.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ t_1 := \sin t_0\\ t_2 := \cos t_0\\ t_3 := \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot t_1\right) \cdot t_2}{x-scale}}{y-scale}\\ \mathbf{if}\;t_3 \cdot t_3 - \left(4 \cdot \frac{\frac{{\left(a \cdot t_1\right)}^{2} + {\left(b \cdot t_2\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot t_2\right)}^{2} + {\left(b \cdot t_1\right)}^{2}}{y-scale}}{y-scale} \leq 0:\\ \;\;\;\;-4 \cdot {\left(\frac{b}{x-scale} \cdot \frac{a}{y-scale}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot {\left(\frac{b \cdot a}{x-scale \cdot y-scale}\right)}^{2}\\ \end{array} \end{array} \]

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)))
   (if (<=
        (-
         (* 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)))
        0.0)
     (* -4.0 (pow (* (/ b x-scale) (/ a y-scale)) 2.0))
     (* -4.0 (pow (/ (* b a) (* x-scale y-scale)) 2.0)))))

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;
	double tmp;
	if (((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))) <= 0.0) {
		tmp = -4.0 * pow(((b / x_45_scale) * (a / y_45_scale)), 2.0);
	} else {
		tmp = -4.0 * pow(((b * a) / (x_45_scale * y_45_scale)), 2.0);
	}
	return tmp;
}

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;
	double tmp;
	if (((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))) <= 0.0) {
		tmp = -4.0 * Math.pow(((b / x_45_scale) * (a / y_45_scale)), 2.0);
	} else {
		tmp = -4.0 * Math.pow(((b * a) / (x_45_scale * y_45_scale)), 2.0);
	}
	return tmp;
}

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
	tmp = 0
	if ((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))) <= 0.0:
		tmp = -4.0 * math.pow(((b / x_45_scale) * (a / y_45_scale)), 2.0)
	else:
		tmp = -4.0 * math.pow(((b * a) / (x_45_scale * y_45_scale)), 2.0)
	return tmp

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)
	tmp = 0.0
	if (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))) <= 0.0)
		tmp = Float64(-4.0 * (Float64(Float64(b / x_45_scale) * Float64(a / y_45_scale)) ^ 2.0));
	else
		tmp = Float64(-4.0 * (Float64(Float64(b * a) / Float64(x_45_scale * y_45_scale)) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = 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 = 0.0;
	if (((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))) <= 0.0)
		tmp = -4.0 * (((b / x_45_scale) * (a / y_45_scale)) ^ 2.0);
	else
		tmp = -4.0 * (((b * a) / (x_45_scale * y_45_scale)) ^ 2.0);
	end
	tmp_2 = tmp;
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]}, If[LessEqual[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], 0.0], N[(-4.0 * N[Power[N[(N[(b / x$45$scale), $MachinePrecision] * N[(a / y$45$scale), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[Power[N[(N[(b * a), $MachinePrecision] / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))) (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) x-scale) y-scale) (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))) (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) x-scale) y-scale)) (*.f64 (*.f64 4 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2)) x-scale) x-scale)) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2)) y-scale) y-scale))) < 0.0

   1. Initial program 72.7%

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

   3. Simplified 59.5%

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

   4. Taylor expanded in angle around 0 61.7%

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

      1. times-frac 64.0%

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

   6. Simplified 64.0%

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

      1. add-sqr-sqrt 64.0%

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

      2. pow2 64.0%

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

      3. frac-times 61.7%

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

      4. *-commutative 61.7%

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

      5. pow-prod-down 61.7%

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

      6. pow-prod-down 74.4%

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

   8. Applied egg-rr 74.4%

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

   9. Taylor expanded in b around 0 92.6%

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

       1. *-commutative 92.6%

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

       2. times-frac 99.6%

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

   11. Applied egg-rr 99.6%

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

   if 0.0 < (-.f64 (*.f64 (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))) (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) x-scale) y-scale) (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))) (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) x-scale) y-scale)) (*.f64 (*.f64 4 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2)) x-scale) x-scale)) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2)) y-scale) y-scale)))

   1. Initial program 0.9%

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

   3. Simplified 1.9%

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

   4. Taylor expanded in angle around 0 34.8%

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

      1. times-frac 32.6%

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

   6. Simplified 32.6%

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

      1. add-sqr-sqrt 32.6%

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

      2. pow2 32.6%

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

      3. frac-times 34.8%

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

      4. *-commutative 34.8%

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

      5. pow-prod-down 51.2%

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

      6. pow-prod-down 73.1%

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

   8. Applied egg-rr 73.1%

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

   9. Taylor expanded in b around 0 94.9%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \leq 0:\\ \;\;\;\;-4 \cdot {\left(\frac{b}{x-scale} \cdot \frac{a}{y-scale}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot {\left(\frac{b \cdot a}{x-scale \cdot y-scale}\right)}^{2}\\ \end{array} \]

Alternative 2: 93.9% accurate, 22.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{x-scale} \cdot \frac{b}{y-scale}\\ \mathbf{if}\;a \leq 6.7 \cdot 10^{+260}:\\ \;\;\;\;t_0 \cdot \left(-4 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot {\left(\frac{b}{x-scale} \cdot \frac{a}{y-scale}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (/ a x-scale) (/ b y-scale))))
   (if (<= a 6.7e+260)
     (* t_0 (* -4.0 t_0))
     (* -4.0 (pow (* (/ b x-scale) (/ a y-scale)) 2.0)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (a / x_45_scale) * (b / y_45_scale);
	double tmp;
	if (a <= 6.7e+260) {
		tmp = t_0 * (-4.0 * t_0);
	} else {
		tmp = -4.0 * pow(((b / x_45_scale) * (a / y_45_scale)), 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (a / x_45scale) * (b / y_45scale)
    if (a <= 6.7d+260) then
        tmp = t_0 * ((-4.0d0) * t_0)
    else
        tmp = (-4.0d0) * (((b / x_45scale) * (a / y_45scale)) ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (a / x_45_scale) * (b / y_45_scale);
	double tmp;
	if (a <= 6.7e+260) {
		tmp = t_0 * (-4.0 * t_0);
	} else {
		tmp = -4.0 * Math.pow(((b / x_45_scale) * (a / y_45_scale)), 2.0);
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (a / x_45_scale) * (b / y_45_scale)
	tmp = 0
	if a <= 6.7e+260:
		tmp = t_0 * (-4.0 * t_0)
	else:
		tmp = -4.0 * math.pow(((b / x_45_scale) * (a / y_45_scale)), 2.0)
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(a / x_45_scale) * Float64(b / y_45_scale))
	tmp = 0.0
	if (a <= 6.7e+260)
		tmp = Float64(t_0 * Float64(-4.0 * t_0));
	else
		tmp = Float64(-4.0 * (Float64(Float64(b / x_45_scale) * Float64(a / y_45_scale)) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (a / x_45_scale) * (b / y_45_scale);
	tmp = 0.0;
	if (a <= 6.7e+260)
		tmp = t_0 * (-4.0 * t_0);
	else
		tmp = -4.0 * (((b / x_45_scale) * (a / y_45_scale)) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(a / x$45$scale), $MachinePrecision] * N[(b / y$45$scale), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 6.7e+260], N[(t$95$0 * N[(-4.0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[Power[N[(N[(b / x$45$scale), $MachinePrecision] * N[(a / y$45$scale), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 6.69999999999999952e260

   1. Initial program 25.9%

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

   3. Simplified 22.1%

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

   4. Taylor expanded in angle around 0 43.8%

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

      1. associate-*r/ 43.9%

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

      2. times-frac 45.1%

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

      3. *-commutative 45.1%

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

   6. Simplified 45.1%

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

      1. frac-times 43.9%

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

      2. pow-prod-down 54.5%

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

      3. pow-prod-down 73.5%

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

   8. Applied egg-rr 73.5%

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

      1. add-cbrt-cube 55.7%

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

      2. pow3 55.7%

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

   10. Applied egg-rr 55.7%

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

       1. rem-cbrt-cube 73.5%

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

       2. *-un-lft-identity 73.5%

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

       3. times-frac 73.5%

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

       4. metadata-eval 73.5%

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

       5. *-commutative 73.5%

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

       6. pow2 73.5%

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

       7. unpow2 73.5%

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

       8. frac-times 94.7%

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

       9. unpow2 94.7%

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

       10. *-commutative 94.7%

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

       11. frac-times 94.8%

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

       12. unpow2 94.8%

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

       13. associate-*l* 94.8%

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

   12. Applied egg-rr 94.8%

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

   if 6.69999999999999952e260 < a

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in angle around 0 41.7%

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

      1. times-frac 33.3%

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

   6. Simplified 33.3%

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

      1. add-sqr-sqrt 33.3%

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

      2. pow2 33.3%

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

      3. frac-times 41.7%

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

      4. *-commutative 41.7%

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

      5. pow-prod-down 58.7%

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

      6. pow-prod-down 74.9%

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

   8. Applied egg-rr 74.9%

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

   9. Taylor expanded in b around 0 83.8%

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

       1. *-commutative 83.8%

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

       2. times-frac 99.9%

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

   11. Applied egg-rr 99.9%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 6.7 \cdot 10^{+260}:\\ \;\;\;\;\left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right) \cdot \left(-4 \cdot \left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot {\left(\frac{b}{x-scale} \cdot \frac{a}{y-scale}\right)}^{2}\\ \end{array} \]

Alternative 3: 91.5% accurate, 146.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return -4.0 * ((a / x_45_scale) * ((b / y_45_scale) * ((a / x_45_scale) * (b / y_45_scale))));
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	return -4.0 * ((a / x_45_scale) * ((b / y_45_scale) * ((a / x_45_scale) * (b / y_45_scale))))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 24.7%

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

3. Simplified 21.1%

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

4. Taylor expanded in angle around 0 43.7%

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

   1. times-frac 43.0%

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

6. Simplified 43.0%

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

   1. add-sqr-sqrt 43.0%

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

   2. pow2 43.0%

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

   3. frac-times 43.7%

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

   4. *-commutative 43.7%

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

   5. pow-prod-down 54.7%

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

   6. pow-prod-down 73.5%

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

8. Applied egg-rr 73.5%

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

9. Taylor expanded in b around 0 94.2%

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

    1. frac-times 93.8%

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

    2. unpow2 93.8%

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

    3. associate-*l* 89.5%

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

11. Applied egg-rr 89.5%

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

12. Final simplification 89.5%

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

Alternative 4: 93.9% accurate, 146.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{x-scale} \cdot \frac{b}{y-scale}\\ t_0 \cdot \left(-4 \cdot t_0\right) \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (/ a x-scale) (/ b y-scale)))) (* t_0 (* -4.0 t_0))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (a / x_45_scale) * (b / y_45_scale);
	return t_0 * (-4.0 * t_0);
}

Fortran

real(8) function code(a, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    t_0 = (a / x_45scale) * (b / y_45scale)
    code = t_0 * ((-4.0d0) * t_0)
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (a / x_45_scale) * (b / y_45_scale);
	return t_0 * (-4.0 * t_0);
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (a / x_45_scale) * (b / y_45_scale)
	return t_0 * (-4.0 * t_0)

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(a / x_45_scale) * Float64(b / y_45_scale))
	return Float64(t_0 * Float64(-4.0 * t_0))
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (a / x_45_scale) * (b / y_45_scale);
	tmp = t_0 * (-4.0 * t_0);
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(a / x$45$scale), $MachinePrecision] * N[(b / y$45$scale), $MachinePrecision]), $MachinePrecision]}, N[(t$95$0 * N[(-4.0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 24.7%

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

3. Simplified 21.1%

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

4. Taylor expanded in angle around 0 43.7%

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

   1. associate-*r/ 43.7%

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

   2. times-frac 44.9%

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

   3. *-commutative 44.9%

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

6. Simplified 44.9%

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

   1. frac-times 43.7%

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

   2. pow-prod-down 54.7%

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

   3. pow-prod-down 73.6%

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

8. Applied egg-rr 73.6%

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

   1. add-cbrt-cube 55.9%

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

   2. pow3 55.9%

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

10. Applied egg-rr 55.9%

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

    1. rem-cbrt-cube 73.6%

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

    2. *-un-lft-identity 73.6%

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

    3. times-frac 73.6%

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

    4. metadata-eval 73.6%

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

    5. *-commutative 73.6%

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

    6. pow2 73.6%

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

    7. unpow2 73.6%

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

    8. frac-times 94.2%

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

    9. unpow2 94.2%

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

    10. *-commutative 94.2%

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

    11. frac-times 93.8%

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

    12. unpow2 93.8%

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

    13. associate-*l* 93.9%

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

12. Applied egg-rr 93.9%

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

13. Final simplification 93.9%

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

Alternative 5: 35.7% accurate, 2485.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale) :precision binary64 0.0)

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return 0.0;
}

Fortran

real(8) function code(a, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    code = 0.0d0
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return 0.0;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	return 0.0

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return 0.0
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := 0.0

Derivation

1. Initial program 24.7%

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

3. Simplified 20.5%

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

4. Taylor expanded in b around 0 20.7%

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

   1. distribute-rgt-out 20.7%

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

   2. metadata-eval 20.7%

      \[\leadsto \frac{{a}^{4} \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}} \cdot \color{blue}{0} \]

   3. mul0-rgt 35.1%

      \[\leadsto \color{blue}{0} \]

6. Simplified 35.1%

   \[\leadsto \color{blue}{0} \]

7. Final simplification 35.1%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023310 
(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))))