Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 25.3% → 93.6%

Time: 1.7min

Alternatives: 4

Speedup: 1693.0×

• Could not converge on a ground truth

  1. reg0 = (bf "1.040297063413071181363879329032167259791e-95")
  2. reg1 = (bf "3.105893030600675814386523849322772127337e261")
  3. reg2 = (bf "4.503032845699468437779276657588446922101e235")
  4. reg3 = (bf "4.878892899227073010478707826658295314368e217")
  5. reg4 = (bf "2.055605563662730422624519227036751910861e167")

• 34.0% 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: 25.3% 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: 93.6% accurate, 14.5× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := a \cdot \frac{b}{x-scale \cdot y-scale}\\ \mathbf{if}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{+90}:\\ \;\;\;\;\frac{-4}{{\left(x-scale \cdot \frac{\frac{y-scale}{a}}{b}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(-4 \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* a (/ b (* x-scale y-scale)))))
   (if (<= (/ angle_m 180.0) 2e+90)
     (/ -4.0 (pow (* x-scale (/ (/ y-scale a) b)) 2.0))
     (* t_0 (* -4.0 t_0)))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m, double x_45_scale, double y_45_scale) {
	double t_0 = a * (b / (x_45_scale * y_45_scale));
	double tmp;
	if ((angle_m / 180.0) <= 2e+90) {
		tmp = -4.0 / pow((x_45_scale * ((y_45_scale / a) / b)), 2.0);
	} else {
		tmp = t_0 * (-4.0 * t_0);
	}
	return tmp;
}

Fortran

angle_m = abs(angle)
real(8) function code(a, b, angle_m, x_45scale, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle_m
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a * (b / (x_45scale * y_45scale))
    if ((angle_m / 180.0d0) <= 2d+90) then
        tmp = (-4.0d0) / ((x_45scale * ((y_45scale / a) / b)) ** 2.0d0)
    else
        tmp = t_0 * ((-4.0d0) * t_0)
    end if
    code = tmp
end function

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m, double x_45_scale, double y_45_scale) {
	double t_0 = a * (b / (x_45_scale * y_45_scale));
	double tmp;
	if ((angle_m / 180.0) <= 2e+90) {
		tmp = -4.0 / Math.pow((x_45_scale * ((y_45_scale / a) / b)), 2.0);
	} else {
		tmp = t_0 * (-4.0 * t_0);
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m, x_45_scale, y_45_scale):
	t_0 = a * (b / (x_45_scale * y_45_scale))
	tmp = 0
	if (angle_m / 180.0) <= 2e+90:
		tmp = -4.0 / math.pow((x_45_scale * ((y_45_scale / a) / b)), 2.0)
	else:
		tmp = t_0 * (-4.0 * t_0)
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m, x_45_scale, y_45_scale)
	t_0 = Float64(a * Float64(b / Float64(x_45_scale * y_45_scale)))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 2e+90)
		tmp = Float64(-4.0 / (Float64(x_45_scale * Float64(Float64(y_45_scale / a) / b)) ^ 2.0));
	else
		tmp = Float64(t_0 * Float64(-4.0 * t_0));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m, x_45_scale, y_45_scale)
	t_0 = a * (b / (x_45_scale * y_45_scale));
	tmp = 0.0;
	if ((angle_m / 180.0) <= 2e+90)
		tmp = -4.0 / ((x_45_scale * ((y_45_scale / a) / b)) ^ 2.0);
	else
		tmp = t_0 * (-4.0 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(a * N[(b / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+90], N[(-4.0 / N[Power[N[(x$45$scale * N[(N[(y$45$scale / a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(-4.0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 1.99999999999999993e90

   1. Initial program 31.8%

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

   3. Simplified 26.9%

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

   5. Taylor expanded in angle around 0 52.4%

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

      1. *-commutative 52.4%

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

      2. unpow2 52.4%

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

      3. unpow2 52.4%

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

      4. swap-sqr 62.7%

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

      5. unpow2 62.7%

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

      6. *-commutative 62.7%

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

      7. unpow2 62.7%

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

      8. unpow2 62.7%

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

      9. swap-sqr 79.3%

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

      10. unpow2 79.3%

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

   7. Simplified 79.3%

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

      1. add-cube-cbrt 79.2%

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

      2. pow3 79.2%

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

      3. div-inv 79.1%

         \[\leadsto {\left(\sqrt[3]{-4 \cdot \color{blue}{\left({\left(a \cdot b\right)}^{2} \cdot \frac{1}{{\left(x-scale \cdot y-scale\right)}^{2}}\right)}}\right)}^{3} \]

      4. *-commutative 79.1%

         \[\leadsto {\left(\sqrt[3]{-4 \cdot \left({\left(a \cdot b\right)}^{2} \cdot \frac{1}{{\color{blue}{\left(y-scale \cdot x-scale\right)}}^{2}}\right)}\right)}^{3} \]

      5. pow-flip 79.1%

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

      6. *-commutative 79.1%

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

      7. metadata-eval 79.1%

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

   9. Applied egg-rr 79.1%

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

       1. *-commutative 79.1%

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

       2. pow-to-exp 38.9%

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

       3. pow-to-exp 23.6%

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

       4. prod-exp 28.6%

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

   11. Applied egg-rr 28.6%

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

       1. rem-cube-cbrt 28.6%

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

       2. exp-sum 23.6%

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

       3. pow-to-exp 38.9%

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

       4. exp-to-pow 79.2%

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

       5. metadata-eval 79.2%

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

       6. pow-flip 79.2%

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

       7. pow2 79.2%

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

       8. associate-/r/ 79.1%

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

       9. div-inv 79.1%

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

       10. add-sqr-sqrt 79.1%

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

       11. pow2 79.1%

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

   13. Applied egg-rr 93.2%

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

       1. associate-/l* 94.5%

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

       2. associate-/r* 95.8%

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

   15. Simplified 95.8%

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

   if 1.99999999999999993e90 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 18.4%

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

   3. Simplified 10.6%

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

   5. Taylor expanded in angle around 0 47.7%

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

      1. *-commutative 47.7%

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

      2. unpow2 47.7%

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

      3. unpow2 47.7%

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

      4. swap-sqr 55.9%

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

      5. unpow2 55.9%

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

      6. *-commutative 55.9%

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

      7. unpow2 55.9%

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

      8. unpow2 55.9%

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

      9. swap-sqr 75.1%

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

      10. unpow2 75.1%

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

   7. Simplified 75.1%

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

      1. add-cube-cbrt 75.1%

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

      2. pow3 75.1%

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

      3. div-inv 75.0%

         \[\leadsto {\left(\sqrt[3]{-4 \cdot \color{blue}{\left({\left(a \cdot b\right)}^{2} \cdot \frac{1}{{\left(x-scale \cdot y-scale\right)}^{2}}\right)}}\right)}^{3} \]

      4. *-commutative 75.0%

         \[\leadsto {\left(\sqrt[3]{-4 \cdot \left({\left(a \cdot b\right)}^{2} \cdot \frac{1}{{\color{blue}{\left(y-scale \cdot x-scale\right)}}^{2}}\right)}\right)}^{3} \]

      5. pow-flip 77.1%

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

      6. *-commutative 77.1%

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

      7. metadata-eval 77.1%

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

   9. Applied egg-rr 77.1%

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

       1. *-commutative 77.1%

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

       2. pow-to-exp 43.4%

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

       3. pow-to-exp 29.4%

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

       4. prod-exp 31.4%

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

   11. Applied egg-rr 31.4%

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

       1. rem-cube-cbrt 31.4%

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

       2. *-commutative 31.4%

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

       3. add-sqr-sqrt 31.4%

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

       4. associate-*l* 31.4%

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

   13. Applied egg-rr 97.7%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{+90}:\\ \;\;\;\;\frac{-4}{{\left(x-scale \cdot \frac{\frac{y-scale}{a}}{b}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \frac{b}{x-scale \cdot y-scale}\right) \cdot \left(-4 \cdot \left(a \cdot \frac{b}{x-scale \cdot y-scale}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.0% accurate, 99.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 29.1%

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

3. Simplified 23.7%

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

5. Taylor expanded in angle around 0 51.4%

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

   1. *-commutative 51.4%

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

   2. unpow2 51.4%

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

   3. unpow2 51.4%

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

   4. swap-sqr 61.3%

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

   5. unpow2 61.3%

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

   6. *-commutative 61.3%

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

   7. unpow2 61.3%

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

   8. unpow2 61.3%

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

   9. swap-sqr 78.4%

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

   10. unpow2 78.4%

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

7. Simplified 78.4%

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

   1. add-cube-cbrt 78.4%

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

   2. pow3 78.4%

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

   3. div-inv 78.3%

      \[\leadsto {\left(\sqrt[3]{-4 \cdot \color{blue}{\left({\left(a \cdot b\right)}^{2} \cdot \frac{1}{{\left(x-scale \cdot y-scale\right)}^{2}}\right)}}\right)}^{3} \]

   4. *-commutative 78.3%

      \[\leadsto {\left(\sqrt[3]{-4 \cdot \left({\left(a \cdot b\right)}^{2} \cdot \frac{1}{{\color{blue}{\left(y-scale \cdot x-scale\right)}}^{2}}\right)}\right)}^{3} \]

   5. pow-flip 78.7%

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

   6. *-commutative 78.7%

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

   7. metadata-eval 78.7%

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

9. Applied egg-rr 78.7%

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

    1. *-commutative 78.7%

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

    2. pow-to-exp 39.8%

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

    3. pow-to-exp 24.8%

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

    4. prod-exp 29.1%

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

11. Applied egg-rr 29.1%

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

    1. rem-cube-cbrt 29.1%

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

    2. *-commutative 29.1%

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

    3. add-sqr-sqrt 29.1%

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

    4. associate-*l* 29.1%

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

13. Applied egg-rr 94.1%

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

14. Final simplification 94.1%

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

Alternative 3: 78.5% accurate, 99.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 29.1%

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

3. Simplified 23.7%

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

5. Taylor expanded in angle around 0 51.4%

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

   1. *-commutative 51.4%

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

   2. unpow2 51.4%

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

   3. unpow2 51.4%

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

   4. swap-sqr 61.3%

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

   5. unpow2 61.3%

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

   6. *-commutative 61.3%

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

   7. unpow2 61.3%

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

   8. unpow2 61.3%

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

   9. swap-sqr 78.4%

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

   10. unpow2 78.4%

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

7. Simplified 78.4%

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

   1. unpow2 78.4%

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

9. Applied egg-rr 78.4%

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

    1. unpow2 78.4%

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

11. Applied egg-rr 78.4%

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

Alternative 4: 36.0% accurate, 1693.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ 0 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m, x_45_scale, y_45_scale):
	return 0.0

Julia

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

MATLAB

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

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_, x$45$scale_, y$45$scale_] := 0.0

Derivation

1. Initial program 29.1%

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

3. Simplified 23.7%

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

5. Taylor expanded in b around 0 26.1%

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

   1. distribute-rgt-out 26.1%

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

   2. metadata-eval 26.1%

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

   3. mul0-rgt 40.6%

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

7. Simplified 40.6%

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

Reproduce

herbie shell --seed 2024143 
(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))))