Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 24.7% → 87.7%

Time: 26.8s

Alternatives: 5

Speedup: 40.5×

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

• could not determine a ground truth

  1. a = -2.0042954123536468e-272
  2. b = 1.7449320091914869e+305
  3. angle = -1.4283131023181906e+118
  4. x-scale = -4366.97028623179
  5. y-scale = -1.378101649871414e-296

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: 24.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 87.7% accurate, 35.9× speedup

Math

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

FPCore

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

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return (a / (x_45_scale * y_45_scale)) * ((b / (x_45_scale * y_45_scale)) * (b * (a * -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 / (x_45scale * y_45scale)) * ((b / (x_45scale * y_45scale)) * (b * (a * (-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 / (x_45_scale * y_45_scale)) * ((b / (x_45_scale * y_45_scale)) * (b * (a * -4.0)));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 28.5%

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

3. Taylor expanded in angle around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. associate-*l* N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 52.1

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

5. Simplified 52.1%

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

   1. lift-*.f64 N/A

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

   2. pow2 N/A

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

   3. lift-pow.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-/l* N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*r* N/A

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

   12. associate-*l* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-/.f64 62.4

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

   17. lift-pow.f64 N/A

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

   18. pow2 N/A

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

   19. lift-*.f64 62.4

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

7. Applied egg-rr 62.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-/l* N/A

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

   10. associate-*l* N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 N/A

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

9. Applied egg-rr 84.6%

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

    1. lift-*.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. lift-*.f64 N/A

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

    5. lift-/.f64 N/A

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

    6. lift-*.f64 N/A

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

    7. associate-*r* N/A

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

    8. *-commutative N/A

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

    9. lift-*.f64 N/A

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

    10. lift-/.f64 N/A

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

    11. associate-*l/ N/A

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

    12. *-commutative N/A

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

    13. lift-*.f64 N/A

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

    14. times-frac N/A

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

    15. associate-*l* N/A

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

    16. lower-*.f64 N/A

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

11. Applied egg-rr 90.4%

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

12. Final simplification 90.4%

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

Alternative 2: 79.9% accurate, 32.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 5.8 \cdot 10^{-163}:\\ \;\;\;\;\frac{\left(b \cdot \left(a \cdot -4\right)\right) \cdot \left(a \cdot b\right)}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \frac{-4}{x-scale \cdot y-scale}\right) \cdot \left(a \cdot \left(b \cdot \frac{b}{x-scale \cdot y-scale}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= a 5.8e-163)
   (/ (* (* b (* a -4.0)) (* a b)) (* (* x-scale y-scale) (* x-scale y-scale)))
   (*
    (* a (/ -4.0 (* x-scale y-scale)))
    (* a (* b (/ b (* 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 <= 5.8e-163) {
		tmp = ((b * (a * -4.0)) * (a * b)) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale));
	} else {
		tmp = (a * (-4.0 / (x_45_scale * y_45_scale))) * (a * (b * (b / (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 <= 5.8d-163) then
        tmp = ((b * (a * (-4.0d0))) * (a * b)) / ((x_45scale * y_45scale) * (x_45scale * y_45scale))
    else
        tmp = (a * ((-4.0d0) / (x_45scale * y_45scale))) * (a * (b * (b / (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 <= 5.8e-163) {
		tmp = ((b * (a * -4.0)) * (a * b)) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale));
	} else {
		tmp = (a * (-4.0 / (x_45_scale * y_45_scale))) * (a * (b * (b / (x_45_scale * y_45_scale))));
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if a <= 5.8e-163:
		tmp = ((b * (a * -4.0)) * (a * b)) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale))
	else:
		tmp = (a * (-4.0 / (x_45_scale * y_45_scale))) * (a * (b * (b / (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 <= 5.8e-163)
		tmp = Float64(Float64(Float64(b * Float64(a * -4.0)) * Float64(a * b)) / Float64(Float64(x_45_scale * y_45_scale) * Float64(x_45_scale * y_45_scale)));
	else
		tmp = Float64(Float64(a * Float64(-4.0 / Float64(x_45_scale * y_45_scale))) * Float64(a * Float64(b * Float64(b / 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 <= 5.8e-163)
		tmp = ((b * (a * -4.0)) * (a * b)) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale));
	else
		tmp = (a * (-4.0 / (x_45_scale * y_45_scale))) * (a * (b * (b / (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, 5.8e-163], N[(N[(N[(b * N[(a * -4.0), $MachinePrecision]), $MachinePrecision] * N[(a * b), $MachinePrecision]), $MachinePrecision] / N[(N[(x$45$scale * y$45$scale), $MachinePrecision] * N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(-4.0 / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * N[(b * N[(b / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 5.8000000000000002e-163

   1. Initial program 34.5%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 53.1

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

   5. Simplified 53.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/r* N/A

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

      13. lower-/.f64 N/A

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

   7. Applied egg-rr 64.8%

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

   9. Applied egg-rr 78.7%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. *-commutative N/A

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

       4. lift-*.f64 N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

       7. lift-*.f64 N/A

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

       8. associate-*r* N/A

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

       9. lift-*.f64 N/A

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

       10. associate-*l* N/A

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

       11. lift-*.f64 N/A

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

       12. lower-*.f64 N/A

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

       13. lower-*.f64 83.0

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

   11. Applied egg-rr 83.0%

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

   if 5.8000000000000002e-163 < a

   1. Initial program 18.0%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 50.4

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

   5. Simplified 50.4%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 60.1

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

      17. lift-pow.f64 N/A

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

      18. pow2 N/A

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

      19. lift-*.f64 60.1

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

   7. Applied egg-rr 60.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

   9. Applied egg-rr 82.7%

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

   11. Applied egg-rr 89.6%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 5.8 \cdot 10^{-163}:\\ \;\;\;\;\frac{\left(b \cdot \left(a \cdot -4\right)\right) \cdot \left(a \cdot b\right)}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \frac{-4}{x-scale \cdot y-scale}\right) \cdot \left(a \cdot \left(b \cdot \frac{b}{x-scale \cdot y-scale}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.7% accurate, 40.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[(N[(a * -4.0), $MachinePrecision] * N[(b * N[(a * N[(b / 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 28.5%

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

3. Taylor expanded in angle around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. associate-*l* N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 52.1

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

5. Simplified 52.1%

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

   1. lift-*.f64 N/A

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

   2. pow2 N/A

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

   3. lift-pow.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-/l* N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*r* N/A

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

   12. associate-*l* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-/.f64 62.4

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

   17. lift-pow.f64 N/A

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

   18. pow2 N/A

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

   19. lift-*.f64 62.4

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

7. Applied egg-rr 62.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-/l* N/A

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

   10. associate-*l* N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 N/A

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

9. Applied egg-rr 84.6%

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

10. Final simplification 84.6%

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

Alternative 4: 77.3% accurate, 40.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 28.5%

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

3. Taylor expanded in angle around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. associate-*l* N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 52.1

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

5. Simplified 52.1%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/r* N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*r* N/A

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

   11. lift-*.f64 N/A

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

   12. associate-/r* N/A

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

   13. lower-/.f64 N/A

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

7. Applied egg-rr 66.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/r* N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*r* N/A

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

   10. lift-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. associate-*r/ N/A

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

   14. lift-/.f64 N/A

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

   15. associate-/r* N/A

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

   16. lift-*.f64 N/A

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

9. Applied egg-rr 71.7%

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

    1. associate-*r* N/A

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

    2. lift-*.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. lift-*.f64 N/A

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

    5. lift-*.f64 N/A

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

    6. lift-/.f64 N/A

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

    7. associate-*l* N/A

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

    8. *-commutative N/A

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

    9. associate-*r* N/A

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

    10. *-commutative N/A

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

    11. lower-*.f64 N/A

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

11. Applied egg-rr 83.2%

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

12. Final simplification 83.2%

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

Alternative 5: 69.3% accurate, 40.5× speedup

Math

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 28.5%

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

3. Taylor expanded in angle around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. associate-*l* N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 52.1

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

5. Simplified 52.1%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/r* N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*r* N/A

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

   11. lift-*.f64 N/A

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

   12. associate-/r* N/A

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

   13. lower-/.f64 N/A

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

7. Applied egg-rr 66.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/r* N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*r* N/A

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

   10. lift-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. associate-*r/ N/A

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

   14. lift-/.f64 N/A

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

   15. associate-/r* N/A

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

   16. lift-*.f64 N/A

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

9. Applied egg-rr 71.7%

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

10. Final simplification 71.7%

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

Reproduce

herbie shell --seed 2024207 
(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))))