Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 25.0% → 93.9%

Time: 27.1s

Alternatives: 5

Speedup: 1693.0×

• 34.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 25.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 93.9% accurate, 99.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x-scale \cdot \frac{y-scale}{b}\\ \frac{-4 \cdot a}{t\_0} \cdot \frac{a}{t\_0} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.9%

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

3. Simplified 22.2%

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

5. Taylor expanded in angle around 0

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

   1. associate-/l* N/A

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

   2. associate-*r* N/A

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

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(-4 \cdot {a}^{2}\right), \color{blue}{\left(\frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)}\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \left({a}^{2}\right)\right), \left(\frac{\color{blue}{{b}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \left(a \cdot a\right)\right), \left(\frac{{b}^{\color{blue}{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(a, a\right)\right), \left(\frac{{b}^{\color{blue}{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\left({b}^{2}\right), \color{blue}{\left({x-scale}^{2} \cdot {y-scale}^{2}\right)}\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\left(b \cdot b\right), \left(\color{blue}{{x-scale}^{2}} \cdot {y-scale}^{2}\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\color{blue}{{x-scale}^{2}} \cdot {y-scale}^{2}\right)\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(x-scale \cdot x-scale\right) \cdot {\color{blue}{y-scale}}^{2}\right)\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(x-scale \cdot \color{blue}{\left(x-scale \cdot {y-scale}^{2}\right)}\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(x-scale, \color{blue}{\left(x-scale \cdot {y-scale}^{2}\right)}\right)\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(x-scale, \mathsf{*.f64}\left(x-scale, \color{blue}{\left({y-scale}^{2}\right)}\right)\right)\right)\right) \]

   14. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(x-scale, \mathsf{*.f64}\left(x-scale, \left(y-scale \cdot \color{blue}{y-scale}\right)\right)\right)\right)\right) \]

   15. *-lowering-*.f64 52.6%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(x-scale, \mathsf{*.f64}\left(x-scale, \mathsf{*.f64}\left(y-scale, \color{blue}{y-scale}\right)\right)\right)\right)\right) \]

7. Simplified 52.6%

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

   1. clear-num N/A

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

   2. un-div-inv N/A

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

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(-4 \cdot \left(a \cdot a\right)\right), \color{blue}{\left(\frac{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}{b \cdot b}\right)}\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \left(a \cdot a\right)\right), \left(\frac{\color{blue}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}}{b \cdot b}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(a, a\right)\right), \left(\frac{x-scale \cdot \color{blue}{\left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}}{b \cdot b}\right)\right) \]

   6. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(a, a\right)\right), \left(\frac{\left(x-scale \cdot x-scale\right) \cdot \left(y-scale \cdot y-scale\right)}{\color{blue}{b} \cdot b}\right)\right) \]

   7. swap-sqr N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(a, a\right)\right), \left(\frac{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}{\color{blue}{b} \cdot b}\right)\right) \]

   8. times-frac N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(a, a\right)\right), \left(\frac{x-scale \cdot y-scale}{b} \cdot \color{blue}{\frac{x-scale \cdot y-scale}{b}}\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{*.f64}\left(\left(\frac{x-scale \cdot y-scale}{b}\right), \color{blue}{\left(\frac{x-scale \cdot y-scale}{b}\right)}\right)\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(x-scale \cdot y-scale\right), b\right), \left(\frac{\color{blue}{x-scale \cdot y-scale}}{b}\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x-scale, y-scale\right), b\right), \left(\frac{\color{blue}{x-scale} \cdot y-scale}{b}\right)\right)\right) \]

   12. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x-scale, y-scale\right), b\right), \mathsf{/.f64}\left(\left(x-scale \cdot y-scale\right), \color{blue}{b}\right)\right)\right) \]

   13. *-lowering-*.f64 72.3%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x-scale, y-scale\right), b\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(x-scale, y-scale\right), b\right)\right)\right) \]

9. Applied egg-rr 72.3%

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

    1. associate-*r* N/A

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

    2. times-frac N/A

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

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{-4 \cdot a}{\frac{x-scale \cdot y-scale}{b}}\right), \color{blue}{\left(\frac{a}{\frac{x-scale \cdot y-scale}{b}}\right)}\right) \]

    4. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(-4 \cdot a\right), \left(\frac{x-scale \cdot y-scale}{b}\right)\right), \left(\frac{\color{blue}{a}}{\frac{x-scale \cdot y-scale}{b}}\right)\right) \]

    5. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, a\right), \left(\frac{x-scale \cdot y-scale}{b}\right)\right), \left(\frac{a}{\frac{x-scale \cdot y-scale}{b}}\right)\right) \]

    6. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, a\right), \mathsf{/.f64}\left(\left(x-scale \cdot y-scale\right), b\right)\right), \left(\frac{a}{\frac{x-scale \cdot y-scale}{b}}\right)\right) \]

    7. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, a\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(x-scale, y-scale\right), b\right)\right), \left(\frac{a}{\frac{x-scale \cdot y-scale}{b}}\right)\right) \]

    8. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, a\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(x-scale, y-scale\right), b\right)\right), \mathsf{/.f64}\left(a, \color{blue}{\left(\frac{x-scale \cdot y-scale}{b}\right)}\right)\right) \]

    9. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, a\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(x-scale, y-scale\right), b\right)\right), \mathsf{/.f64}\left(a, \mathsf{/.f64}\left(\left(x-scale \cdot y-scale\right), \color{blue}{b}\right)\right)\right) \]

    10. *-lowering-*.f64 91.6%

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, a\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(x-scale, y-scale\right), b\right)\right), \mathsf{/.f64}\left(a, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x-scale, y-scale\right), b\right)\right)\right) \]

11. Applied egg-rr 91.6%

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

    1. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{-4 \cdot a}{\frac{x-scale \cdot y-scale}{b}}\right), \color{blue}{\left(\frac{a}{\frac{x-scale \cdot y-scale}{b}}\right)}\right) \]

    2. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(-4 \cdot a\right), \left(\frac{x-scale \cdot y-scale}{b}\right)\right), \left(\frac{\color{blue}{a}}{\frac{x-scale \cdot y-scale}{b}}\right)\right) \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, a\right), \left(\frac{x-scale \cdot y-scale}{b}\right)\right), \left(\frac{a}{\frac{x-scale \cdot y-scale}{b}}\right)\right) \]

    4. associate-/l* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, a\right), \left(x-scale \cdot \frac{y-scale}{b}\right)\right), \left(\frac{a}{\frac{x-scale \cdot y-scale}{b}}\right)\right) \]

    5. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, a\right), \mathsf{*.f64}\left(x-scale, \left(\frac{y-scale}{b}\right)\right)\right), \left(\frac{a}{\frac{x-scale \cdot y-scale}{b}}\right)\right) \]

    6. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, a\right), \mathsf{*.f64}\left(x-scale, \mathsf{/.f64}\left(y-scale, b\right)\right)\right), \left(\frac{a}{\frac{x-scale \cdot y-scale}{b}}\right)\right) \]

    7. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, a\right), \mathsf{*.f64}\left(x-scale, \mathsf{/.f64}\left(y-scale, b\right)\right)\right), \mathsf{/.f64}\left(a, \color{blue}{\left(\frac{x-scale \cdot y-scale}{b}\right)}\right)\right) \]

    8. associate-/l* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, a\right), \mathsf{*.f64}\left(x-scale, \mathsf{/.f64}\left(y-scale, b\right)\right)\right), \mathsf{/.f64}\left(a, \left(x-scale \cdot \color{blue}{\frac{y-scale}{b}}\right)\right)\right) \]

    9. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, a\right), \mathsf{*.f64}\left(x-scale, \mathsf{/.f64}\left(y-scale, b\right)\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(x-scale, \color{blue}{\left(\frac{y-scale}{b}\right)}\right)\right)\right) \]

    10. /-lowering-/.f64 93.2%

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, a\right), \mathsf{*.f64}\left(x-scale, \mathsf{/.f64}\left(y-scale, b\right)\right)\right), \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(x-scale, \mathsf{/.f64}\left(y-scale, \color{blue}{b}\right)\right)\right)\right) \]

13. Applied egg-rr 93.2%

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

Alternative 2: 91.7% accurate, 99.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.9%

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

3. Simplified 22.2%

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

5. Taylor expanded in angle around 0

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

   1. associate-/l* N/A

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

   2. associate-*r* N/A

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

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(-4 \cdot {a}^{2}\right), \color{blue}{\left(\frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)}\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \left({a}^{2}\right)\right), \left(\frac{\color{blue}{{b}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \left(a \cdot a\right)\right), \left(\frac{{b}^{\color{blue}{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(a, a\right)\right), \left(\frac{{b}^{\color{blue}{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\left({b}^{2}\right), \color{blue}{\left({x-scale}^{2} \cdot {y-scale}^{2}\right)}\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\left(b \cdot b\right), \left(\color{blue}{{x-scale}^{2}} \cdot {y-scale}^{2}\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\color{blue}{{x-scale}^{2}} \cdot {y-scale}^{2}\right)\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(x-scale \cdot x-scale\right) \cdot {\color{blue}{y-scale}}^{2}\right)\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(x-scale \cdot \color{blue}{\left(x-scale \cdot {y-scale}^{2}\right)}\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(x-scale, \color{blue}{\left(x-scale \cdot {y-scale}^{2}\right)}\right)\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(x-scale, \mathsf{*.f64}\left(x-scale, \color{blue}{\left({y-scale}^{2}\right)}\right)\right)\right)\right) \]

   14. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(x-scale, \mathsf{*.f64}\left(x-scale, \left(y-scale \cdot \color{blue}{y-scale}\right)\right)\right)\right)\right) \]

   15. *-lowering-*.f64 52.6%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(x-scale, \mathsf{*.f64}\left(x-scale, \mathsf{*.f64}\left(y-scale, \color{blue}{y-scale}\right)\right)\right)\right)\right) \]

7. Simplified 52.6%

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

   1. clear-num N/A

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

   2. un-div-inv N/A

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

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(-4 \cdot \left(a \cdot a\right)\right), \color{blue}{\left(\frac{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}{b \cdot b}\right)}\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \left(a \cdot a\right)\right), \left(\frac{\color{blue}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}}{b \cdot b}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(a, a\right)\right), \left(\frac{x-scale \cdot \color{blue}{\left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}}{b \cdot b}\right)\right) \]

   6. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(a, a\right)\right), \left(\frac{\left(x-scale \cdot x-scale\right) \cdot \left(y-scale \cdot y-scale\right)}{\color{blue}{b} \cdot b}\right)\right) \]

   7. swap-sqr N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(a, a\right)\right), \left(\frac{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}{\color{blue}{b} \cdot b}\right)\right) \]

   8. times-frac N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(a, a\right)\right), \left(\frac{x-scale \cdot y-scale}{b} \cdot \color{blue}{\frac{x-scale \cdot y-scale}{b}}\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{*.f64}\left(\left(\frac{x-scale \cdot y-scale}{b}\right), \color{blue}{\left(\frac{x-scale \cdot y-scale}{b}\right)}\right)\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(x-scale \cdot y-scale\right), b\right), \left(\frac{\color{blue}{x-scale \cdot y-scale}}{b}\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x-scale, y-scale\right), b\right), \left(\frac{\color{blue}{x-scale} \cdot y-scale}{b}\right)\right)\right) \]

   12. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x-scale, y-scale\right), b\right), \mathsf{/.f64}\left(\left(x-scale \cdot y-scale\right), \color{blue}{b}\right)\right)\right) \]

   13. *-lowering-*.f64 72.3%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x-scale, y-scale\right), b\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(x-scale, y-scale\right), b\right)\right)\right) \]

9. Applied egg-rr 72.3%

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

    1. associate-*r* N/A

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

    2. times-frac N/A

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

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{-4 \cdot a}{\frac{x-scale \cdot y-scale}{b}}\right), \color{blue}{\left(\frac{a}{\frac{x-scale \cdot y-scale}{b}}\right)}\right) \]

    4. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(-4 \cdot a\right), \left(\frac{x-scale \cdot y-scale}{b}\right)\right), \left(\frac{\color{blue}{a}}{\frac{x-scale \cdot y-scale}{b}}\right)\right) \]

    5. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, a\right), \left(\frac{x-scale \cdot y-scale}{b}\right)\right), \left(\frac{a}{\frac{x-scale \cdot y-scale}{b}}\right)\right) \]

    6. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, a\right), \mathsf{/.f64}\left(\left(x-scale \cdot y-scale\right), b\right)\right), \left(\frac{a}{\frac{x-scale \cdot y-scale}{b}}\right)\right) \]

    7. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, a\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(x-scale, y-scale\right), b\right)\right), \left(\frac{a}{\frac{x-scale \cdot y-scale}{b}}\right)\right) \]

    8. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, a\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(x-scale, y-scale\right), b\right)\right), \mathsf{/.f64}\left(a, \color{blue}{\left(\frac{x-scale \cdot y-scale}{b}\right)}\right)\right) \]

    9. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, a\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(x-scale, y-scale\right), b\right)\right), \mathsf{/.f64}\left(a, \mathsf{/.f64}\left(\left(x-scale \cdot y-scale\right), \color{blue}{b}\right)\right)\right) \]

    10. *-lowering-*.f64 91.6%

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, a\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(x-scale, y-scale\right), b\right)\right), \mathsf{/.f64}\left(a, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x-scale, y-scale\right), b\right)\right)\right) \]

11. Applied egg-rr 91.6%

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

    1. div-inv N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{-4 \cdot a}{\left(x-scale \cdot y-scale\right) \cdot \frac{1}{b}}\right), \mathsf{/.f64}\left(a, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x-scale, y-scale\right), b\right)\right)\right) \]

    2. times-frac N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{-4}{x-scale \cdot y-scale} \cdot \frac{a}{\frac{1}{b}}\right), \mathsf{/.f64}\left(\color{blue}{a}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x-scale, y-scale\right), b\right)\right)\right) \]

    3. div-inv N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{-4}{x-scale \cdot y-scale} \cdot \left(a \cdot \frac{1}{\frac{1}{b}}\right)\right), \mathsf{/.f64}\left(a, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x-scale, y-scale\right), b\right)\right)\right) \]

    4. clear-num N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{-4}{x-scale \cdot y-scale} \cdot \left(a \cdot \frac{b}{1}\right)\right), \mathsf{/.f64}\left(a, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x-scale, y-scale\right), b\right)\right)\right) \]

    5. /-rgt-identity N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{-4}{x-scale \cdot y-scale} \cdot \left(a \cdot b\right)\right), \mathsf{/.f64}\left(a, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x-scale, y-scale\right), b\right)\right)\right) \]

    6. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{-4}{x-scale \cdot y-scale}\right), \left(a \cdot b\right)\right), \mathsf{/.f64}\left(\color{blue}{a}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x-scale, y-scale\right), b\right)\right)\right) \]

    7. associate-/r* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{-4}{x-scale}}{y-scale}\right), \left(a \cdot b\right)\right), \mathsf{/.f64}\left(a, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x-scale, y-scale\right), b\right)\right)\right) \]

    8. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{-4}{x-scale}\right), y-scale\right), \left(a \cdot b\right)\right), \mathsf{/.f64}\left(a, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x-scale, y-scale\right), b\right)\right)\right) \]

    9. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, x-scale\right), y-scale\right), \left(a \cdot b\right)\right), \mathsf{/.f64}\left(a, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x-scale, y-scale\right), b\right)\right)\right) \]

    10. *-lowering-*.f64 89.6%

        \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, x-scale\right), y-scale\right), \mathsf{*.f64}\left(a, b\right)\right), \mathsf{/.f64}\left(a, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x-scale, y-scale\right), b\right)\right)\right) \]

13. Applied egg-rr 89.6%

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

Alternative 3: 88.1% accurate, 99.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.9%

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

3. Simplified 22.2%

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

5. Taylor expanded in angle around 0

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

   1. associate-/l* N/A

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

   2. associate-*r* N/A

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

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(-4 \cdot {a}^{2}\right), \color{blue}{\left(\frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)}\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \left({a}^{2}\right)\right), \left(\frac{\color{blue}{{b}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \left(a \cdot a\right)\right), \left(\frac{{b}^{\color{blue}{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(a, a\right)\right), \left(\frac{{b}^{\color{blue}{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\left({b}^{2}\right), \color{blue}{\left({x-scale}^{2} \cdot {y-scale}^{2}\right)}\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\left(b \cdot b\right), \left(\color{blue}{{x-scale}^{2}} \cdot {y-scale}^{2}\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\color{blue}{{x-scale}^{2}} \cdot {y-scale}^{2}\right)\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(x-scale \cdot x-scale\right) \cdot {\color{blue}{y-scale}}^{2}\right)\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(x-scale \cdot \color{blue}{\left(x-scale \cdot {y-scale}^{2}\right)}\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(x-scale, \color{blue}{\left(x-scale \cdot {y-scale}^{2}\right)}\right)\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(x-scale, \mathsf{*.f64}\left(x-scale, \color{blue}{\left({y-scale}^{2}\right)}\right)\right)\right)\right) \]

   14. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(x-scale, \mathsf{*.f64}\left(x-scale, \left(y-scale \cdot \color{blue}{y-scale}\right)\right)\right)\right)\right) \]

   15. *-lowering-*.f64 52.6%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(x-scale, \mathsf{*.f64}\left(x-scale, \mathsf{*.f64}\left(y-scale, \color{blue}{y-scale}\right)\right)\right)\right)\right) \]

7. Simplified 52.6%

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. associate-*l* N/A

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

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(b, \color{blue}{\left(\frac{b}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)} \cdot \left(-4 \cdot \left(a \cdot a\right)\right)\right)}\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\left(\frac{b}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}\right), \color{blue}{\left(-4 \cdot \left(a \cdot a\right)\right)}\right)\right) \]

   6. associate-/r* N/A

      \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\left(\frac{\frac{b}{x-scale}}{x-scale \cdot \left(y-scale \cdot y-scale\right)}\right), \left(\color{blue}{-4} \cdot \left(a \cdot a\right)\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{b}{x-scale}\right), \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)\right), \left(\color{blue}{-4} \cdot \left(a \cdot a\right)\right)\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(b, x-scale\right), \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)\right), \left(-4 \cdot \left(a \cdot a\right)\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(b, x-scale\right), \mathsf{*.f64}\left(x-scale, \left(y-scale \cdot y-scale\right)\right)\right), \left(-4 \cdot \left(a \cdot a\right)\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(b, x-scale\right), \mathsf{*.f64}\left(x-scale, \mathsf{*.f64}\left(y-scale, y-scale\right)\right)\right), \left(-4 \cdot \left(a \cdot a\right)\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(b, x-scale\right), \mathsf{*.f64}\left(x-scale, \mathsf{*.f64}\left(y-scale, y-scale\right)\right)\right), \mathsf{*.f64}\left(-4, \color{blue}{\left(a \cdot a\right)}\right)\right)\right) \]

   12. *-lowering-*.f64 62.1%

       \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(b, x-scale\right), \mathsf{*.f64}\left(x-scale, \mathsf{*.f64}\left(y-scale, y-scale\right)\right)\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right)\right) \]

9. Applied egg-rr 62.1%

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

    1. associate-*r* N/A

       \[\leadsto \mathsf{*.f64}\left(b, \left(\left(\frac{\frac{b}{x-scale}}{x-scale \cdot \left(y-scale \cdot y-scale\right)} \cdot -4\right) \cdot \color{blue}{\left(a \cdot a\right)}\right)\right) \]

    2. associate-*r* N/A

       \[\leadsto \mathsf{*.f64}\left(b, \left(\left(\left(\frac{\frac{b}{x-scale}}{x-scale \cdot \left(y-scale \cdot y-scale\right)} \cdot -4\right) \cdot a\right) \cdot \color{blue}{a}\right)\right) \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\left(\left(\frac{\frac{b}{x-scale}}{x-scale \cdot \left(y-scale \cdot y-scale\right)} \cdot -4\right) \cdot a\right), \color{blue}{a}\right)\right) \]

11. Applied egg-rr 82.8%

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

    1. associate-*r/ N/A

       \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\left(\frac{a \cdot -4}{\frac{x-scale \cdot y-scale}{\frac{b}{x-scale \cdot y-scale}}}\right), a\right)\right) \]

    2. associate-/r/ N/A

       \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\left(\frac{a \cdot -4}{\frac{x-scale \cdot y-scale}{b} \cdot \left(x-scale \cdot y-scale\right)}\right), a\right)\right) \]

    3. times-frac N/A

       \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\left(\frac{a}{\frac{x-scale \cdot y-scale}{b}} \cdot \frac{-4}{x-scale \cdot y-scale}\right), a\right)\right) \]

    4. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\left(\frac{-4}{x-scale \cdot y-scale} \cdot \frac{a}{\frac{x-scale \cdot y-scale}{b}}\right), a\right)\right) \]

    5. clear-num N/A

       \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\left(\frac{-4}{x-scale \cdot y-scale} \cdot \frac{1}{\frac{\frac{x-scale \cdot y-scale}{b}}{a}}\right), a\right)\right) \]

    6. un-div-inv N/A

       \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\left(\frac{\frac{-4}{x-scale \cdot y-scale}}{\frac{\frac{x-scale \cdot y-scale}{b}}{a}}\right), a\right)\right) \]

    7. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{-4}{x-scale \cdot y-scale}\right), \left(\frac{\frac{x-scale \cdot y-scale}{b}}{a}\right)\right), a\right)\right) \]

    8. associate-/r* N/A

       \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{-4}{x-scale}}{y-scale}\right), \left(\frac{\frac{x-scale \cdot y-scale}{b}}{a}\right)\right), a\right)\right) \]

    9. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{-4}{x-scale}\right), y-scale\right), \left(\frac{\frac{x-scale \cdot y-scale}{b}}{a}\right)\right), a\right)\right) \]

    10. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, x-scale\right), y-scale\right), \left(\frac{\frac{x-scale \cdot y-scale}{b}}{a}\right)\right), a\right)\right) \]

    11. associate-/l/ N/A

        \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, x-scale\right), y-scale\right), \left(\frac{x-scale \cdot y-scale}{a \cdot b}\right)\right), a\right)\right) \]

    12. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, x-scale\right), y-scale\right), \mathsf{/.f64}\left(\left(x-scale \cdot y-scale\right), \left(a \cdot b\right)\right)\right), a\right)\right) \]

    13. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, x-scale\right), y-scale\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(x-scale, y-scale\right), \left(a \cdot b\right)\right)\right), a\right)\right) \]

    14. *-lowering-*.f64 85.8%

        \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, x-scale\right), y-scale\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(x-scale, y-scale\right), \mathsf{*.f64}\left(a, b\right)\right)\right), a\right)\right) \]

13. Applied egg-rr 85.8%

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

14. Final simplification 85.8%

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

Alternative 4: 83.5% accurate, 99.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.9%

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

3. Simplified 22.2%

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

5. Taylor expanded in angle around 0

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

   1. associate-/l* N/A

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

   2. associate-*r* N/A

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

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(-4 \cdot {a}^{2}\right), \color{blue}{\left(\frac{{b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)}\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \left({a}^{2}\right)\right), \left(\frac{\color{blue}{{b}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \left(a \cdot a\right)\right), \left(\frac{{b}^{\color{blue}{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(a, a\right)\right), \left(\frac{{b}^{\color{blue}{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\left({b}^{2}\right), \color{blue}{\left({x-scale}^{2} \cdot {y-scale}^{2}\right)}\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\left(b \cdot b\right), \left(\color{blue}{{x-scale}^{2}} \cdot {y-scale}^{2}\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\color{blue}{{x-scale}^{2}} \cdot {y-scale}^{2}\right)\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(x-scale \cdot x-scale\right) \cdot {\color{blue}{y-scale}}^{2}\right)\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(x-scale \cdot \color{blue}{\left(x-scale \cdot {y-scale}^{2}\right)}\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(x-scale, \color{blue}{\left(x-scale \cdot {y-scale}^{2}\right)}\right)\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(x-scale, \mathsf{*.f64}\left(x-scale, \color{blue}{\left({y-scale}^{2}\right)}\right)\right)\right)\right) \]

   14. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(x-scale, \mathsf{*.f64}\left(x-scale, \left(y-scale \cdot \color{blue}{y-scale}\right)\right)\right)\right)\right) \]

   15. *-lowering-*.f64 52.6%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(x-scale, \mathsf{*.f64}\left(x-scale, \mathsf{*.f64}\left(y-scale, \color{blue}{y-scale}\right)\right)\right)\right)\right) \]

7. Simplified 52.6%

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. associate-*l* N/A

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

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(b, \color{blue}{\left(\frac{b}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)} \cdot \left(-4 \cdot \left(a \cdot a\right)\right)\right)}\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\left(\frac{b}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}\right), \color{blue}{\left(-4 \cdot \left(a \cdot a\right)\right)}\right)\right) \]

   6. associate-/r* N/A

      \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\left(\frac{\frac{b}{x-scale}}{x-scale \cdot \left(y-scale \cdot y-scale\right)}\right), \left(\color{blue}{-4} \cdot \left(a \cdot a\right)\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{b}{x-scale}\right), \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)\right), \left(\color{blue}{-4} \cdot \left(a \cdot a\right)\right)\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(b, x-scale\right), \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)\right), \left(-4 \cdot \left(a \cdot a\right)\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(b, x-scale\right), \mathsf{*.f64}\left(x-scale, \left(y-scale \cdot y-scale\right)\right)\right), \left(-4 \cdot \left(a \cdot a\right)\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(b, x-scale\right), \mathsf{*.f64}\left(x-scale, \mathsf{*.f64}\left(y-scale, y-scale\right)\right)\right), \left(-4 \cdot \left(a \cdot a\right)\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(b, x-scale\right), \mathsf{*.f64}\left(x-scale, \mathsf{*.f64}\left(y-scale, y-scale\right)\right)\right), \mathsf{*.f64}\left(-4, \color{blue}{\left(a \cdot a\right)}\right)\right)\right) \]

   12. *-lowering-*.f64 62.1%

       \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(b, x-scale\right), \mathsf{*.f64}\left(x-scale, \mathsf{*.f64}\left(y-scale, y-scale\right)\right)\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right)\right) \]

9. Applied egg-rr 62.1%

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

    1. associate-*r* N/A

       \[\leadsto \mathsf{*.f64}\left(b, \left(\left(\frac{\frac{b}{x-scale}}{x-scale \cdot \left(y-scale \cdot y-scale\right)} \cdot -4\right) \cdot \color{blue}{\left(a \cdot a\right)}\right)\right) \]

    2. associate-*r* N/A

       \[\leadsto \mathsf{*.f64}\left(b, \left(\left(\left(\frac{\frac{b}{x-scale}}{x-scale \cdot \left(y-scale \cdot y-scale\right)} \cdot -4\right) \cdot a\right) \cdot \color{blue}{a}\right)\right) \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\left(\left(\frac{\frac{b}{x-scale}}{x-scale \cdot \left(y-scale \cdot y-scale\right)} \cdot -4\right) \cdot a\right), \color{blue}{a}\right)\right) \]

11. Applied egg-rr 82.8%

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

12. Final simplification 82.8%

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

Alternative 5: 34.9% accurate, 1693.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.9%

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

3. Simplified 22.2%

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

5. Taylor expanded in b around 0

   \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{4} \cdot \left({\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}} + 4 \cdot \frac{{a}^{4} \cdot \left({\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
6. Step-by-step derivation

   1. distribute-rgt-out N/A

      \[\leadsto \frac{{a}^{4} \cdot \left({\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}} \cdot \color{blue}{\left(-4 + 4\right)} \]

   2. metadata-eval N/A

      \[\leadsto \frac{{a}^{4} \cdot \left({\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}} \cdot 0 \]

   3. mul0-rgt 34.2%

      \[\leadsto 0 \]

7. Simplified 34.2%

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

Reproduce

herbie shell --seed 2024161 
(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))))