Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 24.9% → 90.7%

Time: 24.5s

Alternatives: 5

Speedup: 40.5×

• 31.5% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. a = -7.738965646237165e+262
  2. b = 8.135366607798327e+77
  3. angle = -1.8824813768600508e+198
  4. x-scale = -2.3982405354820685e-271
  5. y-scale = -3.905674483770569e-241

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: 24.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.7% accurate, 35.9× speedup

Math

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

FPCore

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

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return b * ((a / (x_45_scale * y_45_scale)) * (b * (a * (-4.0 / (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 / (x_45scale * y_45scale)) * (b * (a * ((-4.0d0) / (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 / (x_45_scale * y_45_scale)) * (b * (a * (-4.0 / (x_45_scale * y_45_scale)))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 28.0%

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

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

   2. associate-*l/ N/A

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

   3. /-lowering-/.f64 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. *-commutative N/A

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

   8. *-lowering-*.f64 N/A

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

   9. *-lowering-*.f64 N/A

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

   10. unpow2 N/A

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

   11. *-lowering-*.f64 N/A

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

   12. unpow2 N/A

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

   13. associate-*l* N/A

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

   14. *-lowering-*.f64 N/A

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

   15. *-lowering-*.f64 N/A

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

   16. unpow2 N/A

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

   17. *-lowering-*.f64 55.3

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

5. Simplified 55.3%

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

   1. associate-*l* N/A

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

   2. associate-/l* N/A

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

   3. *-lowering-*.f64 N/A

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

   4. /-lowering-/.f64 N/A

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

   5. *-lowering-*.f64 N/A

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

   6. *-lowering-*.f64 N/A

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

   7. *-lowering-*.f64 N/A

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

   8. *-lowering-*.f64 N/A

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

   9. *-lowering-*.f64 N/A

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

   10. *-lowering-*.f64 58.0

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

7. Applied egg-rr 58.0%

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

   1. associate-*l* N/A

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

   2. associate-/l* N/A

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

   3. *-lowering-*.f64 N/A

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

   4. /-lowering-/.f64 N/A

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

   5. associate-*l* N/A

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

   6. *-lowering-*.f64 N/A

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

   7. *-lowering-*.f64 N/A

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

   8. associate-*r* N/A

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

   9. unswap-sqr N/A

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

   10. *-lowering-*.f64 N/A

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

   11. *-lowering-*.f64 N/A

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

   12. *-lowering-*.f64 65.1

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

9. Applied egg-rr 65.1%

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

    1. *-commutative N/A

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

    2. times-frac N/A

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

    3. associate-*l* N/A

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

    4. *-lowering-*.f64 N/A

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

    5. /-lowering-/.f64 N/A

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

    6. *-lowering-*.f64 N/A

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

    7. *-lowering-*.f64 N/A

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

    8. associate-/l* N/A

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

    9. *-lowering-*.f64 N/A

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

    10. /-lowering-/.f64 N/A

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

    11. *-lowering-*.f64 92.3

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

11. Applied egg-rr 92.3%

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

12. Final simplification 92.3%

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

Alternative 2: 81.8% accurate, 35.9× speedup

Math

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

FPCore

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

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return b * (b * (((a / (x_45_scale * y_45_scale)) * (a * -4.0)) / (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 * (b * (((a / (x_45scale * y_45scale)) * (a * (-4.0d0))) / (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 * (b * (((a / (x_45_scale * y_45_scale)) * (a * -4.0)) / (x_45_scale * y_45_scale)));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 28.0%

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

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

   2. associate-*l/ N/A

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

   3. /-lowering-/.f64 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. *-commutative N/A

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

   8. *-lowering-*.f64 N/A

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

   9. *-lowering-*.f64 N/A

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

   10. unpow2 N/A

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

   11. *-lowering-*.f64 N/A

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

   12. unpow2 N/A

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

   13. associate-*l* N/A

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

   14. *-lowering-*.f64 N/A

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

   15. *-lowering-*.f64 N/A

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

   16. unpow2 N/A

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

   17. *-lowering-*.f64 55.3

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

5. Simplified 55.3%

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

   1. associate-*l* N/A

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

   2. associate-/l* N/A

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

   3. *-lowering-*.f64 N/A

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

   4. /-lowering-/.f64 N/A

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

   5. *-lowering-*.f64 N/A

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

   6. *-lowering-*.f64 N/A

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

   7. *-lowering-*.f64 N/A

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

   8. *-lowering-*.f64 N/A

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

   9. *-lowering-*.f64 N/A

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

   10. *-lowering-*.f64 58.0

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

7. Applied egg-rr 58.0%

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

   1. associate-*l* N/A

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

   2. associate-/l* N/A

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

   3. *-lowering-*.f64 N/A

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

   4. /-lowering-/.f64 N/A

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

   5. associate-*l* N/A

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

   6. *-lowering-*.f64 N/A

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

   7. *-lowering-*.f64 N/A

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

   8. associate-*r* N/A

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

   9. unswap-sqr N/A

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

   10. *-lowering-*.f64 N/A

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

   11. *-lowering-*.f64 N/A

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

   12. *-lowering-*.f64 65.1

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

9. Applied egg-rr 65.1%

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

    1. times-frac N/A

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

    2. associate-*r/ N/A

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

    3. /-lowering-/.f64 N/A

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

    4. *-lowering-*.f64 N/A

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

    5. /-lowering-/.f64 N/A

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

    6. *-lowering-*.f64 N/A

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

    7. *-lowering-*.f64 N/A

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

    8. *-lowering-*.f64 82.2

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

11. Applied egg-rr 82.2%

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

Alternative 3: 79.5% accurate, 40.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = b * ((b * a) * (a * (-4.0 / ((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[(b * N[(N[(b * a), $MachinePrecision] * N[(a * N[(-4.0 / 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 28.0%

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

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

   2. associate-*l/ N/A

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

   3. /-lowering-/.f64 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. *-commutative N/A

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

   8. *-lowering-*.f64 N/A

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

   9. *-lowering-*.f64 N/A

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

   10. unpow2 N/A

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

   11. *-lowering-*.f64 N/A

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

   12. unpow2 N/A

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

   13. associate-*l* N/A

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

   14. *-lowering-*.f64 N/A

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

   15. *-lowering-*.f64 N/A

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

   16. unpow2 N/A

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

   17. *-lowering-*.f64 55.3

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

5. Simplified 55.3%

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

   1. associate-*l* N/A

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

   2. associate-/l* N/A

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

   3. *-lowering-*.f64 N/A

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

   4. /-lowering-/.f64 N/A

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

   5. *-lowering-*.f64 N/A

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

   6. *-lowering-*.f64 N/A

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

   7. *-lowering-*.f64 N/A

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

   8. *-lowering-*.f64 N/A

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

   9. *-lowering-*.f64 N/A

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

   10. *-lowering-*.f64 58.0

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

7. Applied egg-rr 58.0%

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

   1. associate-/l* N/A

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

   2. associate-*r* N/A

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

   3. associate-*l* N/A

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

   4. *-lowering-*.f64 N/A

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

   5. *-lowering-*.f64 N/A

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

   6. *-lowering-*.f64 N/A

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

   7. /-lowering-/.f64 N/A

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

   8. associate-*r* N/A

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

   9. unswap-sqr N/A

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

   10. *-lowering-*.f64 N/A

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

   11. *-lowering-*.f64 N/A

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

   12. *-lowering-*.f64 77.6

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

9. Applied egg-rr 77.6%

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

Alternative 4: 76.8% accurate, 40.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = b * (a * (b * ((a * -4.0) / (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[(b * N[(a * N[(b * N[(N[(a * -4.0), $MachinePrecision] / N[(x$45$scale * N[(y$45$scale * N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 28.0%

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

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

   2. associate-*l/ N/A

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

   3. /-lowering-/.f64 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. *-commutative N/A

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

   8. *-lowering-*.f64 N/A

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

   9. *-lowering-*.f64 N/A

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

   10. unpow2 N/A

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

   11. *-lowering-*.f64 N/A

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

   12. unpow2 N/A

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

   13. associate-*l* N/A

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

   14. *-lowering-*.f64 N/A

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

   15. *-lowering-*.f64 N/A

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

   16. unpow2 N/A

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

   17. *-lowering-*.f64 55.3

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

5. Simplified 55.3%

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

   1. associate-*l* N/A

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

   2. associate-/l* N/A

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

   3. *-lowering-*.f64 N/A

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

   4. /-lowering-/.f64 N/A

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

   5. *-lowering-*.f64 N/A

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

   6. *-lowering-*.f64 N/A

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

   7. *-lowering-*.f64 N/A

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

   8. *-lowering-*.f64 N/A

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

   9. *-lowering-*.f64 N/A

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

   10. *-lowering-*.f64 58.0

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

7. Applied egg-rr 58.0%

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

   1. associate-*l* N/A

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

   2. associate-/l* N/A

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

   3. *-lowering-*.f64 N/A

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

   4. /-lowering-/.f64 N/A

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

   5. associate-*l* N/A

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

   6. *-lowering-*.f64 N/A

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

   7. *-lowering-*.f64 N/A

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

   8. associate-*r* N/A

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

   9. unswap-sqr N/A

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

   10. *-lowering-*.f64 N/A

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

   11. *-lowering-*.f64 N/A

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

   12. *-lowering-*.f64 65.1

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

9. Applied egg-rr 65.1%

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

    1. *-commutative N/A

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

    2. swap-sqr N/A

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

    3. associate-*l* N/A

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

    4. associate-/l* N/A

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

    5. associate-*l* N/A

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

    6. *-lowering-*.f64 N/A

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

    7. *-lowering-*.f64 N/A

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

    8. /-lowering-/.f64 N/A

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

    9. *-lowering-*.f64 N/A

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

    10. associate-*l* N/A

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

    11. swap-sqr N/A

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

    12. associate-*l* N/A

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

    13. *-lowering-*.f64 N/A

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

    14. *-lowering-*.f64 N/A

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

    15. *-lowering-*.f64 74.7

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

11. Applied egg-rr 74.7%

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

12. Final simplification 74.7%

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

Alternative 5: 72.8% accurate, 40.5× speedup

Math

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

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (* a (* (* b (* b a)) (/ -4.0 (* 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 * ((b * (b * a)) * (-4.0 / (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 * ((b * (b * a)) * ((-4.0d0) / (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 * ((b * (b * a)) * (-4.0 / (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 * ((b * (b * a)) * (-4.0 / (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(a * Float64(Float64(b * Float64(b * a)) * Float64(-4.0 / Float64(x_45_scale * Float64(y_45_scale * Float64(x_45_scale * y_45_scale))))))
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = a * ((b * (b * a)) * (-4.0 / (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[(a * N[(N[(b * N[(b * a), $MachinePrecision]), $MachinePrecision] * N[(-4.0 / N[(x$45$scale * N[(y$45$scale * N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 28.0%

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

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

   2. associate-*l/ N/A

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

   3. /-lowering-/.f64 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. *-commutative N/A

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

   8. *-lowering-*.f64 N/A

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

   9. *-lowering-*.f64 N/A

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

   10. unpow2 N/A

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

   11. *-lowering-*.f64 N/A

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

   12. unpow2 N/A

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

   13. associate-*l* N/A

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

   14. *-lowering-*.f64 N/A

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

   15. *-lowering-*.f64 N/A

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

   16. unpow2 N/A

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

   17. *-lowering-*.f64 55.3

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

5. Simplified 55.3%

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

   1. associate-/r* N/A

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

   2. associate-*r* N/A

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

   3. associate-/r* N/A

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

   4. /-lowering-/.f64 N/A

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

   5. /-lowering-/.f64 N/A

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

   6. associate-*l* N/A

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

   7. associate-/l* N/A

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

   8. *-lowering-*.f64 N/A

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

   9. /-lowering-/.f64 N/A

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

   10. *-lowering-*.f64 N/A

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

   11. *-lowering-*.f64 N/A

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

   12. *-lowering-*.f64 N/A

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

   13. *-lowering-*.f64 71.1

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

7. Applied egg-rr 71.1%

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

   1. associate-/l* N/A

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

   2. associate-*r* N/A

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

   3. associate-*l* N/A

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

   4. *-lowering-*.f64 N/A

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

   5. *-lowering-*.f64 N/A

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

   6. *-lowering-*.f64 N/A

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

   7. /-lowering-/.f64 81.1

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

9. Applied egg-rr 81.1%

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

    1. associate-/l/ N/A

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

    2. associate-*r* N/A

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

    3. associate-*r/ N/A

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

    4. associate-*r/ N/A

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

    5. associate-*l* N/A

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

    6. associate-/r* N/A

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

    7. associate-*l* N/A

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

    8. swap-sqr N/A

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

    9. associate-*l* N/A

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

    10. associate-/l* N/A

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

    11. *-commutative N/A

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

    12. associate-*l* N/A

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

    13. *-lowering-*.f64 N/A

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

    14. *-lowering-*.f64 N/A

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

    15. *-lowering-*.f64 N/A

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

    16. *-lowering-*.f64 N/A

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

    17. /-lowering-/.f64 N/A

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

    18. associate-*l* N/A

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

    19. swap-sqr N/A

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

    20. associate-*l* N/A

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

11. Applied egg-rr 70.7%

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

Reproduce

herbie shell --seed 2024205 
(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))))