Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 25.8% → 93.4%

Time: 2.0min

Alternatives: 7

Speedup: 1693.0×

• 33.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 25.8% 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: 93.4% accurate, 76.9× speedup

Math

\[\begin{array}{l} y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;y-scale\_m \leq 1.58 \cdot 10^{-254}:\\ \;\;\;\;\frac{-4}{x-scale} \cdot \left(\frac{b}{y-scale\_m} \cdot \left(a \cdot \left(b \cdot \frac{\frac{a}{x-scale}}{y-scale\_m}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-4 \cdot \left(b \cdot a\right)}{y-scale\_m \cdot x-scale} \cdot \frac{b \cdot a}{y-scale\_m \cdot x-scale}\\ \end{array} \end{array} \]

FPCore

y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale y-scale_m)
 :precision binary64
 (if (<= y-scale_m 1.58e-254)
   (*
    (/ -4.0 x-scale)
    (* (/ b y-scale_m) (* a (* b (/ (/ a x-scale) y-scale_m)))))
   (*
    (/ (* -4.0 (* b a)) (* y-scale_m x-scale))
    (/ (* b a) (* y-scale_m x-scale)))))

C

y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	double tmp;
	if (y_45_scale_m <= 1.58e-254) {
		tmp = (-4.0 / x_45_scale) * ((b / y_45_scale_m) * (a * (b * ((a / x_45_scale) / y_45_scale_m))));
	} else {
		tmp = ((-4.0 * (b * a)) / (y_45_scale_m * x_45_scale)) * ((b * a) / (y_45_scale_m * x_45_scale));
	}
	return tmp;
}

Fortran

y-scale_m = abs(y_45scale)
real(8) function code(a, b, angle, x_45scale, y_45scale_m)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale_m
    real(8) :: tmp
    if (y_45scale_m <= 1.58d-254) then
        tmp = ((-4.0d0) / x_45scale) * ((b / y_45scale_m) * (a * (b * ((a / x_45scale) / y_45scale_m))))
    else
        tmp = (((-4.0d0) * (b * a)) / (y_45scale_m * x_45scale)) * ((b * a) / (y_45scale_m * x_45scale))
    end if
    code = tmp
end function

Java

y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	double tmp;
	if (y_45_scale_m <= 1.58e-254) {
		tmp = (-4.0 / x_45_scale) * ((b / y_45_scale_m) * (a * (b * ((a / x_45_scale) / y_45_scale_m))));
	} else {
		tmp = ((-4.0 * (b * a)) / (y_45_scale_m * x_45_scale)) * ((b * a) / (y_45_scale_m * x_45_scale));
	}
	return tmp;
}

Python

y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale, y_45_scale_m):
	tmp = 0
	if y_45_scale_m <= 1.58e-254:
		tmp = (-4.0 / x_45_scale) * ((b / y_45_scale_m) * (a * (b * ((a / x_45_scale) / y_45_scale_m))))
	else:
		tmp = ((-4.0 * (b * a)) / (y_45_scale_m * x_45_scale)) * ((b * a) / (y_45_scale_m * x_45_scale))
	return tmp

Julia

y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale, y_45_scale_m)
	tmp = 0.0
	if (y_45_scale_m <= 1.58e-254)
		tmp = Float64(Float64(-4.0 / x_45_scale) * Float64(Float64(b / y_45_scale_m) * Float64(a * Float64(b * Float64(Float64(a / x_45_scale) / y_45_scale_m)))));
	else
		tmp = Float64(Float64(Float64(-4.0 * Float64(b * a)) / Float64(y_45_scale_m * x_45_scale)) * Float64(Float64(b * a) / Float64(y_45_scale_m * x_45_scale)));
	end
	return tmp
end

MATLAB

y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale_m)
	tmp = 0.0;
	if (y_45_scale_m <= 1.58e-254)
		tmp = (-4.0 / x_45_scale) * ((b / y_45_scale_m) * (a * (b * ((a / x_45_scale) / y_45_scale_m))));
	else
		tmp = ((-4.0 * (b * a)) / (y_45_scale_m * x_45_scale)) * ((b * a) / (y_45_scale_m * x_45_scale));
	end
	tmp_2 = tmp;
end

Wolfram

y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale_, y$45$scale$95$m_] := If[LessEqual[y$45$scale$95$m, 1.58e-254], N[(N[(-4.0 / x$45$scale), $MachinePrecision] * N[(N[(b / y$45$scale$95$m), $MachinePrecision] * N[(a * N[(b * N[(N[(a / x$45$scale), $MachinePrecision] / y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-4.0 * N[(b * a), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale$95$m * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(b * a), $MachinePrecision] / N[(y$45$scale$95$m * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 1.58000000000000002e-254

   1. Initial program 21.4%

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

   3. Simplified 15.5%

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

   5. Taylor expanded in angle around 0 42.9%

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

      1. associate-*r/ 42.9%

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

      2. *-commutative 42.9%

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

      3. *-commutative 42.9%

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

      4. unpow2 42.9%

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

      5. unpow2 42.9%

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

      6. swap-sqr 53.2%

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

      7. unpow2 53.2%

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

      8. *-commutative 53.2%

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

   7. Simplified 53.2%

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

      1. div-inv 53.1%

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

      2. pow-prod-down 74.7%

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

      3. pow-flip 74.8%

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

      4. metadata-eval 74.8%

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

   9. Applied egg-rr 74.8%

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

       1. unpow2 74.8%

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

   11. Applied egg-rr 74.8%

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

       1. *-commutative 74.8%

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

       2. unpow-prod-down 59.0%

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

   13. Applied egg-rr 59.0%

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

       1. pow-prod-down 74.8%

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

       2. *-commutative 74.8%

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

       3. metadata-eval 74.8%

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

       4. pow-flip 74.7%

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

       5. pow2 74.7%

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

       6. div-inv 74.8%

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

       7. associate-/r* 81.1%

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

       8. pow2 81.1%

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

       9. div-inv 81.1%

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

       10. clear-num 81.1%

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

       11. frac-times 81.2%

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

       12. metadata-eval 81.2%

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

   15. Applied egg-rr 81.2%

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

       1. associate-*l/ 74.8%

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

       2. pow2 74.8%

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

       3. clear-num 74.8%

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

       4. associate-/l/ 81.1%

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

       5. associate-/l* 81.1%

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

       6. times-frac 79.1%

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

       7. div-inv 79.1%

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

       8. *-commutative 79.1%

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

       9. associate-/l/ 79.1%

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

       10. associate-/l/ 79.1%

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

       11. *-commutative 79.1%

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

       12. div-inv 79.1%

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

       13. pow2 79.1%

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

   17. Applied egg-rr 79.1%

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

       1. associate-/l/ 74.3%

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

       2. pow2 74.3%

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

       3. associate-*l* 73.5%

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

       4. times-frac 79.1%

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

   19. Applied egg-rr 79.1%

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

       1. associate-/l* 83.8%

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

       2. associate-/l* 85.4%

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

       3. *-commutative 85.4%

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

       4. associate-/l/ 88.7%

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

   21. Simplified 88.7%

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

   if 1.58000000000000002e-254 < y-scale

   1. Initial program 27.0%

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

   3. Simplified 24.4%

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

   5. Taylor expanded in angle around 0 47.2%

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

      1. associate-*r/ 47.2%

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

      2. *-commutative 47.2%

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

      3. *-commutative 47.2%

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

      4. unpow2 47.2%

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

      5. unpow2 47.2%

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

      6. swap-sqr 55.9%

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

      7. unpow2 55.9%

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

      8. *-commutative 55.9%

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

   7. Simplified 55.9%

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

      1. div-inv 55.9%

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

      2. pow-prod-down 73.3%

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

      3. pow-flip 73.3%

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

      4. metadata-eval 73.3%

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

   9. Applied egg-rr 73.3%

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

       1. unpow2 73.3%

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

   11. Applied egg-rr 73.3%

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

       1. *-commutative 73.3%

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

       2. unpow-prod-down 59.8%

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

   13. Applied egg-rr 59.8%

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

       1. pow-prod-down 73.3%

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

       2. *-commutative 73.3%

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

       3. metadata-eval 73.3%

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

       4. pow-flip 73.3%

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

       5. pow2 73.3%

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

       6. div-inv 73.3%

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

       7. associate-/r* 78.4%

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

       8. pow2 78.4%

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

       9. div-inv 78.4%

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

       10. clear-num 78.3%

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

       11. frac-times 78.4%

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

       12. metadata-eval 78.4%

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

   15. Applied egg-rr 78.4%

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

       1. associate-*l/ 73.3%

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

       2. pow2 73.3%

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

       3. clear-num 73.3%

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

       4. associate-*r* 73.3%

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

       5. times-frac 94.2%

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

   17. Applied egg-rr 94.2%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 1.58 \cdot 10^{-254}:\\ \;\;\;\;\frac{-4}{x-scale} \cdot \left(\frac{b}{y-scale} \cdot \left(a \cdot \left(b \cdot \frac{\frac{a}{x-scale}}{y-scale}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-4 \cdot \left(b \cdot a\right)}{y-scale \cdot x-scale} \cdot \frac{b \cdot a}{y-scale \cdot x-scale}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.9% accurate, 15.4× speedup

Math

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

FPCore

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

C

y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	return pow((b * ((a / y_45_scale_m) / x_45_scale)), 2.0) * -4.0;
}

Fortran

y-scale_m = abs(y_45scale)
real(8) function code(a, b, angle, x_45scale, y_45scale_m)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale_m
    code = ((b * ((a / y_45scale_m) / x_45scale)) ** 2.0d0) * (-4.0d0)
end function

Java

y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	return Math.pow((b * ((a / y_45_scale_m) / x_45_scale)), 2.0) * -4.0;
}

Python

y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale, y_45_scale_m):
	return math.pow((b * ((a / y_45_scale_m) / x_45_scale)), 2.0) * -4.0

Julia

y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale, y_45_scale_m)
	return Float64((Float64(b * Float64(Float64(a / y_45_scale_m) / x_45_scale)) ^ 2.0) * -4.0)
end

MATLAB

y-scale_m = abs(y_45_scale);
function tmp = code(a, b, angle, x_45_scale, y_45_scale_m)
	tmp = ((b * ((a / y_45_scale_m) / x_45_scale)) ^ 2.0) * -4.0;
end

Wolfram

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

Derivation

1. Initial program 24.0%

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

3. Simplified 19.6%

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

5. Taylor expanded in angle around 0 44.9%

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

   1. associate-*r/ 44.9%

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

   2. *-commutative 44.9%

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

   3. *-commutative 44.9%

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

   4. unpow2 44.9%

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

   5. unpow2 44.9%

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

   6. swap-sqr 54.4%

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

   7. unpow2 54.4%

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

   8. *-commutative 54.4%

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

7. Simplified 54.4%

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

   1. div-inv 54.4%

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

   2. pow-prod-down 74.1%

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

   3. pow-flip 74.1%

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

   4. metadata-eval 74.1%

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

9. Applied egg-rr 74.1%

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

    1. unpow2 74.1%

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

11. Applied egg-rr 74.1%

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

    1. *-commutative 74.1%

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

    2. unpow-prod-down 59.4%

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

13. Applied egg-rr 59.4%

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

    1. pow-prod-down 74.1%

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

    2. *-commutative 74.1%

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

    3. metadata-eval 74.1%

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

    4. pow-flip 74.1%

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

    5. pow2 74.1%

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

    6. div-inv 74.1%

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

    7. associate-/r* 79.9%

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

    8. pow2 79.9%

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

    9. div-inv 79.8%

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

    10. clear-num 79.8%

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

    11. frac-times 79.9%

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

    12. metadata-eval 79.9%

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

15. Applied egg-rr 79.9%

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

    1. associate-*l/ 74.1%

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

    2. pow2 74.1%

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

    3. clear-num 74.1%

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

    4. associate-*r* 74.1%

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

    5. times-frac 92.3%

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

17. Applied egg-rr 92.3%

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

    1. pow1 92.3%

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

19. Applied egg-rr 0.0%

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

    1. unpow1 0.0%

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

    2. *-lft-identity 0.0%

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

    3. associate-*l/ 0.0%

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

    4. unpow2 0.0%

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

    5. associate-*r* 0.0%

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

    6. *-commutative 0.0%

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

    7. associate-*r/ 0.0%

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

    8. *-rgt-identity 0.0%

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

    9. associate-/l* 0.0%

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

    10. *-commutative 0.0%

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

    11. associate-/l/ 0.0%

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

    12. associate-*r* 0.0%

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

    13. *-commutative 0.0%

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

    14. associate-*r/ 0.0%

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

    15. *-rgt-identity 0.0%

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

    16. associate-/l* 0.0%

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

    17. *-commutative 0.0%

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

    18. associate-/l/ 0.0%

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

21. Simplified 93.7%

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

Alternative 3: 89.1% accurate, 76.9× speedup

Math

\[\begin{array}{l} y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;angle \leq 3.2 \cdot 10^{+37}:\\ \;\;\;\;\left(\frac{-4}{x-scale} \cdot \left(b \cdot \frac{a}{y-scale\_m}\right)\right) \cdot \frac{\frac{b \cdot a}{x-scale}}{y-scale\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4}{x-scale} \cdot \left(b \cdot \left(\frac{a}{y-scale\_m} \cdot \left(b \cdot \frac{\frac{a}{y-scale\_m}}{x-scale}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale y-scale_m)
 :precision binary64
 (if (<= angle 3.2e+37)
   (*
    (* (/ -4.0 x-scale) (* b (/ a y-scale_m)))
    (/ (/ (* b a) x-scale) y-scale_m))
   (*
    (/ -4.0 x-scale)
    (* b (* (/ a y-scale_m) (* b (/ (/ a y-scale_m) x-scale)))))))

C

y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	double tmp;
	if (angle <= 3.2e+37) {
		tmp = ((-4.0 / x_45_scale) * (b * (a / y_45_scale_m))) * (((b * a) / x_45_scale) / y_45_scale_m);
	} else {
		tmp = (-4.0 / x_45_scale) * (b * ((a / y_45_scale_m) * (b * ((a / y_45_scale_m) / x_45_scale))));
	}
	return tmp;
}

Fortran

y-scale_m = abs(y_45scale)
real(8) function code(a, b, angle, x_45scale, y_45scale_m)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale_m
    real(8) :: tmp
    if (angle <= 3.2d+37) then
        tmp = (((-4.0d0) / x_45scale) * (b * (a / y_45scale_m))) * (((b * a) / x_45scale) / y_45scale_m)
    else
        tmp = ((-4.0d0) / x_45scale) * (b * ((a / y_45scale_m) * (b * ((a / y_45scale_m) / x_45scale))))
    end if
    code = tmp
end function

Java

y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	double tmp;
	if (angle <= 3.2e+37) {
		tmp = ((-4.0 / x_45_scale) * (b * (a / y_45_scale_m))) * (((b * a) / x_45_scale) / y_45_scale_m);
	} else {
		tmp = (-4.0 / x_45_scale) * (b * ((a / y_45_scale_m) * (b * ((a / y_45_scale_m) / x_45_scale))));
	}
	return tmp;
}

Python

y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale, y_45_scale_m):
	tmp = 0
	if angle <= 3.2e+37:
		tmp = ((-4.0 / x_45_scale) * (b * (a / y_45_scale_m))) * (((b * a) / x_45_scale) / y_45_scale_m)
	else:
		tmp = (-4.0 / x_45_scale) * (b * ((a / y_45_scale_m) * (b * ((a / y_45_scale_m) / x_45_scale))))
	return tmp

Julia

y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale, y_45_scale_m)
	tmp = 0.0
	if (angle <= 3.2e+37)
		tmp = Float64(Float64(Float64(-4.0 / x_45_scale) * Float64(b * Float64(a / y_45_scale_m))) * Float64(Float64(Float64(b * a) / x_45_scale) / y_45_scale_m));
	else
		tmp = Float64(Float64(-4.0 / x_45_scale) * Float64(b * Float64(Float64(a / y_45_scale_m) * Float64(b * Float64(Float64(a / y_45_scale_m) / x_45_scale)))));
	end
	return tmp
end

MATLAB

y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale_m)
	tmp = 0.0;
	if (angle <= 3.2e+37)
		tmp = ((-4.0 / x_45_scale) * (b * (a / y_45_scale_m))) * (((b * a) / x_45_scale) / y_45_scale_m);
	else
		tmp = (-4.0 / x_45_scale) * (b * ((a / y_45_scale_m) * (b * ((a / y_45_scale_m) / x_45_scale))));
	end
	tmp_2 = tmp;
end

Wolfram

y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale_, y$45$scale$95$m_] := If[LessEqual[angle, 3.2e+37], N[(N[(N[(-4.0 / x$45$scale), $MachinePrecision] * N[(b * N[(a / y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(b * a), $MachinePrecision] / x$45$scale), $MachinePrecision] / y$45$scale$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(-4.0 / x$45$scale), $MachinePrecision] * N[(b * N[(N[(a / y$45$scale$95$m), $MachinePrecision] * N[(b * N[(N[(a / y$45$scale$95$m), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 3.20000000000000014e37

   1. Initial program 24.3%

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

   3. Simplified 18.8%

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

   5. Taylor expanded in angle around 0 41.9%

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

      1. associate-*r/ 41.9%

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

      2. *-commutative 41.9%

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

      3. *-commutative 41.9%

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

      4. unpow2 41.9%

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

      5. unpow2 41.9%

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

      6. swap-sqr 50.9%

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

      7. unpow2 50.9%

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

      8. *-commutative 50.9%

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

   7. Simplified 50.9%

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

      1. div-inv 50.9%

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

      2. pow-prod-down 72.4%

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

      3. pow-flip 72.5%

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

      4. metadata-eval 72.5%

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

   9. Applied egg-rr 72.5%

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

       1. unpow2 72.5%

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

   11. Applied egg-rr 72.5%

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

       1. *-commutative 72.5%

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

       2. unpow-prod-down 58.7%

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

   13. Applied egg-rr 58.7%

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

       1. pow-prod-down 72.5%

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

       2. *-commutative 72.5%

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

       3. metadata-eval 72.5%

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

       4. pow-flip 72.4%

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

       5. pow2 72.4%

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

       6. div-inv 72.4%

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

       7. associate-/r* 78.7%

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

       8. pow2 78.7%

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

       9. div-inv 78.7%

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

       10. clear-num 78.7%

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

       11. frac-times 78.8%

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

       12. metadata-eval 78.8%

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

   15. Applied egg-rr 78.8%

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

       1. associate-*l/ 72.5%

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

       2. pow2 72.5%

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

       3. clear-num 72.4%

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

       4. associate-*r* 72.4%

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

       5. times-frac 91.3%

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

   17. Applied egg-rr 91.3%

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

       1. times-frac 88.8%

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

       2. associate-/l* 86.9%

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

       3. associate-/l* 88.1%

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

   19. Simplified 88.1%

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

       1. associate-*r/ 86.9%

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

       2. associate-/r* 90.0%

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

   21. Applied egg-rr 90.0%

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

   if 3.20000000000000014e37 < angle

   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 22.6%

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

   5. Taylor expanded in angle around 0 56.3%

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

      1. associate-*r/ 56.3%

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

      2. *-commutative 56.3%

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

      3. *-commutative 56.3%

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

      4. unpow2 56.3%

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

      5. unpow2 56.3%

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

      6. swap-sqr 67.6%

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

      7. unpow2 67.6%

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

      8. *-commutative 67.6%

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

   7. Simplified 67.6%

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

      1. div-inv 67.6%

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

      2. pow-prod-down 80.3%

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

      3. pow-flip 80.3%

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

      4. metadata-eval 80.3%

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

   9. Applied egg-rr 80.3%

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

       1. unpow2 80.3%

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

   11. Applied egg-rr 80.3%

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

       1. *-commutative 80.3%

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

       2. unpow-prod-down 62.0%

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

   13. Applied egg-rr 62.0%

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

       1. pow-prod-down 80.3%

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

       2. *-commutative 80.3%

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

       3. metadata-eval 80.3%

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

       4. pow-flip 80.3%

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

       5. pow2 80.3%

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

       6. div-inv 80.3%

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

       7. associate-/r* 84.0%

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

       8. pow2 84.0%

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

       9. div-inv 84.1%

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

       10. clear-num 84.1%

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

       11. frac-times 84.1%

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

       12. metadata-eval 84.1%

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

   15. Applied egg-rr 84.1%

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

       1. associate-*l/ 80.3%

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

       2. pow2 80.3%

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

       3. clear-num 80.3%

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

       4. associate-/l/ 84.0%

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

       5. associate-/l* 84.0%

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

       6. times-frac 82.4%

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

       7. div-inv 82.4%

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

       8. *-commutative 82.4%

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

       9. associate-/l/ 82.4%

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

       10. associate-/l/ 82.4%

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

       11. *-commutative 82.4%

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

       12. div-inv 82.4%

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

       13. pow2 82.4%

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

   17. Applied egg-rr 82.4%

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

       1. associate-/l/ 80.4%

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

       2. pow2 80.4%

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

       3. times-frac 91.8%

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

       4. associate-*r/ 88.3%

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

       5. associate-*r/ 88.4%

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

       6. associate-/r* 85.0%

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

   19. Applied egg-rr 85.0%

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

       1. associate-*l* 85.0%

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

       2. associate-/l/ 88.5%

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

       3. *-commutative 88.5%

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

       4. associate-/l/ 90.3%

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

   21. Simplified 90.3%

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

Alternative 4: 89.2% accurate, 99.6× speedup

Math

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

FPCore

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

C

y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	return ((-4.0 / x_45_scale) * (b * (a / y_45_scale_m))) * (((b * a) / x_45_scale) / y_45_scale_m);
}

Fortran

y-scale_m = abs(y_45scale)
real(8) function code(a, b, angle, x_45scale, y_45scale_m)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale_m
    code = (((-4.0d0) / x_45scale) * (b * (a / y_45scale_m))) * (((b * a) / x_45scale) / y_45scale_m)
end function

Java

y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	return ((-4.0 / x_45_scale) * (b * (a / y_45_scale_m))) * (((b * a) / x_45_scale) / y_45_scale_m);
}

Python

y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale, y_45_scale_m):
	return ((-4.0 / x_45_scale) * (b * (a / y_45_scale_m))) * (((b * a) / x_45_scale) / y_45_scale_m)

Julia

y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale, y_45_scale_m)
	return Float64(Float64(Float64(-4.0 / x_45_scale) * Float64(b * Float64(a / y_45_scale_m))) * Float64(Float64(Float64(b * a) / x_45_scale) / y_45_scale_m))
end

MATLAB

y-scale_m = abs(y_45_scale);
function tmp = code(a, b, angle, x_45_scale, y_45_scale_m)
	tmp = ((-4.0 / x_45_scale) * (b * (a / y_45_scale_m))) * (((b * a) / x_45_scale) / y_45_scale_m);
end

Wolfram

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

Derivation

1. Initial program 24.0%

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

3. Simplified 19.6%

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

5. Taylor expanded in angle around 0 44.9%

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

   1. associate-*r/ 44.9%

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

   2. *-commutative 44.9%

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

   3. *-commutative 44.9%

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

   4. unpow2 44.9%

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

   5. unpow2 44.9%

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

   6. swap-sqr 54.4%

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

   7. unpow2 54.4%

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

   8. *-commutative 54.4%

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

7. Simplified 54.4%

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

   1. div-inv 54.4%

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

   2. pow-prod-down 74.1%

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

   3. pow-flip 74.1%

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

   4. metadata-eval 74.1%

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

9. Applied egg-rr 74.1%

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

    1. unpow2 74.1%

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

11. Applied egg-rr 74.1%

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

    1. *-commutative 74.1%

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

    2. unpow-prod-down 59.4%

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

13. Applied egg-rr 59.4%

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

    1. pow-prod-down 74.1%

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

    2. *-commutative 74.1%

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

    3. metadata-eval 74.1%

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

    4. pow-flip 74.1%

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

    5. pow2 74.1%

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

    6. div-inv 74.1%

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

    7. associate-/r* 79.9%

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

    8. pow2 79.9%

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

    9. div-inv 79.8%

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

    10. clear-num 79.8%

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

    11. frac-times 79.9%

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

    12. metadata-eval 79.9%

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

15. Applied egg-rr 79.9%

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

    1. associate-*l/ 74.1%

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

    2. pow2 74.1%

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

    3. clear-num 74.1%

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

    4. associate-*r* 74.1%

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

    5. times-frac 92.3%

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

17. Applied egg-rr 92.3%

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

    1. times-frac 90.0%

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

    2. associate-/l* 87.8%

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

    3. associate-/l* 88.7%

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

19. Simplified 88.7%

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

    1. associate-*r/ 87.8%

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

    2. associate-/r* 89.4%

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

21. Applied egg-rr 89.4%

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

Alternative 5: 89.2% accurate, 99.6× speedup

Math

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

FPCore

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

C

y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	return ((-4.0 / x_45_scale) * (b * (a / y_45_scale_m))) * (b * (a / (y_45_scale_m * x_45_scale)));
}

Fortran

y-scale_m = abs(y_45scale)
real(8) function code(a, b, angle, x_45scale, y_45scale_m)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale_m
    code = (((-4.0d0) / x_45scale) * (b * (a / y_45scale_m))) * (b * (a / (y_45scale_m * x_45scale)))
end function

Java

y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	return ((-4.0 / x_45_scale) * (b * (a / y_45_scale_m))) * (b * (a / (y_45_scale_m * x_45_scale)));
}

Python

y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale, y_45_scale_m):
	return ((-4.0 / x_45_scale) * (b * (a / y_45_scale_m))) * (b * (a / (y_45_scale_m * x_45_scale)))

Julia

y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale, y_45_scale_m)
	return Float64(Float64(Float64(-4.0 / x_45_scale) * Float64(b * Float64(a / y_45_scale_m))) * Float64(b * Float64(a / Float64(y_45_scale_m * x_45_scale))))
end

MATLAB

y-scale_m = abs(y_45_scale);
function tmp = code(a, b, angle, x_45_scale, y_45_scale_m)
	tmp = ((-4.0 / x_45_scale) * (b * (a / y_45_scale_m))) * (b * (a / (y_45_scale_m * x_45_scale)));
end

Wolfram

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

Derivation

1. Initial program 24.0%

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

3. Simplified 19.6%

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

5. Taylor expanded in angle around 0 44.9%

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

   1. associate-*r/ 44.9%

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

   2. *-commutative 44.9%

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

   3. *-commutative 44.9%

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

   4. unpow2 44.9%

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

   5. unpow2 44.9%

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

   6. swap-sqr 54.4%

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

   7. unpow2 54.4%

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

   8. *-commutative 54.4%

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

7. Simplified 54.4%

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

   1. div-inv 54.4%

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

   2. pow-prod-down 74.1%

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

   3. pow-flip 74.1%

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

   4. metadata-eval 74.1%

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

9. Applied egg-rr 74.1%

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

    1. unpow2 74.1%

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

11. Applied egg-rr 74.1%

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

    1. *-commutative 74.1%

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

    2. unpow-prod-down 59.4%

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

13. Applied egg-rr 59.4%

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

    1. pow-prod-down 74.1%

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

    2. *-commutative 74.1%

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

    3. metadata-eval 74.1%

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

    4. pow-flip 74.1%

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

    5. pow2 74.1%

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

    6. div-inv 74.1%

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

    7. associate-/r* 79.9%

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

    8. pow2 79.9%

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

    9. div-inv 79.8%

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

    10. clear-num 79.8%

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

    11. frac-times 79.9%

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

    12. metadata-eval 79.9%

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

15. Applied egg-rr 79.9%

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

    1. associate-*l/ 74.1%

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

    2. pow2 74.1%

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

    3. clear-num 74.1%

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

    4. associate-*r* 74.1%

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

    5. times-frac 92.3%

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

17. Applied egg-rr 92.3%

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

    1. times-frac 90.0%

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

    2. associate-/l* 87.8%

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

    3. associate-/l* 88.7%

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

19. Simplified 88.7%

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

20. Final simplification 88.7%

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

Alternative 6: 88.1% accurate, 99.6× speedup

Math

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

FPCore

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

C

y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	return ((b * a) / (y_45_scale_m * x_45_scale)) * ((a / y_45_scale_m) * (b * (-4.0 / x_45_scale)));
}

Fortran

y-scale_m = abs(y_45scale)
real(8) function code(a, b, angle, x_45scale, y_45scale_m)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale_m
    code = ((b * a) / (y_45scale_m * x_45scale)) * ((a / y_45scale_m) * (b * ((-4.0d0) / x_45scale)))
end function

Java

y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	return ((b * a) / (y_45_scale_m * x_45_scale)) * ((a / y_45_scale_m) * (b * (-4.0 / x_45_scale)));
}

Python

y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale, y_45_scale_m):
	return ((b * a) / (y_45_scale_m * x_45_scale)) * ((a / y_45_scale_m) * (b * (-4.0 / x_45_scale)))

Julia

y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale, y_45_scale_m)
	return Float64(Float64(Float64(b * a) / Float64(y_45_scale_m * x_45_scale)) * Float64(Float64(a / y_45_scale_m) * Float64(b * Float64(-4.0 / x_45_scale))))
end

MATLAB

y-scale_m = abs(y_45_scale);
function tmp = code(a, b, angle, x_45_scale, y_45_scale_m)
	tmp = ((b * a) / (y_45_scale_m * x_45_scale)) * ((a / y_45_scale_m) * (b * (-4.0 / x_45_scale)));
end

Wolfram

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

Derivation

1. Initial program 24.0%

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

3. Simplified 19.6%

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

5. Taylor expanded in angle around 0 44.9%

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

   1. associate-*r/ 44.9%

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

   2. *-commutative 44.9%

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

   3. *-commutative 44.9%

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

   4. unpow2 44.9%

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

   5. unpow2 44.9%

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

   6. swap-sqr 54.4%

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

   7. unpow2 54.4%

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

   8. *-commutative 54.4%

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

7. Simplified 54.4%

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

   1. div-inv 54.4%

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

   2. pow-prod-down 74.1%

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

   3. pow-flip 74.1%

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

   4. metadata-eval 74.1%

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

9. Applied egg-rr 74.1%

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

    1. unpow2 74.1%

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

11. Applied egg-rr 74.1%

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

    1. *-commutative 74.1%

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

    2. unpow-prod-down 59.4%

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

13. Applied egg-rr 59.4%

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

    1. pow-prod-down 74.1%

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

    2. *-commutative 74.1%

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

    3. metadata-eval 74.1%

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

    4. pow-flip 74.1%

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

    5. pow2 74.1%

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

    6. div-inv 74.1%

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

    7. associate-/r* 79.9%

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

    8. pow2 79.9%

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

    9. div-inv 79.8%

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

    10. clear-num 79.8%

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

    11. frac-times 79.9%

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

    12. metadata-eval 79.9%

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

15. Applied egg-rr 79.9%

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

    1. associate-*l/ 74.1%

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

    2. pow2 74.1%

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

    3. clear-num 74.1%

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

    4. associate-*r* 74.1%

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

    5. times-frac 92.3%

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

17. Applied egg-rr 92.3%

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

    1. times-frac 90.0%

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

    2. associate-*r/ 87.8%

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

    3. associate-*r* 86.5%

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

19. Applied egg-rr 86.5%

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

20. Final simplification 86.5%

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

Alternative 7: 35.4% accurate, 1693.0× speedup

Math

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

FPCore

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

C

y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	return 0.0;
}

Fortran

y-scale_m = abs(y_45scale)
real(8) function code(a, b, angle, x_45scale, y_45scale_m)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale_m
    code = 0.0d0
end function

Java

y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	return 0.0;
}

Python

y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale, y_45_scale_m):
	return 0.0

Julia

y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale, y_45_scale_m)
	return 0.0
end

MATLAB

y-scale_m = abs(y_45_scale);
function tmp = code(a, b, angle, x_45_scale, y_45_scale_m)
	tmp = 0.0;
end

Wolfram

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

Derivation

1. Initial program 24.0%

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

3. Simplified 21.3%

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

5. Taylor expanded in b around 0 22.4%

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

   1. distribute-rgt-out 22.4%

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

   2. metadata-eval 22.4%

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

   3. mul0-rgt 34.7%

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

7. Simplified 34.7%

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

Reproduce

herbie shell --seed 2024107 
(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))))