Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 24.2% → 94.0%

Time: 31.3s

Alternatives: 6

Speedup: 1693.0×

• 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: 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: 94.0% accurate, 14.7× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ \begin{array}{l} t_0 := \frac{a \cdot b}{x-scale\_m \cdot y-scale}\\ \mathbf{if}\;x-scale\_m \leq 10^{+138}:\\ \;\;\;\;-4 \cdot {\left(\frac{a \cdot \frac{b}{y-scale}}{x-scale\_m}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(-4 \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
(FPCore (a b angle x-scale_m y-scale)
 :precision binary64
 (let* ((t_0 (/ (* a b) (* x-scale_m y-scale))))
   (if (<= x-scale_m 1e+138)
     (* -4.0 (pow (/ (* a (/ b y-scale)) x-scale_m) 2.0))
     (* t_0 (* -4.0 t_0)))))

C

x-scale_m = fabs(x_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	double t_0 = (a * b) / (x_45_scale_m * y_45_scale);
	double tmp;
	if (x_45_scale_m <= 1e+138) {
		tmp = -4.0 * pow(((a * (b / y_45_scale)) / x_45_scale_m), 2.0);
	} else {
		tmp = t_0 * (-4.0 * t_0);
	}
	return tmp;
}

Fortran

x-scale_m = abs(x_45scale)
real(8) function code(a, b, angle, x_45scale_m, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (a * b) / (x_45scale_m * y_45scale)
    if (x_45scale_m <= 1d+138) then
        tmp = (-4.0d0) * (((a * (b / y_45scale)) / x_45scale_m) ** 2.0d0)
    else
        tmp = t_0 * ((-4.0d0) * t_0)
    end if
    code = tmp
end function

Java

x-scale_m = Math.abs(x_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	double t_0 = (a * b) / (x_45_scale_m * y_45_scale);
	double tmp;
	if (x_45_scale_m <= 1e+138) {
		tmp = -4.0 * Math.pow(((a * (b / y_45_scale)) / x_45_scale_m), 2.0);
	} else {
		tmp = t_0 * (-4.0 * t_0);
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale):
	t_0 = (a * b) / (x_45_scale_m * y_45_scale)
	tmp = 0
	if x_45_scale_m <= 1e+138:
		tmp = -4.0 * math.pow(((a * (b / y_45_scale)) / x_45_scale_m), 2.0)
	else:
		tmp = t_0 * (-4.0 * t_0)
	return tmp

Julia

x-scale_m = abs(x_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale)
	t_0 = Float64(Float64(a * b) / Float64(x_45_scale_m * y_45_scale))
	tmp = 0.0
	if (x_45_scale_m <= 1e+138)
		tmp = Float64(-4.0 * (Float64(Float64(a * Float64(b / y_45_scale)) / x_45_scale_m) ^ 2.0));
	else
		tmp = Float64(t_0 * Float64(-4.0 * t_0));
	end
	return tmp
end

MATLAB

x-scale_m = abs(x_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale)
	t_0 = (a * b) / (x_45_scale_m * y_45_scale);
	tmp = 0.0;
	if (x_45_scale_m <= 1e+138)
		tmp = -4.0 * (((a * (b / y_45_scale)) / x_45_scale_m) ^ 2.0);
	else
		tmp = t_0 * (-4.0 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale_] := Block[{t$95$0 = N[(N[(a * b), $MachinePrecision] / N[(x$45$scale$95$m * y$45$scale), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$45$scale$95$m, 1e+138], N[(-4.0 * N[Power[N[(N[(a * N[(b / y$45$scale), $MachinePrecision]), $MachinePrecision] / x$45$scale$95$m), $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 x-scale < 1e138

   1. Initial program 25.8%

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

   3. Simplified 24.1%

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

   5. Taylor expanded in angle around 0 57.0%

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

      1. *-commutative 57.0%

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

      2. unpow2 57.0%

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

      3. unpow2 57.0%

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

      4. swap-sqr 65.8%

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

      5. unpow2 65.8%

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

      6. *-commutative 65.8%

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

      7. unpow2 65.8%

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

      8. unpow2 65.8%

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

      9. swap-sqr 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. associate-*r/ 79.4%

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

   9. Applied egg-rr 79.4%

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

   10. Taylor expanded in a around 0 57.0%

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

       1. unpow2 57.0%

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

       2. unpow2 57.0%

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

       3. swap-sqr 65.8%

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

       4. unpow2 65.8%

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

       5. associate-*r/ 65.8%

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

       6. unpow2 65.8%

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

       7. unpow2 65.8%

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

       8. swap-sqr 79.4%

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

       9. associate-/r* 86.1%

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

       10. associate-/l* 86.1%

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

       11. unpow2 86.1%

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

       12. associate-*l/ 91.5%

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

       13. associate-*r/ 91.5%

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

       14. associate-*r/ 92.8%

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

       15. unpow2 92.8%

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

   12. Simplified 95.8%

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

   if 1e138 < x-scale

   1. Initial program 39.0%

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

   3. Simplified 36.3%

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

   5. Taylor expanded in angle around 0 41.5%

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

      1. *-commutative 41.5%

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

      2. unpow2 41.5%

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

      3. unpow2 41.5%

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

      4. swap-sqr 46.7%

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

      5. unpow2 46.7%

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

      6. *-commutative 46.7%

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

      7. unpow2 46.7%

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

      8. unpow2 46.7%

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

      9. swap-sqr 74.7%

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

      10. unpow2 74.7%

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

   7. Simplified 74.7%

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

      1. associate-*r/ 74.7%

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

   9. Applied egg-rr 74.7%

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

       1. rem-cube-cbrt 74.5%

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

       2. associate-*r/ 74.5%

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

       3. add-sqr-sqrt 74.5%

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

       4. associate-*r* 74.5%

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

   11. Applied egg-rr 97.1%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq 10^{+138}:\\ \;\;\;\;-4 \cdot {\left(\frac{a \cdot \frac{b}{y-scale}}{x-scale}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot b}{x-scale \cdot y-scale} \cdot \left(-4 \cdot \frac{a \cdot b}{x-scale \cdot y-scale}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.1% accurate, 76.9× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ \begin{array}{l} t_0 := \frac{b}{y-scale \cdot \frac{x-scale\_m}{a}}\\ t_1 := \frac{a \cdot b}{x-scale\_m \cdot y-scale}\\ \mathbf{if}\;a \leq 2.1 \cdot 10^{-147}:\\ \;\;\;\;t\_1 \cdot \left(-4 \cdot t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(t\_0 \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
(FPCore (a b angle x-scale_m y-scale)
 :precision binary64
 (let* ((t_0 (/ b (* y-scale (/ x-scale_m a))))
        (t_1 (/ (* a b) (* x-scale_m y-scale))))
   (if (<= a 2.1e-147) (* t_1 (* -4.0 t_1)) (* -4.0 (* t_0 t_0)))))

C

x-scale_m = fabs(x_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	double t_0 = b / (y_45_scale * (x_45_scale_m / a));
	double t_1 = (a * b) / (x_45_scale_m * y_45_scale);
	double tmp;
	if (a <= 2.1e-147) {
		tmp = t_1 * (-4.0 * t_1);
	} else {
		tmp = -4.0 * (t_0 * t_0);
	}
	return tmp;
}

Fortran

x-scale_m = abs(x_45scale)
real(8) function code(a, b, angle, x_45scale_m, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = b / (y_45scale * (x_45scale_m / a))
    t_1 = (a * b) / (x_45scale_m * y_45scale)
    if (a <= 2.1d-147) then
        tmp = t_1 * ((-4.0d0) * t_1)
    else
        tmp = (-4.0d0) * (t_0 * t_0)
    end if
    code = tmp
end function

Java

x-scale_m = Math.abs(x_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	double t_0 = b / (y_45_scale * (x_45_scale_m / a));
	double t_1 = (a * b) / (x_45_scale_m * y_45_scale);
	double tmp;
	if (a <= 2.1e-147) {
		tmp = t_1 * (-4.0 * t_1);
	} else {
		tmp = -4.0 * (t_0 * t_0);
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale):
	t_0 = b / (y_45_scale * (x_45_scale_m / a))
	t_1 = (a * b) / (x_45_scale_m * y_45_scale)
	tmp = 0
	if a <= 2.1e-147:
		tmp = t_1 * (-4.0 * t_1)
	else:
		tmp = -4.0 * (t_0 * t_0)
	return tmp

Julia

x-scale_m = abs(x_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale)
	t_0 = Float64(b / Float64(y_45_scale * Float64(x_45_scale_m / a)))
	t_1 = Float64(Float64(a * b) / Float64(x_45_scale_m * y_45_scale))
	tmp = 0.0
	if (a <= 2.1e-147)
		tmp = Float64(t_1 * Float64(-4.0 * t_1));
	else
		tmp = Float64(-4.0 * Float64(t_0 * t_0));
	end
	return tmp
end

MATLAB

x-scale_m = abs(x_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale)
	t_0 = b / (y_45_scale * (x_45_scale_m / a));
	t_1 = (a * b) / (x_45_scale_m * y_45_scale);
	tmp = 0.0;
	if (a <= 2.1e-147)
		tmp = t_1 * (-4.0 * t_1);
	else
		tmp = -4.0 * (t_0 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale_] := Block[{t$95$0 = N[(b / N[(y$45$scale * N[(x$45$scale$95$m / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(a * b), $MachinePrecision] / N[(x$45$scale$95$m * y$45$scale), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 2.1e-147], N[(t$95$1 * N[(-4.0 * t$95$1), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if a < 2.1e-147

   1. Initial program 31.4%

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

   3. Simplified 27.4%

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

   5. Taylor expanded in angle around 0 54.4%

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

      1. *-commutative 54.4%

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

      2. unpow2 54.4%

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

      3. unpow2 54.4%

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

      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 80.2%

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

      10. unpow2 80.2%

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

   7. Simplified 80.2%

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

      1. associate-*r/ 80.2%

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

   9. Applied egg-rr 80.2%

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

       1. rem-cube-cbrt 80.0%

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

       2. associate-*r/ 80.0%

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

       3. add-sqr-sqrt 80.0%

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

       4. associate-*r* 80.0%

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

   11. Applied egg-rr 90.7%

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

   if 2.1e-147 < a

   1. Initial program 23.0%

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

   3. Simplified 24.0%

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

   5. Taylor expanded in angle around 0 55.0%

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

      1. *-commutative 55.0%

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

      2. unpow2 55.0%

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

      3. unpow2 55.0%

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

      4. swap-sqr 65.1%

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

      5. unpow2 65.1%

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

      6. *-commutative 65.1%

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

      7. unpow2 65.1%

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

      8. unpow2 65.1%

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

      9. swap-sqr 76.7%

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

      10. unpow2 76.7%

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

   7. Simplified 76.7%

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

      1. associate-*r/ 76.7%

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

   9. Applied egg-rr 76.7%

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

   10. Taylor expanded in a around 0 55.0%

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

       1. unpow2 55.0%

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

       2. unpow2 55.0%

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

       3. swap-sqr 65.1%

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

       4. unpow2 65.1%

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

       5. associate-*r/ 65.1%

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

       6. unpow2 65.1%

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

       7. unpow2 65.1%

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

       8. swap-sqr 76.7%

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

       9. associate-/r* 85.0%

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

       10. associate-/l* 85.0%

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

       11. unpow2 85.0%

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

       12. associate-*l/ 93.8%

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

       13. associate-*r/ 93.8%

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

       14. associate-*r/ 97.1%

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

       15. unpow2 97.1%

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

   12. Simplified 95.8%

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

       1. associate-*l/ 94.8%

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

       2. unpow2 94.8%

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

       3. clear-num 94.8%

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

       4. frac-times 94.0%

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

       5. *-un-lft-identity 94.0%

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

       6. clear-num 94.0%

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

       7. frac-times 96.3%

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

       8. *-un-lft-identity 96.3%

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

   14. Applied egg-rr 96.3%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.1 \cdot 10^{-147}:\\ \;\;\;\;\frac{a \cdot b}{x-scale \cdot y-scale} \cdot \left(-4 \cdot \frac{a \cdot b}{x-scale \cdot y-scale}\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\frac{b}{y-scale \cdot \frac{x-scale}{a}} \cdot \frac{b}{y-scale \cdot \frac{x-scale}{a}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.7% accurate, 99.6× speedup

Math

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

FPCore

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

C

x-scale_m = fabs(x_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	double t_0 = b / (y_45_scale * (x_45_scale_m / a));
	return -4.0 * (t_0 * t_0);
}

Fortran

x-scale_m = abs(x_45scale)
real(8) function code(a, b, angle, x_45scale_m, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    t_0 = b / (y_45scale * (x_45scale_m / a))
    code = (-4.0d0) * (t_0 * t_0)
end function

Java

x-scale_m = Math.abs(x_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	double t_0 = b / (y_45_scale * (x_45_scale_m / a));
	return -4.0 * (t_0 * t_0);
}

Python

x-scale_m = math.fabs(x_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale):
	t_0 = b / (y_45_scale * (x_45_scale_m / a))
	return -4.0 * (t_0 * t_0)

Julia

x-scale_m = abs(x_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale)
	t_0 = Float64(b / Float64(y_45_scale * Float64(x_45_scale_m / a)))
	return Float64(-4.0 * Float64(t_0 * t_0))
end

MATLAB

x-scale_m = abs(x_45_scale);
function tmp = code(a, b, angle, x_45_scale_m, y_45_scale)
	t_0 = b / (y_45_scale * (x_45_scale_m / a));
	tmp = -4.0 * (t_0 * t_0);
end

Wolfram

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

Derivation

1. Initial program 27.8%

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

3. Simplified 26.0%

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

5. Taylor expanded in angle around 0 54.6%

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

   1. *-commutative 54.6%

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

   2. unpow2 54.6%

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

   3. unpow2 54.6%

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

   4. swap-sqr 62.9%

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

   5. unpow2 62.9%

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

   6. *-commutative 62.9%

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

   7. unpow2 62.9%

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

   8. unpow2 62.9%

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

   9. swap-sqr 78.7%

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

   10. unpow2 78.7%

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

7. Simplified 78.7%

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

   1. associate-*r/ 78.7%

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

9. Applied egg-rr 78.7%

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

10. Taylor expanded in a around 0 54.6%

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

    1. unpow2 54.6%

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

    2. unpow2 54.6%

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

    3. swap-sqr 62.9%

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

    4. unpow2 62.9%

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

    5. associate-*r/ 62.9%

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

    6. unpow2 62.9%

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

    7. unpow2 62.9%

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

    8. swap-sqr 78.7%

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

    9. associate-/r* 85.2%

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

    10. associate-/l* 85.2%

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

    11. unpow2 85.2%

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

    12. associate-*l/ 90.9%

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

    13. associate-*r/ 90.9%

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

    14. associate-*r/ 93.4%

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

    15. unpow2 93.4%

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

12. Simplified 95.0%

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

    1. associate-*l/ 94.2%

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

    2. unpow2 94.2%

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

    3. clear-num 94.0%

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

    4. frac-times 92.6%

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

    5. *-un-lft-identity 92.6%

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

    6. clear-num 92.6%

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

    7. frac-times 93.7%

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

    8. *-un-lft-identity 93.7%

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

14. Applied egg-rr 93.7%

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

15. Final simplification 93.7%

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

Alternative 4: 90.2% accurate, 99.6× speedup

Math

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

FPCore

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

C

x-scale_m = fabs(x_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	return -4.0 * ((b / y_45_scale) * ((b / (y_45_scale * (x_45_scale_m / a))) * (a / x_45_scale_m)));
}

Fortran

x-scale_m = abs(x_45scale)
real(8) function code(a, b, angle, x_45scale_m, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale
    code = (-4.0d0) * ((b / y_45scale) * ((b / (y_45scale * (x_45scale_m / a))) * (a / x_45scale_m)))
end function

Java

x-scale_m = Math.abs(x_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	return -4.0 * ((b / y_45_scale) * ((b / (y_45_scale * (x_45_scale_m / a))) * (a / x_45_scale_m)));
}

Python

x-scale_m = math.fabs(x_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale):
	return -4.0 * ((b / y_45_scale) * ((b / (y_45_scale * (x_45_scale_m / a))) * (a / x_45_scale_m)))

Julia

x-scale_m = abs(x_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale)
	return Float64(-4.0 * Float64(Float64(b / y_45_scale) * Float64(Float64(b / Float64(y_45_scale * Float64(x_45_scale_m / a))) * Float64(a / x_45_scale_m))))
end

MATLAB

x-scale_m = abs(x_45_scale);
function tmp = code(a, b, angle, x_45_scale_m, y_45_scale)
	tmp = -4.0 * ((b / y_45_scale) * ((b / (y_45_scale * (x_45_scale_m / a))) * (a / x_45_scale_m)));
end

Wolfram

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

Derivation

1. Initial program 27.8%

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

3. Simplified 26.0%

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

5. Taylor expanded in angle around 0 54.6%

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

   1. *-commutative 54.6%

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

   2. unpow2 54.6%

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

   3. unpow2 54.6%

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

   4. swap-sqr 62.9%

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

   5. unpow2 62.9%

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

   6. *-commutative 62.9%

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

   7. unpow2 62.9%

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

   8. unpow2 62.9%

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

   9. swap-sqr 78.7%

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

   10. unpow2 78.7%

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

7. Simplified 78.7%

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

   1. associate-*r/ 78.7%

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

9. Applied egg-rr 78.7%

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

10. Taylor expanded in a around 0 54.6%

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

    1. unpow2 54.6%

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

    2. unpow2 54.6%

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

    3. swap-sqr 62.9%

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

    4. unpow2 62.9%

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

    5. associate-*r/ 62.9%

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

    6. unpow2 62.9%

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

    7. unpow2 62.9%

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

    8. swap-sqr 78.7%

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

    9. associate-/r* 85.2%

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

    10. associate-/l* 85.2%

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

    11. unpow2 85.2%

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

    12. associate-*l/ 90.9%

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

    13. associate-*r/ 90.9%

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

    14. associate-*r/ 93.4%

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

    15. unpow2 93.4%

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

12. Simplified 95.0%

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

    1. associate-*l/ 94.2%

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

    2. unpow2 94.2%

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

    3. associate-*r* 91.2%

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

    4. clear-num 91.2%

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

    5. frac-times 90.1%

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

    6. *-un-lft-identity 90.1%

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

14. Applied egg-rr 90.1%

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

15. Final simplification 90.1%

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

Alternative 5: 83.3% accurate, 99.6× speedup

Math

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

FPCore

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

C

x-scale_m = fabs(x_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	return -4.0 * (((a * (b / y_45_scale)) * (a / (x_45_scale_m * y_45_scale))) * (b / x_45_scale_m));
}

Fortran

x-scale_m = abs(x_45scale)
real(8) function code(a, b, angle, x_45scale_m, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale
    code = (-4.0d0) * (((a * (b / y_45scale)) * (a / (x_45scale_m * y_45scale))) * (b / x_45scale_m))
end function

Java

x-scale_m = Math.abs(x_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	return -4.0 * (((a * (b / y_45_scale)) * (a / (x_45_scale_m * y_45_scale))) * (b / x_45_scale_m));
}

Python

x-scale_m = math.fabs(x_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale):
	return -4.0 * (((a * (b / y_45_scale)) * (a / (x_45_scale_m * y_45_scale))) * (b / x_45_scale_m))

Julia

x-scale_m = abs(x_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale)
	return Float64(-4.0 * Float64(Float64(Float64(a * Float64(b / y_45_scale)) * Float64(a / Float64(x_45_scale_m * y_45_scale))) * Float64(b / x_45_scale_m)))
end

MATLAB

x-scale_m = abs(x_45_scale);
function tmp = code(a, b, angle, x_45_scale_m, y_45_scale)
	tmp = -4.0 * (((a * (b / y_45_scale)) * (a / (x_45_scale_m * y_45_scale))) * (b / x_45_scale_m));
end

Wolfram

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

Derivation

1. Initial program 27.8%

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

3. Simplified 26.0%

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

5. Taylor expanded in angle around 0 54.6%

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

   1. *-commutative 54.6%

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

   2. unpow2 54.6%

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

   3. unpow2 54.6%

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

   4. swap-sqr 62.9%

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

   5. unpow2 62.9%

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

   6. *-commutative 62.9%

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

   7. unpow2 62.9%

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

   8. unpow2 62.9%

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

   9. swap-sqr 78.7%

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

   10. unpow2 78.7%

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

7. Simplified 78.7%

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

   1. associate-*r/ 78.7%

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

9. Applied egg-rr 78.7%

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

10. Taylor expanded in a around 0 54.6%

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

    1. unpow2 54.6%

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

    2. unpow2 54.6%

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

    3. swap-sqr 62.9%

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

    4. unpow2 62.9%

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

    5. associate-*r/ 62.9%

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

    6. unpow2 62.9%

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

    7. unpow2 62.9%

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

    8. swap-sqr 78.7%

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

    9. associate-/r* 85.2%

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

    10. associate-/l* 85.2%

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

    11. unpow2 85.2%

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

    12. associate-*l/ 90.9%

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

    13. associate-*r/ 90.9%

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

    14. associate-*r/ 93.4%

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

    15. unpow2 93.4%

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

12. Simplified 95.0%

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

    1. unpow2 95.0%

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

    2. associate-*l/ 92.0%

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

    3. frac-times 89.5%

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

    4. frac-times 79.1%

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

    5. associate-*l/ 80.9%

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

    6. *-commutative 80.9%

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

    7. times-frac 87.1%

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

    8. associate-*r* 87.1%

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

    9. *-commutative 87.1%

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

14. Applied egg-rr 87.1%

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

15. Final simplification 87.1%

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

Alternative 6: 34.2% accurate, 1693.0× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ 0 \end{array} \]

FPCore

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

C

x-scale_m = fabs(x_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	return 0.0;
}

Fortran

x-scale_m = abs(x_45scale)
real(8) function code(a, b, angle, x_45scale_m, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale
    code = 0.0d0
end function

Java

x-scale_m = Math.abs(x_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	return 0.0;
}

Python

x-scale_m = math.fabs(x_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale):
	return 0.0

Julia

x-scale_m = abs(x_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale)
	return 0.0
end

MATLAB

x-scale_m = abs(x_45_scale);
function tmp = code(a, b, angle, x_45_scale_m, y_45_scale)
	tmp = 0.0;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale_] := 0.0

Derivation

1. Initial program 27.8%

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

3. Simplified 26.0%

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

5. Taylor expanded in b around 0 29.8%

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

   1. distribute-rgt-out 29.8%

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

   2. metadata-eval 29.8%

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

   3. mul0-rgt 40.3%

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

7. Simplified 40.3%

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

Reproduce

herbie shell --seed 2024136 
(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))))