Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 25.1% → 93.5%

Time: 1.8min

Alternatives: 6

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: 25.1% accurate, 1.0× speedup

Math

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

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: 93.5% accurate, 22.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{x-scale} \cdot \frac{b}{y-scale}\\ \mathbf{if}\;y-scale \leq 1.2 \cdot 10^{-178}:\\ \;\;\;\;-4 \cdot {\left(\frac{a}{y-scale \cdot \frac{x-scale}{b}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(t_0 \cdot t_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (/ a x-scale) (/ b y-scale))))
   (if (<= y-scale 1.2e-178)
     (* -4.0 (pow (/ a (* y-scale (/ x-scale b))) 2.0))
     (* -4.0 (* t_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);
	double tmp;
	if (y_45_scale <= 1.2e-178) {
		tmp = -4.0 * pow((a / (y_45_scale * (x_45_scale / b))), 2.0);
	} else {
		tmp = -4.0 * (t_0 * t_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 (y_45scale <= 1.2d-178) then
        tmp = (-4.0d0) * ((a / (y_45scale * (x_45scale / b))) ** 2.0d0)
    else
        tmp = (-4.0d0) * (t_0 * t_0)
    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 (y_45_scale <= 1.2e-178) {
		tmp = -4.0 * Math.pow((a / (y_45_scale * (x_45_scale / b))), 2.0);
	} else {
		tmp = -4.0 * (t_0 * t_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 y_45_scale <= 1.2e-178:
		tmp = -4.0 * math.pow((a / (y_45_scale * (x_45_scale / b))), 2.0)
	else:
		tmp = -4.0 * (t_0 * t_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 (y_45_scale <= 1.2e-178)
		tmp = Float64(-4.0 * (Float64(a / Float64(y_45_scale * Float64(x_45_scale / b))) ^ 2.0));
	else
		tmp = Float64(-4.0 * Float64(t_0 * t_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 (y_45_scale <= 1.2e-178)
		tmp = -4.0 * ((a / (y_45_scale * (x_45_scale / b))) ^ 2.0);
	else
		tmp = -4.0 * (t_0 * t_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[y$45$scale, 1.2e-178], N[(-4.0 * N[Power[N[(a / N[(y$45$scale * N[(x$45$scale / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 1.20000000000000002e-178

   1. Initial program 17.7%

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

   3. Simplified 13.3%

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

   4. Taylor expanded in angle around 0 40.0%

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

      1. times-frac 38.6%

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

      2. unpow2 38.6%

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

      3. unpow2 38.6%

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

      4. unpow2 38.6%

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

      5. unpow2 38.6%

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

   6. Simplified 38.6%

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

      1. add-sqr-sqrt 38.6%

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

      2. pow2 38.6%

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

      3. times-frac 51.8%

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

      4. pow2 51.8%

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

      5. times-frac 67.5%

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

      6. pow2 67.5%

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

   8. Applied egg-rr 67.5%

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

      1. pow1/2 67.5%

         \[\leadsto -4 \cdot {\color{blue}{\left({\left({\left(\frac{a}{x-scale}\right)}^{2} \cdot {\left(\frac{b}{y-scale}\right)}^{2}\right)}^{0.5}\right)}}^{2} \]

      2. pow-prod-down 93.1%

         \[\leadsto -4 \cdot {\left({\color{blue}{\left({\left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right)}^{2}\right)}}^{0.5}\right)}^{2} \]

      3. pow-pow 93.1%

         \[\leadsto -4 \cdot {\color{blue}{\left({\left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right)}^{\left(2 \cdot 0.5\right)}\right)}}^{2} \]

      4. metadata-eval 93.1%

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

      5. pow1 93.1%

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

      6. frac-times 94.9%

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

      7. *-commutative 94.9%

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

      8. frac-times 99.0%

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

      9. clear-num 98.9%

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

      10. frac-times 98.3%

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

      11. *-un-lft-identity 98.3%

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

   10. Applied egg-rr 98.3%

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

   if 1.20000000000000002e-178 < y-scale

   1. Initial program 36.6%

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

   3. Simplified 29.5%

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

   4. Taylor expanded in angle around 0 47.9%

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

      1. *-commutative 47.9%

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

      2. times-frac 49.4%

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

      3. unpow2 49.4%

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

      4. unpow2 49.4%

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

      5. times-frac 64.9%

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

      6. unpow2 64.9%

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

      7. unpow2 64.9%

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

   6. Simplified 64.9%

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

      1. times-frac 79.2%

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

   8. Applied egg-rr 79.2%

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

      1. unswap-sqr 94.4%

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

      2. pow2 94.4%

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

      3. frac-times 88.3%

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

      4. *-commutative 88.3%

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

      5. frac-times 97.8%

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

      6. pow-prod-down 77.4%

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

      7. pow2 77.4%

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

      8. unpow2 77.4%

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

      9. times-frac 60.3%

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

      10. associate-*r/ 59.3%

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

      11. frac-times 59.5%

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

   10. Applied egg-rr 59.5%

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

       1. times-frac 59.3%

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

       2. associate-*r/ 60.3%

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

       3. frac-times 77.4%

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

       4. unswap-sqr 97.8%

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

   12. Applied egg-rr 97.8%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 1.2 \cdot 10^{-178}:\\ \;\;\;\;-4 \cdot {\left(\frac{a}{y-scale \cdot \frac{x-scale}{b}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right) \cdot \left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right)\right)\\ \end{array} \]

Alternative 2: 75.6% accurate, 130.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 6 \cdot 10^{+83}:\\ \;\;\;\;-4 \cdot \left(\left(b \cdot \frac{a}{\frac{x-scale}{a}}\right) \cdot \frac{b}{y-scale \cdot \left(y-scale \cdot x-scale\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right) \cdot \left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= a 6e+83)
   (* -4.0 (* (* b (/ a (/ x-scale a))) (/ b (* y-scale (* y-scale x-scale)))))
   (*
    -4.0
    (* (* (/ a x-scale) (/ a x-scale)) (* (/ b y-scale) (/ b y-scale))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (a <= 6e+83) {
		tmp = -4.0 * ((b * (a / (x_45_scale / a))) * (b / (y_45_scale * (y_45_scale * x_45_scale))));
	} else {
		tmp = -4.0 * (((a / x_45_scale) * (a / x_45_scale)) * ((b / y_45_scale) * (b / y_45_scale)));
	}
	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) :: tmp
    if (a <= 6d+83) then
        tmp = (-4.0d0) * ((b * (a / (x_45scale / a))) * (b / (y_45scale * (y_45scale * x_45scale))))
    else
        tmp = (-4.0d0) * (((a / x_45scale) * (a / x_45scale)) * ((b / y_45scale) * (b / y_45scale)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (a <= 6e+83) {
		tmp = -4.0 * ((b * (a / (x_45_scale / a))) * (b / (y_45_scale * (y_45_scale * x_45_scale))));
	} else {
		tmp = -4.0 * (((a / x_45_scale) * (a / x_45_scale)) * ((b / y_45_scale) * (b / y_45_scale)));
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if a <= 6e+83:
		tmp = -4.0 * ((b * (a / (x_45_scale / a))) * (b / (y_45_scale * (y_45_scale * x_45_scale))))
	else:
		tmp = -4.0 * (((a / x_45_scale) * (a / x_45_scale)) * ((b / y_45_scale) * (b / y_45_scale)))
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (a <= 6e+83)
		tmp = Float64(-4.0 * Float64(Float64(b * Float64(a / Float64(x_45_scale / a))) * Float64(b / Float64(y_45_scale * Float64(y_45_scale * x_45_scale)))));
	else
		tmp = Float64(-4.0 * Float64(Float64(Float64(a / x_45_scale) * Float64(a / x_45_scale)) * Float64(Float64(b / y_45_scale) * Float64(b / y_45_scale))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (a <= 6e+83)
		tmp = -4.0 * ((b * (a / (x_45_scale / a))) * (b / (y_45_scale * (y_45_scale * x_45_scale))));
	else
		tmp = -4.0 * (((a / x_45_scale) * (a / x_45_scale)) * ((b / y_45_scale) * (b / y_45_scale)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[a, 6e+83], N[(-4.0 * N[(N[(b * N[(a / N[(x$45$scale / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b / N[(y$45$scale * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(N[(N[(a / x$45$scale), $MachinePrecision] * N[(a / x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(b / y$45$scale), $MachinePrecision] * N[(b / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 5.9999999999999999e83

   1. Initial program 30.0%

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

   3. Simplified 24.1%

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

   4. Taylor expanded in angle around 0 47.3%

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

      1. *-commutative 47.3%

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

      2. times-frac 47.3%

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

      3. unpow2 47.3%

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

      4. unpow2 47.3%

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

      5. times-frac 62.5%

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

      6. unpow2 62.5%

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

      7. unpow2 62.5%

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

   6. Simplified 62.5%

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

      1. times-frac 80.3%

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

   8. Applied egg-rr 80.3%

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

      1. unswap-sqr 97.7%

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

      2. pow2 97.7%

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

      3. frac-times 93.6%

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

      4. *-commutative 93.6%

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

      5. frac-times 95.5%

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

      6. pow-prod-down 70.3%

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

      7. pow2 70.3%

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

      8. unpow2 70.3%

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

      9. times-frac 55.4%

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

      10. associate-*r/ 53.9%

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

      11. frac-times 58.3%

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

   10. Applied egg-rr 58.3%

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

       1. associate-*r* 63.2%

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

       2. *-un-lft-identity 63.2%

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

       3. times-frac 64.0%

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

       4. associate-*l/ 59.7%

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

   12. Applied egg-rr 59.7%

       \[\leadsto -4 \cdot \color{blue}{\left(\frac{\frac{a \cdot a}{x-scale} \cdot b}{1} \cdot \frac{b}{x-scale \cdot \left(y-scale \cdot y-scale\right)}\right)} \]
   13. Step-by-step derivation

       1. /-rgt-identity 59.7%

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

       2. *-commutative 59.7%

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

       3. associate-/l* 64.0%

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

       4. associate-*r* 71.2%

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

   14. Simplified 71.2%

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

   if 5.9999999999999999e83 < a

   1. Initial program 6.6%

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

   3. Simplified 2.4%

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

   4. Taylor expanded in angle around 0 24.7%

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

      1. times-frac 29.1%

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

      2. unpow2 29.1%

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

      3. unpow2 29.1%

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

      4. unpow2 29.1%

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

      5. unpow2 29.1%

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

   6. Simplified 29.1%

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

      1. times-frac 56.0%

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

   8. Applied egg-rr 56.0%

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

      1. times-frac 78.6%

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

   10. Applied egg-rr 78.6%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 6 \cdot 10^{+83}:\\ \;\;\;\;-4 \cdot \left(\left(b \cdot \frac{a}{\frac{x-scale}{a}}\right) \cdot \frac{b}{y-scale \cdot \left(y-scale \cdot x-scale\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right) \cdot \left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right)\right)\\ \end{array} \]

Alternative 3: 94.0% accurate, 130.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{x-scale} \cdot \frac{b}{y-scale}\\ t_1 := \frac{a}{\frac{y-scale \cdot x-scale}{b}}\\ \mathbf{if}\;y-scale \leq 2 \cdot 10^{-178}:\\ \;\;\;\;-4 \cdot \left(t_1 \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(t_0 \cdot t_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (/ a x-scale) (/ b y-scale)))
        (t_1 (/ a (/ (* y-scale x-scale) b))))
   (if (<= y-scale 2e-178) (* -4.0 (* t_1 t_1)) (* -4.0 (* t_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);
	double t_1 = a / ((y_45_scale * x_45_scale) / b);
	double tmp;
	if (y_45_scale <= 2e-178) {
		tmp = -4.0 * (t_1 * t_1);
	} else {
		tmp = -4.0 * (t_0 * t_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) :: t_1
    real(8) :: tmp
    t_0 = (a / x_45scale) * (b / y_45scale)
    t_1 = a / ((y_45scale * x_45scale) / b)
    if (y_45scale <= 2d-178) then
        tmp = (-4.0d0) * (t_1 * t_1)
    else
        tmp = (-4.0d0) * (t_0 * t_0)
    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 t_1 = a / ((y_45_scale * x_45_scale) / b);
	double tmp;
	if (y_45_scale <= 2e-178) {
		tmp = -4.0 * (t_1 * t_1);
	} else {
		tmp = -4.0 * (t_0 * t_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)
	t_1 = a / ((y_45_scale * x_45_scale) / b)
	tmp = 0
	if y_45_scale <= 2e-178:
		tmp = -4.0 * (t_1 * t_1)
	else:
		tmp = -4.0 * (t_0 * t_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))
	t_1 = Float64(a / Float64(Float64(y_45_scale * x_45_scale) / b))
	tmp = 0.0
	if (y_45_scale <= 2e-178)
		tmp = Float64(-4.0 * Float64(t_1 * t_1));
	else
		tmp = Float64(-4.0 * Float64(t_0 * t_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);
	t_1 = a / ((y_45_scale * x_45_scale) / b);
	tmp = 0.0;
	if (y_45_scale <= 2e-178)
		tmp = -4.0 * (t_1 * t_1);
	else
		tmp = -4.0 * (t_0 * t_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]}, Block[{t$95$1 = N[(a / N[(N[(y$45$scale * x$45$scale), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$45$scale, 2e-178], N[(-4.0 * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 1.9999999999999999e-178

   1. Initial program 17.7%

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

   3. Simplified 13.3%

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

   4. Taylor expanded in angle around 0 40.0%

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

      1. times-frac 38.6%

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

      2. unpow2 38.6%

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

      3. unpow2 38.6%

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

      4. unpow2 38.6%

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

      5. unpow2 38.6%

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

   6. Simplified 38.6%

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

      1. add-sqr-sqrt 38.6%

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

      2. pow2 38.6%

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

      3. times-frac 51.8%

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

      4. pow2 51.8%

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

      5. times-frac 67.5%

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

      6. pow2 67.5%

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

   8. Applied egg-rr 67.5%

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

      1. pow1/2 67.5%

         \[\leadsto -4 \cdot {\color{blue}{\left({\left({\left(\frac{a}{x-scale}\right)}^{2} \cdot {\left(\frac{b}{y-scale}\right)}^{2}\right)}^{0.5}\right)}}^{2} \]

      2. pow-prod-down 93.1%

         \[\leadsto -4 \cdot {\left({\color{blue}{\left({\left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right)}^{2}\right)}}^{0.5}\right)}^{2} \]

      3. pow-pow 93.1%

         \[\leadsto -4 \cdot {\color{blue}{\left({\left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right)}^{\left(2 \cdot 0.5\right)}\right)}}^{2} \]

      4. metadata-eval 93.1%

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

      5. pow1 93.1%

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

      6. frac-times 94.9%

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

      7. *-commutative 94.9%

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

      8. frac-times 99.0%

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

      9. clear-num 98.9%

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

      10. frac-times 98.3%

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

      11. *-un-lft-identity 98.3%

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

   10. Applied egg-rr 98.3%

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

       1. unpow2 98.3%

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

       2. associate-*l/ 95.9%

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

       3. associate-*l/ 95.8%

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

   12. Applied egg-rr 95.8%

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

   if 1.9999999999999999e-178 < y-scale

   1. Initial program 36.6%

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

   3. Simplified 29.5%

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

   4. Taylor expanded in angle around 0 47.9%

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

      1. *-commutative 47.9%

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

      2. times-frac 49.4%

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

      3. unpow2 49.4%

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

      4. unpow2 49.4%

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

      5. times-frac 64.9%

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

      6. unpow2 64.9%

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

      7. unpow2 64.9%

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

   6. Simplified 64.9%

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

      1. times-frac 79.2%

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

   8. Applied egg-rr 79.2%

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

      1. unswap-sqr 94.4%

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

      2. pow2 94.4%

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

      3. frac-times 88.3%

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

      4. *-commutative 88.3%

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

      5. frac-times 97.8%

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

      6. pow-prod-down 77.4%

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

      7. pow2 77.4%

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

      8. unpow2 77.4%

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

      9. times-frac 60.3%

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

      10. associate-*r/ 59.3%

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

      11. frac-times 59.5%

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

   10. Applied egg-rr 59.5%

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

       1. times-frac 59.3%

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

       2. associate-*r/ 60.3%

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

       3. frac-times 77.4%

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

       4. unswap-sqr 97.8%

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

   12. Applied egg-rr 97.8%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 2 \cdot 10^{-178}:\\ \;\;\;\;-4 \cdot \left(\frac{a}{\frac{y-scale \cdot x-scale}{b}} \cdot \frac{a}{\frac{y-scale \cdot x-scale}{b}}\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right) \cdot \left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right)\right)\\ \end{array} \]

Alternative 4: 74.0% accurate, 146.2× speedup

Math

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

FPCore

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

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return -4.0 * ((b * (a / (x_45_scale / a))) * (b / (y_45_scale * (y_45_scale * x_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) * ((b * (a / (x_45scale / a))) * (b / (y_45scale * (y_45scale * x_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 * ((b * (a / (x_45_scale / a))) * (b / (y_45_scale * (y_45_scale * x_45_scale))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 20.3%

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

4. Taylor expanded in angle around 0 43.4%

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

   1. *-commutative 43.4%

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

   2. times-frac 44.1%

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

   3. unpow2 44.1%

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

   4. unpow2 44.1%

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

   5. times-frac 59.4%

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

   6. unpow2 59.4%

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

   7. unpow2 59.4%

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

6. Simplified 59.4%

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

   1. times-frac 79.5%

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

8. Applied egg-rr 79.5%

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

   1. unswap-sqr 97.0%

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

   2. pow2 97.0%

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

   3. frac-times 92.1%

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

   4. *-commutative 92.1%

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

   5. frac-times 95.1%

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

   6. pow-prod-down 71.7%

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

   7. pow2 71.7%

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

   8. unpow2 71.7%

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

   9. times-frac 55.5%

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

   10. associate-*r/ 53.9%

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

   11. frac-times 56.3%

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

10. Applied egg-rr 56.3%

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

    1. associate-*r* 61.3%

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

    2. *-un-lft-identity 61.3%

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

    3. times-frac 62.0%

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

    4. associate-*l/ 57.2%

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

12. Applied egg-rr 57.2%

    \[\leadsto -4 \cdot \color{blue}{\left(\frac{\frac{a \cdot a}{x-scale} \cdot b}{1} \cdot \frac{b}{x-scale \cdot \left(y-scale \cdot y-scale\right)}\right)} \]
13. Step-by-step derivation

    1. /-rgt-identity 57.2%

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

    2. *-commutative 57.2%

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

    3. associate-/l* 62.0%

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

    4. associate-*r* 69.4%

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

14. Simplified 69.4%

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

15. Final simplification 69.4%

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

Alternative 5: 93.6% accurate, 146.2× speedup

Math

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

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (/ a x-scale) (/ b y-scale)))) (* -4.0 (* t_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 -4.0 * (t_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 = (-4.0d0) * (t_0 * 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 -4.0 * (t_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 -4.0 * (t_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(-4.0 * Float64(t_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 = -4.0 * (t_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[(-4.0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

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 20.3%

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

4. Taylor expanded in angle around 0 43.4%

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

   1. *-commutative 43.4%

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

   2. times-frac 44.1%

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

   3. unpow2 44.1%

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

   4. unpow2 44.1%

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

   5. times-frac 59.4%

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

   6. unpow2 59.4%

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

   7. unpow2 59.4%

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

6. Simplified 59.4%

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

   1. times-frac 79.5%

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

8. Applied egg-rr 79.5%

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

   1. unswap-sqr 97.0%

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

   2. pow2 97.0%

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

   3. frac-times 92.1%

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

   4. *-commutative 92.1%

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

   5. frac-times 95.1%

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

   6. pow-prod-down 71.7%

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

   7. pow2 71.7%

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

   8. unpow2 71.7%

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

   9. times-frac 55.5%

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

   10. associate-*r/ 53.9%

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

   11. frac-times 56.3%

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

10. Applied egg-rr 56.3%

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

    1. times-frac 53.9%

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

    2. associate-*r/ 55.5%

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

    3. frac-times 71.7%

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

    4. unswap-sqr 95.1%

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

12. Applied egg-rr 95.1%

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

13. Final simplification 95.1%

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

Alternative 6: 35.1% 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 25.9%

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

3. Simplified 20.1%

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

4. Taylor expanded in b around 0 22.0%

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

   1. distribute-rgt-out 22.0%

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

   2. metadata-eval 22.0%

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

   3. mul0-rgt 33.2%

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

6. Simplified 33.2%

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

7. Final simplification 33.2%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023283 
(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))))