Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 24.1% → 80.0%

Time: 1.5min

Alternatives: 7

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: 24.1% 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: 80.0% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.05 \cdot 10^{+116}:\\ \;\;\;\;\frac{-4 \cdot \left(a \cdot \left(b \cdot \left(a \cdot b\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\\ \mathbf{elif}\;a \leq -6.5 \cdot 10^{-12}:\\ \;\;\;\;-4 \cdot \frac{{\left(\frac{b}{x-scale}\right)}^{2} \cdot \frac{a}{\frac{y-scale}{a}}}{y-scale}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\frac{b}{x-scale} \cdot \frac{b \cdot {\left(\frac{a}{y-scale}\right)}^{2}}{x-scale}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= a -2.05e+116)
   (/ (* -4.0 (* a (* b (* a b)))) (pow (* x-scale y-scale) 2.0))
   (if (<= a -6.5e-12)
     (* -4.0 (/ (* (pow (/ b x-scale) 2.0) (/ a (/ y-scale a))) y-scale))
     (* -4.0 (* (/ b x-scale) (/ (* b (pow (/ a y-scale) 2.0)) x-scale))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (a <= -2.05e+116) {
		tmp = (-4.0 * (a * (b * (a * b)))) / pow((x_45_scale * y_45_scale), 2.0);
	} else if (a <= -6.5e-12) {
		tmp = -4.0 * ((pow((b / x_45_scale), 2.0) * (a / (y_45_scale / a))) / y_45_scale);
	} else {
		tmp = -4.0 * ((b / x_45_scale) * ((b * pow((a / y_45_scale), 2.0)) / x_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 <= (-2.05d+116)) then
        tmp = ((-4.0d0) * (a * (b * (a * b)))) / ((x_45scale * y_45scale) ** 2.0d0)
    else if (a <= (-6.5d-12)) then
        tmp = (-4.0d0) * ((((b / x_45scale) ** 2.0d0) * (a / (y_45scale / a))) / y_45scale)
    else
        tmp = (-4.0d0) * ((b / x_45scale) * ((b * ((a / y_45scale) ** 2.0d0)) / x_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 <= -2.05e+116) {
		tmp = (-4.0 * (a * (b * (a * b)))) / Math.pow((x_45_scale * y_45_scale), 2.0);
	} else if (a <= -6.5e-12) {
		tmp = -4.0 * ((Math.pow((b / x_45_scale), 2.0) * (a / (y_45_scale / a))) / y_45_scale);
	} else {
		tmp = -4.0 * ((b / x_45_scale) * ((b * Math.pow((a / y_45_scale), 2.0)) / x_45_scale));
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if a <= -2.05e+116:
		tmp = (-4.0 * (a * (b * (a * b)))) / math.pow((x_45_scale * y_45_scale), 2.0)
	elif a <= -6.5e-12:
		tmp = -4.0 * ((math.pow((b / x_45_scale), 2.0) * (a / (y_45_scale / a))) / y_45_scale)
	else:
		tmp = -4.0 * ((b / x_45_scale) * ((b * math.pow((a / y_45_scale), 2.0)) / x_45_scale))
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (a <= -2.05e+116)
		tmp = Float64(Float64(-4.0 * Float64(a * Float64(b * Float64(a * b)))) / (Float64(x_45_scale * y_45_scale) ^ 2.0));
	elseif (a <= -6.5e-12)
		tmp = Float64(-4.0 * Float64(Float64((Float64(b / x_45_scale) ^ 2.0) * Float64(a / Float64(y_45_scale / a))) / y_45_scale));
	else
		tmp = Float64(-4.0 * Float64(Float64(b / x_45_scale) * Float64(Float64(b * (Float64(a / y_45_scale) ^ 2.0)) / x_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 <= -2.05e+116)
		tmp = (-4.0 * (a * (b * (a * b)))) / ((x_45_scale * y_45_scale) ^ 2.0);
	elseif (a <= -6.5e-12)
		tmp = -4.0 * ((((b / x_45_scale) ^ 2.0) * (a / (y_45_scale / a))) / y_45_scale);
	else
		tmp = -4.0 * ((b / x_45_scale) * ((b * ((a / y_45_scale) ^ 2.0)) / x_45_scale));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[a, -2.05e+116], N[(N[(-4.0 * N[(a * N[(b * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[(x$45$scale * y$45$scale), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -6.5e-12], N[(-4.0 * N[(N[(N[Power[N[(b / x$45$scale), $MachinePrecision], 2.0], $MachinePrecision] * N[(a / N[(y$45$scale / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(N[(b / x$45$scale), $MachinePrecision] * N[(N[(b * N[Power[N[(a / y$45$scale), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.0499999999999999e116

   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. Taylor expanded in angle around 0 25.0%

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

      1. *-commutative 25.0%

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

      2. *-commutative 25.0%

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

      3. times-frac 25.1%

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

      4. unpow2 25.1%

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

      5. unpow2 25.1%

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

      6. times-frac 39.8%

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

      7. unpow2 39.8%

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

      8. unpow2 39.8%

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

      9. times-frac 64.7%

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

   4. Simplified 64.7%

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

      1. *-commutative 64.7%

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

      2. frac-times 39.8%

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

      3. frac-times 25.1%

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

      4. pow2 25.1%

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

      5. frac-times 25.0%

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

      6. pow2 25.0%

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

      7. pow2 25.0%

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

      8. pow2 25.0%

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

      9. pow-prod-down 46.6%

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

   6. Applied egg-rr 46.6%

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

      1. associate-*r/ 46.6%

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

      2. associate-*l* 58.0%

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

      3. *-commutative 58.0%

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

   8. Applied egg-rr 58.0%

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

      1. pow1 58.0%

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

      2. associate-*r* 89.2%

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

   10. Applied egg-rr 89.2%

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

   if -2.0499999999999999e116 < a < -6.5000000000000002e-12

   1. Initial program 36.7%

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

   2. Taylor expanded in angle around 0 45.9%

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

      1. *-commutative 45.9%

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

      2. *-commutative 45.9%

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

      3. times-frac 48.9%

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

      4. unpow2 48.9%

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

      5. unpow2 48.9%

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

      6. times-frac 74.0%

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

      7. unpow2 74.0%

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

      8. unpow2 74.0%

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

      9. times-frac 80.0%

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

   4. Simplified 80.0%

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

      1. associate-*r/ 77.1%

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

   6. Applied egg-rr 77.1%

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

      1. associate-*r/ 80.0%

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

      2. frac-times 74.0%

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

      3. associate-/l/ 79.9%

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

      4. associate-*r/ 94.0%

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

      5. pow2 94.0%

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

      6. associate-/l* 93.9%

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

   8. Applied egg-rr 93.9%

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

   if -6.5000000000000002e-12 < a

   1. Initial program 32.6%

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

   2. Taylor expanded in angle around 0 52.4%

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

      1. *-commutative 52.4%

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

      2. *-commutative 52.4%

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

      3. times-frac 52.5%

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

      4. unpow2 52.5%

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

      5. unpow2 52.5%

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

      6. times-frac 66.7%

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

      7. unpow2 66.7%

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

      8. unpow2 66.7%

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

      9. times-frac 82.8%

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

   4. Simplified 82.8%

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

      1. frac-times 64.2%

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

      2. associate-/l* 70.6%

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

   6. Applied egg-rr 70.6%

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

      1. associate-*l/ 73.7%

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

      2. pow2 73.7%

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

      3. associate-/l* 81.7%

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

   8. Applied egg-rr 81.7%

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

      1. associate-/r/ 87.2%

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

   10. Simplified 87.2%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.05 \cdot 10^{+116}:\\ \;\;\;\;\frac{-4 \cdot \left(a \cdot \left(b \cdot \left(a \cdot b\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\\ \mathbf{elif}\;a \leq -6.5 \cdot 10^{-12}:\\ \;\;\;\;-4 \cdot \frac{{\left(\frac{b}{x-scale}\right)}^{2} \cdot \frac{a}{\frac{y-scale}{a}}}{y-scale}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\frac{b}{x-scale} \cdot \frac{b \cdot {\left(\frac{a}{y-scale}\right)}^{2}}{x-scale}\right)\\ \end{array} \]

Alternative 2: 80.5% accurate, 11.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.3 \cdot 10^{+116}:\\ \;\;\;\;\frac{-4}{\frac{{\left(x-scale \cdot y-scale\right)}^{2}}{{\left(a \cdot b\right)}^{2}}}\\ \mathbf{elif}\;a \leq -1.36 \cdot 10^{-11}:\\ \;\;\;\;-4 \cdot \frac{{\left(\frac{b}{x-scale}\right)}^{2} \cdot \frac{a}{\frac{y-scale}{a}}}{y-scale}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\frac{b}{x-scale} \cdot \frac{b \cdot {\left(\frac{a}{y-scale}\right)}^{2}}{x-scale}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= a -1.3e+116)
   (/ -4.0 (/ (pow (* x-scale y-scale) 2.0) (pow (* a b) 2.0)))
   (if (<= a -1.36e-11)
     (* -4.0 (/ (* (pow (/ b x-scale) 2.0) (/ a (/ y-scale a))) y-scale))
     (* -4.0 (* (/ b x-scale) (/ (* b (pow (/ a y-scale) 2.0)) x-scale))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (a <= -1.3e+116) {
		tmp = -4.0 / (pow((x_45_scale * y_45_scale), 2.0) / pow((a * b), 2.0));
	} else if (a <= -1.36e-11) {
		tmp = -4.0 * ((pow((b / x_45_scale), 2.0) * (a / (y_45_scale / a))) / y_45_scale);
	} else {
		tmp = -4.0 * ((b / x_45_scale) * ((b * pow((a / y_45_scale), 2.0)) / x_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 <= (-1.3d+116)) then
        tmp = (-4.0d0) / (((x_45scale * y_45scale) ** 2.0d0) / ((a * b) ** 2.0d0))
    else if (a <= (-1.36d-11)) then
        tmp = (-4.0d0) * ((((b / x_45scale) ** 2.0d0) * (a / (y_45scale / a))) / y_45scale)
    else
        tmp = (-4.0d0) * ((b / x_45scale) * ((b * ((a / y_45scale) ** 2.0d0)) / x_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 <= -1.3e+116) {
		tmp = -4.0 / (Math.pow((x_45_scale * y_45_scale), 2.0) / Math.pow((a * b), 2.0));
	} else if (a <= -1.36e-11) {
		tmp = -4.0 * ((Math.pow((b / x_45_scale), 2.0) * (a / (y_45_scale / a))) / y_45_scale);
	} else {
		tmp = -4.0 * ((b / x_45_scale) * ((b * Math.pow((a / y_45_scale), 2.0)) / x_45_scale));
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if a <= -1.3e+116:
		tmp = -4.0 / (math.pow((x_45_scale * y_45_scale), 2.0) / math.pow((a * b), 2.0))
	elif a <= -1.36e-11:
		tmp = -4.0 * ((math.pow((b / x_45_scale), 2.0) * (a / (y_45_scale / a))) / y_45_scale)
	else:
		tmp = -4.0 * ((b / x_45_scale) * ((b * math.pow((a / y_45_scale), 2.0)) / x_45_scale))
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (a <= -1.3e+116)
		tmp = Float64(-4.0 / Float64((Float64(x_45_scale * y_45_scale) ^ 2.0) / (Float64(a * b) ^ 2.0)));
	elseif (a <= -1.36e-11)
		tmp = Float64(-4.0 * Float64(Float64((Float64(b / x_45_scale) ^ 2.0) * Float64(a / Float64(y_45_scale / a))) / y_45_scale));
	else
		tmp = Float64(-4.0 * Float64(Float64(b / x_45_scale) * Float64(Float64(b * (Float64(a / y_45_scale) ^ 2.0)) / x_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 <= -1.3e+116)
		tmp = -4.0 / (((x_45_scale * y_45_scale) ^ 2.0) / ((a * b) ^ 2.0));
	elseif (a <= -1.36e-11)
		tmp = -4.0 * ((((b / x_45_scale) ^ 2.0) * (a / (y_45_scale / a))) / y_45_scale);
	else
		tmp = -4.0 * ((b / x_45_scale) * ((b * ((a / y_45_scale) ^ 2.0)) / x_45_scale));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[a, -1.3e+116], N[(-4.0 / N[(N[Power[N[(x$45$scale * y$45$scale), $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[(a * b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -1.36e-11], N[(-4.0 * N[(N[(N[Power[N[(b / x$45$scale), $MachinePrecision], 2.0], $MachinePrecision] * N[(a / N[(y$45$scale / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(N[(b / x$45$scale), $MachinePrecision] * N[(N[(b * N[Power[N[(a / y$45$scale), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.29999999999999993e116

   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. Taylor expanded in angle around 0 25.0%

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

      1. *-commutative 25.0%

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

      2. *-commutative 25.0%

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

      3. times-frac 25.1%

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

      4. unpow2 25.1%

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

      5. unpow2 25.1%

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

      6. times-frac 39.8%

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

      7. unpow2 39.8%

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

      8. unpow2 39.8%

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

      9. times-frac 64.7%

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

   4. Simplified 64.7%

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

      1. *-commutative 64.7%

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

      2. frac-times 39.8%

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

      3. frac-times 25.1%

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

      4. pow2 25.1%

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

      5. frac-times 25.0%

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

      6. pow2 25.0%

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

      7. pow2 25.0%

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

      8. pow2 25.0%

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

      9. pow-prod-down 46.6%

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

   6. Applied egg-rr 46.6%

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

   7. Taylor expanded in a around 0 46.6%

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

      1. unpow2 46.6%

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

      2. unpow2 46.6%

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

      3. unswap-sqr 89.0%

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

   9. Simplified 89.0%

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

       1. associate-*r/ 89.0%

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

       2. *-commutative 89.0%

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

       3. pow2 89.0%

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

       4. associate-/l* 89.1%

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

       5. pow2 89.1%

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

       6. *-commutative 89.1%

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

       7. pow2 89.1%

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

   11. Applied egg-rr 89.1%

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

   if -1.29999999999999993e116 < a < -1.36e-11

   1. Initial program 36.7%

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

   2. Taylor expanded in angle around 0 45.9%

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

      1. *-commutative 45.9%

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

      2. *-commutative 45.9%

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

      3. times-frac 48.9%

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

      4. unpow2 48.9%

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

      5. unpow2 48.9%

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

      6. times-frac 74.0%

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

      7. unpow2 74.0%

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

      8. unpow2 74.0%

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

      9. times-frac 80.0%

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

   4. Simplified 80.0%

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

      1. associate-*r/ 77.1%

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

   6. Applied egg-rr 77.1%

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

      1. associate-*r/ 80.0%

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

      2. frac-times 74.0%

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

      3. associate-/l/ 79.9%

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

      4. associate-*r/ 94.0%

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

      5. pow2 94.0%

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

      6. associate-/l* 93.9%

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

   8. Applied egg-rr 93.9%

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

   if -1.36e-11 < a

   1. Initial program 32.6%

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

   2. Taylor expanded in angle around 0 52.4%

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

      1. *-commutative 52.4%

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

      2. *-commutative 52.4%

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

      3. times-frac 52.5%

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

      4. unpow2 52.5%

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

      5. unpow2 52.5%

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

      6. times-frac 66.7%

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

      7. unpow2 66.7%

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

      8. unpow2 66.7%

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

      9. times-frac 82.8%

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

   4. Simplified 82.8%

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

      1. frac-times 64.2%

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

      2. associate-/l* 70.6%

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

   6. Applied egg-rr 70.6%

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

      1. associate-*l/ 73.7%

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

      2. pow2 73.7%

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

      3. associate-/l* 81.7%

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

   8. Applied egg-rr 81.7%

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

      1. associate-/r/ 87.2%

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

   10. Simplified 87.2%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.3 \cdot 10^{+116}:\\ \;\;\;\;\frac{-4}{\frac{{\left(x-scale \cdot y-scale\right)}^{2}}{{\left(a \cdot b\right)}^{2}}}\\ \mathbf{elif}\;a \leq -1.36 \cdot 10^{-11}:\\ \;\;\;\;-4 \cdot \frac{{\left(\frac{b}{x-scale}\right)}^{2} \cdot \frac{a}{\frac{y-scale}{a}}}{y-scale}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\frac{b}{x-scale} \cdot \frac{b \cdot {\left(\frac{a}{y-scale}\right)}^{2}}{x-scale}\right)\\ \end{array} \]

Alternative 3: 80.5% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.1 \cdot 10^{+116}:\\ \;\;\;\;-4 \cdot \frac{\left(a \cdot b\right) \cdot \left(a \cdot b\right)}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}\\ \mathbf{elif}\;a \leq -8.4 \cdot 10^{-12}:\\ \;\;\;\;-4 \cdot \frac{{\left(\frac{b}{x-scale}\right)}^{2} \cdot \frac{a}{\frac{y-scale}{a}}}{y-scale}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\frac{b}{x-scale} \cdot \frac{b \cdot {\left(\frac{a}{y-scale}\right)}^{2}}{x-scale}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= a -1.1e+116)
   (* -4.0 (/ (* (* a b) (* a b)) (* (* x-scale y-scale) (* x-scale y-scale))))
   (if (<= a -8.4e-12)
     (* -4.0 (/ (* (pow (/ b x-scale) 2.0) (/ a (/ y-scale a))) y-scale))
     (* -4.0 (* (/ b x-scale) (/ (* b (pow (/ a y-scale) 2.0)) x-scale))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (a <= -1.1e+116) {
		tmp = -4.0 * (((a * b) * (a * b)) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale)));
	} else if (a <= -8.4e-12) {
		tmp = -4.0 * ((pow((b / x_45_scale), 2.0) * (a / (y_45_scale / a))) / y_45_scale);
	} else {
		tmp = -4.0 * ((b / x_45_scale) * ((b * pow((a / y_45_scale), 2.0)) / x_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 <= (-1.1d+116)) then
        tmp = (-4.0d0) * (((a * b) * (a * b)) / ((x_45scale * y_45scale) * (x_45scale * y_45scale)))
    else if (a <= (-8.4d-12)) then
        tmp = (-4.0d0) * ((((b / x_45scale) ** 2.0d0) * (a / (y_45scale / a))) / y_45scale)
    else
        tmp = (-4.0d0) * ((b / x_45scale) * ((b * ((a / y_45scale) ** 2.0d0)) / x_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 <= -1.1e+116) {
		tmp = -4.0 * (((a * b) * (a * b)) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale)));
	} else if (a <= -8.4e-12) {
		tmp = -4.0 * ((Math.pow((b / x_45_scale), 2.0) * (a / (y_45_scale / a))) / y_45_scale);
	} else {
		tmp = -4.0 * ((b / x_45_scale) * ((b * Math.pow((a / y_45_scale), 2.0)) / x_45_scale));
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if a <= -1.1e+116:
		tmp = -4.0 * (((a * b) * (a * b)) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale)))
	elif a <= -8.4e-12:
		tmp = -4.0 * ((math.pow((b / x_45_scale), 2.0) * (a / (y_45_scale / a))) / y_45_scale)
	else:
		tmp = -4.0 * ((b / x_45_scale) * ((b * math.pow((a / y_45_scale), 2.0)) / x_45_scale))
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (a <= -1.1e+116)
		tmp = Float64(-4.0 * Float64(Float64(Float64(a * b) * Float64(a * b)) / Float64(Float64(x_45_scale * y_45_scale) * Float64(x_45_scale * y_45_scale))));
	elseif (a <= -8.4e-12)
		tmp = Float64(-4.0 * Float64(Float64((Float64(b / x_45_scale) ^ 2.0) * Float64(a / Float64(y_45_scale / a))) / y_45_scale));
	else
		tmp = Float64(-4.0 * Float64(Float64(b / x_45_scale) * Float64(Float64(b * (Float64(a / y_45_scale) ^ 2.0)) / x_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 <= -1.1e+116)
		tmp = -4.0 * (((a * b) * (a * b)) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale)));
	elseif (a <= -8.4e-12)
		tmp = -4.0 * ((((b / x_45_scale) ^ 2.0) * (a / (y_45_scale / a))) / y_45_scale);
	else
		tmp = -4.0 * ((b / x_45_scale) * ((b * ((a / y_45_scale) ^ 2.0)) / x_45_scale));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[a, -1.1e+116], N[(-4.0 * N[(N[(N[(a * b), $MachinePrecision] * N[(a * b), $MachinePrecision]), $MachinePrecision] / N[(N[(x$45$scale * y$45$scale), $MachinePrecision] * N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -8.4e-12], N[(-4.0 * N[(N[(N[Power[N[(b / x$45$scale), $MachinePrecision], 2.0], $MachinePrecision] * N[(a / N[(y$45$scale / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(N[(b / x$45$scale), $MachinePrecision] * N[(N[(b * N[Power[N[(a / y$45$scale), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.1e116

   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. Taylor expanded in angle around 0 25.0%

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

      1. *-commutative 25.0%

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

      2. *-commutative 25.0%

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

      3. times-frac 25.1%

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

      4. unpow2 25.1%

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

      5. unpow2 25.1%

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

      6. times-frac 39.8%

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

      7. unpow2 39.8%

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

      8. unpow2 39.8%

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

      9. times-frac 64.7%

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

   4. Simplified 64.7%

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

      1. *-commutative 64.7%

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

      2. frac-times 39.8%

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

      3. frac-times 25.1%

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

      4. pow2 25.1%

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

      5. frac-times 25.0%

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

      6. pow2 25.0%

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

      7. pow2 25.0%

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

      8. pow2 25.0%

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

      9. pow-prod-down 46.6%

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

   6. Applied egg-rr 46.6%

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

   7. Taylor expanded in a around 0 46.6%

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

      1. unpow2 46.6%

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

      2. unpow2 46.6%

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

      3. unswap-sqr 89.0%

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

   9. Simplified 89.0%

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

       1. unpow2 89.0%

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

       2. *-commutative 89.0%

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

       3. *-commutative 89.0%

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

   11. Applied egg-rr 89.0%

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

   if -1.1e116 < a < -8.39999999999999975e-12

   1. Initial program 36.7%

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

   2. Taylor expanded in angle around 0 45.9%

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

      1. *-commutative 45.9%

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

      2. *-commutative 45.9%

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

      3. times-frac 48.9%

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

      4. unpow2 48.9%

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

      5. unpow2 48.9%

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

      6. times-frac 74.0%

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

      7. unpow2 74.0%

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

      8. unpow2 74.0%

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

      9. times-frac 80.0%

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

   4. Simplified 80.0%

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

      1. associate-*r/ 77.1%

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

   6. Applied egg-rr 77.1%

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

      1. associate-*r/ 80.0%

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

      2. frac-times 74.0%

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

      3. associate-/l/ 79.9%

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

      4. associate-*r/ 94.0%

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

      5. pow2 94.0%

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

      6. associate-/l* 93.9%

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

   8. Applied egg-rr 93.9%

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

   if -8.39999999999999975e-12 < a

   1. Initial program 32.6%

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

   2. Taylor expanded in angle around 0 52.4%

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

      1. *-commutative 52.4%

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

      2. *-commutative 52.4%

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

      3. times-frac 52.5%

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

      4. unpow2 52.5%

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

      5. unpow2 52.5%

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

      6. times-frac 66.7%

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

      7. unpow2 66.7%

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

      8. unpow2 66.7%

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

      9. times-frac 82.8%

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

   4. Simplified 82.8%

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

      1. frac-times 64.2%

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

      2. associate-/l* 70.6%

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

   6. Applied egg-rr 70.6%

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

      1. associate-*l/ 73.7%

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

      2. pow2 73.7%

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

      3. associate-/l* 81.7%

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

   8. Applied egg-rr 81.7%

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

      1. associate-/r/ 87.2%

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

   10. Simplified 87.2%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.1 \cdot 10^{+116}:\\ \;\;\;\;-4 \cdot \frac{\left(a \cdot b\right) \cdot \left(a \cdot b\right)}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}\\ \mathbf{elif}\;a \leq -8.4 \cdot 10^{-12}:\\ \;\;\;\;-4 \cdot \frac{{\left(\frac{b}{x-scale}\right)}^{2} \cdot \frac{a}{\frac{y-scale}{a}}}{y-scale}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\frac{b}{x-scale} \cdot \frac{b \cdot {\left(\frac{a}{y-scale}\right)}^{2}}{x-scale}\right)\\ \end{array} \]

Alternative 4: 79.7% accurate, 21.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.15 \cdot 10^{-165}:\\ \;\;\;\;-4 \cdot \frac{\left(a \cdot b\right) \cdot \left(a \cdot b\right)}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\frac{b}{x-scale} \cdot \frac{b \cdot {\left(\frac{a}{y-scale}\right)}^{2}}{x-scale}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= a -1.15e-165)
   (* -4.0 (/ (* (* a b) (* a b)) (* (* x-scale y-scale) (* x-scale y-scale))))
   (* -4.0 (* (/ b x-scale) (/ (* b (pow (/ a y-scale) 2.0)) x-scale)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (a <= -1.15e-165) {
		tmp = -4.0 * (((a * b) * (a * b)) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale)));
	} else {
		tmp = -4.0 * ((b / x_45_scale) * ((b * pow((a / y_45_scale), 2.0)) / x_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 <= (-1.15d-165)) then
        tmp = (-4.0d0) * (((a * b) * (a * b)) / ((x_45scale * y_45scale) * (x_45scale * y_45scale)))
    else
        tmp = (-4.0d0) * ((b / x_45scale) * ((b * ((a / y_45scale) ** 2.0d0)) / x_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 <= -1.15e-165) {
		tmp = -4.0 * (((a * b) * (a * b)) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale)));
	} else {
		tmp = -4.0 * ((b / x_45_scale) * ((b * Math.pow((a / y_45_scale), 2.0)) / x_45_scale));
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if a <= -1.15e-165:
		tmp = -4.0 * (((a * b) * (a * b)) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale)))
	else:
		tmp = -4.0 * ((b / x_45_scale) * ((b * math.pow((a / y_45_scale), 2.0)) / x_45_scale))
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (a <= -1.15e-165)
		tmp = Float64(-4.0 * Float64(Float64(Float64(a * b) * Float64(a * b)) / Float64(Float64(x_45_scale * y_45_scale) * Float64(x_45_scale * y_45_scale))));
	else
		tmp = Float64(-4.0 * Float64(Float64(b / x_45_scale) * Float64(Float64(b * (Float64(a / y_45_scale) ^ 2.0)) / x_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 <= -1.15e-165)
		tmp = -4.0 * (((a * b) * (a * b)) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale)));
	else
		tmp = -4.0 * ((b / x_45_scale) * ((b * ((a / y_45_scale) ^ 2.0)) / x_45_scale));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[a, -1.15e-165], N[(-4.0 * N[(N[(N[(a * b), $MachinePrecision] * N[(a * b), $MachinePrecision]), $MachinePrecision] / N[(N[(x$45$scale * y$45$scale), $MachinePrecision] * N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(N[(b / x$45$scale), $MachinePrecision] * N[(N[(b * N[Power[N[(a / y$45$scale), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.15e-165

   1. Initial program 25.5%

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

   2. Taylor expanded in angle around 0 46.8%

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

      1. *-commutative 46.8%

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

      2. *-commutative 46.8%

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

      3. times-frac 46.7%

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

      4. unpow2 46.7%

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

      5. unpow2 46.7%

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

      6. times-frac 63.4%

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

      7. unpow2 63.4%

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

      8. unpow2 63.4%

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

      9. times-frac 74.2%

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

   4. Simplified 74.2%

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

      1. *-commutative 74.2%

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

      2. frac-times 63.4%

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

      3. frac-times 46.7%

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

      4. pow2 46.7%

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

      5. frac-times 46.8%

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

      6. pow2 46.8%

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

      7. pow2 46.8%

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

      8. pow2 46.8%

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

      9. pow-prod-down 67.6%

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

   6. Applied egg-rr 67.6%

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

   7. Taylor expanded in a around 0 67.6%

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

      1. unpow2 67.6%

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

      2. unpow2 67.6%

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

      3. unswap-sqr 82.7%

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

   9. Simplified 82.7%

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

       1. unpow2 82.7%

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

       2. *-commutative 82.7%

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

       3. *-commutative 82.7%

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

   11. Applied egg-rr 82.7%

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

   if -1.15e-165 < a

   1. Initial program 31.9%

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

   2. Taylor expanded in angle around 0 49.6%

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

      1. *-commutative 49.6%

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

      2. *-commutative 49.6%

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

      3. times-frac 50.3%

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

      4. unpow2 50.3%

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

      5. unpow2 50.3%

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

      6. times-frac 65.5%

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

      7. unpow2 65.5%

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

      8. unpow2 65.5%

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

      9. times-frac 84.0%

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

   4. Simplified 84.0%

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

      1. frac-times 64.3%

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

      2. associate-/l* 71.3%

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

   6. Applied egg-rr 71.3%

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

      1. associate-*l/ 73.7%

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

      2. pow2 73.7%

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

      3. associate-/l* 82.0%

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

   8. Applied egg-rr 82.0%

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

      1. associate-/r/ 88.4%

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

   10. Simplified 88.4%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.15 \cdot 10^{-165}:\\ \;\;\;\;-4 \cdot \frac{\left(a \cdot b\right) \cdot \left(a \cdot b\right)}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\frac{b}{x-scale} \cdot \frac{b \cdot {\left(\frac{a}{y-scale}\right)}^{2}}{x-scale}\right)\\ \end{array} \]

Alternative 5: 77.4% accurate, 130.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;angle \leq 2.05 \cdot 10^{+83}:\\ \;\;\;\;-4 \cdot \frac{\left(a \cdot b\right) \cdot \left(a \cdot b\right)}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right) \cdot \left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= angle 2.05e+83)
   (* -4.0 (/ (* (* a b) (* a b)) (* (* x-scale y-scale) (* x-scale y-scale))))
   (*
    -4.0
    (* (* (/ b x-scale) (/ b x-scale)) (* (/ a y-scale) (/ a y-scale))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (angle <= 2.05e+83) {
		tmp = -4.0 * (((a * b) * (a * b)) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale)));
	} else {
		tmp = -4.0 * (((b / x_45_scale) * (b / x_45_scale)) * ((a / y_45_scale) * (a / 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 (angle <= 2.05d+83) then
        tmp = (-4.0d0) * (((a * b) * (a * b)) / ((x_45scale * y_45scale) * (x_45scale * y_45scale)))
    else
        tmp = (-4.0d0) * (((b / x_45scale) * (b / x_45scale)) * ((a / y_45scale) * (a / 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 (angle <= 2.05e+83) {
		tmp = -4.0 * (((a * b) * (a * b)) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale)));
	} else {
		tmp = -4.0 * (((b / x_45_scale) * (b / x_45_scale)) * ((a / y_45_scale) * (a / y_45_scale)));
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if angle <= 2.05e+83:
		tmp = -4.0 * (((a * b) * (a * b)) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale)))
	else:
		tmp = -4.0 * (((b / x_45_scale) * (b / x_45_scale)) * ((a / y_45_scale) * (a / y_45_scale)))
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (angle <= 2.05e+83)
		tmp = Float64(-4.0 * Float64(Float64(Float64(a * b) * Float64(a * b)) / Float64(Float64(x_45_scale * y_45_scale) * Float64(x_45_scale * y_45_scale))));
	else
		tmp = Float64(-4.0 * Float64(Float64(Float64(b / x_45_scale) * Float64(b / x_45_scale)) * Float64(Float64(a / y_45_scale) * Float64(a / 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 (angle <= 2.05e+83)
		tmp = -4.0 * (((a * b) * (a * b)) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale)));
	else
		tmp = -4.0 * (((b / x_45_scale) * (b / x_45_scale)) * ((a / y_45_scale) * (a / y_45_scale)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[angle, 2.05e+83], N[(-4.0 * N[(N[(N[(a * b), $MachinePrecision] * N[(a * b), $MachinePrecision]), $MachinePrecision] / N[(N[(x$45$scale * y$45$scale), $MachinePrecision] * N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(N[(N[(b / x$45$scale), $MachinePrecision] * N[(b / x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(a / y$45$scale), $MachinePrecision] * N[(a / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 2.05e83

   1. Initial program 31.7%

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

   2. Taylor expanded in angle around 0 48.0%

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

      1. *-commutative 48.0%

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

      2. *-commutative 48.0%

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

      3. times-frac 47.9%

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

      4. unpow2 47.9%

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

      5. unpow2 47.9%

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

      6. times-frac 62.6%

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

      7. unpow2 62.6%

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

      8. unpow2 62.6%

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

      9. times-frac 78.7%

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

   4. Simplified 78.7%

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

      1. *-commutative 78.7%

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

      2. frac-times 62.6%

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

      3. frac-times 47.9%

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

      4. pow2 47.9%

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

      5. frac-times 48.0%

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

      6. pow2 48.0%

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

      7. pow2 48.0%

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

      8. pow2 48.0%

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

      9. pow-prod-down 66.1%

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

   6. Applied egg-rr 66.1%

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

   7. Taylor expanded in a around 0 66.1%

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

      1. unpow2 66.1%

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

      2. unpow2 66.1%

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

      3. unswap-sqr 82.7%

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

   9. Simplified 82.7%

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

       1. unpow2 82.7%

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

       2. *-commutative 82.7%

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

       3. *-commutative 82.7%

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

   11. Applied egg-rr 82.7%

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

   if 2.05e83 < angle

   1. Initial program 20.7%

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

   2. Taylor expanded in angle around 0 51.3%

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

      1. *-commutative 51.3%

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

      2. *-commutative 51.3%

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

      3. times-frac 53.5%

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

      4. unpow2 53.5%

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

      5. unpow2 53.5%

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

      6. times-frac 73.9%

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

      7. unpow2 73.9%

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

      8. unpow2 73.9%

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

      9. times-frac 88.0%

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

   4. Simplified 88.0%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 2.05 \cdot 10^{+83}:\\ \;\;\;\;-4 \cdot \frac{\left(a \cdot b\right) \cdot \left(a \cdot b\right)}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right) \cdot \left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right)\right)\\ \end{array} \]

Alternative 6: 76.9% accurate, 146.2× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 29.6%

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

2. Taylor expanded in angle around 0 48.6%

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

   1. *-commutative 48.6%

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

   2. *-commutative 48.6%

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

   3. times-frac 49.0%

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

   4. unpow2 49.0%

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

   5. unpow2 49.0%

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

   6. times-frac 64.7%

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

   7. unpow2 64.7%

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

   8. unpow2 64.7%

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

   9. times-frac 80.4%

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

4. Simplified 80.4%

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

5. Final simplification 80.4%

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

Alternative 7: 34.6% 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 29.6%

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

   1. fma-neg 29.8%

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

3. Simplified 24.4%

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

4. Taylor expanded in b around 0 26.3%

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

   1. *-commutative 26.3%

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

   2. *-commutative 26.3%

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

   3. *-commutative 26.3%

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

   4. distribute-lft-out 26.3%

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

6. Simplified 35.4%

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

7. Final simplification 35.4%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023187 
(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))))