Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 25.0% → 83.8%

Time: 2.5min

Alternatives: 7

Speedup: 2485.0×

• 33.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 25.0% 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: 83.8% accurate, 21.4× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;a_m \leq 350000000:\\ \;\;\;\;\frac{\frac{-4 \cdot {\left(a_m \cdot b\right)}^{2}}{x-scale \cdot y-scale}}{x-scale \cdot y-scale}\\ \mathbf{else}:\\ \;\;\;\;\left(a_m \cdot b\right) \cdot \left(\left(a_m \cdot b\right) \cdot \left(-4 \cdot {\left(x-scale \cdot y-scale\right)}^{-2}\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (if (<= a_m 350000000.0)
   (/ (/ (* -4.0 (pow (* a_m b) 2.0)) (* x-scale y-scale)) (* x-scale y-scale))
   (* (* a_m b) (* (* a_m b) (* -4.0 (pow (* x-scale y-scale) -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 <= 350000000.0) {
		tmp = ((-4.0 * pow((a_m * b), 2.0)) / (x_45_scale * y_45_scale)) / (x_45_scale * y_45_scale);
	} else {
		tmp = (a_m * b) * ((a_m * b) * (-4.0 * pow((x_45_scale * y_45_scale), -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 <= 350000000.0d0) then
        tmp = (((-4.0d0) * ((a_m * b) ** 2.0d0)) / (x_45scale * y_45scale)) / (x_45scale * y_45scale)
    else
        tmp = (a_m * b) * ((a_m * b) * ((-4.0d0) * ((x_45scale * y_45scale) ** (-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 <= 350000000.0) {
		tmp = ((-4.0 * Math.pow((a_m * b), 2.0)) / (x_45_scale * y_45_scale)) / (x_45_scale * y_45_scale);
	} else {
		tmp = (a_m * b) * ((a_m * b) * (-4.0 * Math.pow((x_45_scale * y_45_scale), -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 <= 350000000.0:
		tmp = ((-4.0 * math.pow((a_m * b), 2.0)) / (x_45_scale * y_45_scale)) / (x_45_scale * y_45_scale)
	else:
		tmp = (a_m * b) * ((a_m * b) * (-4.0 * math.pow((x_45_scale * y_45_scale), -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 <= 350000000.0)
		tmp = Float64(Float64(Float64(-4.0 * (Float64(a_m * b) ^ 2.0)) / Float64(x_45_scale * y_45_scale)) / Float64(x_45_scale * y_45_scale));
	else
		tmp = Float64(Float64(a_m * b) * Float64(Float64(a_m * b) * Float64(-4.0 * (Float64(x_45_scale * y_45_scale) ^ -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 <= 350000000.0)
		tmp = ((-4.0 * ((a_m * b) ^ 2.0)) / (x_45_scale * y_45_scale)) / (x_45_scale * y_45_scale);
	else
		tmp = (a_m * b) * ((a_m * b) * (-4.0 * ((x_45_scale * y_45_scale) ^ -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, 350000000.0], N[(N[(N[(-4.0 * N[Power[N[(a$95$m * b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision], N[(N[(a$95$m * b), $MachinePrecision] * N[(N[(a$95$m * b), $MachinePrecision] * N[(-4.0 * N[Power[N[(x$45$scale * y$45$scale), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 3.5e8

   1. Initial program 27.5%

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

   3. Simplified 23.8%

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

   4. Taylor expanded in angle around 0 44.8%

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

      1. associate-*r/ 44.8%

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

      2. *-commutative 44.8%

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

      3. unpow2 44.8%

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

      4. unpow2 44.8%

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

      5. swap-sqr 56.5%

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

      6. unpow2 56.5%

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

   6. Simplified 56.5%

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

      1. expm1-log1p-u 32.0%

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

      2. expm1-udef 29.3%

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

      3. div-inv 29.3%

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

      4. *-commutative 29.3%

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

      5. pow-prod-down 32.7%

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

      6. pow-flip 32.7%

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

      7. metadata-eval 32.7%

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

   8. Applied egg-rr 32.7%

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

      1. expm1-def 42.9%

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

      2. expm1-log1p 75.4%

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

      3. associate-*l* 75.4%

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

      4. *-commutative 75.4%

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

   10. Simplified 75.4%

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

       1. unpow2 75.4%

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

   12. Applied egg-rr 75.4%

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

       1. associate-*r* 75.4%

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

       2. pow2 75.4%

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

       3. *-commutative 75.4%

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

       4. *-commutative 75.4%

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

       5. metadata-eval 75.4%

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

       6. pow-flip 74.7%

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

       7. div-inv 74.8%

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

       8. pow2 74.8%

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

       9. associate-/r* 82.5%

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

       10. *-commutative 82.5%

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

   14. Applied egg-rr 82.5%

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

   if 3.5e8 < a

   1. Initial program 5.8%

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

   3. Simplified 5.8%

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

   4. Taylor expanded in angle around 0 46.0%

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

      1. associate-*r/ 46.0%

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

      2. *-commutative 46.0%

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

      3. unpow2 46.0%

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

      4. unpow2 46.0%

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

      5. swap-sqr 58.0%

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

      6. unpow2 58.1%

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

   6. Simplified 58.1%

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

      1. pow-prod-down 81.0%

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

   8. Applied egg-rr 81.0%

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

      1. unpow2 81.0%

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

   10. Applied egg-rr 81.0%

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

       1. pow2 81.0%

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

       2. div-inv 81.0%

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

       3. *-commutative 81.0%

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

       4. *-commutative 81.0%

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

       5. pow2 81.0%

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

       6. pow-flip 82.5%

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

       7. metadata-eval 82.5%

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

       8. associate-*r* 82.5%

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

       9. associate-*l* 91.1%

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

   12. Applied egg-rr 91.1%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 350000000:\\ \;\;\;\;\frac{\frac{-4 \cdot {\left(a \cdot b\right)}^{2}}{x-scale \cdot y-scale}}{x-scale \cdot y-scale}\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot b\right) \cdot \left(\left(a \cdot b\right) \cdot \left(-4 \cdot {\left(x-scale \cdot y-scale\right)}^{-2}\right)\right)\\ \end{array} \]

Alternative 2: 84.0% accurate, 21.4× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;a_m \leq 1.06 \cdot 10^{-10}:\\ \;\;\;\;\frac{-4}{x-scale \cdot y-scale} \cdot \frac{{\left(a_m \cdot b\right)}^{2}}{x-scale \cdot y-scale}\\ \mathbf{else}:\\ \;\;\;\;\left(a_m \cdot b\right) \cdot \left(\left(a_m \cdot b\right) \cdot \left(-4 \cdot {\left(x-scale \cdot y-scale\right)}^{-2}\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (if (<= a_m 1.06e-10)
   (* (/ -4.0 (* x-scale y-scale)) (/ (pow (* a_m b) 2.0) (* x-scale y-scale)))
   (* (* a_m b) (* (* a_m b) (* -4.0 (pow (* x-scale y-scale) -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 <= 1.06e-10) {
		tmp = (-4.0 / (x_45_scale * y_45_scale)) * (pow((a_m * b), 2.0) / (x_45_scale * y_45_scale));
	} else {
		tmp = (a_m * b) * ((a_m * b) * (-4.0 * pow((x_45_scale * y_45_scale), -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 <= 1.06d-10) then
        tmp = ((-4.0d0) / (x_45scale * y_45scale)) * (((a_m * b) ** 2.0d0) / (x_45scale * y_45scale))
    else
        tmp = (a_m * b) * ((a_m * b) * ((-4.0d0) * ((x_45scale * y_45scale) ** (-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 <= 1.06e-10) {
		tmp = (-4.0 / (x_45_scale * y_45_scale)) * (Math.pow((a_m * b), 2.0) / (x_45_scale * y_45_scale));
	} else {
		tmp = (a_m * b) * ((a_m * b) * (-4.0 * Math.pow((x_45_scale * y_45_scale), -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 <= 1.06e-10:
		tmp = (-4.0 / (x_45_scale * y_45_scale)) * (math.pow((a_m * b), 2.0) / (x_45_scale * y_45_scale))
	else:
		tmp = (a_m * b) * ((a_m * b) * (-4.0 * math.pow((x_45_scale * y_45_scale), -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 <= 1.06e-10)
		tmp = Float64(Float64(-4.0 / Float64(x_45_scale * y_45_scale)) * Float64((Float64(a_m * b) ^ 2.0) / Float64(x_45_scale * y_45_scale)));
	else
		tmp = Float64(Float64(a_m * b) * Float64(Float64(a_m * b) * Float64(-4.0 * (Float64(x_45_scale * y_45_scale) ^ -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 <= 1.06e-10)
		tmp = (-4.0 / (x_45_scale * y_45_scale)) * (((a_m * b) ^ 2.0) / (x_45_scale * y_45_scale));
	else
		tmp = (a_m * b) * ((a_m * b) * (-4.0 * ((x_45_scale * y_45_scale) ^ -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, 1.06e-10], N[(N[(-4.0 / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(a$95$m * b), $MachinePrecision], 2.0], $MachinePrecision] / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a$95$m * b), $MachinePrecision] * N[(N[(a$95$m * b), $MachinePrecision] * N[(-4.0 * N[Power[N[(x$45$scale * y$45$scale), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.06e-10

   1. Initial program 27.6%

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

   3. Simplified 23.9%

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

   4. Taylor expanded in angle around 0 44.5%

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

      1. associate-*r/ 44.5%

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

      2. *-commutative 44.5%

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

      3. unpow2 44.5%

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

      4. unpow2 44.5%

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

      5. swap-sqr 56.3%

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

      6. unpow2 56.3%

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

   6. Simplified 56.3%

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

      1. pow-prod-down 74.6%

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

   8. Applied egg-rr 74.6%

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

      1. unpow2 74.6%

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

   10. Applied egg-rr 74.6%

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

       1. times-frac 82.3%

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

       2. *-commutative 82.3%

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

   12. Applied egg-rr 82.3%

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

   if 1.06e-10 < a

   1. Initial program 5.7%

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

   3. Simplified 5.7%

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

   4. Taylor expanded in angle around 0 46.8%

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

      1. associate-*r/ 46.8%

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

      2. *-commutative 46.8%

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

      3. unpow2 46.8%

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

      4. unpow2 46.8%

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

      5. swap-sqr 58.6%

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

      6. unpow2 58.7%

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

   6. Simplified 58.7%

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

      1. pow-prod-down 81.3%

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

   8. Applied egg-rr 81.3%

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

      1. unpow2 81.3%

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

   10. Applied egg-rr 81.3%

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

       1. pow2 81.3%

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

       2. div-inv 81.3%

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

       3. *-commutative 81.3%

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

       4. *-commutative 81.3%

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

       5. pow2 81.3%

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

       6. pow-flip 82.8%

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

       7. metadata-eval 82.8%

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

       8. associate-*r* 82.8%

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

       9. associate-*l* 91.3%

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

   12. Applied egg-rr 91.3%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.06 \cdot 10^{-10}:\\ \;\;\;\;\frac{-4}{x-scale \cdot y-scale} \cdot \frac{{\left(a \cdot b\right)}^{2}}{x-scale \cdot y-scale}\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot b\right) \cdot \left(\left(a \cdot b\right) \cdot \left(-4 \cdot {\left(x-scale \cdot y-scale\right)}^{-2}\right)\right)\\ \end{array} \]

Alternative 3: 84.1% accurate, 21.8× speedup

Math

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

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (* (* a_m b) (* (* a_m b) (* -4.0 (pow (* x-scale y-scale) -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 (a_m * b) * ((a_m * b) * (-4.0 * pow((x_45_scale * y_45_scale), -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 = (a_m * b) * ((a_m * b) * ((-4.0d0) * ((x_45scale * y_45scale) ** (-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 (a_m * b) * ((a_m * b) * (-4.0 * Math.pow((x_45_scale * y_45_scale), -2.0)));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 21.7%

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

3. Simplified 19.0%

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

4. Taylor expanded in angle around 0 45.1%

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

   1. associate-*r/ 45.1%

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

   2. *-commutative 45.1%

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

   3. unpow2 45.1%

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

   4. unpow2 45.1%

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

   5. swap-sqr 56.9%

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

   6. unpow2 56.9%

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

6. Simplified 56.9%

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

   1. pow-prod-down 76.4%

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

8. Applied egg-rr 76.4%

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

   1. unpow2 76.4%

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

10. Applied egg-rr 76.4%

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

    1. pow2 76.4%

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

    2. div-inv 76.4%

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

    3. *-commutative 76.4%

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

    4. *-commutative 76.4%

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

    5. pow2 76.4%

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

    6. pow-flip 77.3%

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

    7. metadata-eval 77.3%

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

    8. associate-*r* 77.3%

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

    9. associate-*l* 81.4%

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

12. Applied egg-rr 81.4%

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

13. Final simplification 81.4%

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

Alternative 4: 78.1% accurate, 118.3× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := \frac{1}{x-scale \cdot y-scale}\\ \left(\left(a_m \cdot b\right) \cdot \left(a_m \cdot b\right)\right) \cdot \left(-4 \cdot \left(t_0 \cdot t_0\right)\right) \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* x-scale y-scale))))
   (* (* (* a_m b) (* a_m b)) (* -4.0 (* t_0 t_0)))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 1.0 / (x_45_scale * y_45_scale);
	return ((a_m * b) * (a_m * b)) * (-4.0 * (t_0 * t_0));
}

Fortran

a_m = abs(a)
real(8) function code(a_m, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    t_0 = 1.0d0 / (x_45scale * y_45scale)
    code = ((a_m * b) * (a_m * b)) * ((-4.0d0) * (t_0 * t_0))
end function

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 1.0 / (x_45_scale * y_45_scale);
	return ((a_m * b) * (a_m * b)) * (-4.0 * (t_0 * t_0));
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle, x_45_scale, y_45_scale):
	t_0 = 1.0 / (x_45_scale * y_45_scale)
	return ((a_m * b) * (a_m * b)) * (-4.0 * (t_0 * t_0))

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(1.0 / Float64(x_45_scale * y_45_scale))
	return Float64(Float64(Float64(a_m * b) * Float64(a_m * b)) * Float64(-4.0 * Float64(t_0 * t_0)))
end

MATLAB

a_m = abs(a);
function tmp = code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = 1.0 / (x_45_scale * y_45_scale);
	tmp = ((a_m * b) * (a_m * b)) * (-4.0 * (t_0 * t_0));
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(1.0 / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(a$95$m * b), $MachinePrecision] * N[(a$95$m * b), $MachinePrecision]), $MachinePrecision] * N[(-4.0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 21.7%

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

3. Simplified 19.0%

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

4. Taylor expanded in angle around 0 45.1%

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

   1. associate-*r/ 45.1%

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

   2. *-commutative 45.1%

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

   3. unpow2 45.1%

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

   4. unpow2 45.1%

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

   5. swap-sqr 56.9%

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

   6. unpow2 56.9%

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

6. Simplified 56.9%

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

   1. expm1-log1p-u 26.7%

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

   2. expm1-udef 24.7%

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

   3. div-inv 24.7%

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

   4. *-commutative 24.7%

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

   5. pow-prod-down 28.8%

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

   6. pow-flip 28.8%

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

   7. metadata-eval 28.8%

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

8. Applied egg-rr 28.8%

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

   1. expm1-def 37.3%

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

   2. expm1-log1p 77.3%

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

   3. associate-*l* 77.3%

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

   4. *-commutative 77.3%

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

10. Simplified 77.3%

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

    1. unpow2 77.3%

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

12. Applied egg-rr 77.3%

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

    1. sqr-pow 77.2%

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

    2. metadata-eval 77.2%

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

    3. inv-pow 77.2%

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

    4. metadata-eval 77.2%

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

    5. inv-pow 77.2%

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

14. Applied egg-rr 77.2%

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

15. Final simplification 77.2%

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

Alternative 5: 75.4% accurate, 146.2× speedup

Math

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

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (/
  (* -4.0 (* a_m (* b (* a_m b))))
  (* (* x-scale y-scale) (* 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 * (b * (a_m * b)))) / ((x_45_scale * y_45_scale) * (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 * (b * (a_m * b)))) / ((x_45scale * y_45scale) * (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 * (b * (a_m * b)))) / ((x_45_scale * y_45_scale) * (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 * (b * (a_m * b)))) / ((x_45_scale * y_45_scale) * (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(Float64(-4.0 * Float64(a_m * Float64(b * Float64(a_m * b)))) / Float64(Float64(x_45_scale * y_45_scale) * 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 * (b * (a_m * b)))) / ((x_45_scale * y_45_scale) * (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[(N[(-4.0 * N[(a$95$m * N[(b * N[(a$95$m * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x$45$scale * y$45$scale), $MachinePrecision] * N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 21.7%

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

3. Simplified 19.0%

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

4. Taylor expanded in angle around 0 45.1%

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

   1. associate-*r/ 45.1%

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

   2. *-commutative 45.1%

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

   3. unpow2 45.1%

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

   4. unpow2 45.1%

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

   5. swap-sqr 56.9%

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

   6. unpow2 56.9%

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

6. Simplified 56.9%

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

   1. pow-prod-down 76.4%

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

8. Applied egg-rr 76.4%

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

   1. unpow2 76.4%

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

10. Applied egg-rr 76.4%

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

    1. *-commutative 76.4%

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

    2. pow2 76.4%

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

    3. associate-*l* 72.9%

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

12. Applied egg-rr 72.9%

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

13. Final simplification 72.9%

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

Alternative 6: 78.2% accurate, 146.2× speedup

Math

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

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (/
  (* -4.0 (* (* a_m b) (* a_m b)))
  (* (* x-scale y-scale) (* 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 * b) * (a_m * b))) / ((x_45_scale * y_45_scale) * (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 * b) * (a_m * b))) / ((x_45scale * y_45scale) * (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 * b) * (a_m * b))) / ((x_45_scale * y_45_scale) * (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 * b) * (a_m * b))) / ((x_45_scale * y_45_scale) * (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(Float64(-4.0 * Float64(Float64(a_m * b) * Float64(a_m * b))) / Float64(Float64(x_45_scale * y_45_scale) * 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 * b) * (a_m * b))) / ((x_45_scale * y_45_scale) * (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[(N[(-4.0 * N[(N[(a$95$m * b), $MachinePrecision] * N[(a$95$m * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x$45$scale * y$45$scale), $MachinePrecision] * N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 21.7%

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

3. Simplified 19.0%

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

4. Taylor expanded in angle around 0 45.1%

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

   1. associate-*r/ 45.1%

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

   2. *-commutative 45.1%

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

   3. unpow2 45.1%

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

   4. unpow2 45.1%

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

   5. swap-sqr 56.9%

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

   6. unpow2 56.9%

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

6. Simplified 56.9%

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

   1. pow-prod-down 76.4%

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

8. Applied egg-rr 76.4%

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

   1. unpow2 76.4%

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

10. Applied egg-rr 76.4%

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

    1. *-commutative 76.4%

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

    2. pow2 76.4%

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

12. Applied egg-rr 76.4%

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

13. Final simplification 76.4%

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

Alternative 7: 35.9% accurate, 2485.0× 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 21.7%

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

3. Simplified 18.7%

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

4. Taylor expanded in b around 0 22.9%

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

   1. distribute-rgt-out 22.9%

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

   2. metadata-eval 22.9%

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

   3. mul0-rgt 32.3%

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

6. Simplified 32.3%

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

7. Final simplification 32.3%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023319 
(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))))