Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 24.8% → 92.3%

Time: 26.6s

Alternatives: 6

Speedup: 40.5×

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

• could not determine a ground truth

  1. a = -8.888755512981099e+261
  2. b = -8.789538147998102e-246
  3. angle = -4.084035185094833e+222
  4. x-scale = 1.2146901005970898e-134
  5. y-scale = -1.314832310978326e-295

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 24.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 92.3% accurate, 35.9× speedup

Math

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

FPCore

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

C

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

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
    code = ((a * b) / (x_45scale * y_45scale)) * ((a / (x_45scale * y_45scale)) * (b * (-4.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 24.8%

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

3. Taylor expanded in angle around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. associate-*l* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 50.1

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

5. Simplified 50.1%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/l* N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*r* N/A

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

   11. associate-*l* N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 58.6

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

7. Applied egg-rr 58.6%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. times-frac N/A

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

   4. associate-*r/ N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*r* N/A

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

   9. lift-*.f64 N/A

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

   10. times-frac N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-/.f64 83.7

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

9. Applied egg-rr 83.7%

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

11. Applied egg-rr 92.4%

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

12. Final simplification 92.4%

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

Alternative 2: 82.1% accurate, 29.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(b \cdot -4\right) \cdot \left(a \cdot \left(b \cdot \frac{a}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}\right)\right)\\ \mathbf{if}\;y-scale \leq 2.2 \cdot 10^{-105}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y-scale \leq 4 \cdot 10^{+120}:\\ \;\;\;\;\left(b \cdot -4\right) \cdot \left(\left(b \cdot \frac{a}{x-scale}\right) \cdot \frac{a}{x-scale \cdot \left(y-scale \cdot y-scale\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0
         (*
          (* b -4.0)
          (* a (* b (/ a (* (* x-scale y-scale) (* x-scale y-scale))))))))
   (if (<= y-scale 2.2e-105)
     t_0
     (if (<= y-scale 4e+120)
       (*
        (* b -4.0)
        (* (* b (/ a x-scale)) (/ a (* x-scale (* y-scale y-scale)))))
       t_0))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (b * -4.0) * (a * (b * (a / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale)))));
	double tmp;
	if (y_45_scale <= 2.2e-105) {
		tmp = t_0;
	} else if (y_45_scale <= 4e+120) {
		tmp = (b * -4.0) * ((b * (a / x_45_scale)) * (a / (x_45_scale * (y_45_scale * y_45_scale))));
	} else {
		tmp = 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 = (b * (-4.0d0)) * (a * (b * (a / ((x_45scale * y_45scale) * (x_45scale * y_45scale)))))
    if (y_45scale <= 2.2d-105) then
        tmp = t_0
    else if (y_45scale <= 4d+120) then
        tmp = (b * (-4.0d0)) * ((b * (a / x_45scale)) * (a / (x_45scale * (y_45scale * y_45scale))))
    else
        tmp = 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 = (b * -4.0) * (a * (b * (a / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale)))));
	double tmp;
	if (y_45_scale <= 2.2e-105) {
		tmp = t_0;
	} else if (y_45_scale <= 4e+120) {
		tmp = (b * -4.0) * ((b * (a / x_45_scale)) * (a / (x_45_scale * (y_45_scale * y_45_scale))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (b * -4.0) * (a * (b * (a / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale)))))
	tmp = 0
	if y_45_scale <= 2.2e-105:
		tmp = t_0
	elif y_45_scale <= 4e+120:
		tmp = (b * -4.0) * ((b * (a / x_45_scale)) * (a / (x_45_scale * (y_45_scale * y_45_scale))))
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(b * -4.0) * Float64(a * Float64(b * Float64(a / Float64(Float64(x_45_scale * y_45_scale) * Float64(x_45_scale * y_45_scale))))))
	tmp = 0.0
	if (y_45_scale <= 2.2e-105)
		tmp = t_0;
	elseif (y_45_scale <= 4e+120)
		tmp = Float64(Float64(b * -4.0) * Float64(Float64(b * Float64(a / x_45_scale)) * Float64(a / Float64(x_45_scale * Float64(y_45_scale * y_45_scale)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (b * -4.0) * (a * (b * (a / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale)))));
	tmp = 0.0;
	if (y_45_scale <= 2.2e-105)
		tmp = t_0;
	elseif (y_45_scale <= 4e+120)
		tmp = (b * -4.0) * ((b * (a / x_45_scale)) * (a / (x_45_scale * (y_45_scale * y_45_scale))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(b * -4.0), $MachinePrecision] * N[(a * N[(b * N[(a / N[(N[(x$45$scale * y$45$scale), $MachinePrecision] * N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$45$scale, 2.2e-105], t$95$0, If[LessEqual[y$45$scale, 4e+120], N[(N[(b * -4.0), $MachinePrecision] * N[(N[(b * N[(a / x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(a / N[(x$45$scale * N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 2.20000000000000004e-105 or 3.9999999999999999e120 < y-scale

   1. Initial program 23.4%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 47.3

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

   5. Simplified 47.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 55.9

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

   7. Applied egg-rr 55.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

   9. Applied egg-rr 81.0%

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

   if 2.20000000000000004e-105 < y-scale < 3.9999999999999999e120

   1. Initial program 33.6%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 67.2

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

   5. Simplified 67.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 75.3

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

   7. Applied egg-rr 75.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. times-frac N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 91.9

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

   9. Applied egg-rr 91.9%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 2.2 \cdot 10^{-105}:\\ \;\;\;\;\left(b \cdot -4\right) \cdot \left(a \cdot \left(b \cdot \frac{a}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}\right)\right)\\ \mathbf{elif}\;y-scale \leq 4 \cdot 10^{+120}:\\ \;\;\;\;\left(b \cdot -4\right) \cdot \left(\left(b \cdot \frac{a}{x-scale}\right) \cdot \frac{a}{x-scale \cdot \left(y-scale \cdot y-scale\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot -4\right) \cdot \left(a \cdot \left(b \cdot \frac{a}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.7% accurate, 29.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(b \cdot -4\right) \cdot \left(a \cdot \left(b \cdot \frac{a}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}\right)\right)\\ \mathbf{if}\;y-scale \leq 2 \cdot 10^{-106}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y-scale \leq 2 \cdot 10^{+121}:\\ \;\;\;\;\left(b \cdot -4\right) \cdot \left(a \cdot \left(\frac{a}{x-scale \cdot \left(y-scale \cdot y-scale\right)} \cdot \frac{b}{x-scale}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0
         (*
          (* b -4.0)
          (* a (* b (/ a (* (* x-scale y-scale) (* x-scale y-scale))))))))
   (if (<= y-scale 2e-106)
     t_0
     (if (<= y-scale 2e+121)
       (*
        (* b -4.0)
        (* a (* (/ a (* x-scale (* y-scale y-scale))) (/ b x-scale))))
       t_0))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (b * -4.0) * (a * (b * (a / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale)))));
	double tmp;
	if (y_45_scale <= 2e-106) {
		tmp = t_0;
	} else if (y_45_scale <= 2e+121) {
		tmp = (b * -4.0) * (a * ((a / (x_45_scale * (y_45_scale * y_45_scale))) * (b / x_45_scale)));
	} else {
		tmp = 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 = (b * (-4.0d0)) * (a * (b * (a / ((x_45scale * y_45scale) * (x_45scale * y_45scale)))))
    if (y_45scale <= 2d-106) then
        tmp = t_0
    else if (y_45scale <= 2d+121) then
        tmp = (b * (-4.0d0)) * (a * ((a / (x_45scale * (y_45scale * y_45scale))) * (b / x_45scale)))
    else
        tmp = 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 = (b * -4.0) * (a * (b * (a / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale)))));
	double tmp;
	if (y_45_scale <= 2e-106) {
		tmp = t_0;
	} else if (y_45_scale <= 2e+121) {
		tmp = (b * -4.0) * (a * ((a / (x_45_scale * (y_45_scale * y_45_scale))) * (b / x_45_scale)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (b * -4.0) * (a * (b * (a / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale)))))
	tmp = 0
	if y_45_scale <= 2e-106:
		tmp = t_0
	elif y_45_scale <= 2e+121:
		tmp = (b * -4.0) * (a * ((a / (x_45_scale * (y_45_scale * y_45_scale))) * (b / x_45_scale)))
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(b * -4.0) * Float64(a * Float64(b * Float64(a / Float64(Float64(x_45_scale * y_45_scale) * Float64(x_45_scale * y_45_scale))))))
	tmp = 0.0
	if (y_45_scale <= 2e-106)
		tmp = t_0;
	elseif (y_45_scale <= 2e+121)
		tmp = Float64(Float64(b * -4.0) * Float64(a * Float64(Float64(a / Float64(x_45_scale * Float64(y_45_scale * y_45_scale))) * Float64(b / x_45_scale))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (b * -4.0) * (a * (b * (a / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale)))));
	tmp = 0.0;
	if (y_45_scale <= 2e-106)
		tmp = t_0;
	elseif (y_45_scale <= 2e+121)
		tmp = (b * -4.0) * (a * ((a / (x_45_scale * (y_45_scale * y_45_scale))) * (b / x_45_scale)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(b * -4.0), $MachinePrecision] * N[(a * N[(b * N[(a / N[(N[(x$45$scale * y$45$scale), $MachinePrecision] * N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$45$scale, 2e-106], t$95$0, If[LessEqual[y$45$scale, 2e+121], N[(N[(b * -4.0), $MachinePrecision] * N[(a * N[(N[(a / N[(x$45$scale * N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 1.99999999999999988e-106 or 2.00000000000000007e121 < y-scale

   1. Initial program 23.4%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 47.3

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

   5. Simplified 47.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 55.9

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

   7. Applied egg-rr 55.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

   9. Applied egg-rr 81.0%

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

   if 1.99999999999999988e-106 < y-scale < 2.00000000000000007e121

   1. Initial program 33.6%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 67.2

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

   5. Simplified 67.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 75.3

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

   7. Applied egg-rr 75.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

   9. Applied egg-rr 85.6%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l/ N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lift-*.f64 N/A

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

       8. associate-*l* N/A

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

       9. times-frac N/A

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

       10. associate-/l/ N/A

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

       11. lift-/.f64 N/A

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

       12. lower-*.f64 N/A

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

       13. lower-/.f64 N/A

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

       14. lift-/.f64 N/A

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

       15. associate-/l/ N/A

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

       16. lower-/.f64 N/A

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

       17. *-commutative N/A

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

       18. lift-*.f64 N/A

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

       19. associate-*l* N/A

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

       20. lift-*.f64 N/A

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

       21. lower-*.f64 91.8

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

   11. Applied egg-rr 91.8%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 2 \cdot 10^{-106}:\\ \;\;\;\;\left(b \cdot -4\right) \cdot \left(a \cdot \left(b \cdot \frac{a}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}\right)\right)\\ \mathbf{elif}\;y-scale \leq 2 \cdot 10^{+121}:\\ \;\;\;\;\left(b \cdot -4\right) \cdot \left(a \cdot \left(\frac{a}{x-scale \cdot \left(y-scale \cdot y-scale\right)} \cdot \frac{b}{x-scale}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot -4\right) \cdot \left(a \cdot \left(b \cdot \frac{a}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.0% accurate, 29.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)\\ \mathbf{if}\;\frac{angle}{180} \leq 6 \cdot 10^{+195}:\\ \;\;\;\;\left(b \cdot -4\right) \cdot \left(a \cdot \left(b \cdot \frac{a}{t\_0}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a \cdot b\right) \cdot \left(a \cdot \left(b \cdot -4\right)\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (* x-scale y-scale) (* x-scale y-scale))))
   (if (<= (/ angle 180.0) 6e+195)
     (* (* b -4.0) (* a (* b (/ a t_0))))
     (/ (* (* a b) (* a (* b -4.0))) t_0))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (x_45_scale * y_45_scale) * (x_45_scale * y_45_scale);
	double tmp;
	if ((angle / 180.0) <= 6e+195) {
		tmp = (b * -4.0) * (a * (b * (a / t_0)));
	} else {
		tmp = ((a * b) * (a * (b * -4.0))) / 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 = (x_45scale * y_45scale) * (x_45scale * y_45scale)
    if ((angle / 180.0d0) <= 6d+195) then
        tmp = (b * (-4.0d0)) * (a * (b * (a / t_0)))
    else
        tmp = ((a * b) * (a * (b * (-4.0d0)))) / 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 = (x_45_scale * y_45_scale) * (x_45_scale * y_45_scale);
	double tmp;
	if ((angle / 180.0) <= 6e+195) {
		tmp = (b * -4.0) * (a * (b * (a / t_0)));
	} else {
		tmp = ((a * b) * (a * (b * -4.0))) / t_0;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (x_45_scale * y_45_scale) * (x_45_scale * y_45_scale)
	tmp = 0
	if (angle / 180.0) <= 6e+195:
		tmp = (b * -4.0) * (a * (b * (a / t_0)))
	else:
		tmp = ((a * b) * (a * (b * -4.0))) / t_0
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(x_45_scale * y_45_scale) * Float64(x_45_scale * y_45_scale))
	tmp = 0.0
	if (Float64(angle / 180.0) <= 6e+195)
		tmp = Float64(Float64(b * -4.0) * Float64(a * Float64(b * Float64(a / t_0))));
	else
		tmp = Float64(Float64(Float64(a * b) * Float64(a * Float64(b * -4.0))) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (x_45_scale * y_45_scale) * (x_45_scale * y_45_scale);
	tmp = 0.0;
	if ((angle / 180.0) <= 6e+195)
		tmp = (b * -4.0) * (a * (b * (a / t_0)));
	else
		tmp = ((a * b) * (a * (b * -4.0))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(x$45$scale * y$45$scale), $MachinePrecision] * N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(angle / 180.0), $MachinePrecision], 6e+195], N[(N[(b * -4.0), $MachinePrecision] * N[(a * N[(b * N[(a / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a * b), $MachinePrecision] * N[(a * N[(b * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 6.0000000000000001e195

   1. Initial program 25.2%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 50.5

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

   5. Simplified 50.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 59.5

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

   7. Applied egg-rr 59.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

   9. Applied egg-rr 83.2%

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

   if 6.0000000000000001e195 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 19.0%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 44.1

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

   5. Simplified 44.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/r* N/A

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

      13. lower-/.f64 N/A

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

   7. Applied egg-rr 56.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. associate-/l* N/A

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

      8. associate-/r* N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/r* N/A

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

      15. lift-*.f64 N/A

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

   9. Applied egg-rr 51.3%

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

       1. associate-*r* N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 80.7

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

       10. lift-*.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 80.7

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

   11. Applied egg-rr 80.7%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 6 \cdot 10^{+195}:\\ \;\;\;\;\left(b \cdot -4\right) \cdot \left(a \cdot \left(b \cdot \frac{a}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a \cdot b\right) \cdot \left(a \cdot \left(b \cdot -4\right)\right)}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.1% accurate, 32.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 3.7 \cdot 10^{-213}:\\ \;\;\;\;\left(b \cdot -4\right) \cdot \left(a \cdot \left(b \cdot \frac{a}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot -4\right) \cdot \frac{b \cdot \left(a \cdot \frac{a}{x-scale \cdot y-scale}\right)}{x-scale \cdot y-scale}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= a 3.7e-213)
   (* (* b -4.0) (* a (* b (/ a (* (* x-scale y-scale) (* x-scale y-scale))))))
   (*
    (* b -4.0)
    (/ (* b (* a (/ a (* 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) {
	double tmp;
	if (a <= 3.7e-213) {
		tmp = (b * -4.0) * (a * (b * (a / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale)))));
	} else {
		tmp = (b * -4.0) * ((b * (a * (a / (x_45_scale * y_45_scale)))) / (x_45_scale * y_45_scale));
	}
	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) :: tmp
    if (a <= 3.7d-213) then
        tmp = (b * (-4.0d0)) * (a * (b * (a / ((x_45scale * y_45scale) * (x_45scale * y_45scale)))))
    else
        tmp = (b * (-4.0d0)) * ((b * (a * (a / (x_45scale * y_45scale)))) / (x_45scale * y_45scale))
    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 tmp;
	if (a <= 3.7e-213) {
		tmp = (b * -4.0) * (a * (b * (a / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale)))));
	} else {
		tmp = (b * -4.0) * ((b * (a * (a / (x_45_scale * y_45_scale)))) / (x_45_scale * y_45_scale));
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if a <= 3.7e-213:
		tmp = (b * -4.0) * (a * (b * (a / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale)))))
	else:
		tmp = (b * -4.0) * ((b * (a * (a / (x_45_scale * y_45_scale)))) / (x_45_scale * y_45_scale))
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (a <= 3.7e-213)
		tmp = Float64(Float64(b * -4.0) * Float64(a * Float64(b * Float64(a / Float64(Float64(x_45_scale * y_45_scale) * Float64(x_45_scale * y_45_scale))))));
	else
		tmp = Float64(Float64(b * -4.0) * Float64(Float64(b * Float64(a * Float64(a / Float64(x_45_scale * y_45_scale)))) / Float64(x_45_scale * y_45_scale)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (a <= 3.7e-213)
		tmp = (b * -4.0) * (a * (b * (a / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale)))));
	else
		tmp = (b * -4.0) * ((b * (a * (a / (x_45_scale * y_45_scale)))) / (x_45_scale * y_45_scale));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[a, 3.7e-213], N[(N[(b * -4.0), $MachinePrecision] * N[(a * N[(b * N[(a / N[(N[(x$45$scale * y$45$scale), $MachinePrecision] * N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b * -4.0), $MachinePrecision] * N[(N[(b * N[(a * N[(a / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 3.70000000000000003e-213

   1. Initial program 22.2%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 47.6

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

   5. Simplified 47.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 56.3

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

   7. Applied egg-rr 56.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

   9. Applied egg-rr 80.6%

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

   if 3.70000000000000003e-213 < a

   1. Initial program 28.6%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 53.7

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

   5. Simplified 53.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 61.9

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

   7. Applied egg-rr 61.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

   9. Applied egg-rr 83.1%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l/ N/A

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

       5. *-commutative N/A

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

       6. lift-*.f64 N/A

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

       7. associate-*r/ N/A

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

       8. *-commutative N/A

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

       9. lift-*.f64 N/A

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

       10. associate-*r* N/A

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

       11. lift-*.f64 N/A

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

       12. lift-*.f64 N/A

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

       13. lift-*.f64 N/A

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

       14. associate-/l/ N/A

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

       15. lower-/.f64 N/A

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

   11. Applied egg-rr 88.0%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3.7 \cdot 10^{-213}:\\ \;\;\;\;\left(b \cdot -4\right) \cdot \left(a \cdot \left(b \cdot \frac{a}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot -4\right) \cdot \frac{b \cdot \left(a \cdot \frac{a}{x-scale \cdot y-scale}\right)}{x-scale \cdot y-scale}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.1% accurate, 40.5× speedup

Math

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

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (* (* b -4.0) (* a (* b (/ a (* (* 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 (b * -4.0) * (a * (b * (a / ((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 = (b * (-4.0d0)) * (a * (b * (a / ((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 (b * -4.0) * (a * (b * (a / ((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 (b * -4.0) * (a * (b * (a / ((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(b * -4.0) * Float64(a * Float64(b * Float64(a / Float64(Float64(x_45_scale * 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 = (b * -4.0) * (a * (b * (a / ((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[(b * -4.0), $MachinePrecision] * N[(a * N[(b * N[(a / N[(N[(x$45$scale * y$45$scale), $MachinePrecision] * N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 24.8%

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

3. Taylor expanded in angle around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. associate-*l* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 50.1

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

5. Simplified 50.1%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/l* N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*r* N/A

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

   11. associate-*l* N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 58.6

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

7. Applied egg-rr 58.6%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-/l* N/A

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

   10. associate-*l* N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 N/A

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

9. Applied egg-rr 81.6%

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

10. Final simplification 81.6%

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

Reproduce

herbie shell --seed 2024208 
(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))))