Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 24.2% → 94.3%

Time: 35.5s

Alternatives: 3

Speedup: 1693.0×

• 34.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 24.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 94.3% accurate, 14.7× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := \frac{a\_m \cdot b}{x-scale \cdot y-scale}\\ \mathbf{if}\;a\_m \leq 5.5 \cdot 10^{+181}:\\ \;\;\;\;-4 \cdot {\left(a\_m \cdot \frac{b}{x-scale \cdot y-scale}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(t\_0 \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ (* a_m b) (* x-scale y-scale))))
   (if (<= a_m 5.5e+181)
     (* -4.0 (pow (* a_m (/ b (* x-scale y-scale))) 2.0))
     (* -4.0 (* t_0 t_0)))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (a_m * b) / (x_45_scale * y_45_scale);
	double tmp;
	if (a_m <= 5.5e+181) {
		tmp = -4.0 * pow((a_m * (b / (x_45_scale * y_45_scale))), 2.0);
	} else {
		tmp = -4.0 * (t_0 * t_0);
	}
	return tmp;
}

Fortran

a_m = abs(a)
real(8) function code(a_m, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (a_m * b) / (x_45scale * y_45scale)
    if (a_m <= 5.5d+181) then
        tmp = (-4.0d0) * ((a_m * (b / (x_45scale * y_45scale))) ** 2.0d0)
    else
        tmp = (-4.0d0) * (t_0 * t_0)
    end if
    code = tmp
end function

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (a_m * b) / (x_45_scale * y_45_scale);
	double tmp;
	if (a_m <= 5.5e+181) {
		tmp = -4.0 * Math.pow((a_m * (b / (x_45_scale * y_45_scale))), 2.0);
	} else {
		tmp = -4.0 * (t_0 * t_0);
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle, x_45_scale, y_45_scale):
	t_0 = (a_m * b) / (x_45_scale * y_45_scale)
	tmp = 0
	if a_m <= 5.5e+181:
		tmp = -4.0 * math.pow((a_m * (b / (x_45_scale * y_45_scale))), 2.0)
	else:
		tmp = -4.0 * (t_0 * t_0)
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(a_m * b) / Float64(x_45_scale * y_45_scale))
	tmp = 0.0
	if (a_m <= 5.5e+181)
		tmp = Float64(-4.0 * (Float64(a_m * Float64(b / Float64(x_45_scale * y_45_scale))) ^ 2.0));
	else
		tmp = Float64(-4.0 * Float64(t_0 * t_0));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = (a_m * b) / (x_45_scale * y_45_scale);
	tmp = 0.0;
	if (a_m <= 5.5e+181)
		tmp = -4.0 * ((a_m * (b / (x_45_scale * y_45_scale))) ^ 2.0);
	else
		tmp = -4.0 * (t_0 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(a$95$m * b), $MachinePrecision] / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a$95$m, 5.5e+181], N[(-4.0 * N[Power[N[(a$95$m * N[(b / N[(x$45$scale * y$45$scale), $MachinePrecision]), $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 a < 5.49999999999999991e181

   1. Initial program 24.8%

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

   3. Simplified 20.4%

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

   5. Taylor expanded in angle around 0 45.8%

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

      1. *-commutative 45.8%

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

      2. unpow2 45.8%

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

      3. unpow2 45.8%

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

      4. swap-sqr 61.0%

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

      5. unpow2 61.0%

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

      6. *-commutative 61.0%

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

      7. unpow2 61.0%

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

      8. unpow2 61.0%

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

      9. swap-sqr 79.4%

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

      10. unpow2 79.4%

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

   7. Simplified 79.4%

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

      1. div-inv 79.0%

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

      2. *-commutative 79.0%

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

      3. pow-flip 79.4%

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

      4. *-commutative 79.4%

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

      5. metadata-eval 79.4%

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

   9. Applied egg-rr 79.4%

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

       1. metadata-eval 79.4%

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

       2. pow-flip 79.0%

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

       3. pow2 79.0%

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

       4. div-inv 79.4%

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

       5. pow2 79.4%

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

       6. pow-to-exp 38.9%

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

       7. pow-to-exp 19.0%

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

       8. div-exp 22.9%

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

   11. Applied egg-rr 22.9%

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

   12. Taylor expanded in a around 0 7.1%

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

       1. exp-diff 6.7%

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

       2. distribute-lft-in 6.7%

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

       3. exp-sum 4.3%

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

       4. *-commutative 4.3%

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

       5. exp-to-pow 13.0%

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

       6. *-commutative 13.0%

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

       7. exp-to-pow 29.0%

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

       8. unpow2 29.0%

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

       9. unpow2 29.0%

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

       10. swap-sqr 39.2%

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

       11. unpow2 39.2%

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

       12. *-commutative 39.2%

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

       13. exp-to-pow 79.4%

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

       14. unpow2 79.4%

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

       15. swap-sqr 61.0%

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

       16. unpow2 61.0%

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

       17. unpow2 61.0%

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

   14. Simplified 94.4%

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

   if 5.49999999999999991e181 < a

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in angle around 0 46.3%

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

      1. *-commutative 46.3%

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

      2. unpow2 46.3%

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

      3. unpow2 46.3%

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

      4. swap-sqr 63.9%

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

      5. unpow2 63.9%

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

      6. *-commutative 63.9%

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

      7. unpow2 63.9%

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

      8. unpow2 63.9%

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

      9. swap-sqr 91.1%

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

      10. unpow2 91.1%

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

   7. Simplified 91.1%

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

      1. div-inv 91.1%

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

      2. *-commutative 91.1%

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

      3. pow-flip 91.2%

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

      4. *-commutative 91.2%

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

      5. metadata-eval 91.2%

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

   9. Applied egg-rr 91.2%

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

       1. metadata-eval 91.2%

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

       2. pow-flip 91.1%

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

       3. pow2 91.1%

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

       4. div-inv 91.1%

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

       5. pow2 91.1%

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

       6. pow-to-exp 49.1%

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

       7. pow-to-exp 26.7%

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

       8. div-exp 26.7%

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

   11. Applied egg-rr 26.7%

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

       1. exp-diff 26.7%

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

       2. pow-to-exp 45.1%

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

       3. pow2 45.1%

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

       4. pow-to-exp 91.1%

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

       5. pow2 91.1%

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

       6. times-frac 99.6%

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

   13. Applied egg-rr 99.6%

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

Alternative 2: 93.9% accurate, 99.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(a$95$m * 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 22.6%

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

3. Simplified 18.6%

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

5. Taylor expanded in angle around 0 45.8%

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

   1. *-commutative 45.8%

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

   2. unpow2 45.8%

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

   3. unpow2 45.8%

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

   4. swap-sqr 61.2%

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

   5. unpow2 61.2%

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

   6. *-commutative 61.2%

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

   7. unpow2 61.2%

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

   8. unpow2 61.2%

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

   9. swap-sqr 80.4%

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

   10. unpow2 80.4%

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

7. Simplified 80.4%

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

   1. div-inv 80.0%

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

   2. *-commutative 80.0%

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

   3. pow-flip 80.4%

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

   4. *-commutative 80.4%

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

   5. metadata-eval 80.4%

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

9. Applied egg-rr 80.4%

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

    1. metadata-eval 80.4%

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

    2. pow-flip 80.0%

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

    3. pow2 80.0%

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

    4. div-inv 80.4%

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

    5. pow2 80.4%

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

    6. pow-to-exp 39.7%

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

    7. pow-to-exp 19.7%

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

    8. div-exp 23.2%

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

11. Applied egg-rr 23.2%

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

    1. exp-diff 19.7%

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

    2. pow-to-exp 39.7%

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

    3. pow2 39.7%

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

    4. pow-to-exp 80.4%

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

    5. pow2 80.4%

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

    6. times-frac 94.4%

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

13. Applied egg-rr 94.4%

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

Alternative 3: 35.2% accurate, 1693.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ 0 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := 0.0

Derivation

1. Initial program 22.6%

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

3. Simplified 18.6%

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

5. Taylor expanded in b around 0 16.9%

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

   1. distribute-rgt-out 16.9%

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

   2. metadata-eval 16.9%

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

   3. mul0-rgt 32.9%

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

7. Simplified 32.9%

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

Reproduce

herbie shell --seed 2024141 
(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))))