Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 26.0% → 84.4%

Time: 26.1s

Alternatives: 10

Speedup: 40.5×

• 32.5% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. a = -4.634132096838933e+306
  2. b = -4.8151550327467606e-172
  3. angle = 2.6083154861046734e+68
  4. x-scale = -6.1189244482908766e-21
  5. y-scale = 8.788455832981605e-289

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: 26.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.4% accurate, 29.3× speedup

Math

\[\begin{array}{l} y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := \left(y-scale\_m \cdot x-scale\right) \cdot -0.25\\ \mathbf{if}\;y-scale\_m \leq 8 \cdot 10^{+42}:\\ \;\;\;\;\frac{a \cdot \left(a \cdot b\right)}{t\_0} \cdot \frac{b}{y-scale\_m \cdot x-scale}\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \frac{\frac{a \cdot b}{x-scale}}{t\_0}\right) \cdot \frac{b}{y-scale\_m}\\ \end{array} \end{array} \]

FPCore

y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale y-scale_m)
 :precision binary64
 (let* ((t_0 (* (* y-scale_m x-scale) -0.25)))
   (if (<= y-scale_m 8e+42)
     (* (/ (* a (* a b)) t_0) (/ b (* y-scale_m x-scale)))
     (* (* a (/ (/ (* a b) x-scale) t_0)) (/ b y-scale_m)))))

C

y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	double t_0 = (y_45_scale_m * x_45_scale) * -0.25;
	double tmp;
	if (y_45_scale_m <= 8e+42) {
		tmp = ((a * (a * b)) / t_0) * (b / (y_45_scale_m * x_45_scale));
	} else {
		tmp = (a * (((a * b) / x_45_scale) / t_0)) * (b / y_45_scale_m);
	}
	return tmp;
}

Fortran

y-scale_m = abs(y_45scale)
real(8) function code(a, b, angle, x_45scale, y_45scale_m)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y_45scale_m * x_45scale) * (-0.25d0)
    if (y_45scale_m <= 8d+42) then
        tmp = ((a * (a * b)) / t_0) * (b / (y_45scale_m * x_45scale))
    else
        tmp = (a * (((a * b) / x_45scale) / t_0)) * (b / y_45scale_m)
    end if
    code = tmp
end function

Java

y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	double t_0 = (y_45_scale_m * x_45_scale) * -0.25;
	double tmp;
	if (y_45_scale_m <= 8e+42) {
		tmp = ((a * (a * b)) / t_0) * (b / (y_45_scale_m * x_45_scale));
	} else {
		tmp = (a * (((a * b) / x_45_scale) / t_0)) * (b / y_45_scale_m);
	}
	return tmp;
}

Python

y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale, y_45_scale_m):
	t_0 = (y_45_scale_m * x_45_scale) * -0.25
	tmp = 0
	if y_45_scale_m <= 8e+42:
		tmp = ((a * (a * b)) / t_0) * (b / (y_45_scale_m * x_45_scale))
	else:
		tmp = (a * (((a * b) / x_45_scale) / t_0)) * (b / y_45_scale_m)
	return tmp

Julia

y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale, y_45_scale_m)
	t_0 = Float64(Float64(y_45_scale_m * x_45_scale) * -0.25)
	tmp = 0.0
	if (y_45_scale_m <= 8e+42)
		tmp = Float64(Float64(Float64(a * Float64(a * b)) / t_0) * Float64(b / Float64(y_45_scale_m * x_45_scale)));
	else
		tmp = Float64(Float64(a * Float64(Float64(Float64(a * b) / x_45_scale) / t_0)) * Float64(b / y_45_scale_m));
	end
	return tmp
end

MATLAB

y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale_m)
	t_0 = (y_45_scale_m * x_45_scale) * -0.25;
	tmp = 0.0;
	if (y_45_scale_m <= 8e+42)
		tmp = ((a * (a * b)) / t_0) * (b / (y_45_scale_m * x_45_scale));
	else
		tmp = (a * (((a * b) / x_45_scale) / t_0)) * (b / y_45_scale_m);
	end
	tmp_2 = tmp;
end

Wolfram

y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale_, y$45$scale$95$m_] := Block[{t$95$0 = N[(N[(y$45$scale$95$m * x$45$scale), $MachinePrecision] * -0.25), $MachinePrecision]}, If[LessEqual[y$45$scale$95$m, 8e+42], N[(N[(N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(b / N[(y$45$scale$95$m * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[(N[(a * b), $MachinePrecision] / x$45$scale), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] * N[(b / y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 8.00000000000000036e42

   1. Initial program 27.1%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 57.1

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

   5. Applied rewrites 57.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. times-frac N/A

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

      16. lower-*.f64 N/A

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

   7. Applied rewrites 72.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 84.0

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

   9. Applied rewrites 84.0%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-/.f64 N/A

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

       5. associate-*l/ N/A

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

       6. clear-num N/A

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

       7. frac-times N/A

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

       8. div-inv N/A

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

       9. *-rgt-identity N/A

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

       10. *-commutative N/A

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

       11. associate-/r/ N/A

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

       12. lift-/.f64 N/A

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

       13. clear-num N/A

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

       14. lift-/.f64 N/A

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

       15. associate-/l/ N/A

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

       16. *-commutative N/A

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

       17. lift-*.f64 N/A

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

       18. lower-*.f64 N/A

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

   11. Applied rewrites 89.3%

       \[\leadsto \color{blue}{\frac{a \cdot \left(a \cdot b\right)}{\left(x-scale \cdot y-scale\right) \cdot -0.25} \cdot \frac{b}{x-scale \cdot y-scale}} \]

   if 8.00000000000000036e42 < y-scale

   1. Initial program 37.5%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 61.8

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

   5. Applied rewrites 61.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. times-frac N/A

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

      16. lower-*.f64 N/A

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

   7. Applied rewrites 80.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 86.3

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

   9. Applied rewrites 86.3%

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

       1. lift-*.f64 N/A

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

       2. associate-/l* N/A

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

       3. lift-*.f64 N/A

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

       4. lift-/.f64 N/A

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

       5. associate-*l* N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. lift-/.f64 N/A

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

       9. clear-num N/A

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

       10. un-div-inv N/A

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

       11. lower-/.f64 N/A

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

       12. lower-/.f64 N/A

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

       13. div-inv N/A

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

       14. lower-*.f64 N/A

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

       15. metadata-eval 93.8

           \[\leadsto \left(\frac{\frac{a \cdot b}{x-scale}}{\left(x-scale \cdot y-scale\right) \cdot \color{blue}{-0.25}} \cdot a\right) \cdot \frac{b}{y-scale} \]

   11. Applied rewrites 93.8%

       \[\leadsto \color{blue}{\left(\frac{\frac{a \cdot b}{x-scale}}{\left(x-scale \cdot y-scale\right) \cdot -0.25} \cdot a\right)} \cdot \frac{b}{y-scale} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 8 \cdot 10^{+42}:\\ \;\;\;\;\frac{a \cdot \left(a \cdot b\right)}{\left(y-scale \cdot x-scale\right) \cdot -0.25} \cdot \frac{b}{y-scale \cdot x-scale}\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \frac{\frac{a \cdot b}{x-scale}}{\left(y-scale \cdot x-scale\right) \cdot -0.25}\right) \cdot \frac{b}{y-scale}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 77.5% accurate, 29.3× speedup

Math

\[\begin{array}{l} y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := a \cdot \left(a \cdot b\right)\\ \mathbf{if}\;b \leq 2.2 \cdot 10^{-206}:\\ \;\;\;\;b \cdot \frac{t\_0}{y-scale\_m \cdot \left(x-scale \cdot \left(\left(y-scale\_m \cdot x-scale\right) \cdot -0.25\right)\right)}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+147}:\\ \;\;\;\;\frac{a \cdot \left(-4 \cdot \left(b \cdot b\right)\right)}{y-scale\_m \cdot x-scale} \cdot \frac{a}{y-scale\_m \cdot x-scale}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{b \cdot t\_0}{x-scale \cdot \left(y-scale\_m \cdot \left(y-scale\_m \cdot x-scale\right)\right)}\\ \end{array} \end{array} \]

FPCore

y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale y-scale_m)
 :precision binary64
 (let* ((t_0 (* a (* a b))))
   (if (<= b 2.2e-206)
     (* b (/ t_0 (* y-scale_m (* x-scale (* (* y-scale_m x-scale) -0.25)))))
     (if (<= b 2e+147)
       (*
        (/ (* a (* -4.0 (* b b))) (* y-scale_m x-scale))
        (/ a (* y-scale_m x-scale)))
       (*
        -4.0
        (/ (* b t_0) (* x-scale (* y-scale_m (* y-scale_m x-scale)))))))))

C

y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	double t_0 = a * (a * b);
	double tmp;
	if (b <= 2.2e-206) {
		tmp = b * (t_0 / (y_45_scale_m * (x_45_scale * ((y_45_scale_m * x_45_scale) * -0.25))));
	} else if (b <= 2e+147) {
		tmp = ((a * (-4.0 * (b * b))) / (y_45_scale_m * x_45_scale)) * (a / (y_45_scale_m * x_45_scale));
	} else {
		tmp = -4.0 * ((b * t_0) / (x_45_scale * (y_45_scale_m * (y_45_scale_m * x_45_scale))));
	}
	return tmp;
}

Fortran

y-scale_m = abs(y_45scale)
real(8) function code(a, b, angle, x_45scale, y_45scale_m)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a * (a * b)
    if (b <= 2.2d-206) then
        tmp = b * (t_0 / (y_45scale_m * (x_45scale * ((y_45scale_m * x_45scale) * (-0.25d0)))))
    else if (b <= 2d+147) then
        tmp = ((a * ((-4.0d0) * (b * b))) / (y_45scale_m * x_45scale)) * (a / (y_45scale_m * x_45scale))
    else
        tmp = (-4.0d0) * ((b * t_0) / (x_45scale * (y_45scale_m * (y_45scale_m * x_45scale))))
    end if
    code = tmp
end function

Java

y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	double t_0 = a * (a * b);
	double tmp;
	if (b <= 2.2e-206) {
		tmp = b * (t_0 / (y_45_scale_m * (x_45_scale * ((y_45_scale_m * x_45_scale) * -0.25))));
	} else if (b <= 2e+147) {
		tmp = ((a * (-4.0 * (b * b))) / (y_45_scale_m * x_45_scale)) * (a / (y_45_scale_m * x_45_scale));
	} else {
		tmp = -4.0 * ((b * t_0) / (x_45_scale * (y_45_scale_m * (y_45_scale_m * x_45_scale))));
	}
	return tmp;
}

Python

y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale, y_45_scale_m):
	t_0 = a * (a * b)
	tmp = 0
	if b <= 2.2e-206:
		tmp = b * (t_0 / (y_45_scale_m * (x_45_scale * ((y_45_scale_m * x_45_scale) * -0.25))))
	elif b <= 2e+147:
		tmp = ((a * (-4.0 * (b * b))) / (y_45_scale_m * x_45_scale)) * (a / (y_45_scale_m * x_45_scale))
	else:
		tmp = -4.0 * ((b * t_0) / (x_45_scale * (y_45_scale_m * (y_45_scale_m * x_45_scale))))
	return tmp

Julia

y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale, y_45_scale_m)
	t_0 = Float64(a * Float64(a * b))
	tmp = 0.0
	if (b <= 2.2e-206)
		tmp = Float64(b * Float64(t_0 / Float64(y_45_scale_m * Float64(x_45_scale * Float64(Float64(y_45_scale_m * x_45_scale) * -0.25)))));
	elseif (b <= 2e+147)
		tmp = Float64(Float64(Float64(a * Float64(-4.0 * Float64(b * b))) / Float64(y_45_scale_m * x_45_scale)) * Float64(a / Float64(y_45_scale_m * x_45_scale)));
	else
		tmp = Float64(-4.0 * Float64(Float64(b * t_0) / Float64(x_45_scale * Float64(y_45_scale_m * Float64(y_45_scale_m * x_45_scale)))));
	end
	return tmp
end

MATLAB

y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale_m)
	t_0 = a * (a * b);
	tmp = 0.0;
	if (b <= 2.2e-206)
		tmp = b * (t_0 / (y_45_scale_m * (x_45_scale * ((y_45_scale_m * x_45_scale) * -0.25))));
	elseif (b <= 2e+147)
		tmp = ((a * (-4.0 * (b * b))) / (y_45_scale_m * x_45_scale)) * (a / (y_45_scale_m * x_45_scale));
	else
		tmp = -4.0 * ((b * t_0) / (x_45_scale * (y_45_scale_m * (y_45_scale_m * x_45_scale))));
	end
	tmp_2 = tmp;
end

Wolfram

y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale_, y$45$scale$95$m_] := Block[{t$95$0 = N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 2.2e-206], N[(b * N[(t$95$0 / N[(y$45$scale$95$m * N[(x$45$scale * N[(N[(y$45$scale$95$m * x$45$scale), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2e+147], N[(N[(N[(a * N[(-4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale$95$m * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(a / N[(y$45$scale$95$m * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(N[(b * t$95$0), $MachinePrecision] / N[(x$45$scale * N[(y$45$scale$95$m * N[(y$45$scale$95$m * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < 2.1999999999999999e-206

   1. Initial program 31.9%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 53.5

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

   5. Applied rewrites 53.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. times-frac N/A

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

      16. lower-*.f64 N/A

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

   7. Applied rewrites 72.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 84.2

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

   9. Applied rewrites 84.2%

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

   11. Applied rewrites 77.7%

       \[\leadsto \color{blue}{\frac{a \cdot \left(a \cdot b\right)}{y-scale \cdot \left(x-scale \cdot \left(\left(x-scale \cdot y-scale\right) \cdot -0.25\right)\right)} \cdot b} \]

   if 2.1999999999999999e-206 < b < 2e147

   1. Initial program 34.4%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 70.1

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

   5. Applied rewrites 70.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. times-frac N/A

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

      16. lower-*.f64 N/A

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

   7. Applied rewrites 80.7%

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

   9. Applied rewrites 89.5%

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

   if 2e147 < b

   1. Initial program 6.9%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 55.2

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

   5. Applied rewrites 55.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

   7. Applied rewrites 65.9%

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 65.9

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 79.5

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

   9. Applied rewrites 79.5%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.2 \cdot 10^{-206}:\\ \;\;\;\;b \cdot \frac{a \cdot \left(a \cdot b\right)}{y-scale \cdot \left(x-scale \cdot \left(\left(y-scale \cdot x-scale\right) \cdot -0.25\right)\right)}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+147}:\\ \;\;\;\;\frac{a \cdot \left(-4 \cdot \left(b \cdot b\right)\right)}{y-scale \cdot x-scale} \cdot \frac{a}{y-scale \cdot x-scale}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{b \cdot \left(a \cdot \left(a \cdot b\right)\right)}{x-scale \cdot \left(y-scale \cdot \left(y-scale \cdot x-scale\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.1% accurate, 29.3× speedup

Math

\[\begin{array}{l} y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := a \cdot \left(a \cdot b\right)\\ \mathbf{if}\;y-scale\_m \leq 3.7 \cdot 10^{-173}:\\ \;\;\;\;b \cdot \frac{t\_0}{y-scale\_m \cdot \left(x-scale \cdot \left(\left(y-scale\_m \cdot x-scale\right) \cdot -0.25\right)\right)}\\ \mathbf{elif}\;y-scale\_m \leq 3.9 \cdot 10^{+155}:\\ \;\;\;\;\frac{a}{x-scale} \cdot \left(-4 \cdot \left(a \cdot \left(b \cdot \frac{b}{x-scale \cdot \left(y-scale\_m \cdot y-scale\_m\right)}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{b \cdot t\_0}{y-scale\_m \cdot \left(x-scale \cdot \left(y-scale\_m \cdot x-scale\right)\right)}\\ \end{array} \end{array} \]

FPCore

y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale y-scale_m)
 :precision binary64
 (let* ((t_0 (* a (* a b))))
   (if (<= y-scale_m 3.7e-173)
     (* b (/ t_0 (* y-scale_m (* x-scale (* (* y-scale_m x-scale) -0.25)))))
     (if (<= y-scale_m 3.9e+155)
       (*
        (/ a x-scale)
        (* -4.0 (* a (* b (/ b (* x-scale (* y-scale_m y-scale_m)))))))
       (*
        -4.0
        (/ (* b t_0) (* y-scale_m (* x-scale (* y-scale_m x-scale)))))))))

C

y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	double t_0 = a * (a * b);
	double tmp;
	if (y_45_scale_m <= 3.7e-173) {
		tmp = b * (t_0 / (y_45_scale_m * (x_45_scale * ((y_45_scale_m * x_45_scale) * -0.25))));
	} else if (y_45_scale_m <= 3.9e+155) {
		tmp = (a / x_45_scale) * (-4.0 * (a * (b * (b / (x_45_scale * (y_45_scale_m * y_45_scale_m))))));
	} else {
		tmp = -4.0 * ((b * t_0) / (y_45_scale_m * (x_45_scale * (y_45_scale_m * x_45_scale))));
	}
	return tmp;
}

Fortran

y-scale_m = abs(y_45scale)
real(8) function code(a, b, angle, x_45scale, y_45scale_m)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a * (a * b)
    if (y_45scale_m <= 3.7d-173) then
        tmp = b * (t_0 / (y_45scale_m * (x_45scale * ((y_45scale_m * x_45scale) * (-0.25d0)))))
    else if (y_45scale_m <= 3.9d+155) then
        tmp = (a / x_45scale) * ((-4.0d0) * (a * (b * (b / (x_45scale * (y_45scale_m * y_45scale_m))))))
    else
        tmp = (-4.0d0) * ((b * t_0) / (y_45scale_m * (x_45scale * (y_45scale_m * x_45scale))))
    end if
    code = tmp
end function

Java

y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	double t_0 = a * (a * b);
	double tmp;
	if (y_45_scale_m <= 3.7e-173) {
		tmp = b * (t_0 / (y_45_scale_m * (x_45_scale * ((y_45_scale_m * x_45_scale) * -0.25))));
	} else if (y_45_scale_m <= 3.9e+155) {
		tmp = (a / x_45_scale) * (-4.0 * (a * (b * (b / (x_45_scale * (y_45_scale_m * y_45_scale_m))))));
	} else {
		tmp = -4.0 * ((b * t_0) / (y_45_scale_m * (x_45_scale * (y_45_scale_m * x_45_scale))));
	}
	return tmp;
}

Python

y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale, y_45_scale_m):
	t_0 = a * (a * b)
	tmp = 0
	if y_45_scale_m <= 3.7e-173:
		tmp = b * (t_0 / (y_45_scale_m * (x_45_scale * ((y_45_scale_m * x_45_scale) * -0.25))))
	elif y_45_scale_m <= 3.9e+155:
		tmp = (a / x_45_scale) * (-4.0 * (a * (b * (b / (x_45_scale * (y_45_scale_m * y_45_scale_m))))))
	else:
		tmp = -4.0 * ((b * t_0) / (y_45_scale_m * (x_45_scale * (y_45_scale_m * x_45_scale))))
	return tmp

Julia

y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale, y_45_scale_m)
	t_0 = Float64(a * Float64(a * b))
	tmp = 0.0
	if (y_45_scale_m <= 3.7e-173)
		tmp = Float64(b * Float64(t_0 / Float64(y_45_scale_m * Float64(x_45_scale * Float64(Float64(y_45_scale_m * x_45_scale) * -0.25)))));
	elseif (y_45_scale_m <= 3.9e+155)
		tmp = Float64(Float64(a / x_45_scale) * Float64(-4.0 * Float64(a * Float64(b * Float64(b / Float64(x_45_scale * Float64(y_45_scale_m * y_45_scale_m)))))));
	else
		tmp = Float64(-4.0 * Float64(Float64(b * t_0) / Float64(y_45_scale_m * Float64(x_45_scale * Float64(y_45_scale_m * x_45_scale)))));
	end
	return tmp
end

MATLAB

y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale_m)
	t_0 = a * (a * b);
	tmp = 0.0;
	if (y_45_scale_m <= 3.7e-173)
		tmp = b * (t_0 / (y_45_scale_m * (x_45_scale * ((y_45_scale_m * x_45_scale) * -0.25))));
	elseif (y_45_scale_m <= 3.9e+155)
		tmp = (a / x_45_scale) * (-4.0 * (a * (b * (b / (x_45_scale * (y_45_scale_m * y_45_scale_m))))));
	else
		tmp = -4.0 * ((b * t_0) / (y_45_scale_m * (x_45_scale * (y_45_scale_m * x_45_scale))));
	end
	tmp_2 = tmp;
end

Wolfram

y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale_, y$45$scale$95$m_] := Block[{t$95$0 = N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$45$scale$95$m, 3.7e-173], N[(b * N[(t$95$0 / N[(y$45$scale$95$m * N[(x$45$scale * N[(N[(y$45$scale$95$m * x$45$scale), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale$95$m, 3.9e+155], N[(N[(a / x$45$scale), $MachinePrecision] * N[(-4.0 * N[(a * N[(b * N[(b / N[(x$45$scale * N[(y$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(N[(b * t$95$0), $MachinePrecision] / N[(y$45$scale$95$m * N[(x$45$scale * N[(y$45$scale$95$m * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y-scale < 3.7e-173

   1. Initial program 30.6%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 53.4

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

   5. Applied rewrites 53.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. times-frac N/A

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

      16. lower-*.f64 N/A

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

   7. Applied rewrites 72.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 82.4

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

   9. Applied rewrites 82.4%

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

   11. Applied rewrites 80.0%

       \[\leadsto \color{blue}{\frac{a \cdot \left(a \cdot b\right)}{y-scale \cdot \left(x-scale \cdot \left(\left(x-scale \cdot y-scale\right) \cdot -0.25\right)\right)} \cdot b} \]

   if 3.7e-173 < y-scale < 3.8999999999999998e155

   1. Initial program 25.7%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 74.5

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

   5. Applied rewrites 74.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. times-frac N/A

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

      16. lower-*.f64 N/A

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

   7. Applied rewrites 78.4%

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

   9. Applied rewrites 89.2%

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

   if 3.8999999999999998e155 < y-scale

   1. Initial program 34.7%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 46.0

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

   5. Applied rewrites 46.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

   7. Applied rewrites 74.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. lower-*.f64 73.1

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

   9. Applied rewrites 73.1%

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

       1. associate-*r* N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*r* N/A

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

       4. lift-*.f64 N/A

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

       5. lower-*.f64 78.5

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

   11. Applied rewrites 78.5%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 3.7 \cdot 10^{-173}:\\ \;\;\;\;b \cdot \frac{a \cdot \left(a \cdot b\right)}{y-scale \cdot \left(x-scale \cdot \left(\left(y-scale \cdot x-scale\right) \cdot -0.25\right)\right)}\\ \mathbf{elif}\;y-scale \leq 3.9 \cdot 10^{+155}:\\ \;\;\;\;\frac{a}{x-scale} \cdot \left(-4 \cdot \left(a \cdot \left(b \cdot \frac{b}{x-scale \cdot \left(y-scale \cdot y-scale\right)}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{b \cdot \left(a \cdot \left(a \cdot b\right)\right)}{y-scale \cdot \left(x-scale \cdot \left(y-scale \cdot x-scale\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.7% accurate, 32.3× speedup

Math

\[\begin{array}{l} y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := x-scale \cdot \left(y-scale\_m \cdot \left(y-scale\_m \cdot x-scale\right)\right)\\ \mathbf{if}\;a \leq 1.5 \cdot 10^{-50}:\\ \;\;\;\;-4 \cdot \frac{b \cdot \left(a \cdot \left(a \cdot b\right)\right)}{y-scale\_m \cdot \left(x-scale \cdot \left(y-scale\_m \cdot x-scale\right)\right)}\\ \mathbf{elif}\;a \leq 1.9 \cdot 10^{+148}:\\ \;\;\;\;\left(-4 \cdot \left(b \cdot \left(a \cdot a\right)\right)\right) \cdot \frac{b}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{a \cdot \left(b \cdot \left(a \cdot b\right)\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale y-scale_m)
 :precision binary64
 (let* ((t_0 (* x-scale (* y-scale_m (* y-scale_m x-scale)))))
   (if (<= a 1.5e-50)
     (*
      -4.0
      (/ (* b (* a (* a b))) (* y-scale_m (* x-scale (* y-scale_m x-scale)))))
     (if (<= a 1.9e+148)
       (* (* -4.0 (* b (* a a))) (/ b t_0))
       (* -4.0 (/ (* a (* b (* a b))) t_0))))))

C

y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	double t_0 = x_45_scale * (y_45_scale_m * (y_45_scale_m * x_45_scale));
	double tmp;
	if (a <= 1.5e-50) {
		tmp = -4.0 * ((b * (a * (a * b))) / (y_45_scale_m * (x_45_scale * (y_45_scale_m * x_45_scale))));
	} else if (a <= 1.9e+148) {
		tmp = (-4.0 * (b * (a * a))) * (b / t_0);
	} else {
		tmp = -4.0 * ((a * (b * (a * b))) / t_0);
	}
	return tmp;
}

Fortran

y-scale_m = abs(y_45scale)
real(8) function code(a, b, angle, x_45scale, y_45scale_m)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x_45scale * (y_45scale_m * (y_45scale_m * x_45scale))
    if (a <= 1.5d-50) then
        tmp = (-4.0d0) * ((b * (a * (a * b))) / (y_45scale_m * (x_45scale * (y_45scale_m * x_45scale))))
    else if (a <= 1.9d+148) then
        tmp = ((-4.0d0) * (b * (a * a))) * (b / t_0)
    else
        tmp = (-4.0d0) * ((a * (b * (a * b))) / t_0)
    end if
    code = tmp
end function

Java

y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	double t_0 = x_45_scale * (y_45_scale_m * (y_45_scale_m * x_45_scale));
	double tmp;
	if (a <= 1.5e-50) {
		tmp = -4.0 * ((b * (a * (a * b))) / (y_45_scale_m * (x_45_scale * (y_45_scale_m * x_45_scale))));
	} else if (a <= 1.9e+148) {
		tmp = (-4.0 * (b * (a * a))) * (b / t_0);
	} else {
		tmp = -4.0 * ((a * (b * (a * b))) / t_0);
	}
	return tmp;
}

Python

y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale, y_45_scale_m):
	t_0 = x_45_scale * (y_45_scale_m * (y_45_scale_m * x_45_scale))
	tmp = 0
	if a <= 1.5e-50:
		tmp = -4.0 * ((b * (a * (a * b))) / (y_45_scale_m * (x_45_scale * (y_45_scale_m * x_45_scale))))
	elif a <= 1.9e+148:
		tmp = (-4.0 * (b * (a * a))) * (b / t_0)
	else:
		tmp = -4.0 * ((a * (b * (a * b))) / t_0)
	return tmp

Julia

y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale, y_45_scale_m)
	t_0 = Float64(x_45_scale * Float64(y_45_scale_m * Float64(y_45_scale_m * x_45_scale)))
	tmp = 0.0
	if (a <= 1.5e-50)
		tmp = Float64(-4.0 * Float64(Float64(b * Float64(a * Float64(a * b))) / Float64(y_45_scale_m * Float64(x_45_scale * Float64(y_45_scale_m * x_45_scale)))));
	elseif (a <= 1.9e+148)
		tmp = Float64(Float64(-4.0 * Float64(b * Float64(a * a))) * Float64(b / t_0));
	else
		tmp = Float64(-4.0 * Float64(Float64(a * Float64(b * Float64(a * b))) / t_0));
	end
	return tmp
end

MATLAB

y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale_m)
	t_0 = x_45_scale * (y_45_scale_m * (y_45_scale_m * x_45_scale));
	tmp = 0.0;
	if (a <= 1.5e-50)
		tmp = -4.0 * ((b * (a * (a * b))) / (y_45_scale_m * (x_45_scale * (y_45_scale_m * x_45_scale))));
	elseif (a <= 1.9e+148)
		tmp = (-4.0 * (b * (a * a))) * (b / t_0);
	else
		tmp = -4.0 * ((a * (b * (a * b))) / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale_, y$45$scale$95$m_] := Block[{t$95$0 = N[(x$45$scale * N[(y$45$scale$95$m * N[(y$45$scale$95$m * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 1.5e-50], N[(-4.0 * N[(N[(b * N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale$95$m * N[(x$45$scale * N[(y$45$scale$95$m * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.9e+148], N[(N[(-4.0 * N[(b * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b / t$95$0), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(N[(a * N[(b * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < 1.49999999999999995e-50

   1. Initial program 35.6%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 59.9

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

   5. Applied rewrites 59.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

   7. Applied rewrites 72.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. lower-*.f64 74.9

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

   9. Applied rewrites 74.9%

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

       1. associate-*r* N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*r* N/A

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

       4. lift-*.f64 N/A

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

       5. lower-*.f64 80.9

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

   11. Applied rewrites 80.9%

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

   if 1.49999999999999995e-50 < a < 1.8999999999999999e148

   1. Initial program 23.2%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 59.3

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

   5. Applied rewrites 59.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*l* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

   7. Applied rewrites 87.6%

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

   if 1.8999999999999999e148 < a

   1. Initial program 0.0%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 46.4

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

   5. Applied rewrites 46.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

   7. Applied rewrites 64.6%

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 75.2

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

   9. Applied rewrites 75.2%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.5 \cdot 10^{-50}:\\ \;\;\;\;-4 \cdot \frac{b \cdot \left(a \cdot \left(a \cdot b\right)\right)}{y-scale \cdot \left(x-scale \cdot \left(y-scale \cdot x-scale\right)\right)}\\ \mathbf{elif}\;a \leq 1.9 \cdot 10^{+148}:\\ \;\;\;\;\left(-4 \cdot \left(b \cdot \left(a \cdot a\right)\right)\right) \cdot \frac{b}{x-scale \cdot \left(y-scale \cdot \left(y-scale \cdot x-scale\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{a \cdot \left(b \cdot \left(a \cdot b\right)\right)}{x-scale \cdot \left(y-scale \cdot \left(y-scale \cdot x-scale\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.8% accurate, 35.9× speedup

Math

\[\begin{array}{l} y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;y-scale\_m \leq 2.9 \cdot 10^{-185}:\\ \;\;\;\;-4 \cdot \frac{b \cdot \left(a \cdot \left(a \cdot b\right)\right)}{y-scale\_m \cdot \left(x-scale \cdot \left(y-scale\_m \cdot x-scale\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{a \cdot \left(b \cdot \left(a \cdot b\right)\right)}{x-scale \cdot \left(y-scale\_m \cdot \left(y-scale\_m \cdot x-scale\right)\right)}\\ \end{array} \end{array} \]

FPCore

y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale y-scale_m)
 :precision binary64
 (if (<= y-scale_m 2.9e-185)
   (*
    -4.0
    (/ (* b (* a (* a b))) (* y-scale_m (* x-scale (* y-scale_m x-scale)))))
   (*
    -4.0
    (/ (* a (* b (* a b))) (* x-scale (* y-scale_m (* y-scale_m x-scale)))))))

C

y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	double tmp;
	if (y_45_scale_m <= 2.9e-185) {
		tmp = -4.0 * ((b * (a * (a * b))) / (y_45_scale_m * (x_45_scale * (y_45_scale_m * x_45_scale))));
	} else {
		tmp = -4.0 * ((a * (b * (a * b))) / (x_45_scale * (y_45_scale_m * (y_45_scale_m * x_45_scale))));
	}
	return tmp;
}

Fortran

y-scale_m = abs(y_45scale)
real(8) function code(a, b, angle, x_45scale, y_45scale_m)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale_m
    real(8) :: tmp
    if (y_45scale_m <= 2.9d-185) then
        tmp = (-4.0d0) * ((b * (a * (a * b))) / (y_45scale_m * (x_45scale * (y_45scale_m * x_45scale))))
    else
        tmp = (-4.0d0) * ((a * (b * (a * b))) / (x_45scale * (y_45scale_m * (y_45scale_m * x_45scale))))
    end if
    code = tmp
end function

Java

y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	double tmp;
	if (y_45_scale_m <= 2.9e-185) {
		tmp = -4.0 * ((b * (a * (a * b))) / (y_45_scale_m * (x_45_scale * (y_45_scale_m * x_45_scale))));
	} else {
		tmp = -4.0 * ((a * (b * (a * b))) / (x_45_scale * (y_45_scale_m * (y_45_scale_m * x_45_scale))));
	}
	return tmp;
}

Python

y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale, y_45_scale_m):
	tmp = 0
	if y_45_scale_m <= 2.9e-185:
		tmp = -4.0 * ((b * (a * (a * b))) / (y_45_scale_m * (x_45_scale * (y_45_scale_m * x_45_scale))))
	else:
		tmp = -4.0 * ((a * (b * (a * b))) / (x_45_scale * (y_45_scale_m * (y_45_scale_m * x_45_scale))))
	return tmp

Julia

y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale, y_45_scale_m)
	tmp = 0.0
	if (y_45_scale_m <= 2.9e-185)
		tmp = Float64(-4.0 * Float64(Float64(b * Float64(a * Float64(a * b))) / Float64(y_45_scale_m * Float64(x_45_scale * Float64(y_45_scale_m * x_45_scale)))));
	else
		tmp = Float64(-4.0 * Float64(Float64(a * Float64(b * Float64(a * b))) / Float64(x_45_scale * Float64(y_45_scale_m * Float64(y_45_scale_m * x_45_scale)))));
	end
	return tmp
end

MATLAB

y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale_m)
	tmp = 0.0;
	if (y_45_scale_m <= 2.9e-185)
		tmp = -4.0 * ((b * (a * (a * b))) / (y_45_scale_m * (x_45_scale * (y_45_scale_m * x_45_scale))));
	else
		tmp = -4.0 * ((a * (b * (a * b))) / (x_45_scale * (y_45_scale_m * (y_45_scale_m * x_45_scale))));
	end
	tmp_2 = tmp;
end

Wolfram

y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale_, y$45$scale$95$m_] := If[LessEqual[y$45$scale$95$m, 2.9e-185], N[(-4.0 * N[(N[(b * N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale$95$m * N[(x$45$scale * N[(y$45$scale$95$m * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(N[(a * N[(b * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * N[(y$45$scale$95$m * N[(y$45$scale$95$m * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 2.89999999999999995e-185

   1. Initial program 30.3%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 53.4

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

   5. Applied rewrites 53.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

   7. Applied rewrites 68.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. lower-*.f64 70.4

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

   9. Applied rewrites 70.4%

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

       1. associate-*r* N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*r* N/A

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

       4. lift-*.f64 N/A

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

       5. lower-*.f64 77.8

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

   11. Applied rewrites 77.8%

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

   if 2.89999999999999995e-185 < y-scale

   1. Initial program 29.0%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 64.9

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

   5. Applied rewrites 64.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

   7. Applied rewrites 78.3%

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 83.0

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

   9. Applied rewrites 83.0%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 2.9 \cdot 10^{-185}:\\ \;\;\;\;-4 \cdot \frac{b \cdot \left(a \cdot \left(a \cdot b\right)\right)}{y-scale \cdot \left(x-scale \cdot \left(y-scale \cdot x-scale\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{a \cdot \left(b \cdot \left(a \cdot b\right)\right)}{x-scale \cdot \left(y-scale \cdot \left(y-scale \cdot x-scale\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.8% accurate, 35.9× speedup

Math

\[\begin{array}{l} y-scale_m = \left|y-scale\right| \\ \frac{a \cdot \left(a \cdot b\right)}{\left(y-scale\_m \cdot x-scale\right) \cdot -0.25} \cdot \frac{b}{y-scale\_m \cdot x-scale} \end{array} \]

FPCore

y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale y-scale_m)
 :precision binary64
 (*
  (/ (* a (* a b)) (* (* y-scale_m x-scale) -0.25))
  (/ b (* y-scale_m x-scale))))

C

y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	return ((a * (a * b)) / ((y_45_scale_m * x_45_scale) * -0.25)) * (b / (y_45_scale_m * x_45_scale));
}

Fortran

y-scale_m = abs(y_45scale)
real(8) function code(a, b, angle, x_45scale, y_45scale_m)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale_m
    code = ((a * (a * b)) / ((y_45scale_m * x_45scale) * (-0.25d0))) * (b / (y_45scale_m * x_45scale))
end function

Java

y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	return ((a * (a * b)) / ((y_45_scale_m * x_45_scale) * -0.25)) * (b / (y_45_scale_m * x_45_scale));
}

Python

y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale, y_45_scale_m):
	return ((a * (a * b)) / ((y_45_scale_m * x_45_scale) * -0.25)) * (b / (y_45_scale_m * x_45_scale))

Julia

y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale, y_45_scale_m)
	return Float64(Float64(Float64(a * Float64(a * b)) / Float64(Float64(y_45_scale_m * x_45_scale) * -0.25)) * Float64(b / Float64(y_45_scale_m * x_45_scale)))
end

MATLAB

y-scale_m = abs(y_45_scale);
function tmp = code(a, b, angle, x_45_scale, y_45_scale_m)
	tmp = ((a * (a * b)) / ((y_45_scale_m * x_45_scale) * -0.25)) * (b / (y_45_scale_m * x_45_scale));
end

Wolfram

y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale_, y$45$scale$95$m_] := N[(N[(N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision] / N[(N[(y$45$scale$95$m * x$45$scale), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision] * N[(b / N[(y$45$scale$95$m * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 29.8%

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

3. Taylor expanded in angle around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. associate-*l* N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 58.3

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

5. Applied rewrites 58.3%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*r* N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*r* N/A

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

   11. lift-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. associate-*r* N/A

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

   15. times-frac N/A

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

   16. lower-*.f64 N/A

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

7. Applied rewrites 74.6%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. times-frac N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*l* N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-/.f64 84.6

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

9. Applied rewrites 84.6%

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

    1. lift-*.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. lift-/.f64 N/A

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

    5. associate-*l/ N/A

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

    6. clear-num N/A

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

    7. frac-times N/A

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

    8. div-inv N/A

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

    9. *-rgt-identity N/A

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

    10. *-commutative N/A

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

    11. associate-/r/ N/A

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

    12. lift-/.f64 N/A

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

    13. clear-num N/A

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

    14. lift-/.f64 N/A

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

    15. associate-/l/ N/A

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

    16. *-commutative N/A

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

    17. lift-*.f64 N/A

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

    18. lower-*.f64 N/A

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

11. Applied rewrites 89.3%

    \[\leadsto \color{blue}{\frac{a \cdot \left(a \cdot b\right)}{\left(x-scale \cdot y-scale\right) \cdot -0.25} \cdot \frac{b}{x-scale \cdot y-scale}} \]

12. Final simplification 89.3%

    \[\leadsto \frac{a \cdot \left(a \cdot b\right)}{\left(y-scale \cdot x-scale\right) \cdot -0.25} \cdot \frac{b}{y-scale \cdot x-scale} \]
13. Add Preprocessing

Alternative 7: 73.5% accurate, 40.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 29.8%

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

3. Taylor expanded in angle around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. associate-*l* N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 58.3

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

5. Applied rewrites 58.3%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*l* N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*r* N/A

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

   11. lift-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. associate-/l* N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 N/A

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

7. Applied rewrites 72.5%

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

   1. associate-*r* N/A

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

   2. associate-*r* N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lower-*.f64 69.5

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 78.4

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

9. Applied rewrites 78.4%

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

10. Final simplification 78.4%

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

Alternative 8: 73.1% accurate, 40.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 29.8%

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

3. Taylor expanded in angle around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. associate-*l* N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 58.3

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

5. Applied rewrites 58.3%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*l* N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*r* N/A

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

   11. lift-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. associate-/l* N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 N/A

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

7. Applied rewrites 72.5%

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 78.4

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

9. Applied rewrites 78.4%

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

10. Final simplification 78.4%

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

Alternative 9: 65.5% accurate, 40.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 29.8%

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

3. Taylor expanded in angle around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. associate-*l* N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

       \[\leadsto \frac{\left(-4 \cdot \left(a \cdot a\right)\right) \cdot \left(b \cdot b\right)}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale\right)}\right)} \]

   15. lower-*.f64 58.3

       \[\leadsto \frac{\left(-4 \cdot \left(a \cdot a\right)\right) \cdot \left(b \cdot b\right)}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale\right)}\right)} \]

5. Applied rewrites 58.3%

   \[\leadsto \color{blue}{\frac{\left(-4 \cdot \left(a \cdot a\right)\right) \cdot \left(b \cdot b\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\left(-4 \cdot \color{blue}{\left(a \cdot a\right)}\right) \cdot \left(b \cdot b\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(-4 \cdot \left(a \cdot a\right)\right)} \cdot \left(b \cdot b\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\left(-4 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(b \cdot b\right)}}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\left(-4 \cdot \left(a \cdot a\right)\right) \cdot \left(b \cdot b\right)}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale\right)}\right)} \]

   5. associate-*r* N/A

      \[\leadsto \frac{\left(-4 \cdot \left(a \cdot a\right)\right) \cdot \left(b \cdot b\right)}{\color{blue}{\left(x-scale \cdot x-scale\right) \cdot \left(y-scale \cdot y-scale\right)}} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\left(-4 \cdot \left(a \cdot a\right)\right) \cdot \left(b \cdot b\right)}{\color{blue}{\left(x-scale \cdot x-scale\right)} \cdot \left(y-scale \cdot y-scale\right)} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(-4 \cdot \left(a \cdot a\right)\right)} \cdot \left(b \cdot b\right)}{\left(x-scale \cdot x-scale\right) \cdot \left(y-scale \cdot y-scale\right)} \]

   8. associate-*l* N/A

      \[\leadsto \frac{\color{blue}{-4 \cdot \left(\left(a \cdot a\right) \cdot \left(b \cdot b\right)\right)}}{\left(x-scale \cdot x-scale\right) \cdot \left(y-scale \cdot y-scale\right)} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{-4 \cdot \left(\left(a \cdot a\right) \cdot \left(b \cdot b\right)\right)}{\color{blue}{\left(x-scale \cdot x-scale\right)} \cdot \left(y-scale \cdot y-scale\right)} \]

   10. associate-*r* N/A

       \[\leadsto \frac{-4 \cdot \left(\left(a \cdot a\right) \cdot \left(b \cdot b\right)\right)}{\color{blue}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}} \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{-4 \cdot \left(\left(a \cdot a\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot \color{blue}{\left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}} \]

   12. lift-*.f64 N/A

       \[\leadsto \frac{-4 \cdot \left(\left(a \cdot a\right) \cdot \left(b \cdot b\right)\right)}{\color{blue}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}} \]

   13. associate-/l* N/A

       \[\leadsto \color{blue}{-4 \cdot \frac{\left(a \cdot a\right) \cdot \left(b \cdot b\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}} \]

   14. lower-*.f64 N/A

       \[\leadsto \color{blue}{-4 \cdot \frac{\left(a \cdot a\right) \cdot \left(b \cdot b\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}} \]

   15. lower-/.f64 N/A

       \[\leadsto -4 \cdot \color{blue}{\frac{\left(a \cdot a\right) \cdot \left(b \cdot b\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}} \]

7. Applied rewrites 72.5%

   \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot \left(a \cdot \left(b \cdot b\right)\right)}{x-scale \cdot \left(y-scale \cdot \left(x-scale \cdot y-scale\right)\right)}} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto -4 \cdot \frac{a \cdot \left(a \cdot \left(b \cdot b\right)\right)}{x-scale \cdot \left(y-scale \cdot \color{blue}{\left(x-scale \cdot y-scale\right)}\right)} \]

   2. *-commutative N/A

      \[\leadsto -4 \cdot \frac{a \cdot \left(a \cdot \left(b \cdot b\right)\right)}{x-scale \cdot \color{blue}{\left(\left(x-scale \cdot y-scale\right) \cdot y-scale\right)}} \]

   3. associate-*r* N/A

      \[\leadsto -4 \cdot \frac{a \cdot \left(a \cdot \left(b \cdot b\right)\right)}{\color{blue}{\left(x-scale \cdot \left(x-scale \cdot y-scale\right)\right) \cdot y-scale}} \]

   4. lift-*.f64 N/A

      \[\leadsto -4 \cdot \frac{a \cdot \left(a \cdot \left(b \cdot b\right)\right)}{\color{blue}{\left(x-scale \cdot \left(x-scale \cdot y-scale\right)\right)} \cdot y-scale} \]

   5. lower-*.f64 73.2

      \[\leadsto -4 \cdot \frac{a \cdot \left(a \cdot \left(b \cdot b\right)\right)}{\color{blue}{\left(x-scale \cdot \left(x-scale \cdot y-scale\right)\right) \cdot y-scale}} \]

9. Applied rewrites 73.2%

   \[\leadsto -4 \cdot \frac{a \cdot \left(a \cdot \left(b \cdot b\right)\right)}{\color{blue}{\left(x-scale \cdot \left(x-scale \cdot y-scale\right)\right) \cdot y-scale}} \]

10. Final simplification 73.2%

    \[\leadsto -4 \cdot \frac{a \cdot \left(a \cdot \left(b \cdot b\right)\right)}{y-scale \cdot \left(x-scale \cdot \left(y-scale \cdot x-scale\right)\right)} \]
11. Add Preprocessing

Alternative 10: 65.4% accurate, 40.5× speedup

Math

\[\begin{array}{l} y-scale_m = \left|y-scale\right| \\ -4 \cdot \frac{a \cdot \left(a \cdot \left(b \cdot b\right)\right)}{x-scale \cdot \left(y-scale\_m \cdot \left(y-scale\_m \cdot x-scale\right)\right)} \end{array} \]

FPCore

y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale y-scale_m)
 :precision binary64
 (*
  -4.0
  (/ (* a (* a (* b b))) (* x-scale (* y-scale_m (* y-scale_m x-scale))))))

C

y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	return -4.0 * ((a * (a * (b * b))) / (x_45_scale * (y_45_scale_m * (y_45_scale_m * x_45_scale))));
}

Fortran

y-scale_m = abs(y_45scale)
real(8) function code(a, b, angle, x_45scale, y_45scale_m)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale_m
    code = (-4.0d0) * ((a * (a * (b * b))) / (x_45scale * (y_45scale_m * (y_45scale_m * x_45scale))))
end function

Java

y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	return -4.0 * ((a * (a * (b * b))) / (x_45_scale * (y_45_scale_m * (y_45_scale_m * x_45_scale))));
}

Python

y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale, y_45_scale_m):
	return -4.0 * ((a * (a * (b * b))) / (x_45_scale * (y_45_scale_m * (y_45_scale_m * x_45_scale))))

Julia

y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale, y_45_scale_m)
	return Float64(-4.0 * Float64(Float64(a * Float64(a * Float64(b * b))) / Float64(x_45_scale * Float64(y_45_scale_m * Float64(y_45_scale_m * x_45_scale)))))
end

MATLAB

y-scale_m = abs(y_45_scale);
function tmp = code(a, b, angle, x_45_scale, y_45_scale_m)
	tmp = -4.0 * ((a * (a * (b * b))) / (x_45_scale * (y_45_scale_m * (y_45_scale_m * x_45_scale))));
end

Wolfram

y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale_, y$45$scale$95$m_] := N[(-4.0 * N[(N[(a * N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * N[(y$45$scale$95$m * N[(y$45$scale$95$m * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 29.8%

   \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
2. Add Preprocessing

3. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

   2. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

   3. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(-4 \cdot {a}^{2}\right) \cdot {b}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(-4 \cdot {a}^{2}\right) \cdot {b}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(-4 \cdot {a}^{2}\right)} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

   6. unpow2 N/A

      \[\leadsto \frac{\left(-4 \cdot \color{blue}{\left(a \cdot a\right)}\right) \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\left(-4 \cdot \color{blue}{\left(a \cdot a\right)}\right) \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

   8. unpow2 N/A

      \[\leadsto \frac{\left(-4 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(b \cdot b\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\left(-4 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(b \cdot b\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

   10. unpow2 N/A

       \[\leadsto \frac{\left(-4 \cdot \left(a \cdot a\right)\right) \cdot \left(b \cdot b\right)}{\color{blue}{\left(x-scale \cdot x-scale\right)} \cdot {y-scale}^{2}} \]

   11. associate-*l* N/A

       \[\leadsto \frac{\left(-4 \cdot \left(a \cdot a\right)\right) \cdot \left(b \cdot b\right)}{\color{blue}{x-scale \cdot \left(x-scale \cdot {y-scale}^{2}\right)}} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{\left(-4 \cdot \left(a \cdot a\right)\right) \cdot \left(b \cdot b\right)}{\color{blue}{x-scale \cdot \left(x-scale \cdot {y-scale}^{2}\right)}} \]

   13. lower-*.f64 N/A

       \[\leadsto \frac{\left(-4 \cdot \left(a \cdot a\right)\right) \cdot \left(b \cdot b\right)}{x-scale \cdot \color{blue}{\left(x-scale \cdot {y-scale}^{2}\right)}} \]

   14. unpow2 N/A

       \[\leadsto \frac{\left(-4 \cdot \left(a \cdot a\right)\right) \cdot \left(b \cdot b\right)}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale\right)}\right)} \]

   15. lower-*.f64 58.3

       \[\leadsto \frac{\left(-4 \cdot \left(a \cdot a\right)\right) \cdot \left(b \cdot b\right)}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale\right)}\right)} \]

5. Applied rewrites 58.3%

   \[\leadsto \color{blue}{\frac{\left(-4 \cdot \left(a \cdot a\right)\right) \cdot \left(b \cdot b\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\left(-4 \cdot \color{blue}{\left(a \cdot a\right)}\right) \cdot \left(b \cdot b\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(-4 \cdot \left(a \cdot a\right)\right)} \cdot \left(b \cdot b\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\left(-4 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(b \cdot b\right)}}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\left(-4 \cdot \left(a \cdot a\right)\right) \cdot \left(b \cdot b\right)}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale\right)}\right)} \]

   5. associate-*r* N/A

      \[\leadsto \frac{\left(-4 \cdot \left(a \cdot a\right)\right) \cdot \left(b \cdot b\right)}{\color{blue}{\left(x-scale \cdot x-scale\right) \cdot \left(y-scale \cdot y-scale\right)}} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\left(-4 \cdot \left(a \cdot a\right)\right) \cdot \left(b \cdot b\right)}{\color{blue}{\left(x-scale \cdot x-scale\right)} \cdot \left(y-scale \cdot y-scale\right)} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(-4 \cdot \left(a \cdot a\right)\right)} \cdot \left(b \cdot b\right)}{\left(x-scale \cdot x-scale\right) \cdot \left(y-scale \cdot y-scale\right)} \]

   8. associate-*l* N/A

      \[\leadsto \frac{\color{blue}{-4 \cdot \left(\left(a \cdot a\right) \cdot \left(b \cdot b\right)\right)}}{\left(x-scale \cdot x-scale\right) \cdot \left(y-scale \cdot y-scale\right)} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{-4 \cdot \left(\left(a \cdot a\right) \cdot \left(b \cdot b\right)\right)}{\color{blue}{\left(x-scale \cdot x-scale\right)} \cdot \left(y-scale \cdot y-scale\right)} \]

   10. associate-*r* N/A

       \[\leadsto \frac{-4 \cdot \left(\left(a \cdot a\right) \cdot \left(b \cdot b\right)\right)}{\color{blue}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}} \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{-4 \cdot \left(\left(a \cdot a\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot \color{blue}{\left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}} \]

   12. lift-*.f64 N/A

       \[\leadsto \frac{-4 \cdot \left(\left(a \cdot a\right) \cdot \left(b \cdot b\right)\right)}{\color{blue}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}} \]

   13. associate-/l* N/A

       \[\leadsto \color{blue}{-4 \cdot \frac{\left(a \cdot a\right) \cdot \left(b \cdot b\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}} \]

   14. lower-*.f64 N/A

       \[\leadsto \color{blue}{-4 \cdot \frac{\left(a \cdot a\right) \cdot \left(b \cdot b\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}} \]

   15. lower-/.f64 N/A

       \[\leadsto -4 \cdot \color{blue}{\frac{\left(a \cdot a\right) \cdot \left(b \cdot b\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}} \]

7. Applied rewrites 72.5%

   \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot \left(a \cdot \left(b \cdot b\right)\right)}{x-scale \cdot \left(y-scale \cdot \left(x-scale \cdot y-scale\right)\right)}} \]

8. Final simplification 72.5%

   \[\leadsto -4 \cdot \frac{a \cdot \left(a \cdot \left(b \cdot b\right)\right)}{x-scale \cdot \left(y-scale \cdot \left(y-scale \cdot x-scale\right)\right)} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024214 
(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))))