Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 24.9% → 93.6%

Time: 31.5s

Alternatives: 4

Speedup: 1693.0×

• 33.5% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. a = -1.2445470438960444e-67
  2. b = -3.4141289832385855e+253
  3. angle = -9.21228383894567e+198
  4. x-scale = 9.267479774965895e-243
  5. y-scale = -4.115864016060701e-223

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: 93.6% accurate, 14.5× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \frac{a \cdot b}{x-scale \cdot y-scale}\\ \mathbf{if}\;\frac{angle\_m}{180} \leq 10^{+100}:\\ \;\;\;\;-4 \cdot {\left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(t\_0 \cdot t\_0\right)\\ \end{array} \end{array} \]

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) 1e+100)
     (* -4.0 (pow (* (/ a x-scale) (/ b y-scale)) 2.0))
     (* -4.0 (* t_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) <= 1e+100) {
		tmp = -4.0 * pow(((a / x_45_scale) * (b / y_45_scale)), 2.0);
	} else {
		tmp = -4.0 * (t_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) <= 1d+100) then
        tmp = (-4.0d0) * (((a / x_45scale) * (b / y_45scale)) ** 2.0d0)
    else
        tmp = (-4.0d0) * (t_0 * 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) <= 1e+100) {
		tmp = -4.0 * Math.pow(((a / x_45_scale) * (b / y_45_scale)), 2.0);
	} else {
		tmp = -4.0 * (t_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) <= 1e+100:
		tmp = -4.0 * math.pow(((a / x_45_scale) * (b / y_45_scale)), 2.0)
	else:
		tmp = -4.0 * (t_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(Float64(a * b) / Float64(x_45_scale * y_45_scale))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 1e+100)
		tmp = Float64(-4.0 * (Float64(Float64(a / x_45_scale) * Float64(b / y_45_scale)) ^ 2.0));
	else
		tmp = Float64(-4.0 * Float64(t_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) <= 1e+100)
		tmp = -4.0 * (((a / x_45_scale) * (b / y_45_scale)) ^ 2.0);
	else
		tmp = -4.0 * (t_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[(N[(a * b), $MachinePrecision] / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+100], N[(-4.0 * N[Power[N[(N[(a / x$45$scale), $MachinePrecision] * N[(b / y$45$scale), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 1.00000000000000002e100

   1. Initial program 27.4%

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

   3. Simplified 21.2%

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

   5. Taylor expanded in angle around 0 44.0%

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

      1. *-commutative 44.0%

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

      2. unpow2 44.0%

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

      3. unpow2 44.0%

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

      4. swap-sqr 54.1%

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

      5. unpow2 54.1%

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

      6. *-commutative 54.1%

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

      7. unpow2 54.1%

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

      8. unpow2 54.1%

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

      9. swap-sqr 77.7%

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

      10. unpow2 77.7%

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

   7. Simplified 77.7%

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

      1. unpow2 77.7%

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

   9. Applied egg-rr 77.7%

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

       1. add-cbrt-cube 72.7%

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

       2. pow3 72.7%

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

   11. Applied egg-rr 72.7%

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

   12. Taylor expanded in a around 0 44.0%

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

       1. associate-*r/ 44.0%

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

       2. unpow2 44.0%

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

       3. unpow2 44.0%

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

       4. swap-sqr 54.1%

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

       5. unpow2 54.1%

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

       6. associate-/l* 54.1%

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

       7. unpow2 54.1%

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

       8. unpow2 54.1%

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

       9. swap-sqr 77.7%

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

       10. associate-/l/ 81.6%

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

       11. associate-/r* 80.0%

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

       12. associate-/r* 81.6%

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

       13. unpow2 81.6%

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

       14. associate-*r/ 91.5%

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

       15. associate-*l/ 93.5%

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

       16. unpow2 93.5%

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

       17. times-frac 94.0%

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

   14. Simplified 94.0%

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

   if 1.00000000000000002e100 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 23.4%

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

   3. Simplified 18.1%

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

   5. Taylor expanded in angle around 0 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 59.0%

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

      5. unpow2 59.0%

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

      6. *-commutative 59.0%

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

      7. unpow2 59.0%

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

      8. unpow2 59.0%

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

      9. swap-sqr 78.3%

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

      10. unpow2 78.3%

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

   7. Simplified 78.3%

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

      1. unpow2 78.3%

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

   9. Applied egg-rr 78.3%

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

       1. add-cbrt-cube 75.6%

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

       2. pow3 75.6%

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

   11. Applied egg-rr 75.6%

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

       1. rem-cbrt-cube 78.3%

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

       2. pow2 78.3%

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

       3. times-frac 96.9%

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

   13. Applied egg-rr 96.9%

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

Alternative 2: 93.9% accurate, 99.6× speedup

Math

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

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)))) (* -4.0 (* t_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 -4.0 * (t_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 = (-4.0d0) * (t_0 * 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 -4.0 * (t_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 -4.0 * (t_0 * t_0)

Julia

angle_m = abs(angle)
function code(a, b, angle_m, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(a * b) / Float64(x_45_scale * y_45_scale))
	return Float64(-4.0 * Float64(t_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 = -4.0 * (t_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[(N[(a * b), $MachinePrecision] / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]}, N[(-4.0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 26.8%

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

3. Simplified 20.8%

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

5. Taylor expanded in angle around 0 44.5%

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

   1. *-commutative 44.5%

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

   2. unpow2 44.5%

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

   3. unpow2 44.5%

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

   4. swap-sqr 54.8%

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

   5. unpow2 54.8%

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

   6. *-commutative 54.8%

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

   7. unpow2 54.8%

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

   8. unpow2 54.8%

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

   9. swap-sqr 77.8%

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

   10. unpow2 77.8%

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

7. Simplified 77.8%

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

   1. unpow2 77.8%

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

9. Applied egg-rr 77.8%

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

    1. add-cbrt-cube 73.1%

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

    2. pow3 73.1%

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

11. Applied egg-rr 73.1%

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

    1. rem-cbrt-cube 77.8%

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

    2. pow2 77.8%

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

    3. times-frac 94.0%

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

13. Applied egg-rr 94.0%

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

Alternative 3: 83.8% accurate, 99.6× speedup

Math

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

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(a * b) * Float64(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[(a * b), $MachinePrecision] * N[(N[(a * b), $MachinePrecision] / N[(N[(x$45$scale * y$45$scale), $MachinePrecision] * N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 26.8%

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

3. Simplified 20.8%

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

5. Taylor expanded in angle around 0 44.5%

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

   1. *-commutative 44.5%

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

   2. unpow2 44.5%

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

   3. unpow2 44.5%

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

   4. swap-sqr 54.8%

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

   5. unpow2 54.8%

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

   6. *-commutative 54.8%

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

   7. unpow2 54.8%

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

   8. unpow2 54.8%

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

   9. swap-sqr 77.8%

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

   10. unpow2 77.8%

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

7. Simplified 77.8%

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

   1. unpow2 77.8%

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

9. Applied egg-rr 77.8%

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

    1. associate-/l* 83.6%

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

11. Applied egg-rr 83.6%

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

    1. unpow2 83.6%

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

13. Applied egg-rr 83.6%

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

Alternative 4: 34.7% 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 26.8%

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

3. Simplified 20.8%

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

5. Taylor expanded in b around 0 23.4%

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

   1. distribute-rgt-out 23.4%

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

   2. metadata-eval 23.4%

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

   3. mul0-rgt 39.6%

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

7. Simplified 39.6%

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

Reproduce

herbie shell --seed 2024188 
(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))))