Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 25.4% → 94.3%

Time: 31.2s

Alternatives: 7

Speedup: 1693.0×

• Could not converge on a ground truth

  1. a = 1.0459249181480811e-81
  2. b = 2.24934573480039e+261
  3. angle = 7.788570920384362e+179
  4. x-scale = 4.2663536889344375e+129
  5. y-scale = 6.240892952342099e-268

• 33.3% 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: 25.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 94.3% accurate, 7.4× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := b\_m \cdot \frac{\frac{a\_m}{y-scale\_m}}{x-scale\_m}\\ t_1 := a\_m \cdot \frac{\frac{b\_m}{x-scale\_m}}{y-scale\_m}\\ t_2 := \sqrt{t\_1}\\ \mathbf{if}\;a\_m \leq 8.5 \cdot 10^{+209}:\\ \;\;\;\;-4 \cdot \left(t\_2 \cdot \left(t\_1 \cdot t\_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(-4 \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a_m b_m angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (* b_m (/ (/ a_m y-scale_m) x-scale_m)))
        (t_1 (* a_m (/ (/ b_m x-scale_m) y-scale_m)))
        (t_2 (sqrt t_1)))
   (if (<= a_m 8.5e+209) (* -4.0 (* t_2 (* t_1 t_2))) (* t_0 (* -4.0 t_0)))))

C

a_m = fabs(a);
b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = b_m * ((a_m / y_45_scale_m) / x_45_scale_m);
	double t_1 = a_m * ((b_m / x_45_scale_m) / y_45_scale_m);
	double t_2 = sqrt(t_1);
	double tmp;
	if (a_m <= 8.5e+209) {
		tmp = -4.0 * (t_2 * (t_1 * t_2));
	} else {
		tmp = t_0 * (-4.0 * t_0);
	}
	return tmp;
}

Fortran

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45scale)
y-scale_m = abs(y_45scale)
real(8) function code(a_m, b_m, angle, x_45scale_m, y_45scale_m)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = b_m * ((a_m / y_45scale_m) / x_45scale_m)
    t_1 = a_m * ((b_m / x_45scale_m) / y_45scale_m)
    t_2 = sqrt(t_1)
    if (a_m <= 8.5d+209) then
        tmp = (-4.0d0) * (t_2 * (t_1 * t_2))
    else
        tmp = t_0 * ((-4.0d0) * t_0)
    end if
    code = tmp
end function

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = b_m * ((a_m / y_45_scale_m) / x_45_scale_m);
	double t_1 = a_m * ((b_m / x_45_scale_m) / y_45_scale_m);
	double t_2 = Math.sqrt(t_1);
	double tmp;
	if (a_m <= 8.5e+209) {
		tmp = -4.0 * (t_2 * (t_1 * t_2));
	} else {
		tmp = t_0 * (-4.0 * t_0);
	}
	return tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m):
	t_0 = b_m * ((a_m / y_45_scale_m) / x_45_scale_m)
	t_1 = a_m * ((b_m / x_45_scale_m) / y_45_scale_m)
	t_2 = math.sqrt(t_1)
	tmp = 0
	if a_m <= 8.5e+209:
		tmp = -4.0 * (t_2 * (t_1 * t_2))
	else:
		tmp = t_0 * (-4.0 * t_0)
	return tmp

Julia

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(b_m * Float64(Float64(a_m / y_45_scale_m) / x_45_scale_m))
	t_1 = Float64(a_m * Float64(Float64(b_m / x_45_scale_m) / y_45_scale_m))
	t_2 = sqrt(t_1)
	tmp = 0.0
	if (a_m <= 8.5e+209)
		tmp = Float64(-4.0 * Float64(t_2 * Float64(t_1 * t_2)));
	else
		tmp = Float64(t_0 * Float64(-4.0 * t_0));
	end
	return tmp
end

MATLAB

a_m = abs(a);
b_m = abs(b);
x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = b_m * ((a_m / y_45_scale_m) / x_45_scale_m);
	t_1 = a_m * ((b_m / x_45_scale_m) / y_45_scale_m);
	t_2 = sqrt(t_1);
	tmp = 0.0;
	if (a_m <= 8.5e+209)
		tmp = -4.0 * (t_2 * (t_1 * t_2));
	else
		tmp = t_0 * (-4.0 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(b$95$m * N[(N[(a$95$m / y$45$scale$95$m), $MachinePrecision] / x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(a$95$m * N[(N[(b$95$m / x$45$scale$95$m), $MachinePrecision] / y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[t$95$1], $MachinePrecision]}, If[LessEqual[a$95$m, 8.5e+209], N[(-4.0 * N[(t$95$2 * N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(-4.0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if a < 8.50000000000000062e209

   1. Initial program 24.9%

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

   3. Simplified 23.2%

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

   5. Taylor expanded in angle around 0 47.1%

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

      1. *-commutative 47.1%

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

      2. unpow2 47.1%

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

      3. unpow2 47.1%

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

      4. swap-sqr 64.1%

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

      5. unpow2 64.1%

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

      6. *-commutative 64.1%

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

      7. unpow2 64.1%

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

      8. unpow2 64.1%

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

      9. swap-sqr 76.5%

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

      10. unpow2 76.5%

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

   7. Simplified 76.5%

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

      1. unpow2 76.5%

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

   9. Applied egg-rr 76.5%

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

       1. unpow2 76.5%

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

   11. Applied egg-rr 76.5%

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

       1. frac-times 92.3%

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

       2. add-sqr-sqrt 60.7%

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

       3. associate-*r* 60.7%

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

       4. associate-/l* 60.3%

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

       5. associate-/r* 59.3%

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

       6. associate-/l* 58.9%

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

       7. associate-/r* 58.9%

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

       8. associate-/l* 60.6%

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

       9. associate-/r* 63.9%

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

   13. Applied egg-rr 63.9%

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

   if 8.50000000000000062e209 < a

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in angle around 0 37.7%

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

      1. *-commutative 37.7%

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

      2. unpow2 37.7%

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

      3. unpow2 37.7%

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

      4. swap-sqr 42.8%

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

      5. unpow2 42.8%

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

      6. *-commutative 42.8%

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

      7. unpow2 42.8%

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

      8. unpow2 42.8%

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

      9. swap-sqr 55.3%

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

      10. unpow2 55.3%

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

   7. Simplified 55.3%

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

      1. add-log-exp 54.4%

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

      2. add-log-exp 54.4%

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

      3. exp-to-pow 54.4%

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

   9. Applied egg-rr 54.4%

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

       1. pow-exp 54.4%

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

       2. add-log-exp 55.3%

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

       3. pow2 55.3%

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

       4. add-sqr-sqrt 55.3%

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

       5. sqrt-div 55.3%

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

       6. sqrt-pow1 21.8%

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

       7. metadata-eval 21.8%

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

       8. pow1 21.8%

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

       9. sqrt-prod 13.1%

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

       10. add-sqr-sqrt 34.3%

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

       11. sqrt-div 34.3%

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

       12. sqrt-pow1 34.4%

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

       13. metadata-eval 34.4%

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

       14. pow1 34.4%

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

       15. sqrt-prod 49.7%

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

       16. add-sqr-sqrt 91.2%

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

   11. Applied egg-rr 91.2%

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

       1. associate-*r* 91.2%

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

       2. associate-*r/ 87.4%

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

       3. associate-/l* 76.1%

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

       4. associate-*r* 76.1%

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

       5. associate-/r* 76.1%

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

   13. Applied egg-rr 76.1%

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

       1. associate-/l* 79.9%

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

       2. associate-*l* 79.9%

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

       3. associate-/l/ 79.9%

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

       4. *-commutative 79.9%

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

       5. associate-*r/ 91.2%

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

       6. associate-*l/ 91.4%

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

       7. *-commutative 91.4%

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

       8. *-lft-identity 91.4%

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

       9. associate-*l/ 91.4%

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

       10. *-commutative 91.4%

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

       11. associate-/l/ 91.3%

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

       12. *-rgt-identity 91.3%

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

       13. associate-*r/ 91.3%

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

       14. associate-*l/ 91.3%

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

       15. *-lft-identity 91.3%

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

       16. associate-*l/ 91.3%

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

       17. associate-*l/ 91.2%

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

       18. *-lft-identity 91.2%

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

       19. associate-*l/ 91.6%

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

       20. *-commutative 91.6%

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

   15. Simplified 99.4%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 8.5 \cdot 10^{+209}:\\ \;\;\;\;-4 \cdot \left(\sqrt{a \cdot \frac{\frac{b}{x-scale}}{y-scale}} \cdot \left(\left(a \cdot \frac{\frac{b}{x-scale}}{y-scale}\right) \cdot \sqrt{a \cdot \frac{\frac{b}{x-scale}}{y-scale}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot \frac{\frac{a}{y-scale}}{x-scale}\right) \cdot \left(-4 \cdot \left(b \cdot \frac{\frac{a}{y-scale}}{x-scale}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.3% accurate, 62.6× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 1.8 \cdot 10^{-231}:\\ \;\;\;\;-4 \cdot \left(\frac{a\_m}{x-scale\_m} \cdot \left(\left(a\_m \cdot \frac{\frac{b\_m}{x-scale\_m}}{y-scale\_m}\right) \cdot \frac{b\_m}{y-scale\_m}\right)\right)\\ \mathbf{elif}\;b\_m \leq 6.2 \cdot 10^{+259}:\\ \;\;\;\;-4 \cdot \left(\frac{a\_m \cdot b\_m}{x-scale\_m \cdot y-scale\_m} \cdot \left(b\_m \cdot \frac{a\_m}{x-scale\_m \cdot y-scale\_m}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(a\_m \cdot \frac{\frac{b\_m}{x-scale\_m}}{y-scale\_m \cdot \left(\frac{x-scale\_m}{a\_m} \cdot \frac{y-scale\_m}{b\_m}\right)}\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a_m b_m angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= b_m 1.8e-231)
   (*
    -4.0
    (*
     (/ a_m x-scale_m)
     (* (* a_m (/ (/ b_m x-scale_m) y-scale_m)) (/ b_m y-scale_m))))
   (if (<= b_m 6.2e+259)
     (*
      -4.0
      (*
       (/ (* a_m b_m) (* x-scale_m y-scale_m))
       (* b_m (/ a_m (* x-scale_m y-scale_m)))))
     (*
      -4.0
      (*
       a_m
       (/
        (/ b_m x-scale_m)
        (* y-scale_m (* (/ x-scale_m a_m) (/ y-scale_m b_m)))))))))

C

a_m = fabs(a);
b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (b_m <= 1.8e-231) {
		tmp = -4.0 * ((a_m / x_45_scale_m) * ((a_m * ((b_m / x_45_scale_m) / y_45_scale_m)) * (b_m / y_45_scale_m)));
	} else if (b_m <= 6.2e+259) {
		tmp = -4.0 * (((a_m * b_m) / (x_45_scale_m * y_45_scale_m)) * (b_m * (a_m / (x_45_scale_m * y_45_scale_m))));
	} else {
		tmp = -4.0 * (a_m * ((b_m / x_45_scale_m) / (y_45_scale_m * ((x_45_scale_m / a_m) * (y_45_scale_m / b_m)))));
	}
	return tmp;
}

Fortran

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45scale)
y-scale_m = abs(y_45scale)
real(8) function code(a_m, b_m, angle, x_45scale_m, y_45scale_m)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale_m
    real(8) :: tmp
    if (b_m <= 1.8d-231) then
        tmp = (-4.0d0) * ((a_m / x_45scale_m) * ((a_m * ((b_m / x_45scale_m) / y_45scale_m)) * (b_m / y_45scale_m)))
    else if (b_m <= 6.2d+259) then
        tmp = (-4.0d0) * (((a_m * b_m) / (x_45scale_m * y_45scale_m)) * (b_m * (a_m / (x_45scale_m * y_45scale_m))))
    else
        tmp = (-4.0d0) * (a_m * ((b_m / x_45scale_m) / (y_45scale_m * ((x_45scale_m / a_m) * (y_45scale_m / b_m)))))
    end if
    code = tmp
end function

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (b_m <= 1.8e-231) {
		tmp = -4.0 * ((a_m / x_45_scale_m) * ((a_m * ((b_m / x_45_scale_m) / y_45_scale_m)) * (b_m / y_45_scale_m)));
	} else if (b_m <= 6.2e+259) {
		tmp = -4.0 * (((a_m * b_m) / (x_45_scale_m * y_45_scale_m)) * (b_m * (a_m / (x_45_scale_m * y_45_scale_m))));
	} else {
		tmp = -4.0 * (a_m * ((b_m / x_45_scale_m) / (y_45_scale_m * ((x_45_scale_m / a_m) * (y_45_scale_m / b_m)))));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if b_m <= 1.8e-231:
		tmp = -4.0 * ((a_m / x_45_scale_m) * ((a_m * ((b_m / x_45_scale_m) / y_45_scale_m)) * (b_m / y_45_scale_m)))
	elif b_m <= 6.2e+259:
		tmp = -4.0 * (((a_m * b_m) / (x_45_scale_m * y_45_scale_m)) * (b_m * (a_m / (x_45_scale_m * y_45_scale_m))))
	else:
		tmp = -4.0 * (a_m * ((b_m / x_45_scale_m) / (y_45_scale_m * ((x_45_scale_m / a_m) * (y_45_scale_m / b_m)))))
	return tmp

Julia

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (b_m <= 1.8e-231)
		tmp = Float64(-4.0 * Float64(Float64(a_m / x_45_scale_m) * Float64(Float64(a_m * Float64(Float64(b_m / x_45_scale_m) / y_45_scale_m)) * Float64(b_m / y_45_scale_m))));
	elseif (b_m <= 6.2e+259)
		tmp = Float64(-4.0 * Float64(Float64(Float64(a_m * b_m) / Float64(x_45_scale_m * y_45_scale_m)) * Float64(b_m * Float64(a_m / Float64(x_45_scale_m * y_45_scale_m)))));
	else
		tmp = Float64(-4.0 * Float64(a_m * Float64(Float64(b_m / x_45_scale_m) / Float64(y_45_scale_m * Float64(Float64(x_45_scale_m / a_m) * Float64(y_45_scale_m / b_m))))));
	end
	return tmp
end

MATLAB

a_m = abs(a);
b_m = abs(b);
x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0;
	if (b_m <= 1.8e-231)
		tmp = -4.0 * ((a_m / x_45_scale_m) * ((a_m * ((b_m / x_45_scale_m) / y_45_scale_m)) * (b_m / y_45_scale_m)));
	elseif (b_m <= 6.2e+259)
		tmp = -4.0 * (((a_m * b_m) / (x_45_scale_m * y_45_scale_m)) * (b_m * (a_m / (x_45_scale_m * y_45_scale_m))));
	else
		tmp = -4.0 * (a_m * ((b_m / x_45_scale_m) / (y_45_scale_m * ((x_45_scale_m / a_m) * (y_45_scale_m / b_m)))));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[b$95$m, 1.8e-231], N[(-4.0 * N[(N[(a$95$m / x$45$scale$95$m), $MachinePrecision] * N[(N[(a$95$m * N[(N[(b$95$m / x$45$scale$95$m), $MachinePrecision] / y$45$scale$95$m), $MachinePrecision]), $MachinePrecision] * N[(b$95$m / y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 6.2e+259], N[(-4.0 * N[(N[(N[(a$95$m * b$95$m), $MachinePrecision] / N[(x$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision] * N[(b$95$m * N[(a$95$m / N[(x$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(a$95$m * N[(N[(b$95$m / x$45$scale$95$m), $MachinePrecision] / N[(y$45$scale$95$m * N[(N[(x$45$scale$95$m / a$95$m), $MachinePrecision] * N[(y$45$scale$95$m / b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 1.79999999999999987e-231

   1. Initial program 23.8%

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

   3. Simplified 24.5%

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

   5. Taylor expanded in angle around 0 47.9%

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

      1. *-commutative 47.9%

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

      2. unpow2 47.9%

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

      3. unpow2 47.9%

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

      4. swap-sqr 64.0%

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

      5. unpow2 64.0%

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

      6. *-commutative 64.0%

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

      7. unpow2 64.0%

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

      8. unpow2 64.0%

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

      9. swap-sqr 72.1%

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

      10. unpow2 72.1%

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

   7. Simplified 72.1%

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

      1. unpow2 72.1%

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

   9. Applied egg-rr 72.1%

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

       1. unpow2 72.1%

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

   11. Applied egg-rr 72.1%

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

       1. frac-times 91.5%

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

       2. times-frac 88.3%

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

       3. associate-*l* 87.6%

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

       4. associate-/l* 89.7%

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

       5. associate-/r* 93.1%

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

   13. Applied egg-rr 93.1%

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

   if 1.79999999999999987e-231 < b < 6.2000000000000007e259

   1. Initial program 23.4%

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

   3. Simplified 18.3%

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

   5. Taylor expanded in angle around 0 43.8%

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

      1. *-commutative 43.8%

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

      2. unpow2 43.8%

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

      3. unpow2 43.8%

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

      4. swap-sqr 59.2%

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

      5. unpow2 59.2%

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

      6. *-commutative 59.2%

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

      7. unpow2 59.2%

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

      8. unpow2 59.2%

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

      9. swap-sqr 79.2%

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

      10. unpow2 79.2%

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

   7. Simplified 79.2%

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

      1. add-log-exp 55.4%

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

      2. add-log-exp 52.4%

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

      3. exp-to-pow 52.4%

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

   9. Applied egg-rr 52.4%

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

       1. pow-exp 55.4%

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

       2. add-log-exp 79.2%

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

       3. pow2 79.2%

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

       4. add-sqr-sqrt 79.1%

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

       5. sqrt-div 79.2%

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

       6. sqrt-pow1 53.2%

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

       7. metadata-eval 53.2%

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

       8. pow1 53.2%

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

       9. sqrt-prod 28.6%

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

       10. add-sqr-sqrt 57.2%

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

       11. sqrt-div 57.1%

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

       12. sqrt-pow1 54.3%

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

       13. metadata-eval 54.3%

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

       14. pow1 54.3%

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

       15. sqrt-prod 46.3%

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

       16. add-sqr-sqrt 95.4%

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

   11. Applied egg-rr 95.4%

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

       1. *-commutative 95.4%

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

       2. associate-/l* 95.4%

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

   13. Applied egg-rr 95.4%

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

   if 6.2000000000000007e259 < b

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in angle around 0 45.5%

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

      1. *-commutative 45.5%

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

      2. unpow2 45.5%

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

      3. unpow2 45.5%

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

      4. swap-sqr 64.0%

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

      5. unpow2 64.0%

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

      6. *-commutative 64.0%

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

      7. unpow2 64.0%

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

      8. unpow2 64.0%

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

      9. swap-sqr 64.0%

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

      10. unpow2 64.0%

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

   7. Simplified 64.0%

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

      1. div-inv 64.0%

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

      2. pow-flip 64.0%

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

      3. metadata-eval 64.0%

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

   9. Applied egg-rr 64.0%

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

       1. sqr-pow 64.0%

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

       2. metadata-eval 64.0%

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

       3. inv-pow 64.0%

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

       4. metadata-eval 64.0%

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

       5. inv-pow 64.0%

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

   11. Applied egg-rr 64.0%

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

       1. pow2 64.0%

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

       2. swap-sqr 73.7%

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

       3. div-inv 73.7%

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

       4. div-inv 73.7%

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

       5. clear-num 73.7%

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

       6. frac-times 73.7%

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

       7. *-commutative 73.7%

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

       8. *-un-lft-identity 73.7%

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

       9. times-frac 65.1%

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

   13. Applied egg-rr 65.1%

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

       1. associate-/l* 73.7%

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

       2. *-commutative 73.7%

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

       3. associate-/l/ 73.7%

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

       4. *-commutative 73.7%

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

       5. associate-/l/ 91.0%

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

       6. associate-/l/ 91.1%

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

       7. *-commutative 91.1%

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

   15. Simplified 91.1%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.8 \cdot 10^{-231}:\\ \;\;\;\;-4 \cdot \left(\frac{a}{x-scale} \cdot \left(\left(a \cdot \frac{\frac{b}{x-scale}}{y-scale}\right) \cdot \frac{b}{y-scale}\right)\right)\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{+259}:\\ \;\;\;\;-4 \cdot \left(\frac{a \cdot b}{x-scale \cdot y-scale} \cdot \left(b \cdot \frac{a}{x-scale \cdot y-scale}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(a \cdot \frac{\frac{b}{x-scale}}{y-scale \cdot \left(\frac{x-scale}{a} \cdot \frac{y-scale}{b}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.0% accurate, 76.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := \frac{a\_m \cdot b\_m}{x-scale\_m \cdot y-scale\_m}\\ \mathbf{if}\;b\_m \leq 2 \cdot 10^{+259}:\\ \;\;\;\;-4 \cdot \left(t\_0 \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(a\_m \cdot \frac{\frac{b\_m}{x-scale\_m}}{y-scale\_m \cdot \left(\frac{x-scale\_m}{a\_m} \cdot \frac{y-scale\_m}{b\_m}\right)}\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a_m b_m angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (/ (* a_m b_m) (* x-scale_m y-scale_m))))
   (if (<= b_m 2e+259)
     (* -4.0 (* t_0 t_0))
     (*
      -4.0
      (*
       a_m
       (/
        (/ b_m x-scale_m)
        (* y-scale_m (* (/ x-scale_m a_m) (/ y-scale_m b_m)))))))))

C

a_m = fabs(a);
b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = (a_m * b_m) / (x_45_scale_m * y_45_scale_m);
	double tmp;
	if (b_m <= 2e+259) {
		tmp = -4.0 * (t_0 * t_0);
	} else {
		tmp = -4.0 * (a_m * ((b_m / x_45_scale_m) / (y_45_scale_m * ((x_45_scale_m / a_m) * (y_45_scale_m / b_m)))));
	}
	return tmp;
}

Fortran

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45scale)
y-scale_m = abs(y_45scale)
real(8) function code(a_m, b_m, angle, x_45scale_m, y_45scale_m)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (a_m * b_m) / (x_45scale_m * y_45scale_m)
    if (b_m <= 2d+259) then
        tmp = (-4.0d0) * (t_0 * t_0)
    else
        tmp = (-4.0d0) * (a_m * ((b_m / x_45scale_m) / (y_45scale_m * ((x_45scale_m / a_m) * (y_45scale_m / b_m)))))
    end if
    code = tmp
end function

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = (a_m * b_m) / (x_45_scale_m * y_45_scale_m);
	double tmp;
	if (b_m <= 2e+259) {
		tmp = -4.0 * (t_0 * t_0);
	} else {
		tmp = -4.0 * (a_m * ((b_m / x_45_scale_m) / (y_45_scale_m * ((x_45_scale_m / a_m) * (y_45_scale_m / b_m)))));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m):
	t_0 = (a_m * b_m) / (x_45_scale_m * y_45_scale_m)
	tmp = 0
	if b_m <= 2e+259:
		tmp = -4.0 * (t_0 * t_0)
	else:
		tmp = -4.0 * (a_m * ((b_m / x_45_scale_m) / (y_45_scale_m * ((x_45_scale_m / a_m) * (y_45_scale_m / b_m)))))
	return tmp

Julia

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(Float64(a_m * b_m) / Float64(x_45_scale_m * y_45_scale_m))
	tmp = 0.0
	if (b_m <= 2e+259)
		tmp = Float64(-4.0 * Float64(t_0 * t_0));
	else
		tmp = Float64(-4.0 * Float64(a_m * Float64(Float64(b_m / x_45_scale_m) / Float64(y_45_scale_m * Float64(Float64(x_45_scale_m / a_m) * Float64(y_45_scale_m / b_m))))));
	end
	return tmp
end

MATLAB

a_m = abs(a);
b_m = abs(b);
x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = (a_m * b_m) / (x_45_scale_m * y_45_scale_m);
	tmp = 0.0;
	if (b_m <= 2e+259)
		tmp = -4.0 * (t_0 * t_0);
	else
		tmp = -4.0 * (a_m * ((b_m / x_45_scale_m) / (y_45_scale_m * ((x_45_scale_m / a_m) * (y_45_scale_m / b_m)))));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(N[(a$95$m * b$95$m), $MachinePrecision] / N[(x$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b$95$m, 2e+259], N[(-4.0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(a$95$m * N[(N[(b$95$m / x$45$scale$95$m), $MachinePrecision] / N[(y$45$scale$95$m * N[(N[(x$45$scale$95$m / a$95$m), $MachinePrecision] * N[(y$45$scale$95$m / b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 2e259

   1. Initial program 23.6%

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

   3. Simplified 22.0%

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

   5. Taylor expanded in angle around 0 46.3%

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

      1. *-commutative 46.3%

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

      2. unpow2 46.3%

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

      3. unpow2 46.3%

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

      4. swap-sqr 62.0%

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

      5. unpow2 62.0%

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

      6. *-commutative 62.0%

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

      7. unpow2 62.0%

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

      8. unpow2 62.0%

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

      9. swap-sqr 75.0%

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

      10. unpow2 75.0%

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

   7. Simplified 75.0%

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

      1. add-log-exp 54.6%

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

      2. add-log-exp 49.3%

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

      3. exp-to-pow 49.3%

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

   9. Applied egg-rr 49.3%

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

       1. pow-exp 54.6%

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

       2. add-log-exp 75.0%

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

       3. pow2 75.0%

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

       4. add-sqr-sqrt 74.9%

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

       5. sqrt-div 75.0%

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

       6. sqrt-pow1 47.7%

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

       7. metadata-eval 47.7%

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

       8. pow1 47.7%

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

       9. sqrt-prod 26.0%

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

       10. add-sqr-sqrt 55.5%

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

       11. sqrt-div 55.5%

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

       12. sqrt-pow1 55.5%

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

       13. metadata-eval 55.5%

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

       14. pow1 55.5%

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

       15. sqrt-prod 46.0%

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

       16. add-sqr-sqrt 93.0%

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

   11. Applied egg-rr 93.0%

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

   if 2e259 < b

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in angle around 0 45.5%

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

      1. *-commutative 45.5%

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

      2. unpow2 45.5%

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

      3. unpow2 45.5%

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

      4. swap-sqr 64.0%

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

      5. unpow2 64.0%

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

      6. *-commutative 64.0%

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

      7. unpow2 64.0%

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

      8. unpow2 64.0%

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

      9. swap-sqr 64.0%

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

      10. unpow2 64.0%

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

   7. Simplified 64.0%

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

      1. div-inv 64.0%

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

      2. pow-flip 64.0%

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

      3. metadata-eval 64.0%

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

   9. Applied egg-rr 64.0%

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

       1. sqr-pow 64.0%

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

       2. metadata-eval 64.0%

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

       3. inv-pow 64.0%

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

       4. metadata-eval 64.0%

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

       5. inv-pow 64.0%

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

   11. Applied egg-rr 64.0%

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

       1. pow2 64.0%

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

       2. swap-sqr 73.7%

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

       3. div-inv 73.7%

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

       4. div-inv 73.7%

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

       5. clear-num 73.7%

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

       6. frac-times 73.7%

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

       7. *-commutative 73.7%

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

       8. *-un-lft-identity 73.7%

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

       9. times-frac 65.1%

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

   13. Applied egg-rr 65.1%

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

       1. associate-/l* 73.7%

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

       2. *-commutative 73.7%

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

       3. associate-/l/ 73.7%

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

       4. *-commutative 73.7%

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

       5. associate-/l/ 91.0%

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

       6. associate-/l/ 91.1%

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

       7. *-commutative 91.1%

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

   15. Simplified 91.1%

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

Alternative 4: 93.8% accurate, 99.6× speedup

Math

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

FPCore

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

C

a_m = fabs(a);
b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = b_m * ((a_m / y_45_scale_m) / x_45_scale_m);
	return t_0 * (-4.0 * t_0);
}

Fortran

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45scale)
y-scale_m = abs(y_45scale)
real(8) function code(a_m, b_m, angle, x_45scale_m, y_45scale_m)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale_m
    real(8) :: t_0
    t_0 = b_m * ((a_m / y_45scale_m) / x_45scale_m)
    code = t_0 * ((-4.0d0) * t_0)
end function

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = b_m * ((a_m / y_45_scale_m) / x_45_scale_m);
	return t_0 * (-4.0 * t_0);
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m):
	t_0 = b_m * ((a_m / y_45_scale_m) / x_45_scale_m)
	return t_0 * (-4.0 * t_0)

Julia

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(b_m * Float64(Float64(a_m / y_45_scale_m) / x_45_scale_m))
	return Float64(t_0 * Float64(-4.0 * t_0))
end

MATLAB

a_m = abs(a);
b_m = abs(b);
x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp = code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = b_m * ((a_m / y_45_scale_m) / x_45_scale_m);
	tmp = t_0 * (-4.0 * t_0);
end

Wolfram

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

Derivation

1. Initial program 22.6%

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

3. Simplified 21.0%

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

5. Taylor expanded in angle around 0 46.2%

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

   1. *-commutative 46.2%

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

   2. unpow2 46.2%

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

   3. unpow2 46.2%

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

   4. swap-sqr 62.1%

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

   5. unpow2 62.1%

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

   6. *-commutative 62.1%

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

   7. unpow2 62.1%

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

   8. unpow2 62.1%

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

   9. swap-sqr 74.5%

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

   10. unpow2 74.5%

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

7. Simplified 74.5%

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

   1. add-log-exp 54.6%

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

   2. add-log-exp 49.1%

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

   3. exp-to-pow 49.1%

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

9. Applied egg-rr 49.1%

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

    1. pow-exp 54.6%

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

    2. add-log-exp 74.5%

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

    3. pow2 74.5%

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

    4. add-sqr-sqrt 74.5%

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

    5. sqrt-div 74.5%

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

    6. sqrt-pow1 46.8%

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

    7. metadata-eval 46.8%

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

    8. pow1 46.8%

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

    9. sqrt-prod 25.7%

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

    10. add-sqr-sqrt 54.3%

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

    11. sqrt-div 54.3%

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

    12. sqrt-pow1 55.1%

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

    13. metadata-eval 55.1%

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

    14. pow1 55.1%

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

    15. sqrt-prod 46.4%

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

    16. add-sqr-sqrt 92.2%

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

11. Applied egg-rr 92.2%

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

    1. associate-*r* 92.2%

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

    2. associate-*r/ 89.9%

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

    3. associate-/l* 87.5%

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

    4. associate-*r* 87.5%

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

    5. associate-/r* 85.9%

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

13. Applied egg-rr 85.9%

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

    1. associate-/l* 88.1%

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

    2. associate-*l* 88.1%

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

    3. associate-/l/ 89.9%

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

    4. *-commutative 89.9%

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

    5. associate-*r/ 92.2%

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

    6. associate-*l/ 90.9%

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

    7. *-commutative 90.9%

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

    8. *-lft-identity 90.9%

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

    9. associate-*l/ 90.7%

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

    10. *-commutative 90.7%

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

    11. associate-/l/ 90.7%

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

    12. *-rgt-identity 90.7%

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

    13. associate-*r/ 90.6%

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

    14. associate-*l/ 90.7%

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

    15. *-lft-identity 90.7%

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

    16. associate-*l/ 89.0%

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

    17. associate-*l/ 89.0%

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

    18. *-lft-identity 89.0%

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

    19. associate-*l/ 90.9%

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

    20. *-commutative 90.9%

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

15. Simplified 95.0%

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

16. Final simplification 95.0%

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

Alternative 5: 87.3% accurate, 99.6× speedup

Math

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

FPCore

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

C

a_m = fabs(a);
b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	return -4.0 * ((a_m / x_45_scale_m) * ((a_m * ((b_m / x_45_scale_m) / y_45_scale_m)) * (b_m / y_45_scale_m)));
}

Fortran

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45scale)
y-scale_m = abs(y_45scale)
real(8) function code(a_m, b_m, angle, x_45scale_m, y_45scale_m)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale_m
    code = (-4.0d0) * ((a_m / x_45scale_m) * ((a_m * ((b_m / x_45scale_m) / y_45scale_m)) * (b_m / y_45scale_m)))
end function

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	return -4.0 * ((a_m / x_45_scale_m) * ((a_m * ((b_m / x_45_scale_m) / y_45_scale_m)) * (b_m / y_45_scale_m)));
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m):
	return -4.0 * ((a_m / x_45_scale_m) * ((a_m * ((b_m / x_45_scale_m) / y_45_scale_m)) * (b_m / y_45_scale_m)))

Julia

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	return Float64(-4.0 * Float64(Float64(a_m / x_45_scale_m) * Float64(Float64(a_m * Float64(Float64(b_m / x_45_scale_m) / y_45_scale_m)) * Float64(b_m / y_45_scale_m))))
end

MATLAB

a_m = abs(a);
b_m = abs(b);
x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp = code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = -4.0 * ((a_m / x_45_scale_m) * ((a_m * ((b_m / x_45_scale_m) / y_45_scale_m)) * (b_m / y_45_scale_m)));
end

Wolfram

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

Derivation

1. Initial program 22.6%

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

3. Simplified 21.0%

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

5. Taylor expanded in angle around 0 46.2%

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

   1. *-commutative 46.2%

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

   2. unpow2 46.2%

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

   3. unpow2 46.2%

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

   4. swap-sqr 62.1%

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

   5. unpow2 62.1%

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

   6. *-commutative 62.1%

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

   7. unpow2 62.1%

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

   8. unpow2 62.1%

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

   9. swap-sqr 74.5%

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

   10. unpow2 74.5%

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

7. Simplified 74.5%

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

   1. unpow2 74.5%

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

9. Applied egg-rr 74.5%

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

    1. unpow2 74.5%

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

11. Applied egg-rr 74.5%

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

    1. frac-times 92.2%

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

    2. times-frac 85.3%

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

    3. associate-*l* 83.1%

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

    4. associate-/l* 84.6%

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

    5. associate-/r* 87.8%

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

13. Applied egg-rr 87.8%

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

14. Final simplification 87.8%

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

Alternative 6: 85.5% accurate, 99.6× speedup

Math

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

FPCore

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

C

a_m = fabs(a);
b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	return -4.0 * (a_m * ((b_m / x_45_scale_m) / (y_45_scale_m * ((x_45_scale_m / a_m) * (y_45_scale_m / b_m)))));
}

Fortran

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45scale)
y-scale_m = abs(y_45scale)
real(8) function code(a_m, b_m, angle, x_45scale_m, y_45scale_m)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale_m
    code = (-4.0d0) * (a_m * ((b_m / x_45scale_m) / (y_45scale_m * ((x_45scale_m / a_m) * (y_45scale_m / b_m)))))
end function

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	return -4.0 * (a_m * ((b_m / x_45_scale_m) / (y_45_scale_m * ((x_45_scale_m / a_m) * (y_45_scale_m / b_m)))));
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m):
	return -4.0 * (a_m * ((b_m / x_45_scale_m) / (y_45_scale_m * ((x_45_scale_m / a_m) * (y_45_scale_m / b_m)))))

Julia

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	return Float64(-4.0 * Float64(a_m * Float64(Float64(b_m / x_45_scale_m) / Float64(y_45_scale_m * Float64(Float64(x_45_scale_m / a_m) * Float64(y_45_scale_m / b_m))))))
end

MATLAB

a_m = abs(a);
b_m = abs(b);
x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp = code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = -4.0 * (a_m * ((b_m / x_45_scale_m) / (y_45_scale_m * ((x_45_scale_m / a_m) * (y_45_scale_m / b_m)))));
end

Wolfram

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

Derivation

1. Initial program 22.6%

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

3. Simplified 21.0%

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

5. Taylor expanded in angle around 0 46.2%

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

   1. *-commutative 46.2%

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

   2. unpow2 46.2%

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

   3. unpow2 46.2%

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

   4. swap-sqr 62.1%

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

   5. unpow2 62.1%

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

   6. *-commutative 62.1%

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

   7. unpow2 62.1%

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

   8. unpow2 62.1%

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

   9. swap-sqr 74.5%

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

   10. unpow2 74.5%

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

7. Simplified 74.5%

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

   1. div-inv 74.1%

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

   2. pow-flip 74.3%

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

   3. metadata-eval 74.3%

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

9. Applied egg-rr 74.3%

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

    1. sqr-pow 74.3%

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

    2. metadata-eval 74.3%

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

    3. inv-pow 74.3%

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

    4. metadata-eval 74.3%

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

    5. inv-pow 74.3%

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

11. Applied egg-rr 74.3%

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

    1. pow2 74.3%

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

    2. swap-sqr 91.8%

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

    3. div-inv 91.8%

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

    4. div-inv 92.2%

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

    5. clear-num 92.2%

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

    6. frac-times 89.9%

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

    7. *-commutative 89.9%

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

    8. *-un-lft-identity 89.9%

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

    9. times-frac 83.6%

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

13. Applied egg-rr 83.6%

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

    1. associate-/l* 84.4%

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

    2. *-commutative 84.4%

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

    3. associate-/l/ 85.4%

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

    4. *-commutative 85.4%

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

    5. associate-/l/ 87.7%

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

    6. associate-/l/ 86.2%

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

    7. *-commutative 86.2%

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

15. Simplified 86.2%

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

Alternative 7: 35.8% accurate, 1693.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ 0 \end{array} \]

FPCore

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

C

a_m = fabs(a);
b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	return 0.0;
}

Fortran

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45scale)
y-scale_m = abs(y_45scale)
real(8) function code(a_m, b_m, angle, x_45scale_m, y_45scale_m)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale_m
    code = 0.0d0
end function

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	return 0.0;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m):
	return 0.0

Julia

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	return 0.0
end

MATLAB

a_m = abs(a);
b_m = abs(b);
x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp = code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := 0.0

Derivation

1. Initial program 22.6%

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

3. Simplified 21.0%

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

5. Taylor expanded in b around 0 24.1%

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

   1. distribute-rgt-out 24.1%

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

   2. metadata-eval 24.1%

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

   3. mul0-rgt 34.5%

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

7. Simplified 34.5%

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

Reproduce

herbie shell --seed 2024177 
(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))))