Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 24.6% → 94.1%

Time: 1.9min

Alternatives: 5

Speedup: 99.6×

• 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: 24.6% 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: 94.1% accurate, 14.7× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := b \cdot \frac{a_m}{x-scale}\\ \mathbf{if}\;a_m \leq 2.25 \cdot 10^{-189}:\\ \;\;\;\;-4 \cdot \frac{t_0}{y-scale \cdot \frac{y-scale}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot {\left(a_m \cdot \frac{\frac{b}{x-scale}}{y-scale}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* b (/ a_m x-scale))))
   (if (<= a_m 2.25e-189)
     (* -4.0 (/ t_0 (* y-scale (/ y-scale t_0))))
     (* -4.0 (pow (* a_m (/ (/ b 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 t_0 = b * (a_m / x_45_scale);
	double tmp;
	if (a_m <= 2.25e-189) {
		tmp = -4.0 * (t_0 / (y_45_scale * (y_45_scale / t_0)));
	} else {
		tmp = -4.0 * pow((a_m * ((b / 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) :: t_0
    real(8) :: tmp
    t_0 = b * (a_m / x_45scale)
    if (a_m <= 2.25d-189) then
        tmp = (-4.0d0) * (t_0 / (y_45scale * (y_45scale / t_0)))
    else
        tmp = (-4.0d0) * ((a_m * ((b / 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 t_0 = b * (a_m / x_45_scale);
	double tmp;
	if (a_m <= 2.25e-189) {
		tmp = -4.0 * (t_0 / (y_45_scale * (y_45_scale / t_0)));
	} else {
		tmp = -4.0 * Math.pow((a_m * ((b / 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):
	t_0 = b * (a_m / x_45_scale)
	tmp = 0
	if a_m <= 2.25e-189:
		tmp = -4.0 * (t_0 / (y_45_scale * (y_45_scale / t_0)))
	else:
		tmp = -4.0 * math.pow((a_m * ((b / 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)
	t_0 = Float64(b * Float64(a_m / x_45_scale))
	tmp = 0.0
	if (a_m <= 2.25e-189)
		tmp = Float64(-4.0 * Float64(t_0 / Float64(y_45_scale * Float64(y_45_scale / t_0))));
	else
		tmp = Float64(-4.0 * (Float64(a_m * Float64(Float64(b / 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)
	t_0 = b * (a_m / x_45_scale);
	tmp = 0.0;
	if (a_m <= 2.25e-189)
		tmp = -4.0 * (t_0 / (y_45_scale * (y_45_scale / t_0)));
	else
		tmp = -4.0 * ((a_m * ((b / 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_] := Block[{t$95$0 = N[(b * N[(a$95$m / x$45$scale), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a$95$m, 2.25e-189], N[(-4.0 * N[(t$95$0 / N[(y$45$scale * N[(y$45$scale / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[Power[N[(a$95$m * N[(N[(b / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 2.2499999999999998e-189

   1. Initial program 32.5%

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

   3. Simplified 28.9%

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

   4. Taylor expanded in angle around 0 49.6%

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

      1. *-commutative 49.6%

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

      2. associate-/l* 51.0%

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

   6. Simplified 51.0%

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

      1. associate-/l* 49.6%

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

      2. div-inv 49.6%

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

      3. pow-prod-down 66.1%

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

      4. pow-prod-down 79.1%

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

   8. Applied egg-rr 79.1%

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

      1. associate-*r/ 79.1%

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

      2. associate-*l/ 79.1%

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

      3. *-rgt-identity 79.1%

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

      4. *-commutative 79.1%

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

   10. Simplified 79.1%

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

       1. add-sqr-sqrt 79.1%

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

       2. unpow2 79.1%

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

       3. times-frac 81.3%

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

       4. *-commutative 81.3%

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

       5. sqrt-pow1 65.4%

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

       6. metadata-eval 65.4%

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

       7. pow1 65.4%

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

       8. *-commutative 65.4%

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

       9. sqrt-pow1 92.4%

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

       10. metadata-eval 92.4%

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

       11. pow1 92.4%

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

   12. Applied egg-rr 92.4%

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

       1. clear-num 92.4%

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

       2. associate-/r* 91.8%

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

       3. frac-times 91.3%

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

       4. *-un-lft-identity 91.3%

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

       5. *-un-lft-identity 91.3%

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

       6. times-frac 88.8%

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

       7. /-rgt-identity 88.8%

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

       8. *-commutative 88.8%

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

       9. associate-/l* 91.4%

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

       10. *-un-lft-identity 91.4%

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

       11. times-frac 95.9%

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

       12. /-rgt-identity 95.9%

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

   14. Applied egg-rr 95.9%

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

   if 2.2499999999999998e-189 < a

   1. Initial program 18.8%

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

   3. Simplified 16.9%

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

   4. Taylor expanded in angle around 0 44.1%

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

      1. *-commutative 44.1%

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

      2. associate-/l* 46.8%

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

   6. Simplified 46.8%

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

      1. associate-/l* 44.1%

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

      2. div-inv 44.1%

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

      3. pow-prod-down 59.5%

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

      4. pow-prod-down 76.6%

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

   8. Applied egg-rr 76.6%

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

      1. associate-*r/ 76.6%

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

      2. associate-*l/ 76.6%

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

      3. *-rgt-identity 76.6%

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

      4. *-commutative 76.6%

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

   10. Simplified 76.6%

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

       1. add-sqr-sqrt 76.6%

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

       2. unpow2 76.6%

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

       3. times-frac 83.0%

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

       4. *-commutative 83.0%

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

       5. sqrt-pow1 56.8%

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

       6. metadata-eval 56.8%

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

       7. pow1 56.8%

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

       8. *-commutative 56.8%

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

       9. sqrt-pow1 93.9%

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

       10. metadata-eval 93.9%

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

       11. pow1 93.9%

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

   12. Applied egg-rr 93.9%

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

       1. frac-times 76.6%

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

       2. unpow2 76.6%

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

       3. unpow2 76.6%

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

       4. clear-num 76.6%

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

       5. unpow-prod-down 55.2%

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

       6. associate-/l/ 57.8%

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

       7. unpow-prod-down 46.8%

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

       8. unpow2 46.8%

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

       9. frac-times 51.3%

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

       10. *-commutative 51.3%

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

       11. clear-num 51.3%

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

       12. frac-times 46.8%

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

       13. *-commutative 46.8%

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

       14. unpow-prod-down 57.8%

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

       15. unpow2 57.8%

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

       16. associate-/l* 55.2%

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

   14. Applied egg-rr 95.6%

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

       1. remove-double-div 95.6%

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

       2. associate-*r/ 93.9%

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

       3. associate-*l/ 93.9%

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

       4. *-commutative 93.9%

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

       5. associate-/r* 93.6%

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

   16. Simplified 93.6%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.25 \cdot 10^{-189}:\\ \;\;\;\;-4 \cdot \frac{b \cdot \frac{a}{x-scale}}{y-scale \cdot \frac{y-scale}{b \cdot \frac{a}{x-scale}}}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot {\left(a \cdot \frac{\frac{b}{x-scale}}{y-scale}\right)}^{2}\\ \end{array} \]

Alternative 2: 92.4% accurate, 76.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;x-scale \leq 1.2 \cdot 10^{-127}:\\ \;\;\;\;-4 \cdot \left(\frac{a_m \cdot b}{x-scale \cdot y-scale} \cdot \frac{b}{\frac{x-scale \cdot y-scale}{a_m}}\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\left(a_m \cdot \frac{\frac{b}{x-scale}}{y-scale}\right) \cdot \left(\frac{b}{x-scale} \cdot \frac{a_m}{y-scale}\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (if (<= x-scale 1.2e-127)
   (*
    -4.0
    (* (/ (* a_m b) (* x-scale y-scale)) (/ b (/ (* x-scale y-scale) a_m))))
   (*
    -4.0
    (* (* a_m (/ (/ b x-scale) y-scale)) (* (/ b x-scale) (/ a_m y-scale))))))

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 (x_45_scale <= 1.2e-127) {
		tmp = -4.0 * (((a_m * b) / (x_45_scale * y_45_scale)) * (b / ((x_45_scale * y_45_scale) / a_m)));
	} else {
		tmp = -4.0 * ((a_m * ((b / x_45_scale) / y_45_scale)) * ((b / x_45_scale) * (a_m / y_45_scale)));
	}
	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 (x_45scale <= 1.2d-127) then
        tmp = (-4.0d0) * (((a_m * b) / (x_45scale * y_45scale)) * (b / ((x_45scale * y_45scale) / a_m)))
    else
        tmp = (-4.0d0) * ((a_m * ((b / x_45scale) / y_45scale)) * ((b / x_45scale) * (a_m / y_45scale)))
    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 (x_45_scale <= 1.2e-127) {
		tmp = -4.0 * (((a_m * b) / (x_45_scale * y_45_scale)) * (b / ((x_45_scale * y_45_scale) / a_m)));
	} else {
		tmp = -4.0 * ((a_m * ((b / x_45_scale) / y_45_scale)) * ((b / x_45_scale) * (a_m / y_45_scale)));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if x_45_scale <= 1.2e-127:
		tmp = -4.0 * (((a_m * b) / (x_45_scale * y_45_scale)) * (b / ((x_45_scale * y_45_scale) / a_m)))
	else:
		tmp = -4.0 * ((a_m * ((b / x_45_scale) / y_45_scale)) * ((b / x_45_scale) * (a_m / y_45_scale)))
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (x_45_scale <= 1.2e-127)
		tmp = Float64(-4.0 * Float64(Float64(Float64(a_m * b) / Float64(x_45_scale * y_45_scale)) * Float64(b / Float64(Float64(x_45_scale * y_45_scale) / a_m))));
	else
		tmp = Float64(-4.0 * Float64(Float64(a_m * Float64(Float64(b / x_45_scale) / y_45_scale)) * Float64(Float64(b / x_45_scale) * Float64(a_m / y_45_scale))));
	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 (x_45_scale <= 1.2e-127)
		tmp = -4.0 * (((a_m * b) / (x_45_scale * y_45_scale)) * (b / ((x_45_scale * y_45_scale) / a_m)));
	else
		tmp = -4.0 * ((a_m * ((b / x_45_scale) / y_45_scale)) * ((b / x_45_scale) * (a_m / y_45_scale)));
	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[x$45$scale, 1.2e-127], N[(-4.0 * N[(N[(N[(a$95$m * b), $MachinePrecision] / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] * N[(b / N[(N[(x$45$scale * y$45$scale), $MachinePrecision] / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(N[(a$95$m * N[(N[(b / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(b / x$45$scale), $MachinePrecision] * N[(a$95$m / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 1.19999999999999991e-127

   1. Initial program 21.3%

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

   3. Simplified 16.6%

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

   4. Taylor expanded in angle around 0 41.8%

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

      1. *-commutative 41.8%

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

      2. associate-/l* 43.1%

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

   6. Simplified 43.1%

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

      1. associate-/l* 41.8%

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

      2. div-inv 41.8%

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

      3. pow-prod-down 57.6%

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

      4. pow-prod-down 77.5%

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

   8. Applied egg-rr 77.5%

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

      1. associate-*r/ 77.5%

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

      2. associate-*l/ 77.5%

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

      3. *-rgt-identity 77.5%

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

      4. *-commutative 77.5%

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

   10. Simplified 77.5%

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

       1. add-sqr-sqrt 77.5%

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

       2. unpow2 77.5%

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

       3. times-frac 80.9%

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

       4. *-commutative 80.9%

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

       5. sqrt-pow1 58.1%

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

       6. metadata-eval 58.1%

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

       7. pow1 58.1%

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

       8. *-commutative 58.1%

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

       9. sqrt-pow1 94.3%

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

       10. metadata-eval 94.3%

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

       11. pow1 94.3%

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

   12. Applied egg-rr 94.3%

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

       1. associate-/l* 93.2%

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

       2. div-inv 93.1%

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

       3. *-commutative 93.1%

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

       4. *-un-lft-identity 93.1%

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

       5. times-frac 88.3%

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

       6. /-rgt-identity 88.3%

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

   14. Applied egg-rr 88.3%

       \[\leadsto -4 \cdot \left(\frac{b \cdot a}{x-scale \cdot y-scale} \cdot \color{blue}{\left(b \cdot \frac{1}{y-scale \cdot \frac{x-scale}{a}}\right)}\right) \]
   15. Step-by-step derivation

       1. associate-*r/ 88.3%

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

       2. *-rgt-identity 88.3%

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

       3. associate-*r/ 93.2%

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

       4. *-commutative 93.2%

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

   16. Simplified 93.2%

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

   if 1.19999999999999991e-127 < x-scale

   1. Initial program 34.6%

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

   3. Simplified 34.6%

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

   4. Taylor expanded in angle around 0 55.4%

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

      1. *-commutative 55.4%

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

      2. associate-/l* 58.3%

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

   6. Simplified 58.3%

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

      1. associate-/l* 55.4%

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

      2. *-un-lft-identity 55.4%

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

      3. add-sqr-sqrt 55.3%

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

      4. times-frac 55.3%

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

      5. pow-prod-down 55.4%

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

      6. pow-prod-down 71.7%

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

      7. pow-prod-down 78.7%

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

   8. Applied egg-rr 78.7%

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

      1. *-commutative 78.7%

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

      2. times-frac 78.7%

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

      3. *-un-lft-identity 78.7%

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

      4. unpow2 78.7%

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

      5. add-sqr-sqrt 78.7%

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

      6. associate-/l* 83.8%

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

      7. *-commutative 83.8%

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

      8. *-commutative 83.8%

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

   10. Applied egg-rr 83.8%

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

       1. associate-/l* 78.7%

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

       2. unpow2 78.7%

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

       3. frac-times 91.2%

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

       4. times-frac 87.4%

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

       5. associate-*l* 85.5%

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

       6. *-un-lft-identity 85.5%

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

       7. times-frac 86.8%

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

       8. /-rgt-identity 86.8%

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

   12. Applied egg-rr 86.8%

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

       1. associate-*r* 88.8%

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

       2. associate-*r/ 87.4%

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

       3. associate-*l/ 88.8%

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

       4. *-commutative 88.8%

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

       5. associate-/r* 94.6%

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

   14. Simplified 94.6%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq 1.2 \cdot 10^{-127}:\\ \;\;\;\;-4 \cdot \left(\frac{a \cdot b}{x-scale \cdot y-scale} \cdot \frac{b}{\frac{x-scale \cdot y-scale}{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\left(a \cdot \frac{\frac{b}{x-scale}}{y-scale}\right) \cdot \left(\frac{b}{x-scale} \cdot \frac{a}{y-scale}\right)\right)\\ \end{array} \]

Alternative 3: 93.5% accurate, 76.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := \frac{a_m \cdot b}{x-scale \cdot y-scale}\\ \mathbf{if}\;x-scale \leq 1.04 \cdot 10^{-13}:\\ \;\;\;\;-4 \cdot \left(t_0 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\left(a_m \cdot \frac{\frac{b}{x-scale}}{y-scale}\right) \cdot \left(\frac{b}{x-scale} \cdot \frac{a_m}{y-scale}\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ (* a_m b) (* x-scale y-scale))))
   (if (<= x-scale 1.04e-13)
     (* -4.0 (* t_0 t_0))
     (*
      -4.0
      (*
       (* a_m (/ (/ b x-scale) y-scale))
       (* (/ b x-scale) (/ a_m y-scale)))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (a_m * b) / (x_45_scale * y_45_scale);
	double tmp;
	if (x_45_scale <= 1.04e-13) {
		tmp = -4.0 * (t_0 * t_0);
	} else {
		tmp = -4.0 * ((a_m * ((b / x_45_scale) / y_45_scale)) * ((b / x_45_scale) * (a_m / y_45_scale)));
	}
	return tmp;
}

Fortran

a_m = abs(a)
real(8) function code(a_m, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (a_m * b) / (x_45scale * y_45scale)
    if (x_45scale <= 1.04d-13) then
        tmp = (-4.0d0) * (t_0 * t_0)
    else
        tmp = (-4.0d0) * ((a_m * ((b / x_45scale) / y_45scale)) * ((b / x_45scale) * (a_m / y_45scale)))
    end if
    code = tmp
end function

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (a_m * b) / (x_45_scale * y_45_scale);
	double tmp;
	if (x_45_scale <= 1.04e-13) {
		tmp = -4.0 * (t_0 * t_0);
	} else {
		tmp = -4.0 * ((a_m * ((b / x_45_scale) / y_45_scale)) * ((b / x_45_scale) * (a_m / y_45_scale)));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle, x_45_scale, y_45_scale):
	t_0 = (a_m * b) / (x_45_scale * y_45_scale)
	tmp = 0
	if x_45_scale <= 1.04e-13:
		tmp = -4.0 * (t_0 * t_0)
	else:
		tmp = -4.0 * ((a_m * ((b / x_45_scale) / y_45_scale)) * ((b / x_45_scale) * (a_m / y_45_scale)))
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(a_m * b) / Float64(x_45_scale * y_45_scale))
	tmp = 0.0
	if (x_45_scale <= 1.04e-13)
		tmp = Float64(-4.0 * Float64(t_0 * t_0));
	else
		tmp = Float64(-4.0 * Float64(Float64(a_m * Float64(Float64(b / x_45_scale) / y_45_scale)) * Float64(Float64(b / x_45_scale) * Float64(a_m / y_45_scale))));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = (a_m * b) / (x_45_scale * y_45_scale);
	tmp = 0.0;
	if (x_45_scale <= 1.04e-13)
		tmp = -4.0 * (t_0 * t_0);
	else
		tmp = -4.0 * ((a_m * ((b / x_45_scale) / y_45_scale)) * ((b / x_45_scale) * (a_m / y_45_scale)));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(a$95$m * b), $MachinePrecision] / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$45$scale, 1.04e-13], N[(-4.0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(N[(a$95$m * N[(N[(b / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(b / x$45$scale), $MachinePrecision] * N[(a$95$m / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 1.03999999999999999e-13

   1. Initial program 22.7%

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

   3. Simplified 18.0%

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

   4. Taylor expanded in angle around 0 43.6%

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

      1. *-commutative 43.6%

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

      2. associate-/l* 45.2%

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

   6. Simplified 45.2%

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

      1. associate-/l* 43.6%

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

      2. div-inv 43.6%

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

      3. pow-prod-down 60.2%

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

      4. pow-prod-down 78.1%

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

   8. Applied egg-rr 78.1%

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

      1. associate-*r/ 78.2%

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

      2. associate-*l/ 78.2%

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

      3. *-rgt-identity 78.2%

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

      4. *-commutative 78.2%

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

   10. Simplified 78.2%

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

       1. add-sqr-sqrt 78.2%

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

       2. unpow2 78.2%

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

       3. times-frac 81.7%

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

       4. *-commutative 81.7%

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

       5. sqrt-pow1 58.3%

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

       6. metadata-eval 58.3%

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

       7. pow1 58.3%

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

       8. *-commutative 58.3%

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

       9. sqrt-pow1 93.9%

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

       10. metadata-eval 93.9%

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

       11. pow1 93.9%

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

   12. Applied egg-rr 93.9%

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

   if 1.03999999999999999e-13 < x-scale

   1. Initial program 35.2%

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

   3. Simplified 36.4%

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

   4. Taylor expanded in angle around 0 55.3%

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

      1. *-commutative 55.3%

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

      2. associate-/l* 57.8%

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

   6. Simplified 57.8%

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

      1. associate-/l* 55.3%

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

      2. *-un-lft-identity 55.3%

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

      3. add-sqr-sqrt 55.3%

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

      4. times-frac 55.3%

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

      5. pow-prod-down 55.3%

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

      6. pow-prod-down 69.9%

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

      7. pow-prod-down 77.7%

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

   8. Applied egg-rr 77.7%

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

      1. *-commutative 77.7%

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

      2. times-frac 77.6%

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

      3. *-un-lft-identity 77.6%

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

      4. unpow2 77.6%

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

      5. add-sqr-sqrt 77.6%

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

      6. associate-/l* 83.0%

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

      7. *-commutative 83.0%

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

      8. *-commutative 83.0%

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

   10. Applied egg-rr 83.0%

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

       1. associate-/l* 77.6%

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

       2. unpow2 77.6%

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

       3. frac-times 91.2%

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

       4. times-frac 87.6%

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

       5. associate-*l* 86.4%

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

       6. *-un-lft-identity 86.4%

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

       7. times-frac 87.7%

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

       8. /-rgt-identity 87.7%

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

   12. Applied egg-rr 87.7%

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

       1. associate-*r* 88.9%

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

       2. associate-*r/ 87.6%

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

       3. associate-*l/ 88.9%

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

       4. *-commutative 88.9%

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

       5. associate-/r* 95.5%

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

   14. Simplified 95.5%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq 1.04 \cdot 10^{-13}:\\ \;\;\;\;-4 \cdot \left(\frac{a \cdot b}{x-scale \cdot y-scale} \cdot \frac{a \cdot b}{x-scale \cdot y-scale}\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\left(a \cdot \frac{\frac{b}{x-scale}}{y-scale}\right) \cdot \left(\frac{b}{x-scale} \cdot \frac{a}{y-scale}\right)\right)\\ \end{array} \]

Alternative 4: 92.9% accurate, 76.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := b \cdot \frac{a_m}{x-scale}\\ \mathbf{if}\;a_m \leq 2 \cdot 10^{-190}:\\ \;\;\;\;-4 \cdot \frac{t_0}{y-scale \cdot \frac{y-scale}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\left(a_m \cdot \frac{\frac{b}{x-scale}}{y-scale}\right) \cdot \left(\frac{b}{x-scale} \cdot \frac{a_m}{y-scale}\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* b (/ a_m x-scale))))
   (if (<= a_m 2e-190)
     (* -4.0 (/ t_0 (* y-scale (/ y-scale t_0))))
     (*
      -4.0
      (*
       (* a_m (/ (/ b x-scale) y-scale))
       (* (/ b x-scale) (/ a_m y-scale)))))))

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 = b * (a_m / x_45_scale);
	double tmp;
	if (a_m <= 2e-190) {
		tmp = -4.0 * (t_0 / (y_45_scale * (y_45_scale / t_0)));
	} else {
		tmp = -4.0 * ((a_m * ((b / x_45_scale) / y_45_scale)) * ((b / x_45_scale) * (a_m / y_45_scale)));
	}
	return tmp;
}

Fortran

a_m = abs(a)
real(8) function code(a_m, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: tmp
    t_0 = b * (a_m / x_45scale)
    if (a_m <= 2d-190) then
        tmp = (-4.0d0) * (t_0 / (y_45scale * (y_45scale / t_0)))
    else
        tmp = (-4.0d0) * ((a_m * ((b / x_45scale) / y_45scale)) * ((b / x_45scale) * (a_m / y_45scale)))
    end if
    code = tmp
end function

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = b * (a_m / x_45_scale);
	double tmp;
	if (a_m <= 2e-190) {
		tmp = -4.0 * (t_0 / (y_45_scale * (y_45_scale / t_0)));
	} else {
		tmp = -4.0 * ((a_m * ((b / x_45_scale) / y_45_scale)) * ((b / x_45_scale) * (a_m / y_45_scale)));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle, x_45_scale, y_45_scale):
	t_0 = b * (a_m / x_45_scale)
	tmp = 0
	if a_m <= 2e-190:
		tmp = -4.0 * (t_0 / (y_45_scale * (y_45_scale / t_0)))
	else:
		tmp = -4.0 * ((a_m * ((b / x_45_scale) / y_45_scale)) * ((b / x_45_scale) * (a_m / y_45_scale)))
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(b * Float64(a_m / x_45_scale))
	tmp = 0.0
	if (a_m <= 2e-190)
		tmp = Float64(-4.0 * Float64(t_0 / Float64(y_45_scale * Float64(y_45_scale / t_0))));
	else
		tmp = Float64(-4.0 * Float64(Float64(a_m * Float64(Float64(b / x_45_scale) / y_45_scale)) * Float64(Float64(b / x_45_scale) * Float64(a_m / y_45_scale))));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = b * (a_m / x_45_scale);
	tmp = 0.0;
	if (a_m <= 2e-190)
		tmp = -4.0 * (t_0 / (y_45_scale * (y_45_scale / t_0)));
	else
		tmp = -4.0 * ((a_m * ((b / x_45_scale) / y_45_scale)) * ((b / x_45_scale) * (a_m / y_45_scale)));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(b * N[(a$95$m / x$45$scale), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a$95$m, 2e-190], N[(-4.0 * N[(t$95$0 / N[(y$45$scale * N[(y$45$scale / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(N[(a$95$m * N[(N[(b / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(b / x$45$scale), $MachinePrecision] * N[(a$95$m / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 2e-190

   1. Initial program 32.5%

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

   3. Simplified 28.9%

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

   4. Taylor expanded in angle around 0 49.6%

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

      1. *-commutative 49.6%

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

      2. associate-/l* 51.0%

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

   6. Simplified 51.0%

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

      1. associate-/l* 49.6%

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

      2. div-inv 49.6%

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

      3. pow-prod-down 66.1%

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

      4. pow-prod-down 79.1%

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

   8. Applied egg-rr 79.1%

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

      1. associate-*r/ 79.1%

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

      2. associate-*l/ 79.1%

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

      3. *-rgt-identity 79.1%

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

      4. *-commutative 79.1%

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

   10. Simplified 79.1%

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

       1. add-sqr-sqrt 79.1%

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

       2. unpow2 79.1%

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

       3. times-frac 81.3%

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

       4. *-commutative 81.3%

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

       5. sqrt-pow1 65.4%

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

       6. metadata-eval 65.4%

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

       7. pow1 65.4%

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

       8. *-commutative 65.4%

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

       9. sqrt-pow1 92.4%

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

       10. metadata-eval 92.4%

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

       11. pow1 92.4%

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

   12. Applied egg-rr 92.4%

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

       1. clear-num 92.4%

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

       2. associate-/r* 91.8%

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

       3. frac-times 91.3%

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

       4. *-un-lft-identity 91.3%

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

       5. *-un-lft-identity 91.3%

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

       6. times-frac 88.8%

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

       7. /-rgt-identity 88.8%

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

       8. *-commutative 88.8%

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

       9. associate-/l* 91.4%

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

       10. *-un-lft-identity 91.4%

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

       11. times-frac 95.9%

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

       12. /-rgt-identity 95.9%

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

   14. Applied egg-rr 95.9%

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

   if 2e-190 < a

   1. Initial program 18.8%

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

   3. Simplified 16.9%

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

   4. Taylor expanded in angle around 0 44.1%

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

      1. *-commutative 44.1%

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

      2. associate-/l* 46.8%

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

   6. Simplified 46.8%

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

      1. associate-/l* 44.1%

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

      2. *-un-lft-identity 44.1%

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

      3. add-sqr-sqrt 44.0%

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

      4. times-frac 44.0%

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

      5. pow-prod-down 44.1%

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

      6. pow-prod-down 59.4%

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

      7. pow-prod-down 76.6%

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

   8. Applied egg-rr 76.6%

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

      1. *-commutative 76.6%

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

      2. times-frac 76.6%

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

      3. *-un-lft-identity 76.6%

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

      4. unpow2 76.6%

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

      5. add-sqr-sqrt 76.6%

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

      6. associate-/l* 83.0%

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

      7. *-commutative 83.0%

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

      8. *-commutative 83.0%

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

   10. Applied egg-rr 83.0%

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

       1. associate-/l* 76.6%

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

       2. unpow2 76.6%

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

       3. frac-times 93.9%

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

       4. times-frac 89.7%

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

       5. associate-*l* 87.1%

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

       6. *-un-lft-identity 87.1%

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

       7. times-frac 88.8%

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

       8. /-rgt-identity 88.8%

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

   12. Applied egg-rr 88.8%

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

       1. associate-*r* 91.4%

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

       2. associate-*r/ 89.7%

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

       3. associate-*l/ 89.7%

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

       4. *-commutative 89.7%

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

       5. associate-/r* 92.8%

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

   14. Simplified 92.8%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2 \cdot 10^{-190}:\\ \;\;\;\;-4 \cdot \frac{b \cdot \frac{a}{x-scale}}{y-scale \cdot \frac{y-scale}{b \cdot \frac{a}{x-scale}}}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\left(a \cdot \frac{\frac{b}{x-scale}}{y-scale}\right) \cdot \left(\frac{b}{x-scale} \cdot \frac{a}{y-scale}\right)\right)\\ \end{array} \]

Alternative 5: 92.5% accurate, 99.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 26.6%

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

3. Simplified 23.7%

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

4. Taylor expanded in angle around 0 47.2%

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

   1. *-commutative 47.2%

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

   2. associate-/l* 49.2%

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

6. Simplified 49.2%

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

   1. associate-/l* 47.2%

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

   2. *-un-lft-identity 47.2%

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

   3. add-sqr-sqrt 47.2%

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

   4. times-frac 47.2%

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

   5. pow-prod-down 47.2%

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

   6. pow-prod-down 63.2%

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

   7. pow-prod-down 78.0%

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

8. Applied egg-rr 78.0%

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

   1. *-commutative 78.0%

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

   2. times-frac 78.0%

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

   3. *-un-lft-identity 78.0%

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

   4. unpow2 78.0%

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

   5. add-sqr-sqrt 78.0%

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

   6. associate-/l* 82.0%

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

   7. *-commutative 82.0%

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

   8. *-commutative 82.0%

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

10. Applied egg-rr 82.0%

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

    1. associate-/l* 78.0%

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

    2. unpow2 78.0%

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

    3. frac-times 93.1%

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

    4. times-frac 88.3%

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

    5. associate-*l* 86.4%

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

    6. *-un-lft-identity 86.4%

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

    7. times-frac 87.2%

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

    8. /-rgt-identity 87.2%

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

12. Applied egg-rr 87.2%

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

    1. associate-*r* 88.7%

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

    2. associate-*r/ 88.3%

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

    3. associate-*l/ 89.0%

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

    4. *-commutative 89.0%

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

    5. associate-/r* 92.5%

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

14. Simplified 92.5%

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

15. Final simplification 92.5%

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

Reproduce

herbie shell --seed 2023336 
(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))))