Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 24.9% → 93.7%

Time: 35.9s

Alternatives: 4

Speedup: 1693.0×

• Could not converge on a ground truth

  1. reg0 = (bf "2.209698560237519769895403794257435763051e-244")
  2. reg1 = (bf "2.292604679083602994442749238500627675467e-292")
  3. reg2 = (bf "5.958706400296666360189157263016802033889e-191")
  4. reg3 = (bf #e58165295.887272410094738006591796875)
  5. reg4 = (bf "1.195995966977784512059329082335796847485e-117")

• 33.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 24.9% 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: 93.7% accurate, 99.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 23.5%

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

3. Simplified 20.0%

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

5. Taylor expanded in angle around 0 49.5%

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

   1. *-commutative 49.5%

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

   2. unpow2 49.5%

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

   3. unpow2 49.5%

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

   4. swap-sqr 66.6%

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

   5. unpow2 66.6%

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

   6. *-commutative 66.6%

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

   7. unpow2 66.6%

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

   8. unpow2 66.6%

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

   9. swap-sqr 83.7%

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

   10. unpow2 83.7%

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

7. Simplified 83.7%

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

   1. clear-num 83.7%

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

   2. inv-pow 83.7%

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

9. Applied egg-rr 83.7%

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

    1. unpow-1 83.7%

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

11. Simplified 83.7%

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

    1. unpow2 83.7%

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

13. Applied egg-rr 83.7%

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

    1. un-div-inv 83.7%

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

    2. add-sqr-sqrt 83.6%

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

    3. associate-/r* 83.6%

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

    4. sqrt-div 83.7%

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

    5. sqrt-pow1 62.7%

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

    6. metadata-eval 62.7%

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

    7. pow1 62.7%

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

    8. sqrt-prod 30.7%

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

    9. add-sqr-sqrt 59.4%

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

    10. sqrt-div 59.4%

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

    11. sqrt-pow1 59.4%

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

    12. metadata-eval 59.4%

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

    13. pow1 59.4%

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

    14. sqrt-prod 47.1%

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

    15. add-sqr-sqrt 97.0%

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

15. Applied egg-rr 97.0%

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

Alternative 2: 89.3% accurate, 76.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{x-scale} \cdot \frac{b}{y-scale}\\ \mathbf{if}\;a \leq 2.4 \cdot 10^{-163}:\\ \;\;\;\;-4 \cdot \left(\left(a \cdot b\right) \cdot \frac{\frac{b \cdot \frac{a}{y-scale}}{x-scale}}{x-scale \cdot y-scale}\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(t\_0 \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (/ a x-scale) (/ b y-scale))))
   (if (<= a 2.4e-163)
     (*
      -4.0
      (* (* a b) (/ (/ (* b (/ a y-scale)) x-scale) (* x-scale y-scale))))
     (* -4.0 (* t_0 t_0)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (a / x_45_scale) * (b / y_45_scale);
	double tmp;
	if (a <= 2.4e-163) {
		tmp = -4.0 * ((a * b) * (((b * (a / y_45_scale)) / x_45_scale) / (x_45_scale * y_45_scale)));
	} else {
		tmp = -4.0 * (t_0 * t_0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (a / x_45scale) * (b / y_45scale)
    if (a <= 2.4d-163) then
        tmp = (-4.0d0) * ((a * b) * (((b * (a / y_45scale)) / x_45scale) / (x_45scale * y_45scale)))
    else
        tmp = (-4.0d0) * (t_0 * t_0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (a / x_45_scale) * (b / y_45_scale);
	double tmp;
	if (a <= 2.4e-163) {
		tmp = -4.0 * ((a * b) * (((b * (a / y_45_scale)) / x_45_scale) / (x_45_scale * y_45_scale)));
	} else {
		tmp = -4.0 * (t_0 * t_0);
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (a / x_45_scale) * (b / y_45_scale)
	tmp = 0
	if a <= 2.4e-163:
		tmp = -4.0 * ((a * b) * (((b * (a / y_45_scale)) / x_45_scale) / (x_45_scale * y_45_scale)))
	else:
		tmp = -4.0 * (t_0 * t_0)
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(a / x_45_scale) * Float64(b / y_45_scale))
	tmp = 0.0
	if (a <= 2.4e-163)
		tmp = Float64(-4.0 * Float64(Float64(a * b) * Float64(Float64(Float64(b * Float64(a / y_45_scale)) / x_45_scale) / Float64(x_45_scale * y_45_scale))));
	else
		tmp = Float64(-4.0 * Float64(t_0 * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (a / x_45_scale) * (b / y_45_scale);
	tmp = 0.0;
	if (a <= 2.4e-163)
		tmp = -4.0 * ((a * b) * (((b * (a / y_45_scale)) / x_45_scale) / (x_45_scale * y_45_scale)));
	else
		tmp = -4.0 * (t_0 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(a / x$45$scale), $MachinePrecision] * N[(b / y$45$scale), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 2.4e-163], N[(-4.0 * N[(N[(a * b), $MachinePrecision] * N[(N[(N[(b * N[(a / y$45$scale), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 2.4000000000000001e-163

   1. Initial program 27.0%

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

   3. Simplified 22.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(\frac{angle \cdot \pi}{180}\right) \cdot \frac{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}{x-scale \cdot y-scale}, \cos \left(\frac{angle \cdot \pi}{180}\right) \cdot \frac{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}{x-scale \cdot y-scale}, \frac{{\left(a \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{{y-scale}^{2}} \cdot \left(\frac{{\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{{x-scale}^{2}} \cdot -4\right)\right)} \]
   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 67.2%

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

      5. unpow2 67.2%

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

      6. *-commutative 67.2%

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

      7. unpow2 67.2%

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

      8. unpow2 67.2%

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

      9. swap-sqr 85.1%

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

      10. unpow2 85.1%

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

   7. Simplified 85.1%

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

      1. add-sqr-sqrt 85.1%

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

      2. pow2 85.1%

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

      3. div-inv 84.4%

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

      4. pow-flip 84.4%

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

      5. metadata-eval 84.4%

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

   9. Applied egg-rr 84.4%

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

       1. sqrt-pow2 84.5%

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

       2. pow2 84.5%

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

       3. *-commutative 84.5%

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

       4. metadata-eval 84.5%

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

       5. pow1 84.5%

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

       6. metadata-eval 84.5%

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

       7. pow-flip 84.5%

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

       8. associate-/r/ 85.1%

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

       9. add-sqr-sqrt 85.1%

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

       10. associate-/r* 85.0%

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

   11. Applied egg-rr 88.0%

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

       1. associate-/r/ 88.0%

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

       2. times-frac 97.8%

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

       3. associate-/r* 95.4%

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

       4. associate-/l* 93.6%

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

       5. associate-*l/ 93.1%

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

       6. *-commutative 93.1%

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

       7. associate-*l/ 94.8%

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

   13. Simplified 94.8%

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

   if 2.4000000000000001e-163 < a

   1. Initial program 18.1%

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

   3. Simplified 16.2%

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

   5. Taylor expanded in angle around 0 50.8%

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

      1. *-commutative 50.8%

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

      2. unpow2 50.8%

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

      3. unpow2 50.8%

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

      4. swap-sqr 65.6%

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

      5. unpow2 65.6%

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

      6. *-commutative 65.6%

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

      7. unpow2 65.6%

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

      8. unpow2 65.6%

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

      9. swap-sqr 81.5%

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

      10. unpow2 81.5%

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

   7. Simplified 81.5%

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

      1. add-cube-cbrt 81.2%

         \[\leadsto -4 \cdot \frac{{\color{blue}{\left(\left(\sqrt[3]{a \cdot b} \cdot \sqrt[3]{a \cdot b}\right) \cdot \sqrt[3]{a \cdot b}\right)}}^{2}}{{\left(x-scale \cdot y-scale\right)}^{2}} \]

      2. pow3 81.2%

         \[\leadsto -4 \cdot \frac{{\color{blue}{\left({\left(\sqrt[3]{a \cdot b}\right)}^{3}\right)}}^{2}}{{\left(x-scale \cdot y-scale\right)}^{2}} \]

   9. Applied egg-rr 81.2%

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

       1. pow-pow 81.2%

          \[\leadsto -4 \cdot \frac{\color{blue}{{\left(\sqrt[3]{a \cdot b}\right)}^{\left(3 \cdot 2\right)}}}{{\left(x-scale \cdot y-scale\right)}^{2}} \]

       2. pow-to-exp 42.1%

          \[\leadsto -4 \cdot \frac{\color{blue}{e^{\log \left(\sqrt[3]{a \cdot b}\right) \cdot \left(3 \cdot 2\right)}}}{{\left(x-scale \cdot y-scale\right)}^{2}} \]

       3. metadata-eval 42.1%

          \[\leadsto -4 \cdot \frac{e^{\log \left(\sqrt[3]{a \cdot b}\right) \cdot \color{blue}{6}}}{{\left(x-scale \cdot y-scale\right)}^{2}} \]

   11. Applied egg-rr 42.1%

       \[\leadsto -4 \cdot \frac{\color{blue}{e^{\log \left(\sqrt[3]{a \cdot b}\right) \cdot 6}}}{{\left(x-scale \cdot y-scale\right)}^{2}} \]
   12. Step-by-step derivation

       1. exp-to-pow 81.2%

          \[\leadsto -4 \cdot \frac{\color{blue}{{\left(\sqrt[3]{a \cdot b}\right)}^{6}}}{{\left(x-scale \cdot y-scale\right)}^{2}} \]

   13. Simplified 81.2%

       \[\leadsto -4 \cdot \frac{\color{blue}{{\left(\sqrt[3]{a \cdot b}\right)}^{6}}}{{\left(x-scale \cdot y-scale\right)}^{2}} \]
   14. Step-by-step derivation

       1. pow1/3 47.0%

          \[\leadsto -4 \cdot \frac{{\color{blue}{\left({\left(a \cdot b\right)}^{0.3333333333333333}\right)}}^{6}}{{\left(x-scale \cdot y-scale\right)}^{2}} \]

       2. pow-pow 81.5%

          \[\leadsto -4 \cdot \frac{\color{blue}{{\left(a \cdot b\right)}^{\left(0.3333333333333333 \cdot 6\right)}}}{{\left(x-scale \cdot y-scale\right)}^{2}} \]

       3. metadata-eval 81.5%

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

       4. pow2 81.5%

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

       5. pow2 81.5%

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

       6. times-frac 95.9%

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

       7. times-frac 91.2%

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

       8. times-frac 93.9%

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

   15. Applied egg-rr 93.9%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.4 \cdot 10^{-163}:\\ \;\;\;\;-4 \cdot \left(\left(a \cdot b\right) \cdot \frac{\frac{b \cdot \frac{a}{y-scale}}{x-scale}}{x-scale \cdot y-scale}\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right) \cdot \left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.1% accurate, 99.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 23.5%

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

3. Simplified 20.0%

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

5. Taylor expanded in angle around 0 49.5%

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

   1. *-commutative 49.5%

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

   2. unpow2 49.5%

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

   3. unpow2 49.5%

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

   4. swap-sqr 66.6%

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

   5. unpow2 66.6%

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

   6. *-commutative 66.6%

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

   7. unpow2 66.6%

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

   8. unpow2 66.6%

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

   9. swap-sqr 83.7%

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

   10. unpow2 83.7%

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

7. Simplified 83.7%

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

   1. add-cube-cbrt 83.3%

      \[\leadsto -4 \cdot \frac{{\color{blue}{\left(\left(\sqrt[3]{a \cdot b} \cdot \sqrt[3]{a \cdot b}\right) \cdot \sqrt[3]{a \cdot b}\right)}}^{2}}{{\left(x-scale \cdot y-scale\right)}^{2}} \]

   2. pow3 83.3%

      \[\leadsto -4 \cdot \frac{{\color{blue}{\left({\left(\sqrt[3]{a \cdot b}\right)}^{3}\right)}}^{2}}{{\left(x-scale \cdot y-scale\right)}^{2}} \]

9. Applied egg-rr 83.3%

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

    1. pow-pow 83.4%

       \[\leadsto -4 \cdot \frac{\color{blue}{{\left(\sqrt[3]{a \cdot b}\right)}^{\left(3 \cdot 2\right)}}}{{\left(x-scale \cdot y-scale\right)}^{2}} \]

    2. pow-to-exp 38.9%

       \[\leadsto -4 \cdot \frac{\color{blue}{e^{\log \left(\sqrt[3]{a \cdot b}\right) \cdot \left(3 \cdot 2\right)}}}{{\left(x-scale \cdot y-scale\right)}^{2}} \]

    3. metadata-eval 38.9%

       \[\leadsto -4 \cdot \frac{e^{\log \left(\sqrt[3]{a \cdot b}\right) \cdot \color{blue}{6}}}{{\left(x-scale \cdot y-scale\right)}^{2}} \]

11. Applied egg-rr 38.9%

    \[\leadsto -4 \cdot \frac{\color{blue}{e^{\log \left(\sqrt[3]{a \cdot b}\right) \cdot 6}}}{{\left(x-scale \cdot y-scale\right)}^{2}} \]
12. Step-by-step derivation

    1. exp-to-pow 83.4%

       \[\leadsto -4 \cdot \frac{\color{blue}{{\left(\sqrt[3]{a \cdot b}\right)}^{6}}}{{\left(x-scale \cdot y-scale\right)}^{2}} \]

13. Simplified 83.4%

    \[\leadsto -4 \cdot \frac{\color{blue}{{\left(\sqrt[3]{a \cdot b}\right)}^{6}}}{{\left(x-scale \cdot y-scale\right)}^{2}} \]
14. Step-by-step derivation

    1. pow1/3 43.2%

       \[\leadsto -4 \cdot \frac{{\color{blue}{\left({\left(a \cdot b\right)}^{0.3333333333333333}\right)}}^{6}}{{\left(x-scale \cdot y-scale\right)}^{2}} \]

    2. pow-pow 83.7%

       \[\leadsto -4 \cdot \frac{\color{blue}{{\left(a \cdot b\right)}^{\left(0.3333333333333333 \cdot 6\right)}}}{{\left(x-scale \cdot y-scale\right)}^{2}} \]

    3. metadata-eval 83.7%

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

    4. pow2 83.7%

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

    5. pow2 83.7%

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

    6. times-frac 97.0%

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

    7. times-frac 89.2%

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

    8. times-frac 91.4%

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

15. Applied egg-rr 91.4%

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

Alternative 4: 35.6% 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 23.5%

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

3. Simplified 20.0%

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

5. Taylor expanded in b around 0 24.3%

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

   1. distribute-rgt-out 24.3%

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

   2. metadata-eval 24.3%

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

   3. mul0-rgt 38.4%

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

7. Simplified 38.4%

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

Reproduce

herbie shell --seed 2024150 
(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))))