Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 24.9% → 94.1%

Time: 35.6s

Alternatives: 6

Speedup: 1693.0×

• 32.5% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. a = 1.1859770980468529e+302
  2. b = -7.581684186034608e-174
  3. angle = -4.58561414034925e+150
  4. x-scale = 2.8505097168340834e-290
  5. y-scale = 9.349920060620077e-127

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ t_1 := \sin t\_0\\ t_2 := \cos t\_0\\ t_3 := \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot t\_1\right) \cdot t\_2}{x-scale}}{y-scale}\\ t\_3 \cdot t\_3 - \left(4 \cdot \frac{\frac{{\left(a \cdot t\_1\right)}^{2} + {\left(b \cdot t\_2\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot t\_2\right)}^{2} + {\left(b \cdot t\_1\right)}^{2}}{y-scale}}{y-scale} \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 24.9% 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: 94.1% accurate, 14.7× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ \begin{array}{l} \mathbf{if}\;x-scale\_m \leq 6.5 \cdot 10^{+159}:\\ \;\;\;\;-4 \cdot {\left(x-scale\_m \cdot \frac{\frac{y-scale}{a}}{b}\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4}{{\left(\frac{\frac{x-scale\_m \cdot y-scale}{a}}{b}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
(FPCore (a b angle x-scale_m y-scale)
 :precision binary64
 (if (<= x-scale_m 6.5e+159)
   (* -4.0 (pow (* x-scale_m (/ (/ y-scale a) b)) -2.0))
   (/ -4.0 (pow (/ (/ (* x-scale_m y-scale) a) b) 2.0))))

C

x-scale_m = fabs(x_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	double tmp;
	if (x_45_scale_m <= 6.5e+159) {
		tmp = -4.0 * pow((x_45_scale_m * ((y_45_scale / a) / b)), -2.0);
	} else {
		tmp = -4.0 / pow((((x_45_scale_m * y_45_scale) / a) / b), 2.0);
	}
	return tmp;
}

Fortran

x-scale_m = abs(x_45scale)
real(8) function code(a, b, angle, x_45scale_m, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale
    real(8) :: tmp
    if (x_45scale_m <= 6.5d+159) then
        tmp = (-4.0d0) * ((x_45scale_m * ((y_45scale / a) / b)) ** (-2.0d0))
    else
        tmp = (-4.0d0) / ((((x_45scale_m * y_45scale) / a) / b) ** 2.0d0)
    end if
    code = tmp
end function

Java

x-scale_m = Math.abs(x_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	double tmp;
	if (x_45_scale_m <= 6.5e+159) {
		tmp = -4.0 * Math.pow((x_45_scale_m * ((y_45_scale / a) / b)), -2.0);
	} else {
		tmp = -4.0 / Math.pow((((x_45_scale_m * y_45_scale) / a) / b), 2.0);
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale):
	tmp = 0
	if x_45_scale_m <= 6.5e+159:
		tmp = -4.0 * math.pow((x_45_scale_m * ((y_45_scale / a) / b)), -2.0)
	else:
		tmp = -4.0 / math.pow((((x_45_scale_m * y_45_scale) / a) / b), 2.0)
	return tmp

Julia

x-scale_m = abs(x_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale)
	tmp = 0.0
	if (x_45_scale_m <= 6.5e+159)
		tmp = Float64(-4.0 * (Float64(x_45_scale_m * Float64(Float64(y_45_scale / a) / b)) ^ -2.0));
	else
		tmp = Float64(-4.0 / (Float64(Float64(Float64(x_45_scale_m * y_45_scale) / a) / b) ^ 2.0));
	end
	return tmp
end

MATLAB

x-scale_m = abs(x_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale)
	tmp = 0.0;
	if (x_45_scale_m <= 6.5e+159)
		tmp = -4.0 * ((x_45_scale_m * ((y_45_scale / a) / b)) ^ -2.0);
	else
		tmp = -4.0 / ((((x_45_scale_m * y_45_scale) / a) / b) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale_] := If[LessEqual[x$45$scale$95$m, 6.5e+159], N[(-4.0 * N[Power[N[(x$45$scale$95$m * N[(N[(y$45$scale / a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], N[(-4.0 / N[Power[N[(N[(N[(x$45$scale$95$m * y$45$scale), $MachinePrecision] / a), $MachinePrecision] / b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 6.5000000000000001e159

   1. Initial program 24.2%

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

   3. Simplified 24.1%

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

   5. Taylor expanded in angle around 0 48.9%

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

      1. *-commutative 48.9%

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

      2. unpow2 48.9%

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

      3. unpow2 48.9%

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

      4. swap-sqr 64.4%

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

      5. unpow2 64.4%

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

      6. *-commutative 64.4%

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

      7. unpow2 64.4%

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

      8. unpow2 64.4%

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

      9. swap-sqr 79.4%

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

      10. unpow2 79.4%

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

   7. Simplified 79.4%

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

      1. clear-num 79.4%

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

      2. inv-pow 79.4%

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

   9. Applied egg-rr 79.4%

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

       1. unpow-1 79.4%

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

   11. Simplified 79.4%

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

       1. inv-pow 79.4%

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

       2. add-sqr-sqrt 79.3%

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

       3. unpow-prod-down 79.3%

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

       4. div-inv 78.7%

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

       5. pow-flip 78.7%

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

       6. metadata-eval 78.7%

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

       7. div-inv 78.7%

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

       8. pow-flip 78.7%

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

       9. metadata-eval 78.7%

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

   13. Applied egg-rr 78.7%

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

       1. pow-sqr 78.7%

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

       2. metadata-eval 78.7%

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

   15. Simplified 78.7%

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

   16. Taylor expanded in x-scale around 0 92.2%

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

       1. associate-/l* 93.8%

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

       2. associate-/r* 95.4%

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

   18. Simplified 95.4%

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

   if 6.5000000000000001e159 < x-scale

   1. Initial program 48.8%

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

   3. Simplified 41.1%

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

   5. Taylor expanded in angle around 0 46.2%

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

      1. *-commutative 46.2%

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

      2. unpow2 46.2%

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

      3. unpow2 46.2%

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

      4. swap-sqr 54.0%

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

      5. unpow2 54.0%

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

      6. *-commutative 54.0%

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

      7. unpow2 54.0%

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

      8. unpow2 54.0%

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

      9. swap-sqr 84.9%

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

      10. unpow2 84.9%

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

   7. Simplified 84.9%

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

      1. clear-num 84.9%

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

      2. inv-pow 84.9%

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

   9. Applied egg-rr 84.9%

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

       1. unpow-1 84.9%

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

   11. Simplified 84.9%

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

       1. unpow2 84.9%

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

   13. Applied egg-rr 84.9%

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

       1. un-div-inv 84.9%

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

       2. associate-/r* 95.9%

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

       3. associate-/r* 84.9%

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

       4. div-inv 84.9%

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

       5. pow2 84.9%

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

       6. pow-flip 84.9%

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

       7. metadata-eval 84.9%

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

       8. add-sqr-sqrt 84.9%

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

       9. add-sqr-sqrt 84.9%

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

       10. metadata-eval 84.9%

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

       11. pow-flip 84.9%

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

       12. pow2 84.9%

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

       13. div-inv 84.9%

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

       14. add-sqr-sqrt 84.8%

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

       15. pow2 84.8%

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

   15. Applied egg-rr 95.9%

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

       1. associate-/r* 96.1%

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

   17. Simplified 96.1%

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

Alternative 2: 93.9% accurate, 14.7× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ \begin{array}{l} t_0 := \frac{x-scale\_m \cdot y-scale}{a \cdot b}\\ \mathbf{if}\;x-scale\_m \leq 9 \cdot 10^{+159}:\\ \;\;\;\;-4 \cdot {\left(x-scale\_m \cdot \frac{\frac{y-scale}{a}}{b}\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{1}{t\_0 \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
(FPCore (a b angle x-scale_m y-scale)
 :precision binary64
 (let* ((t_0 (/ (* x-scale_m y-scale) (* a b))))
   (if (<= x-scale_m 9e+159)
     (* -4.0 (pow (* x-scale_m (/ (/ y-scale a) b)) -2.0))
     (* -4.0 (/ 1.0 (* t_0 t_0))))))

C

x-scale_m = fabs(x_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	double t_0 = (x_45_scale_m * y_45_scale) / (a * b);
	double tmp;
	if (x_45_scale_m <= 9e+159) {
		tmp = -4.0 * pow((x_45_scale_m * ((y_45_scale / a) / b)), -2.0);
	} else {
		tmp = -4.0 * (1.0 / (t_0 * t_0));
	}
	return tmp;
}

Fortran

x-scale_m = abs(x_45scale)
real(8) function code(a, b, angle, x_45scale_m, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x_45scale_m * y_45scale) / (a * b)
    if (x_45scale_m <= 9d+159) then
        tmp = (-4.0d0) * ((x_45scale_m * ((y_45scale / a) / b)) ** (-2.0d0))
    else
        tmp = (-4.0d0) * (1.0d0 / (t_0 * t_0))
    end if
    code = tmp
end function

Java

x-scale_m = Math.abs(x_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	double t_0 = (x_45_scale_m * y_45_scale) / (a * b);
	double tmp;
	if (x_45_scale_m <= 9e+159) {
		tmp = -4.0 * Math.pow((x_45_scale_m * ((y_45_scale / a) / b)), -2.0);
	} else {
		tmp = -4.0 * (1.0 / (t_0 * t_0));
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale):
	t_0 = (x_45_scale_m * y_45_scale) / (a * b)
	tmp = 0
	if x_45_scale_m <= 9e+159:
		tmp = -4.0 * math.pow((x_45_scale_m * ((y_45_scale / a) / b)), -2.0)
	else:
		tmp = -4.0 * (1.0 / (t_0 * t_0))
	return tmp

Julia

x-scale_m = abs(x_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale)
	t_0 = Float64(Float64(x_45_scale_m * y_45_scale) / Float64(a * b))
	tmp = 0.0
	if (x_45_scale_m <= 9e+159)
		tmp = Float64(-4.0 * (Float64(x_45_scale_m * Float64(Float64(y_45_scale / a) / b)) ^ -2.0));
	else
		tmp = Float64(-4.0 * Float64(1.0 / Float64(t_0 * t_0)));
	end
	return tmp
end

MATLAB

x-scale_m = abs(x_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale)
	t_0 = (x_45_scale_m * y_45_scale) / (a * b);
	tmp = 0.0;
	if (x_45_scale_m <= 9e+159)
		tmp = -4.0 * ((x_45_scale_m * ((y_45_scale / a) / b)) ^ -2.0);
	else
		tmp = -4.0 * (1.0 / (t_0 * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale_] := Block[{t$95$0 = N[(N[(x$45$scale$95$m * y$45$scale), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$45$scale$95$m, 9e+159], N[(-4.0 * N[Power[N[(x$45$scale$95$m * N[(N[(y$45$scale / a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 9.00000000000000053e159

   1. Initial program 24.2%

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

   3. Simplified 24.1%

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

   5. Taylor expanded in angle around 0 48.9%

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

      1. *-commutative 48.9%

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

      2. unpow2 48.9%

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

      3. unpow2 48.9%

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

      4. swap-sqr 64.4%

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

      5. unpow2 64.4%

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

      6. *-commutative 64.4%

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

      7. unpow2 64.4%

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

      8. unpow2 64.4%

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

      9. swap-sqr 79.4%

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

      10. unpow2 79.4%

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

   7. Simplified 79.4%

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

      1. clear-num 79.4%

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

      2. inv-pow 79.4%

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

   9. Applied egg-rr 79.4%

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

       1. unpow-1 79.4%

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

   11. Simplified 79.4%

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

       1. inv-pow 79.4%

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

       2. add-sqr-sqrt 79.3%

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

       3. unpow-prod-down 79.3%

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

       4. div-inv 78.7%

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

       5. pow-flip 78.7%

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

       6. metadata-eval 78.7%

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

       7. div-inv 78.7%

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

       8. pow-flip 78.7%

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

       9. metadata-eval 78.7%

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

   13. Applied egg-rr 78.7%

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

       1. pow-sqr 78.7%

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

       2. metadata-eval 78.7%

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

   15. Simplified 78.7%

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

   16. Taylor expanded in x-scale around 0 92.2%

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

       1. associate-/l* 93.8%

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

       2. associate-/r* 95.4%

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

   18. Simplified 95.4%

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

   if 9.00000000000000053e159 < x-scale

   1. Initial program 48.8%

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

   3. Simplified 41.1%

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

   5. Taylor expanded in angle around 0 46.2%

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

      1. *-commutative 46.2%

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

      2. unpow2 46.2%

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

      3. unpow2 46.2%

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

      4. swap-sqr 54.0%

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

      5. unpow2 54.0%

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

      6. *-commutative 54.0%

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

      7. unpow2 54.0%

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

      8. unpow2 54.0%

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

      9. swap-sqr 84.9%

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

      10. unpow2 84.9%

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

   7. Simplified 84.9%

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

      1. clear-num 84.9%

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

      2. inv-pow 84.9%

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

   9. Applied egg-rr 84.9%

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

       1. unpow-1 84.9%

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

   11. Simplified 84.9%

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

       1. add-log-exp 70.3%

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

       2. div-inv 70.3%

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

       3. exp-prod 70.3%

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

       4. pow-flip 70.3%

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

       5. metadata-eval 70.3%

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

   13. Applied egg-rr 70.3%

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

       1. pow-exp 70.3%

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

       2. add-log-exp 84.9%

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

       3. metadata-eval 84.9%

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

       4. pow-flip 84.9%

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

       5. pow2 84.9%

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

       6. div-inv 84.9%

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

       7. pow2 84.9%

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

       8. times-frac 95.9%

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

   15. Applied egg-rr 95.9%

       \[\leadsto -4 \cdot \frac{1}{\color{blue}{\frac{x-scale \cdot y-scale}{a \cdot b} \cdot \frac{x-scale \cdot y-scale}{a \cdot b}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 93.7% accurate, 99.6× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ \begin{array}{l} t_0 := a \cdot \frac{b}{x-scale\_m \cdot y-scale}\\ -4 \cdot \left(t\_0 \cdot t\_0\right) \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
(FPCore (a b angle x-scale_m y-scale)
 :precision binary64
 (let* ((t_0 (* a (/ b (* x-scale_m y-scale))))) (* -4.0 (* t_0 t_0))))

C

x-scale_m = fabs(x_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	double t_0 = a * (b / (x_45_scale_m * y_45_scale));
	return -4.0 * (t_0 * t_0);
}

Fortran

x-scale_m = abs(x_45scale)
real(8) function code(a, b, angle, x_45scale_m, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    t_0 = a * (b / (x_45scale_m * y_45scale))
    code = (-4.0d0) * (t_0 * t_0)
end function

Java

x-scale_m = Math.abs(x_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	double t_0 = a * (b / (x_45_scale_m * y_45_scale));
	return -4.0 * (t_0 * t_0);
}

Python

x-scale_m = math.fabs(x_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale):
	t_0 = a * (b / (x_45_scale_m * y_45_scale))
	return -4.0 * (t_0 * t_0)

Julia

x-scale_m = abs(x_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale)
	t_0 = Float64(a * Float64(b / Float64(x_45_scale_m * y_45_scale)))
	return Float64(-4.0 * Float64(t_0 * t_0))
end

MATLAB

x-scale_m = abs(x_45_scale);
function tmp = code(a, b, angle, x_45_scale_m, y_45_scale)
	t_0 = a * (b / (x_45_scale_m * y_45_scale));
	tmp = -4.0 * (t_0 * t_0);
end

Wolfram

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

Derivation

1. Initial program 26.7%

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

3. Simplified 25.8%

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

5. Taylor expanded in angle around 0 48.6%

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

   1. *-commutative 48.6%

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

   2. unpow2 48.6%

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

   3. unpow2 48.6%

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

   4. swap-sqr 63.4%

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

   5. unpow2 63.4%

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

   6. *-commutative 63.4%

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

   7. unpow2 63.4%

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

   8. unpow2 63.4%

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

   9. swap-sqr 79.9%

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

   10. unpow2 79.9%

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

7. Simplified 79.9%

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

   1. clear-num 80.0%

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

   2. inv-pow 80.0%

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

9. Applied egg-rr 80.0%

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

    1. unpow-1 80.0%

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

11. Simplified 80.0%

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

    1. inv-pow 80.0%

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

    2. add-sqr-sqrt 79.9%

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

    3. unpow-prod-down 79.9%

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

    4. div-inv 79.3%

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

    5. pow-flip 79.3%

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

    6. metadata-eval 79.3%

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

    7. div-inv 79.3%

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

    8. pow-flip 79.3%

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

    9. metadata-eval 79.3%

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

13. Applied egg-rr 79.3%

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

    1. pow-sqr 79.3%

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

    2. metadata-eval 79.3%

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

15. Simplified 79.3%

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

    1. sqrt-pow2 79.4%

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

    2. metadata-eval 79.4%

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

    3. unpow-prod-down 79.4%

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

    4. metadata-eval 79.4%

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

    5. pow-flip 79.4%

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

    6. pow2 79.4%

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

    7. unpow-prod-down 79.4%

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

    8. div-inv 80.0%

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

    9. inv-pow 80.0%

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

    10. clear-num 79.9%

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

    11. pow2 79.9%

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

    12. times-frac 92.6%

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

    13. associate-/l* 91.3%

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

    14. associate-/l* 94.6%

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

17. Applied egg-rr 94.6%

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

Alternative 4: 89.3% accurate, 99.6× speedup

Math

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

FPCore

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

C

x-scale_m = fabs(x_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	return -4.0 * (a * ((b / (x_45_scale_m * y_45_scale)) / (((x_45_scale_m * y_45_scale) / a) / b)));
}

Fortran

x-scale_m = abs(x_45scale)
real(8) function code(a, b, angle, x_45scale_m, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale
    code = (-4.0d0) * (a * ((b / (x_45scale_m * y_45scale)) / (((x_45scale_m * y_45scale) / a) / b)))
end function

Java

x-scale_m = Math.abs(x_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	return -4.0 * (a * ((b / (x_45_scale_m * y_45_scale)) / (((x_45_scale_m * y_45_scale) / a) / b)));
}

Python

x-scale_m = math.fabs(x_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale):
	return -4.0 * (a * ((b / (x_45_scale_m * y_45_scale)) / (((x_45_scale_m * y_45_scale) / a) / b)))

Julia

x-scale_m = abs(x_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale)
	return Float64(-4.0 * Float64(a * Float64(Float64(b / Float64(x_45_scale_m * y_45_scale)) / Float64(Float64(Float64(x_45_scale_m * y_45_scale) / a) / b))))
end

MATLAB

x-scale_m = abs(x_45_scale);
function tmp = code(a, b, angle, x_45_scale_m, y_45_scale)
	tmp = -4.0 * (a * ((b / (x_45_scale_m * y_45_scale)) / (((x_45_scale_m * y_45_scale) / a) / b)));
end

Wolfram

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

Derivation

1. Initial program 26.7%

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

3. Simplified 25.8%

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

5. Taylor expanded in angle around 0 48.6%

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

   1. *-commutative 48.6%

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

   2. unpow2 48.6%

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

   3. unpow2 48.6%

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

   4. swap-sqr 63.4%

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

   5. unpow2 63.4%

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

   6. *-commutative 63.4%

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

   7. unpow2 63.4%

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

   8. unpow2 63.4%

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

   9. swap-sqr 79.9%

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

   10. unpow2 79.9%

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

7. Simplified 79.9%

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

   1. clear-num 80.0%

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

   2. inv-pow 80.0%

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

9. Applied egg-rr 80.0%

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

    1. unpow-1 80.0%

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

11. Simplified 80.0%

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

    1. inv-pow 80.0%

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

    2. add-sqr-sqrt 79.9%

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

    3. unpow-prod-down 79.9%

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

    4. div-inv 79.3%

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

    5. pow-flip 79.3%

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

    6. metadata-eval 79.3%

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

    7. div-inv 79.3%

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

    8. pow-flip 79.3%

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

    9. metadata-eval 79.3%

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

13. Applied egg-rr 79.3%

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

    1. pow-sqr 79.3%

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

    2. metadata-eval 79.3%

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

15. Simplified 79.3%

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

    1. sqrt-pow2 79.4%

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

    2. metadata-eval 79.4%

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

    3. unpow-prod-down 79.4%

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

    4. metadata-eval 79.4%

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

    5. pow-flip 79.4%

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

    6. pow2 79.4%

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

    7. unpow-prod-down 79.4%

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

    8. div-inv 80.0%

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

    9. inv-pow 80.0%

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

    10. add-sqr-sqrt 79.9%

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

    11. associate-/r* 79.9%

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

17. Applied egg-rr 91.3%

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

    1. associate-/l* 87.0%

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

    2. associate-/r* 88.2%

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

19. Simplified 88.2%

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

Alternative 5: 86.5% accurate, 99.6× speedup

Math

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

FPCore

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

C

x-scale_m = fabs(x_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	return -4.0 * (a * ((b / (x_45_scale_m * y_45_scale)) / (x_45_scale_m * ((y_45_scale / a) / b))));
}

Fortran

x-scale_m = abs(x_45scale)
real(8) function code(a, b, angle, x_45scale_m, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale
    code = (-4.0d0) * (a * ((b / (x_45scale_m * y_45scale)) / (x_45scale_m * ((y_45scale / a) / b))))
end function

Java

x-scale_m = Math.abs(x_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	return -4.0 * (a * ((b / (x_45_scale_m * y_45_scale)) / (x_45_scale_m * ((y_45_scale / a) / b))));
}

Python

x-scale_m = math.fabs(x_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale):
	return -4.0 * (a * ((b / (x_45_scale_m * y_45_scale)) / (x_45_scale_m * ((y_45_scale / a) / b))))

Julia

x-scale_m = abs(x_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale)
	return Float64(-4.0 * Float64(a * Float64(Float64(b / Float64(x_45_scale_m * y_45_scale)) / Float64(x_45_scale_m * Float64(Float64(y_45_scale / a) / b)))))
end

MATLAB

x-scale_m = abs(x_45_scale);
function tmp = code(a, b, angle, x_45_scale_m, y_45_scale)
	tmp = -4.0 * (a * ((b / (x_45_scale_m * y_45_scale)) / (x_45_scale_m * ((y_45_scale / a) / b))));
end

Wolfram

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

Derivation

1. Initial program 26.7%

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

3. Simplified 25.8%

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

5. Taylor expanded in angle around 0 48.6%

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

   1. *-commutative 48.6%

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

   2. unpow2 48.6%

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

   3. unpow2 48.6%

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

   4. swap-sqr 63.4%

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

   5. unpow2 63.4%

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

   6. *-commutative 63.4%

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

   7. unpow2 63.4%

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

   8. unpow2 63.4%

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

   9. swap-sqr 79.9%

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

   10. unpow2 79.9%

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

7. Simplified 79.9%

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

   1. clear-num 80.0%

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

   2. inv-pow 80.0%

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

9. Applied egg-rr 80.0%

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

    1. unpow-1 80.0%

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

11. Simplified 80.0%

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

    1. inv-pow 80.0%

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

    2. add-sqr-sqrt 79.9%

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

    3. unpow-prod-down 79.9%

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

    4. div-inv 79.3%

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

    5. pow-flip 79.3%

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

    6. metadata-eval 79.3%

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

    7. div-inv 79.3%

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

    8. pow-flip 79.3%

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

    9. metadata-eval 79.3%

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

13. Applied egg-rr 79.3%

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

    1. pow-sqr 79.3%

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

    2. metadata-eval 79.3%

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

15. Simplified 79.3%

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

    1. sqrt-pow2 79.4%

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

    2. metadata-eval 79.4%

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

    3. unpow-prod-down 79.4%

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

    4. metadata-eval 79.4%

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

    5. pow-flip 79.4%

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

    6. pow2 79.4%

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

    7. unpow-prod-down 79.4%

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

    8. div-inv 80.0%

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

    9. inv-pow 80.0%

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

    10. add-sqr-sqrt 79.9%

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

    11. associate-/r* 79.9%

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

17. Applied egg-rr 91.3%

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

    1. associate-/l* 87.0%

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

    2. associate-/l* 86.5%

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

    3. associate-/r* 87.8%

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

19. Simplified 87.8%

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

Alternative 6: 35.9% accurate, 1693.0× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ 0 \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
(FPCore (a b angle x-scale_m y-scale) :precision binary64 0.0)

C

x-scale_m = fabs(x_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	return 0.0;
}

Fortran

x-scale_m = abs(x_45scale)
real(8) function code(a, b, angle, x_45scale_m, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale
    code = 0.0d0
end function

Java

x-scale_m = Math.abs(x_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	return 0.0;
}

Python

x-scale_m = math.fabs(x_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale):
	return 0.0

Julia

x-scale_m = abs(x_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale)
	return 0.0
end

MATLAB

x-scale_m = abs(x_45_scale);
function tmp = code(a, b, angle, x_45_scale_m, y_45_scale)
	tmp = 0.0;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale_] := 0.0

Derivation

1. Initial program 26.7%

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

3. Simplified 25.8%

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

5. Taylor expanded in b around 0 26.5%

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

   1. distribute-rgt-out 26.5%

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

   2. metadata-eval 26.5%

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

   3. mul0-rgt 36.7%

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

7. Simplified 36.7%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024185 
(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))))