Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 25.7% → 90.8%

Time: 23.5s

Alternatives: 6

Speedup: 40.5×

• 32.2% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. a = 1.0772528672627757e+287
  2. b = 1.0578085404004912e-214
  3. angle = 1.3364632804593287e+108
  4. x-scale = -3.151065804902357e-192
  5. y-scale = 4.528490416072285e-207

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 25.7% 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: 90.8% accurate, 35.9× speedup

Math

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

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ b (* x-scale y-scale)))) (* t_0 (* (* -4.0 a) (* a t_0)))))

C

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

Fortran

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

Java

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

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = b / (x_45_scale * y_45_scale)
	return t_0 * ((-4.0 * a) * (a * t_0))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 22.7%

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

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. associate-*l* N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 49.6

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

5. Simplified 49.6%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lift-*.f64 N/A

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

   6. unswap-sqr N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. times-frac N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 75.0

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

7. Applied egg-rr 75.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-*r* N/A

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

   8. lower-*.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. associate-*r/ N/A

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

   11. lower-/.f64 N/A

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

   12. lower-*.f64 76.4

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

9. Applied egg-rr 76.4%

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

    1. lift-*.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. associate-/l* N/A

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

    5. lift-/.f64 N/A

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

    6. lift-*.f64 N/A

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

    7. lift-*.f64 N/A

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

    8. associate-*r* N/A

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

    9. associate-*l* N/A

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

    10. lower-*.f64 N/A

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

    11. lower-*.f64 N/A

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

    12. lower-*.f64 91.4

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

11. Applied egg-rr 91.4%

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

12. Final simplification 91.4%

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

Alternative 2: 76.9% accurate, 29.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y-scale \cdot \left(x-scale \cdot \left(x-scale \cdot y-scale\right)\right)\\ t_1 := \frac{b}{x-scale \cdot y-scale}\\ \mathbf{if}\;a \leq 6.8 \cdot 10^{-161}:\\ \;\;\;\;\frac{-4 \cdot \left(b \cdot \left(a \cdot \left(a \cdot b\right)\right)\right)}{t\_0}\\ \mathbf{elif}\;a \leq 4.6 \cdot 10^{+145}:\\ \;\;\;\;\left(-4 \cdot \left(a \cdot a\right)\right) \cdot \left(t\_1 \cdot t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(-4 \cdot a\right) \cdot \left(b \cdot \frac{b}{t\_0}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* y-scale (* x-scale (* x-scale y-scale))))
        (t_1 (/ b (* x-scale y-scale))))
   (if (<= a 6.8e-161)
     (/ (* -4.0 (* b (* a (* a b)))) t_0)
     (if (<= a 4.6e+145)
       (* (* -4.0 (* a a)) (* t_1 t_1))
       (* a (* (* -4.0 a) (* b (/ b t_0))))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = y_45_scale * (x_45_scale * (x_45_scale * y_45_scale));
	double t_1 = b / (x_45_scale * y_45_scale);
	double tmp;
	if (a <= 6.8e-161) {
		tmp = (-4.0 * (b * (a * (a * b)))) / t_0;
	} else if (a <= 4.6e+145) {
		tmp = (-4.0 * (a * a)) * (t_1 * t_1);
	} else {
		tmp = a * ((-4.0 * a) * (b * (b / t_0)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, angle, x_45scale, y_45scale)
    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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = y_45scale * (x_45scale * (x_45scale * y_45scale))
    t_1 = b / (x_45scale * y_45scale)
    if (a <= 6.8d-161) then
        tmp = ((-4.0d0) * (b * (a * (a * b)))) / t_0
    else if (a <= 4.6d+145) then
        tmp = ((-4.0d0) * (a * a)) * (t_1 * t_1)
    else
        tmp = a * (((-4.0d0) * a) * (b * (b / t_0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = y_45_scale * (x_45_scale * (x_45_scale * y_45_scale));
	double t_1 = b / (x_45_scale * y_45_scale);
	double tmp;
	if (a <= 6.8e-161) {
		tmp = (-4.0 * (b * (a * (a * b)))) / t_0;
	} else if (a <= 4.6e+145) {
		tmp = (-4.0 * (a * a)) * (t_1 * t_1);
	} else {
		tmp = a * ((-4.0 * a) * (b * (b / t_0)));
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = y_45_scale * (x_45_scale * (x_45_scale * y_45_scale))
	t_1 = b / (x_45_scale * y_45_scale)
	tmp = 0
	if a <= 6.8e-161:
		tmp = (-4.0 * (b * (a * (a * b)))) / t_0
	elif a <= 4.6e+145:
		tmp = (-4.0 * (a * a)) * (t_1 * t_1)
	else:
		tmp = a * ((-4.0 * a) * (b * (b / t_0)))
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(y_45_scale * Float64(x_45_scale * Float64(x_45_scale * y_45_scale)))
	t_1 = Float64(b / Float64(x_45_scale * y_45_scale))
	tmp = 0.0
	if (a <= 6.8e-161)
		tmp = Float64(Float64(-4.0 * Float64(b * Float64(a * Float64(a * b)))) / t_0);
	elseif (a <= 4.6e+145)
		tmp = Float64(Float64(-4.0 * Float64(a * a)) * Float64(t_1 * t_1));
	else
		tmp = Float64(a * Float64(Float64(-4.0 * a) * Float64(b * Float64(b / t_0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = y_45_scale * (x_45_scale * (x_45_scale * y_45_scale));
	t_1 = b / (x_45_scale * y_45_scale);
	tmp = 0.0;
	if (a <= 6.8e-161)
		tmp = (-4.0 * (b * (a * (a * b)))) / t_0;
	elseif (a <= 4.6e+145)
		tmp = (-4.0 * (a * a)) * (t_1 * t_1);
	else
		tmp = a * ((-4.0 * a) * (b * (b / t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(y$45$scale * N[(x$45$scale * N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 6.8e-161], N[(N[(-4.0 * N[(b * N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[a, 4.6e+145], N[(N[(-4.0 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(-4.0 * a), $MachinePrecision] * N[(b * N[(b / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if a < 6.79999999999999964e-161

   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-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 47.6

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

   5. Simplified 47.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. unswap-sqr N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 71.8

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

   7. Applied egg-rr 71.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

   9. Applied egg-rr 56.8%

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

   10. Taylor expanded in b around 0

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

       1. lower-*.f64 N/A

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. associate-*l* N/A

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

       9. lower-*.f64 N/A

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

       10. lower-*.f64 74.6

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

   12. Simplified 74.6%

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

   if 6.79999999999999964e-161 < a < 4.6e145

   1. Initial program 17.5%

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

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 60.9

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

   5. Simplified 60.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. unswap-sqr N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 88.7

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

   7. Applied egg-rr 88.7%

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

   if 4.6e145 < 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-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 36.7

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

   5. Simplified 36.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. unswap-sqr N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 63.5

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

   7. Applied egg-rr 63.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   9. Applied egg-rr 68.0%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 6.8 \cdot 10^{-161}:\\ \;\;\;\;\frac{-4 \cdot \left(b \cdot \left(a \cdot \left(a \cdot b\right)\right)\right)}{y-scale \cdot \left(x-scale \cdot \left(x-scale \cdot y-scale\right)\right)}\\ \mathbf{elif}\;a \leq 4.6 \cdot 10^{+145}:\\ \;\;\;\;\left(-4 \cdot \left(a \cdot a\right)\right) \cdot \left(\frac{b}{x-scale \cdot y-scale} \cdot \frac{b}{x-scale \cdot y-scale}\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(-4 \cdot a\right) \cdot \left(b \cdot \frac{b}{y-scale \cdot \left(x-scale \cdot \left(x-scale \cdot y-scale\right)\right)}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.1% accurate, 35.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y-scale \cdot \left(x-scale \cdot \left(x-scale \cdot y-scale\right)\right)\\ \mathbf{if}\;a \leq 2.8 \cdot 10^{+232}:\\ \;\;\;\;\frac{-4 \cdot \left(b \cdot \left(a \cdot \left(a \cdot b\right)\right)\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(-4 \cdot a\right) \cdot \left(b \cdot \frac{b}{t\_0}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* y-scale (* x-scale (* x-scale y-scale)))))
   (if (<= a 2.8e+232)
     (/ (* -4.0 (* b (* a (* a b)))) t_0)
     (* a (* (* -4.0 a) (* b (/ b t_0)))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = y_45_scale * (x_45_scale * (x_45_scale * y_45_scale));
	double tmp;
	if (a <= 2.8e+232) {
		tmp = (-4.0 * (b * (a * (a * b)))) / t_0;
	} else {
		tmp = a * ((-4.0 * a) * (b * (b / t_0)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, angle, x_45scale, y_45scale)
    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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y_45scale * (x_45scale * (x_45scale * y_45scale))
    if (a <= 2.8d+232) then
        tmp = ((-4.0d0) * (b * (a * (a * b)))) / t_0
    else
        tmp = a * (((-4.0d0) * a) * (b * (b / t_0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = y_45_scale * (x_45_scale * (x_45_scale * y_45_scale));
	double tmp;
	if (a <= 2.8e+232) {
		tmp = (-4.0 * (b * (a * (a * b)))) / t_0;
	} else {
		tmp = a * ((-4.0 * a) * (b * (b / t_0)));
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = y_45_scale * (x_45_scale * (x_45_scale * y_45_scale))
	tmp = 0
	if a <= 2.8e+232:
		tmp = (-4.0 * (b * (a * (a * b)))) / t_0
	else:
		tmp = a * ((-4.0 * a) * (b * (b / t_0)))
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(y_45_scale * Float64(x_45_scale * Float64(x_45_scale * y_45_scale)))
	tmp = 0.0
	if (a <= 2.8e+232)
		tmp = Float64(Float64(-4.0 * Float64(b * Float64(a * Float64(a * b)))) / t_0);
	else
		tmp = Float64(a * Float64(Float64(-4.0 * a) * Float64(b * Float64(b / t_0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = y_45_scale * (x_45_scale * (x_45_scale * y_45_scale));
	tmp = 0.0;
	if (a <= 2.8e+232)
		tmp = (-4.0 * (b * (a * (a * b)))) / t_0;
	else
		tmp = a * ((-4.0 * a) * (b * (b / t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(y$45$scale * N[(x$45$scale * N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 2.8e+232], N[(N[(-4.0 * N[(b * N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(a * N[(N[(-4.0 * a), $MachinePrecision] * N[(b * N[(b / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 2.7999999999999999e232

   1. Initial program 24.2%

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

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 51.0

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

   5. Simplified 51.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. unswap-sqr N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 76.3

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

   7. Applied egg-rr 76.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

   9. Applied egg-rr 60.5%

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

   10. Taylor expanded in b around 0

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

       1. lower-*.f64 N/A

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. associate-*l* N/A

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

       9. lower-*.f64 N/A

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

       10. lower-*.f64 74.7

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

   12. Simplified 74.7%

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

   if 2.7999999999999999e232 < 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-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 26.7

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

   5. Simplified 26.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. unswap-sqr N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 53.7

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

   7. Applied egg-rr 53.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   9. Applied egg-rr 61.5%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.8 \cdot 10^{+232}:\\ \;\;\;\;\frac{-4 \cdot \left(b \cdot \left(a \cdot \left(a \cdot b\right)\right)\right)}{y-scale \cdot \left(x-scale \cdot \left(x-scale \cdot y-scale\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(-4 \cdot a\right) \cdot \left(b \cdot \frac{b}{y-scale \cdot \left(x-scale \cdot \left(x-scale \cdot y-scale\right)\right)}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.5% accurate, 40.5× speedup

Math

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

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (* a (* (* -4.0 a) (* b (/ b (* y-scale (* x-scale (* x-scale y-scale))))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return a * ((-4.0 * a) * (b * (b / (y_45_scale * (x_45_scale * (x_45_scale * y_45_scale))))));
}

Fortran

real(8) function code(a, b, angle, x_45scale, y_45scale)
    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
    code = a * (((-4.0d0) * a) * (b * (b / (y_45scale * (x_45scale * (x_45scale * y_45scale))))))
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return a * ((-4.0 * a) * (b * (b / (y_45_scale * (x_45_scale * (x_45_scale * y_45_scale))))));
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	return a * ((-4.0 * a) * (b * (b / (y_45_scale * (x_45_scale * (x_45_scale * y_45_scale))))))

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return Float64(a * Float64(Float64(-4.0 * a) * Float64(b * Float64(b / Float64(y_45_scale * Float64(x_45_scale * Float64(x_45_scale * y_45_scale)))))))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 22.7%

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

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. associate-*l* N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 49.6

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

5. Simplified 49.6%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lift-*.f64 N/A

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

   6. unswap-sqr N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. times-frac N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 75.0

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

7. Applied egg-rr 75.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*r* N/A

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

   12. associate-*r* N/A

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

   13. lower-*.f64 N/A

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

9. Applied egg-rr 73.0%

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

10. Final simplification 73.0%

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

Alternative 5: 66.5% accurate, 40.5× speedup

Math

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

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (* (* -4.0 (* a a)) (* b (/ b (* x-scale (* y-scale (* x-scale y-scale)))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return (-4.0 * (a * a)) * (b * (b / (x_45_scale * (y_45_scale * (x_45_scale * y_45_scale)))));
}

Fortran

real(8) function code(a, b, angle, x_45scale, y_45scale)
    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
    code = ((-4.0d0) * (a * a)) * (b * (b / (x_45scale * (y_45scale * (x_45scale * y_45scale)))))
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return (-4.0 * (a * a)) * (b * (b / (x_45_scale * (y_45_scale * (x_45_scale * y_45_scale)))));
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	return (-4.0 * (a * a)) * (b * (b / (x_45_scale * (y_45_scale * (x_45_scale * y_45_scale)))))

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return Float64(Float64(-4.0 * Float64(a * a)) * Float64(b * Float64(b / Float64(x_45_scale * Float64(y_45_scale * Float64(x_45_scale * y_45_scale))))))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 22.7%

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

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. associate-*l* N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 49.6

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

5. Simplified 49.6%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 58.1

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

7. Applied egg-rr 58.1%

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

   1. associate-*r* N/A

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

   2. lift-*.f64 N/A

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

   3. lower-*.f64 66.8

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

9. Applied egg-rr 66.8%

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

10. Final simplification 66.8%

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

Alternative 6: 60.7% accurate, 40.5× speedup

Math

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

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (* (* -4.0 (* a a)) (* b (/ b (* x-scale (* x-scale (* y-scale y-scale)))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return (-4.0 * (a * a)) * (b * (b / (x_45_scale * (x_45_scale * (y_45_scale * y_45_scale)))));
}

Fortran

real(8) function code(a, b, angle, x_45scale, y_45scale)
    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
    code = ((-4.0d0) * (a * a)) * (b * (b / (x_45scale * (x_45scale * (y_45scale * y_45scale)))))
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return (-4.0 * (a * a)) * (b * (b / (x_45_scale * (x_45_scale * (y_45_scale * y_45_scale)))));
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	return (-4.0 * (a * a)) * (b * (b / (x_45_scale * (x_45_scale * (y_45_scale * y_45_scale)))))

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return Float64(Float64(-4.0 * Float64(a * a)) * Float64(b * Float64(b / Float64(x_45_scale * Float64(x_45_scale * Float64(y_45_scale * y_45_scale))))))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 22.7%

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

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. associate-*l* N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 49.6

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

5. Simplified 49.6%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 58.1

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

7. Applied egg-rr 58.1%

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

8. Final simplification 58.1%

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

Reproduce

herbie shell --seed 2024215 
(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))))