Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 25.0% → 93.4%

Time: 1.7min

Alternatives: 6

Speedup: 2485.0×

• 33.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 25.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 93.4% accurate, 21.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y-scale \leq 2 \cdot 10^{-87}:\\ \;\;\;\;\frac{-4}{{\left(\frac{x-scale}{b} \cdot \frac{y-scale}{a}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot {\left(\left(b \cdot a\right) \cdot \frac{1}{y-scale \cdot x-scale}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= y-scale 2e-87)
   (/ -4.0 (pow (* (/ x-scale b) (/ y-scale a)) 2.0))
   (* -4.0 (pow (* (* b a) (/ 1.0 (* y-scale x-scale))) 2.0))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (y_45_scale <= 2e-87) {
		tmp = -4.0 / pow(((x_45_scale / b) * (y_45_scale / a)), 2.0);
	} else {
		tmp = -4.0 * pow(((b * a) * (1.0 / (y_45_scale * x_45_scale))), 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: tmp
    if (y_45scale <= 2d-87) then
        tmp = (-4.0d0) / (((x_45scale / b) * (y_45scale / a)) ** 2.0d0)
    else
        tmp = (-4.0d0) * (((b * a) * (1.0d0 / (y_45scale * x_45scale))) ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (y_45_scale <= 2e-87) {
		tmp = -4.0 / Math.pow(((x_45_scale / b) * (y_45_scale / a)), 2.0);
	} else {
		tmp = -4.0 * Math.pow(((b * a) * (1.0 / (y_45_scale * x_45_scale))), 2.0);
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if y_45_scale <= 2e-87:
		tmp = -4.0 / math.pow(((x_45_scale / b) * (y_45_scale / a)), 2.0)
	else:
		tmp = -4.0 * math.pow(((b * a) * (1.0 / (y_45_scale * x_45_scale))), 2.0)
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (y_45_scale <= 2e-87)
		tmp = Float64(-4.0 / (Float64(Float64(x_45_scale / b) * Float64(y_45_scale / a)) ^ 2.0));
	else
		tmp = Float64(-4.0 * (Float64(Float64(b * a) * Float64(1.0 / Float64(y_45_scale * x_45_scale))) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (y_45_scale <= 2e-87)
		tmp = -4.0 / (((x_45_scale / b) * (y_45_scale / a)) ^ 2.0);
	else
		tmp = -4.0 * (((b * a) * (1.0 / (y_45_scale * x_45_scale))) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[y$45$scale, 2e-87], N[(-4.0 / N[Power[N[(N[(x$45$scale / b), $MachinePrecision] * N[(y$45$scale / a), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[Power[N[(N[(b * a), $MachinePrecision] * N[(1.0 / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 2.00000000000000004e-87

   1. Initial program 19.7%

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

   3. Simplified 13.9%

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

   4. Taylor expanded in angle around 0 40.3%

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

      1. *-commutative 40.3%

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

      2. unpow2 40.3%

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

      3. unpow2 40.3%

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

      4. swap-sqr 54.1%

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

      5. unpow2 54.1%

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

   6. Simplified 54.1%

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

   7. Taylor expanded in b around 0 40.3%

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

      1. associate-*r/ 40.3%

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

      2. unpow2 40.3%

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

      3. unpow2 40.3%

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

      4. swap-sqr 52.1%

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

      5. unpow2 52.1%

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

      6. *-commutative 52.1%

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

      7. associate-/l* 51.8%

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

      8. unpow2 51.8%

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

      9. unpow2 51.8%

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

      10. swap-sqr 73.0%

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

      11. unpow2 73.0%

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

      12. *-commutative 73.0%

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

   9. Simplified 73.0%

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

       1. unpow2 73.0%

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

   11. Applied egg-rr 73.0%

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

   12. Taylor expanded in x-scale around 0 40.3%

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

       1. unpow2 40.3%

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

       2. unpow2 40.3%

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

       3. swap-sqr 54.1%

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

       4. unpow2 54.1%

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

       5. unpow2 54.1%

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

       6. swap-sqr 73.0%

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

       7. times-frac 89.2%

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

       8. unpow2 89.2%

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

       9. *-commutative 89.2%

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

       10. times-frac 94.6%

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

   14. Simplified 94.6%

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

   if 2.00000000000000004e-87 < y-scale

   1. Initial program 47.6%

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

   3. Simplified 43.9%

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

   4. Taylor expanded in angle around 0 57.1%

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

      1. *-commutative 57.1%

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

      2. unpow2 57.1%

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

      3. unpow2 57.1%

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

      4. swap-sqr 63.8%

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

      5. unpow2 63.8%

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

   6. Simplified 63.8%

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

      1. *-commutative 63.8%

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

      2. unpow-prod-down 57.1%

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

      3. frac-times 59.4%

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

      4. *-commutative 59.4%

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

      5. div-inv 59.4%

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

      6. pow-flip 59.4%

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

      7. metadata-eval 59.4%

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

      8. div-inv 59.4%

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

      9. pow-flip 59.4%

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

      10. metadata-eval 59.4%

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

   8. Applied egg-rr 59.4%

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

      1. associate-*r* 59.3%

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

      2. *-commutative 59.3%

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

      3. associate-*r* 59.5%

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

      4. unpow2 59.5%

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

      5. unpow2 59.5%

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

      6. swap-sqr 72.2%

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

      7. unpow2 72.2%

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

   10. Simplified 72.2%

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

       1. add-sqr-sqrt 72.2%

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

       2. pow2 72.2%

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

       3. pow2 72.2%

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

       4. associate-*l* 69.8%

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

       5. pow-prod-down 83.8%

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

       6. *-commutative 83.8%

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

       7. sqrt-prod 83.8%

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

       8. sqrt-prod 55.1%

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

       9. add-sqr-sqrt 92.4%

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

       10. sqrt-pow1 98.5%

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

       11. metadata-eval 98.5%

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

       12. unpow-1 98.5%

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

   12. Applied egg-rr 98.5%

       \[\leadsto -4 \cdot \color{blue}{{\left(\left(a \cdot b\right) \cdot \frac{1}{x-scale \cdot y-scale}\right)}^{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 2 \cdot 10^{-87}:\\ \;\;\;\;\frac{-4}{{\left(\frac{x-scale}{b} \cdot \frac{y-scale}{a}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot {\left(\left(b \cdot a\right) \cdot \frac{1}{y-scale \cdot x-scale}\right)}^{2}\\ \end{array} \]

Alternative 2: 93.3% accurate, 22.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.2 \cdot 10^{+141}:\\ \;\;\;\;\frac{-4}{{\left(\frac{x-scale}{a} \cdot \frac{y-scale}{b}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot {\left(\frac{b \cdot a}{y-scale \cdot x-scale}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= b 5.2e+141)
   (/ -4.0 (pow (* (/ x-scale a) (/ y-scale b)) 2.0))
   (* -4.0 (pow (/ (* b a) (* y-scale x-scale)) 2.0))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b <= 5.2e+141) {
		tmp = -4.0 / pow(((x_45_scale / a) * (y_45_scale / b)), 2.0);
	} else {
		tmp = -4.0 * pow(((b * a) / (y_45_scale * x_45_scale)), 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: tmp
    if (b <= 5.2d+141) then
        tmp = (-4.0d0) / (((x_45scale / a) * (y_45scale / b)) ** 2.0d0)
    else
        tmp = (-4.0d0) * (((b * a) / (y_45scale * x_45scale)) ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b <= 5.2e+141) {
		tmp = -4.0 / Math.pow(((x_45_scale / a) * (y_45_scale / b)), 2.0);
	} else {
		tmp = -4.0 * Math.pow(((b * a) / (y_45_scale * x_45_scale)), 2.0);
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if b <= 5.2e+141:
		tmp = -4.0 / math.pow(((x_45_scale / a) * (y_45_scale / b)), 2.0)
	else:
		tmp = -4.0 * math.pow(((b * a) / (y_45_scale * x_45_scale)), 2.0)
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (b <= 5.2e+141)
		tmp = Float64(-4.0 / (Float64(Float64(x_45_scale / a) * Float64(y_45_scale / b)) ^ 2.0));
	else
		tmp = Float64(-4.0 * (Float64(Float64(b * a) / Float64(y_45_scale * x_45_scale)) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (b <= 5.2e+141)
		tmp = -4.0 / (((x_45_scale / a) * (y_45_scale / b)) ^ 2.0);
	else
		tmp = -4.0 * (((b * a) / (y_45_scale * x_45_scale)) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[b, 5.2e+141], N[(-4.0 / N[Power[N[(N[(x$45$scale / a), $MachinePrecision] * N[(y$45$scale / b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[Power[N[(N[(b * a), $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.1999999999999999e141

   1. Initial program 31.8%

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

   3. Simplified 26.3%

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

   4. Taylor expanded in angle around 0 46.8%

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

      1. *-commutative 46.8%

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

      2. unpow2 46.8%

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

      3. unpow2 46.8%

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

      4. swap-sqr 56.1%

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

      5. unpow2 56.1%

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

   6. Simplified 56.1%

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

   7. Taylor expanded in b around 0 46.8%

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

      1. associate-*r/ 46.8%

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

      2. unpow2 46.8%

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

      3. unpow2 46.8%

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

      4. swap-sqr 57.5%

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

      5. unpow2 57.5%

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

      6. *-commutative 57.5%

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

      7. associate-/l* 57.3%

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

      8. unpow2 57.3%

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

      9. unpow2 57.3%

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

      10. swap-sqr 74.3%

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

      11. unpow2 74.3%

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

      12. *-commutative 74.3%

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

   9. Simplified 74.3%

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

       1. unpow2 74.3%

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

   11. Applied egg-rr 74.3%

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

       1. expm1-log1p-u 46.9%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-4}{\frac{{\left(x-scale \cdot y-scale\right)}^{2}}{\left(a \cdot b\right) \cdot \left(a \cdot b\right)}}\right)\right)} \]

       2. expm1-udef 40.3%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-4}{\frac{{\left(x-scale \cdot y-scale\right)}^{2}}{\left(a \cdot b\right) \cdot \left(a \cdot b\right)}}\right)} - 1} \]

       3. add-sqr-sqrt 40.3%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{-4}{\color{blue}{\sqrt{\frac{{\left(x-scale \cdot y-scale\right)}^{2}}{\left(a \cdot b\right) \cdot \left(a \cdot b\right)}} \cdot \sqrt{\frac{{\left(x-scale \cdot y-scale\right)}^{2}}{\left(a \cdot b\right) \cdot \left(a \cdot b\right)}}}}\right)} - 1 \]

       4. pow2 40.3%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{-4}{\color{blue}{{\left(\sqrt{\frac{{\left(x-scale \cdot y-scale\right)}^{2}}{\left(a \cdot b\right) \cdot \left(a \cdot b\right)}}\right)}^{2}}}\right)} - 1 \]

       5. sqrt-div 40.3%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{-4}{{\color{blue}{\left(\frac{\sqrt{{\left(x-scale \cdot y-scale\right)}^{2}}}{\sqrt{\left(a \cdot b\right) \cdot \left(a \cdot b\right)}}\right)}}^{2}}\right)} - 1 \]

       6. unpow2 40.3%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{-4}{{\left(\frac{\sqrt{\color{blue}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}}}{\sqrt{\left(a \cdot b\right) \cdot \left(a \cdot b\right)}}\right)}^{2}}\right)} - 1 \]

       7. sqrt-prod 18.9%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{-4}{{\left(\frac{\color{blue}{\sqrt{x-scale \cdot y-scale} \cdot \sqrt{x-scale \cdot y-scale}}}{\sqrt{\left(a \cdot b\right) \cdot \left(a \cdot b\right)}}\right)}^{2}}\right)} - 1 \]

       8. add-sqr-sqrt 41.3%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{-4}{{\left(\frac{\color{blue}{x-scale \cdot y-scale}}{\sqrt{\left(a \cdot b\right) \cdot \left(a \cdot b\right)}}\right)}^{2}}\right)} - 1 \]

       9. sqrt-prod 29.0%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{-4}{{\left(\frac{x-scale \cdot y-scale}{\color{blue}{\sqrt{a \cdot b} \cdot \sqrt{a \cdot b}}}\right)}^{2}}\right)} - 1 \]

       10. add-sqr-sqrt 44.4%

           \[\leadsto e^{\mathsf{log1p}\left(\frac{-4}{{\left(\frac{x-scale \cdot y-scale}{\color{blue}{a \cdot b}}\right)}^{2}}\right)} - 1 \]

   13. Applied egg-rr 44.4%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-4}{{\left(\frac{x-scale \cdot y-scale}{a \cdot b}\right)}^{2}}\right)} - 1} \]
   14. Step-by-step derivation

       1. expm1-def 57.4%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-4}{{\left(\frac{x-scale \cdot y-scale}{a \cdot b}\right)}^{2}}\right)\right)} \]

       2. expm1-log1p 91.0%

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

       3. times-frac 91.1%

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

   15. Simplified 91.1%

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

   if 5.1999999999999999e141 < b

   1. Initial program 5.8%

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

   3. Simplified 2.9%

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

   4. Taylor expanded in angle around 0 37.2%

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

      1. *-commutative 37.2%

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

      2. unpow2 37.2%

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

      3. unpow2 37.2%

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

      4. swap-sqr 63.0%

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

      5. unpow2 63.0%

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

   6. Simplified 63.0%

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

      1. expm1-log1p-u 5.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-4 \cdot \frac{{b}^{2} \cdot {a}^{2}}{{\left(x-scale \cdot y-scale\right)}^{2}}\right)\right)} \]

      2. expm1-udef 5.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-4 \cdot \frac{{b}^{2} \cdot {a}^{2}}{{\left(x-scale \cdot y-scale\right)}^{2}}\right)} - 1} \]

      3. *-commutative 5.7%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{{b}^{2} \cdot {a}^{2}}{{\left(x-scale \cdot y-scale\right)}^{2}} \cdot -4}\right)} - 1 \]

      4. div-inv 5.7%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\left({b}^{2} \cdot {a}^{2}\right) \cdot \frac{1}{{\left(x-scale \cdot y-scale\right)}^{2}}\right)} \cdot -4\right)} - 1 \]

      5. pow-prod-down 12.3%

         \[\leadsto e^{\mathsf{log1p}\left(\left(\color{blue}{{\left(b \cdot a\right)}^{2}} \cdot \frac{1}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot -4\right)} - 1 \]

      6. pow-flip 12.3%

         \[\leadsto e^{\mathsf{log1p}\left(\left({\left(b \cdot a\right)}^{2} \cdot \color{blue}{{\left(x-scale \cdot y-scale\right)}^{\left(-2\right)}}\right) \cdot -4\right)} - 1 \]

      7. metadata-eval 12.3%

         \[\leadsto e^{\mathsf{log1p}\left(\left({\left(b \cdot a\right)}^{2} \cdot {\left(x-scale \cdot y-scale\right)}^{\color{blue}{-2}}\right) \cdot -4\right)} - 1 \]

   8. Applied egg-rr 12.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left({\left(b \cdot a\right)}^{2} \cdot {\left(x-scale \cdot y-scale\right)}^{-2}\right) \cdot -4\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 25.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({\left(b \cdot a\right)}^{2} \cdot {\left(x-scale \cdot y-scale\right)}^{-2}\right) \cdot -4\right)\right)} \]

      2. expm1-log1p 88.3%

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

      3. associate-*l* 88.3%

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

      4. *-commutative 88.3%

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

   10. Simplified 88.3%

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

       1. unpow2 88.4%

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

   12. Applied egg-rr 88.3%

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

   13. Taylor expanded in a around 0 37.2%

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

       1. unpow2 37.2%

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

       2. unpow2 37.2%

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

       3. swap-sqr 57.3%

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

       4. unpow2 57.3%

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

       5. unpow2 57.3%

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

       6. unpow2 57.3%

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

       7. swap-sqr 88.4%

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

       8. unpow2 88.4%

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

       9. unpow2 88.4%

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

       10. unpow2 88.4%

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

       11. times-frac 97.0%

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

       12. *-rgt-identity 97.0%

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

       13. associate-*r/ 97.0%

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

       14. *-rgt-identity 97.0%

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

       15. associate-*r/ 96.9%

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

       16. unpow2 96.9%

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

       17. associate-*r/ 97.0%

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

       18. *-rgt-identity 97.0%

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

   15. Simplified 97.0%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.2 \cdot 10^{+141}:\\ \;\;\;\;\frac{-4}{{\left(\frac{x-scale}{a} \cdot \frac{y-scale}{b}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot {\left(\frac{b \cdot a}{y-scale \cdot x-scale}\right)}^{2}\\ \end{array} \]

Alternative 3: 93.5% accurate, 22.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y-scale \leq 1.35 \cdot 10^{-87}:\\ \;\;\;\;\frac{-4}{{\left(\frac{x-scale}{b} \cdot \frac{y-scale}{a}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot {\left(\frac{b \cdot a}{y-scale \cdot x-scale}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= y-scale 1.35e-87)
   (/ -4.0 (pow (* (/ x-scale b) (/ y-scale a)) 2.0))
   (* -4.0 (pow (/ (* b a) (* y-scale x-scale)) 2.0))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (y_45_scale <= 1.35e-87) {
		tmp = -4.0 / pow(((x_45_scale / b) * (y_45_scale / a)), 2.0);
	} else {
		tmp = -4.0 * pow(((b * a) / (y_45_scale * x_45_scale)), 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: tmp
    if (y_45scale <= 1.35d-87) then
        tmp = (-4.0d0) / (((x_45scale / b) * (y_45scale / a)) ** 2.0d0)
    else
        tmp = (-4.0d0) * (((b * a) / (y_45scale * x_45scale)) ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (y_45_scale <= 1.35e-87) {
		tmp = -4.0 / Math.pow(((x_45_scale / b) * (y_45_scale / a)), 2.0);
	} else {
		tmp = -4.0 * Math.pow(((b * a) / (y_45_scale * x_45_scale)), 2.0);
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if y_45_scale <= 1.35e-87:
		tmp = -4.0 / math.pow(((x_45_scale / b) * (y_45_scale / a)), 2.0)
	else:
		tmp = -4.0 * math.pow(((b * a) / (y_45_scale * x_45_scale)), 2.0)
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (y_45_scale <= 1.35e-87)
		tmp = Float64(-4.0 / (Float64(Float64(x_45_scale / b) * Float64(y_45_scale / a)) ^ 2.0));
	else
		tmp = Float64(-4.0 * (Float64(Float64(b * a) / Float64(y_45_scale * x_45_scale)) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (y_45_scale <= 1.35e-87)
		tmp = -4.0 / (((x_45_scale / b) * (y_45_scale / a)) ^ 2.0);
	else
		tmp = -4.0 * (((b * a) / (y_45_scale * x_45_scale)) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[y$45$scale, 1.35e-87], N[(-4.0 / N[Power[N[(N[(x$45$scale / b), $MachinePrecision] * N[(y$45$scale / a), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[Power[N[(N[(b * a), $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 1.34999999999999992e-87

   1. Initial program 19.7%

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

   3. Simplified 13.9%

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

   4. Taylor expanded in angle around 0 40.3%

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

      1. *-commutative 40.3%

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

      2. unpow2 40.3%

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

      3. unpow2 40.3%

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

      4. swap-sqr 54.1%

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

      5. unpow2 54.1%

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

   6. Simplified 54.1%

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

   7. Taylor expanded in b around 0 40.3%

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

      1. associate-*r/ 40.3%

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

      2. unpow2 40.3%

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

      3. unpow2 40.3%

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

      4. swap-sqr 52.1%

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

      5. unpow2 52.1%

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

      6. *-commutative 52.1%

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

      7. associate-/l* 51.8%

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

      8. unpow2 51.8%

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

      9. unpow2 51.8%

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

      10. swap-sqr 73.0%

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

      11. unpow2 73.0%

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

      12. *-commutative 73.0%

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

   9. Simplified 73.0%

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

       1. unpow2 73.0%

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

   11. Applied egg-rr 73.0%

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

   12. Taylor expanded in x-scale around 0 40.3%

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

       1. unpow2 40.3%

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

       2. unpow2 40.3%

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

       3. swap-sqr 54.1%

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

       4. unpow2 54.1%

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

       5. unpow2 54.1%

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

       6. swap-sqr 73.0%

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

       7. times-frac 89.2%

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

       8. unpow2 89.2%

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

       9. *-commutative 89.2%

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

       10. times-frac 94.6%

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

   14. Simplified 94.6%

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

   if 1.34999999999999992e-87 < y-scale

   1. Initial program 47.6%

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

   3. Simplified 43.9%

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

   4. Taylor expanded in angle around 0 57.1%

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

      1. *-commutative 57.1%

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

      2. unpow2 57.1%

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

      3. unpow2 57.1%

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

      4. swap-sqr 63.8%

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

      5. unpow2 63.8%

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

   6. Simplified 63.8%

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

      1. expm1-log1p-u 48.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-4 \cdot \frac{{b}^{2} \cdot {a}^{2}}{{\left(x-scale \cdot y-scale\right)}^{2}}\right)\right)} \]

      2. expm1-udef 46.8%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-4 \cdot \frac{{b}^{2} \cdot {a}^{2}}{{\left(x-scale \cdot y-scale\right)}^{2}}\right)} - 1} \]

      3. *-commutative 46.8%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{{b}^{2} \cdot {a}^{2}}{{\left(x-scale \cdot y-scale\right)}^{2}} \cdot -4}\right)} - 1 \]

      4. div-inv 46.8%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\left({b}^{2} \cdot {a}^{2}\right) \cdot \frac{1}{{\left(x-scale \cdot y-scale\right)}^{2}}\right)} \cdot -4\right)} - 1 \]

      5. pow-prod-down 54.6%

         \[\leadsto e^{\mathsf{log1p}\left(\left(\color{blue}{{\left(b \cdot a\right)}^{2}} \cdot \frac{1}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot -4\right)} - 1 \]

      6. pow-flip 54.6%

         \[\leadsto e^{\mathsf{log1p}\left(\left({\left(b \cdot a\right)}^{2} \cdot \color{blue}{{\left(x-scale \cdot y-scale\right)}^{\left(-2\right)}}\right) \cdot -4\right)} - 1 \]

      7. metadata-eval 54.6%

         \[\leadsto e^{\mathsf{log1p}\left(\left({\left(b \cdot a\right)}^{2} \cdot {\left(x-scale \cdot y-scale\right)}^{\color{blue}{-2}}\right) \cdot -4\right)} - 1 \]

   8. Applied egg-rr 54.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left({\left(b \cdot a\right)}^{2} \cdot {\left(x-scale \cdot y-scale\right)}^{-2}\right) \cdot -4\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 63.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({\left(b \cdot a\right)}^{2} \cdot {\left(x-scale \cdot y-scale\right)}^{-2}\right) \cdot -4\right)\right)} \]

      2. expm1-log1p 83.8%

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

      3. associate-*l* 83.8%

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

      4. *-commutative 83.8%

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

   10. Simplified 83.8%

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

       1. unpow2 83.5%

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

   12. Applied egg-rr 83.8%

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

   13. Taylor expanded in a around 0 57.1%

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

       1. unpow2 57.1%

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

       2. unpow2 57.1%

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

       3. swap-sqr 69.9%

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

       4. unpow2 69.9%

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

       5. unpow2 69.9%

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

       6. unpow2 69.9%

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

       7. swap-sqr 84.4%

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

       8. unpow2 84.4%

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

       9. unpow2 84.4%

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

       10. unpow2 84.4%

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

       11. times-frac 98.5%

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

       12. *-rgt-identity 98.5%

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

       13. associate-*r/ 98.6%

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

       14. *-rgt-identity 98.6%

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

       15. associate-*r/ 98.5%

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

       16. unpow2 98.5%

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

       17. associate-*r/ 98.5%

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

       18. *-rgt-identity 98.5%

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

   15. Simplified 98.5%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 1.35 \cdot 10^{-87}:\\ \;\;\;\;\frac{-4}{{\left(\frac{x-scale}{b} \cdot \frac{y-scale}{a}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot {\left(\frac{b \cdot a}{y-scale \cdot x-scale}\right)}^{2}\\ \end{array} \]

Alternative 4: 94.1% accurate, 22.6× speedup

Math

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

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (* -4.0 (pow (/ (* b a) (* y-scale x-scale)) 2.0)))

C

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

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 = (-4.0d0) * (((b * a) / (y_45scale * x_45scale)) ** 2.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 28.2%

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

3. Simplified 23.1%

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

4. Taylor expanded in angle around 0 45.5%

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

   1. *-commutative 45.5%

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

   2. unpow2 45.5%

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

   3. unpow2 45.5%

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

   4. swap-sqr 57.1%

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

   5. unpow2 57.1%

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

6. Simplified 57.1%

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

   1. expm1-log1p-u 32.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-4 \cdot \frac{{b}^{2} \cdot {a}^{2}}{{\left(x-scale \cdot y-scale\right)}^{2}}\right)\right)} \]

   2. expm1-udef 30.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-4 \cdot \frac{{b}^{2} \cdot {a}^{2}}{{\left(x-scale \cdot y-scale\right)}^{2}}\right)} - 1} \]

   3. *-commutative 30.7%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{{b}^{2} \cdot {a}^{2}}{{\left(x-scale \cdot y-scale\right)}^{2}} \cdot -4}\right)} - 1 \]

   4. div-inv 30.7%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\left({b}^{2} \cdot {a}^{2}\right) \cdot \frac{1}{{\left(x-scale \cdot y-scale\right)}^{2}}\right)} \cdot -4\right)} - 1 \]

   5. pow-prod-down 36.4%

      \[\leadsto e^{\mathsf{log1p}\left(\left(\color{blue}{{\left(b \cdot a\right)}^{2}} \cdot \frac{1}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot -4\right)} - 1 \]

   6. pow-flip 36.4%

      \[\leadsto e^{\mathsf{log1p}\left(\left({\left(b \cdot a\right)}^{2} \cdot \color{blue}{{\left(x-scale \cdot y-scale\right)}^{\left(-2\right)}}\right) \cdot -4\right)} - 1 \]

   7. metadata-eval 36.4%

      \[\leadsto e^{\mathsf{log1p}\left(\left({\left(b \cdot a\right)}^{2} \cdot {\left(x-scale \cdot y-scale\right)}^{\color{blue}{-2}}\right) \cdot -4\right)} - 1 \]

8. Applied egg-rr 36.4%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left({\left(b \cdot a\right)}^{2} \cdot {\left(x-scale \cdot y-scale\right)}^{-2}\right) \cdot -4\right)} - 1} \]
9. Step-by-step derivation

   1. expm1-def 44.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({\left(b \cdot a\right)}^{2} \cdot {\left(x-scale \cdot y-scale\right)}^{-2}\right) \cdot -4\right)\right)} \]

   2. expm1-log1p 77.1%

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

   3. associate-*l* 77.1%

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

   4. *-commutative 77.1%

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

10. Simplified 77.1%

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

    1. unpow2 76.2%

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

12. Applied egg-rr 77.1%

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

13. Taylor expanded in a around 0 45.5%

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

    1. unpow2 45.5%

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

    2. unpow2 45.5%

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

    3. swap-sqr 57.5%

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

    4. unpow2 57.5%

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

    5. unpow2 57.5%

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

    6. unpow2 57.5%

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

    7. swap-sqr 76.9%

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

    8. unpow2 76.9%

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

    9. unpow2 76.9%

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

    10. unpow2 76.9%

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

    11. times-frac 92.5%

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

    12. *-rgt-identity 92.5%

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

    13. associate-*r/ 92.5%

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

    14. *-rgt-identity 92.5%

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

    15. associate-*r/ 92.5%

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

    16. unpow2 92.5%

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

    17. associate-*r/ 92.5%

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

    18. *-rgt-identity 92.5%

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

15. Simplified 92.5%

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

16. Final simplification 92.5%

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

Alternative 5: 93.8% accurate, 146.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 28.2%

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

3. Simplified 23.1%

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

4. Taylor expanded in angle around 0 45.5%

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

   1. *-commutative 45.5%

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

   2. unpow2 45.5%

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

   3. unpow2 45.5%

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

   4. swap-sqr 57.1%

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

   5. unpow2 57.1%

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

6. Simplified 57.1%

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

7. Taylor expanded in b around 0 45.5%

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

   1. associate-*r/ 45.5%

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

   2. unpow2 45.5%

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

   3. unpow2 45.5%

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

   4. swap-sqr 57.5%

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

   5. unpow2 57.5%

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

   6. *-commutative 57.5%

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

   7. associate-/l* 57.3%

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

   8. unpow2 57.3%

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

   9. unpow2 57.3%

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

   10. swap-sqr 76.2%

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

   11. unpow2 76.2%

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

   12. *-commutative 76.2%

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

9. Simplified 76.2%

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

    1. unpow2 76.2%

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

11. Applied egg-rr 76.2%

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

    1. unpow2 76.2%

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

    2. times-frac 91.8%

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

13. Applied egg-rr 91.8%

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

14. Final simplification 91.8%

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

Alternative 6: 34.7% accurate, 2485.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 28.2%

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

3. Simplified 22.8%

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

4. Taylor expanded in b around 0 24.4%

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

   1. distribute-rgt-out 24.4%

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

   2. metadata-eval 24.4%

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

   3. mul0-rgt 40.1%

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

6. Simplified 40.1%

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

7. Final simplification 40.1%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023334 
(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))))