Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 24.4% → 94.2%

Time: 2.1min

Alternatives: 5

Speedup: 1777.0×

• 33.9% 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: 24.4% 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: 94.2% accurate, 14.4× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;a_m \leq 1.2 \cdot 10^{+235}:\\ \;\;\;\;-4 \cdot {\left(\frac{a_m}{x-scale \cdot y-scale} \cdot b\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4}{{\left(\frac{x-scale}{a_m} \cdot \frac{y-scale}{b}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (if (<= a_m 1.2e+235)
   (* -4.0 (pow (* (/ a_m (* x-scale y-scale)) b) 2.0))
   (/ -4.0 (pow (* (/ x-scale a_m) (/ y-scale b)) 2.0))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (a_m <= 1.2e+235) {
		tmp = -4.0 * pow(((a_m / (x_45_scale * y_45_scale)) * b), 2.0);
	} else {
		tmp = -4.0 / pow(((x_45_scale / a_m) * (y_45_scale / b)), 2.0);
	}
	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) :: tmp
    if (a_m <= 1.2d+235) then
        tmp = (-4.0d0) * (((a_m / (x_45scale * y_45scale)) * b) ** 2.0d0)
    else
        tmp = (-4.0d0) / (((x_45scale / a_m) * (y_45scale / b)) ** 2.0d0)
    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 tmp;
	if (a_m <= 1.2e+235) {
		tmp = -4.0 * Math.pow(((a_m / (x_45_scale * y_45_scale)) * b), 2.0);
	} else {
		tmp = -4.0 / Math.pow(((x_45_scale / a_m) * (y_45_scale / b)), 2.0);
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if a_m <= 1.2e+235:
		tmp = -4.0 * math.pow(((a_m / (x_45_scale * y_45_scale)) * b), 2.0)
	else:
		tmp = -4.0 / math.pow(((x_45_scale / a_m) * (y_45_scale / b)), 2.0)
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (a_m <= 1.2e+235)
		tmp = Float64(-4.0 * (Float64(Float64(a_m / Float64(x_45_scale * y_45_scale)) * b) ^ 2.0));
	else
		tmp = Float64(-4.0 / (Float64(Float64(x_45_scale / a_m) * Float64(y_45_scale / b)) ^ 2.0));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (a_m <= 1.2e+235)
		tmp = -4.0 * (((a_m / (x_45_scale * y_45_scale)) * b) ^ 2.0);
	else
		tmp = -4.0 / (((x_45_scale / a_m) * (y_45_scale / b)) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[a$95$m, 1.2e+235], N[(-4.0 * N[Power[N[(N[(a$95$m / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(-4.0 / N[Power[N[(N[(x$45$scale / a$95$m), $MachinePrecision] * N[(y$45$scale / b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.2e235

   1. Initial program 26.2%

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

   3. Simplified 24.1%

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

   5. Taylor expanded in angle around 0 47.9%

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

      1. associate-*r/ 47.9%

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

      2. *-commutative 47.9%

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

      3. unpow2 47.9%

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

      4. unpow2 47.9%

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

      5. swap-sqr 60.5%

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

      6. unpow2 60.5%

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

   7. Simplified 60.5%

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

      1. pow-prod-down 47.9%

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

      2. associate-*r/ 47.9%

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

      3. expm1-log1p-u 26.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-4 \cdot \frac{{b}^{2} \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)\right)} \]

      4. expm1-udef 25.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-4 \cdot \frac{{b}^{2} \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)} - 1} \]

      5. div-inv 25.5%

         \[\leadsto e^{\mathsf{log1p}\left(-4 \cdot \color{blue}{\left(\left({b}^{2} \cdot {a}^{2}\right) \cdot \frac{1}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)}\right)} - 1 \]

      6. *-commutative 25.5%

         \[\leadsto e^{\mathsf{log1p}\left(-4 \cdot \left(\color{blue}{\left({a}^{2} \cdot {b}^{2}\right)} \cdot \frac{1}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)\right)} - 1 \]

      7. pow-prod-down 28.4%

         \[\leadsto e^{\mathsf{log1p}\left(-4 \cdot \left(\color{blue}{{\left(a \cdot b\right)}^{2}} \cdot \frac{1}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)\right)} - 1 \]

      8. pow-prod-down 32.5%

         \[\leadsto e^{\mathsf{log1p}\left(-4 \cdot \left({\left(a \cdot b\right)}^{2} \cdot \frac{1}{\color{blue}{{\left(x-scale \cdot y-scale\right)}^{2}}}\right)\right)} - 1 \]

      9. *-commutative 32.5%

         \[\leadsto e^{\mathsf{log1p}\left(-4 \cdot \left({\left(a \cdot b\right)}^{2} \cdot \frac{1}{{\color{blue}{\left(y-scale \cdot x-scale\right)}}^{2}}\right)\right)} - 1 \]

      10. pow-flip 32.5%

          \[\leadsto e^{\mathsf{log1p}\left(-4 \cdot \left({\left(a \cdot b\right)}^{2} \cdot \color{blue}{{\left(y-scale \cdot x-scale\right)}^{\left(-2\right)}}\right)\right)} - 1 \]

      11. *-commutative 32.5%

          \[\leadsto e^{\mathsf{log1p}\left(-4 \cdot \left({\left(a \cdot b\right)}^{2} \cdot {\color{blue}{\left(x-scale \cdot y-scale\right)}}^{\left(-2\right)}\right)\right)} - 1 \]

      12. metadata-eval 32.5%

          \[\leadsto e^{\mathsf{log1p}\left(-4 \cdot \left({\left(a \cdot b\right)}^{2} \cdot {\left(x-scale \cdot y-scale\right)}^{\color{blue}{-2}}\right)\right)} - 1 \]

   9. Applied egg-rr 32.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-4 \cdot \left({\left(a \cdot b\right)}^{2} \cdot {\left(x-scale \cdot y-scale\right)}^{-2}\right)\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 39.9%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-4 \cdot \left({\left(a \cdot b\right)}^{2} \cdot {\left(x-scale \cdot y-scale\right)}^{-2}\right)\right)\right)} \]

       2. expm1-log1p 78.3%

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

       3. *-commutative 78.3%

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

       4. associate-*l* 78.3%

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

   11. Simplified 78.3%

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

       1. unpow2 78.3%

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

   13. Applied egg-rr 78.3%

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

   14. Taylor expanded in a around 0 47.9%

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

       1. unpow2 47.9%

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

       2. unpow2 47.9%

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

       3. swap-sqr 59.3%

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

       4. unpow2 59.3%

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

       5. associate-*r/ 59.3%

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

       6. unpow2 59.3%

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

       7. unpow2 59.3%

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

       8. swap-sqr 78.2%

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

       9. unpow2 78.2%

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

       10. associate-/l* 77.9%

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

       11. unpow2 77.9%

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

       12. associate-/l* 83.4%

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

       13. *-rgt-identity 83.4%

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

       14. associate-*r/ 83.4%

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

       15. associate-/l* 83.7%

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

   16. Simplified 94.4%

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

   if 1.2e235 < 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%

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

   5. Taylor expanded in angle around 0 50.0%

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

      1. *-commutative 50.0%

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

   7. Simplified 50.0%

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

      1. associate-*r/ 50.0%

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

      2. associate-/l* 50.0%

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

      3. pow-prod-down 58.7%

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

      4. *-commutative 58.7%

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

      5. pow-prod-down 75.6%

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

   9. Applied egg-rr 75.6%

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

       1. unpow2 75.6%

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

   11. Applied egg-rr 75.6%

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

   12. Taylor expanded in x-scale around 0 50.0%

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

       1. unpow2 50.0%

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

       2. unpow2 50.0%

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

       3. swap-sqr 58.7%

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

       4. unpow2 58.7%

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

       5. unpow2 58.7%

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

       6. unpow2 58.7%

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

       7. swap-sqr 75.6%

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

       8. associate-/l/ 83.6%

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

       9. unpow2 83.6%

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

       10. associate-*r/ 99.6%

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

       11. associate-*l/ 99.7%

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

       12. unpow2 99.7%

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

       13. times-frac 97.0%

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

   14. Simplified 97.0%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.2 \cdot 10^{+235}:\\ \;\;\;\;-4 \cdot {\left(\frac{a}{x-scale \cdot y-scale} \cdot b\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4}{{\left(\frac{x-scale}{a} \cdot \frac{y-scale}{b}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.1% accurate, 15.1× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ -4 \cdot {\left(\frac{a_m}{x-scale \cdot y-scale} \cdot b\right)}^{2} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (* -4.0 (pow (* (/ a_m (* x-scale y-scale)) b) 2.0)))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	return -4.0 * pow(((a_m / (x_45_scale * y_45_scale)) * b), 2.0);
}

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 = (-4.0d0) * (((a_m / (x_45scale * y_45scale)) * b) ** 2.0d0)
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 -4.0 * Math.pow(((a_m / (x_45_scale * y_45_scale)) * b), 2.0);
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle, x_45_scale, y_45_scale):
	return -4.0 * math.pow(((a_m / (x_45_scale * y_45_scale)) * b), 2.0)

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	return Float64(-4.0 * (Float64(Float64(a_m / Float64(x_45_scale * y_45_scale)) * b) ^ 2.0))
end

MATLAB

a_m = abs(a);
function tmp = code(a_m, b, angle, x_45_scale, y_45_scale)
	tmp = -4.0 * (((a_m / (x_45_scale * y_45_scale)) * b) ^ 2.0);
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := N[(-4.0 * N[Power[N[(N[(a$95$m / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 25.0%

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

3. Simplified 23.0%

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

5. Taylor expanded in angle around 0 48.0%

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

   1. associate-*r/ 48.0%

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

   2. *-commutative 48.0%

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

   3. unpow2 48.0%

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

   4. unpow2 48.0%

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

   5. swap-sqr 60.4%

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

   6. unpow2 60.4%

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

7. Simplified 60.4%

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

   1. pow-prod-down 48.0%

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

   2. associate-*r/ 48.0%

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

   3. expm1-log1p-u 24.8%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-4 \cdot \frac{{b}^{2} \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)\right)} \]

   4. expm1-udef 24.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-4 \cdot \frac{{b}^{2} \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)} - 1} \]

   5. div-inv 24.4%

      \[\leadsto e^{\mathsf{log1p}\left(-4 \cdot \color{blue}{\left(\left({b}^{2} \cdot {a}^{2}\right) \cdot \frac{1}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)}\right)} - 1 \]

   6. *-commutative 24.4%

      \[\leadsto e^{\mathsf{log1p}\left(-4 \cdot \left(\color{blue}{\left({a}^{2} \cdot {b}^{2}\right)} \cdot \frac{1}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)\right)} - 1 \]

   7. pow-prod-down 27.5%

      \[\leadsto e^{\mathsf{log1p}\left(-4 \cdot \left(\color{blue}{{\left(a \cdot b\right)}^{2}} \cdot \frac{1}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)\right)} - 1 \]

   8. pow-prod-down 31.4%

      \[\leadsto e^{\mathsf{log1p}\left(-4 \cdot \left({\left(a \cdot b\right)}^{2} \cdot \frac{1}{\color{blue}{{\left(x-scale \cdot y-scale\right)}^{2}}}\right)\right)} - 1 \]

   9. *-commutative 31.4%

      \[\leadsto e^{\mathsf{log1p}\left(-4 \cdot \left({\left(a \cdot b\right)}^{2} \cdot \frac{1}{{\color{blue}{\left(y-scale \cdot x-scale\right)}}^{2}}\right)\right)} - 1 \]

   10. pow-flip 31.4%

       \[\leadsto e^{\mathsf{log1p}\left(-4 \cdot \left({\left(a \cdot b\right)}^{2} \cdot \color{blue}{{\left(y-scale \cdot x-scale\right)}^{\left(-2\right)}}\right)\right)} - 1 \]

   11. *-commutative 31.4%

       \[\leadsto e^{\mathsf{log1p}\left(-4 \cdot \left({\left(a \cdot b\right)}^{2} \cdot {\color{blue}{\left(x-scale \cdot y-scale\right)}}^{\left(-2\right)}\right)\right)} - 1 \]

   12. metadata-eval 31.4%

       \[\leadsto e^{\mathsf{log1p}\left(-4 \cdot \left({\left(a \cdot b\right)}^{2} \cdot {\left(x-scale \cdot y-scale\right)}^{\color{blue}{-2}}\right)\right)} - 1 \]

9. Applied egg-rr 31.4%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-4 \cdot \left({\left(a \cdot b\right)}^{2} \cdot {\left(x-scale \cdot y-scale\right)}^{-2}\right)\right)} - 1} \]
10. Step-by-step derivation

    1. expm1-def 38.5%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-4 \cdot \left({\left(a \cdot b\right)}^{2} \cdot {\left(x-scale \cdot y-scale\right)}^{-2}\right)\right)\right)} \]

    2. expm1-log1p 78.1%

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

    3. *-commutative 78.1%

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

    4. associate-*l* 78.1%

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

11. Simplified 78.1%

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

    1. unpow2 78.1%

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

13. Applied egg-rr 78.1%

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

14. Taylor expanded in a around 0 48.0%

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

    1. unpow2 48.0%

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

    2. unpow2 48.0%

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

    3. swap-sqr 59.6%

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

    4. unpow2 59.6%

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

    5. associate-*r/ 59.6%

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

    6. unpow2 59.6%

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

    7. unpow2 59.6%

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

    8. swap-sqr 78.0%

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

    9. unpow2 78.0%

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

    10. associate-/l* 77.8%

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

    11. unpow2 77.8%

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

    12. associate-/l* 83.8%

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

    13. *-rgt-identity 83.8%

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

    14. associate-*r/ 83.8%

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

    15. associate-/l* 84.1%

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

16. Simplified 94.3%

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

17. Final simplification 94.3%

    \[\leadsto -4 \cdot {\left(\frac{a}{x-scale \cdot y-scale} \cdot b\right)}^{2} \]
18. Add Preprocessing

Alternative 3: 83.2% accurate, 53.8× speedup

Math

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

FPCore

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

C

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

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 = (-4.0d0) * (((a_m * (a_m * b)) / (x_45scale * y_45scale)) * (b / (x_45scale * y_45scale)))
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 -4.0 * (((a_m * (a_m * b)) / (x_45_scale * y_45_scale)) * (b / (x_45_scale * y_45_scale)));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.0%

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

3. Simplified 23.0%

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

5. Taylor expanded in angle around 0 48.0%

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

   1. *-commutative 48.0%

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

7. Simplified 48.0%

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

   1. pow-prod-down 59.6%

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

9. Applied egg-rr 59.6%

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

    1. *-commutative 59.6%

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

    2. pow2 59.6%

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

    3. associate-*r* 57.7%

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

    4. pow-prod-down 75.0%

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

    5. unpow2 75.0%

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

    6. times-frac 83.6%

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

11. Applied egg-rr 83.6%

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

12. Final simplification 83.6%

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

Alternative 4: 93.8% accurate, 53.8× speedup

Math

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

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)))) (* t_0 (* -4.0 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 t_0 * (-4.0 * 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 = t_0 * ((-4.0d0) * 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 t_0 * (-4.0 * 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 t_0 * (-4.0 * 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(t_0 * Float64(-4.0 * 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 = t_0 * (-4.0 * 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[(t$95$0 * N[(-4.0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 25.0%

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

3. Simplified 23.0%

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

5. Taylor expanded in angle around 0 48.0%

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

   1. *-commutative 48.0%

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

7. Simplified 48.0%

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

   1. associate-*r/ 48.0%

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

   2. associate-/l* 48.0%

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

   3. pow-prod-down 60.4%

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

   4. *-commutative 60.4%

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

   5. pow-prod-down 77.8%

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

9. Applied egg-rr 77.8%

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

    1. unpow2 78.1%

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

11. Applied egg-rr 77.8%

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

    1. div-inv 77.8%

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

    2. pow-prod-down 59.6%

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

    3. pow2 59.6%

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

    4. *-commutative 59.6%

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

    5. clear-num 59.6%

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

    6. add-sqr-sqrt 59.6%

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

    7. associate-*r* 59.6%

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

13. Applied egg-rr 93.8%

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

14. Final simplification 93.8%

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

Alternative 5: 35.3% accurate, 1777.0× 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 25.0%

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

3. Simplified 24.7%

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

5. Taylor expanded in b around 0 24.4%

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

   1. distribute-rgt-out 24.4%

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

   2. metadata-eval 24.4%

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

   3. mul0-rgt 34.8%

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

7. Simplified 34.8%

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

8. Final simplification 34.8%

   \[\leadsto 0 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024011 
(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))))