Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 25.3% → 92.5%

Time: 25.4s

Alternatives: 5

Speedup: 40.5×

• 33.2% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. a = 2.248963891840242e+248
  2. b = -7.832029783850953e+264
  3. angle = 2.0989525327009281e+80
  4. x-scale = -2.6914698863249054e-233
  5. y-scale = 3.4055012743460314e-295

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 25.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 92.5% accurate, 35.9× speedup

Math

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

FPCore

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

Julia

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

Wolfram

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

Derivation

1. Initial program 23.9%

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

3. Taylor expanded in angle around 0

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

   1. associate-*r/ N/A

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

   2. /-lowering-/.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

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

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

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

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

   7. unpow2 N/A

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

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

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

   9. unpow2 N/A

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

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

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

   11. unpow2 N/A

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

   12. associate-*l* N/A

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

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

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

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

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

   15. unpow2 N/A

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

   16. *-lowering-*.f64 57.0

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

5. Simplified 57.0%

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

   1. associate-*r* N/A

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

   2. associate-*r* N/A

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

   3. unswap-sqr N/A

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

   4. times-frac N/A

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

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

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

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

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

   7. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

   13. *-lowering-*.f64 79.6

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

7. Applied egg-rr 79.6%

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

   1. associate-*l* N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

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

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

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

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

   6. *-commutative N/A

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

   7. *-lowering-*.f64 87.0

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

9. Applied egg-rr 87.0%

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

    1. *-commutative N/A

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

    2. associate-/l* N/A

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

    3. associate-*r* N/A

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

    4. *-commutative N/A

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

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

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

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

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

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

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

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

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

    9. *-commutative N/A

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

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

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

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

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

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

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

    13. *-lowering-*.f64 96.4

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

11. Applied egg-rr 96.4%

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

12. Final simplification 96.4%

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

Alternative 2: 80.2% accurate, 29.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(\left(b \cdot -4\right) \cdot \frac{b \cdot a}{y-scale \cdot \left(x-scale \cdot \left(x-scale \cdot y-scale\right)\right)}\right)\\ \mathbf{if}\;b \leq 3.8 \cdot 10^{-163}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 5.9 \cdot 10^{+121}:\\ \;\;\;\;a \cdot \left(\frac{a}{x-scale \cdot y-scale} \cdot \frac{-4 \cdot \left(b \cdot b\right)}{x-scale \cdot y-scale}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0
         (*
          a
          (*
           (* b -4.0)
           (/ (* b a) (* y-scale (* x-scale (* x-scale y-scale))))))))
   (if (<= b 3.8e-163)
     t_0
     (if (<= b 5.9e+121)
       (*
        a
        (* (/ a (* x-scale y-scale)) (/ (* -4.0 (* b b)) (* x-scale y-scale))))
       t_0))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = a * ((b * -4.0) * ((b * a) / (y_45_scale * (x_45_scale * (x_45_scale * y_45_scale)))));
	double tmp;
	if (b <= 3.8e-163) {
		tmp = t_0;
	} else if (b <= 5.9e+121) {
		tmp = a * ((a / (x_45_scale * y_45_scale)) * ((-4.0 * (b * b)) / (x_45_scale * y_45_scale)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a * ((b * (-4.0d0)) * ((b * a) / (y_45scale * (x_45scale * (x_45scale * y_45scale)))))
    if (b <= 3.8d-163) then
        tmp = t_0
    else if (b <= 5.9d+121) then
        tmp = a * ((a / (x_45scale * y_45scale)) * (((-4.0d0) * (b * b)) / (x_45scale * y_45scale)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = a * ((b * -4.0) * ((b * a) / (y_45_scale * (x_45_scale * (x_45_scale * y_45_scale)))));
	double tmp;
	if (b <= 3.8e-163) {
		tmp = t_0;
	} else if (b <= 5.9e+121) {
		tmp = a * ((a / (x_45_scale * y_45_scale)) * ((-4.0 * (b * b)) / (x_45_scale * y_45_scale)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = a * ((b * -4.0) * ((b * a) / (y_45_scale * (x_45_scale * (x_45_scale * y_45_scale)))))
	tmp = 0
	if b <= 3.8e-163:
		tmp = t_0
	elif b <= 5.9e+121:
		tmp = a * ((a / (x_45_scale * y_45_scale)) * ((-4.0 * (b * b)) / (x_45_scale * y_45_scale)))
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(a * Float64(Float64(b * -4.0) * Float64(Float64(b * a) / Float64(y_45_scale * Float64(x_45_scale * Float64(x_45_scale * y_45_scale))))))
	tmp = 0.0
	if (b <= 3.8e-163)
		tmp = t_0;
	elseif (b <= 5.9e+121)
		tmp = Float64(a * Float64(Float64(a / Float64(x_45_scale * y_45_scale)) * Float64(Float64(-4.0 * Float64(b * b)) / Float64(x_45_scale * y_45_scale))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = a * ((b * -4.0) * ((b * a) / (y_45_scale * (x_45_scale * (x_45_scale * y_45_scale)))));
	tmp = 0.0;
	if (b <= 3.8e-163)
		tmp = t_0;
	elseif (b <= 5.9e+121)
		tmp = a * ((a / (x_45_scale * y_45_scale)) * ((-4.0 * (b * b)) / (x_45_scale * y_45_scale)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(a * N[(N[(b * -4.0), $MachinePrecision] * N[(N[(b * a), $MachinePrecision] / N[(y$45$scale * N[(x$45$scale * N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 3.8e-163], t$95$0, If[LessEqual[b, 5.9e+121], N[(a * N[(N[(a / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(-4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < 3.8e-163 or 5.90000000000000014e121 < b

   1. Initial program 20.9%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

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

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

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

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

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

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 51.6

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

   5. Simplified 51.6%

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. unswap-sqr N/A

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

      4. times-frac N/A

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

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

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

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

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

      7. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

      13. *-lowering-*.f64 74.6

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

   7. Applied egg-rr 74.6%

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

      1. frac-times N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

      9. associate-*r* N/A

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

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

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

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

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

      12. *-lowering-*.f64 67.8

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

   9. Applied egg-rr 67.8%

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

       1. associate-*l* N/A

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

       2. associate-*l* N/A

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

       3. associate-/l* N/A

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

       4. *-commutative N/A

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

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

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

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

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

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

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

       8. *-commutative N/A

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

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

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

       10. *-commutative N/A

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

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

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

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

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

       13. *-lowering-*.f64 81.6

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

   11. Applied egg-rr 81.6%

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

   if 3.8e-163 < b < 5.90000000000000014e121

   1. Initial program 33.3%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

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

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

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

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

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

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 74.1

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

   5. Simplified 74.1%

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. unswap-sqr N/A

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

      4. times-frac N/A

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

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

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

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

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

      7. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

      13. *-lowering-*.f64 95.0

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

   7. Applied egg-rr 95.0%

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

      1. frac-times N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. frac-times N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      15. *-lowering-*.f64 92.9

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

   9. Applied egg-rr 92.9%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.8 \cdot 10^{-163}:\\ \;\;\;\;a \cdot \left(\left(b \cdot -4\right) \cdot \frac{b \cdot a}{y-scale \cdot \left(x-scale \cdot \left(x-scale \cdot y-scale\right)\right)}\right)\\ \mathbf{elif}\;b \leq 5.9 \cdot 10^{+121}:\\ \;\;\;\;a \cdot \left(\frac{a}{x-scale \cdot y-scale} \cdot \frac{-4 \cdot \left(b \cdot b\right)}{x-scale \cdot y-scale}\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(b \cdot -4\right) \cdot \frac{b \cdot a}{y-scale \cdot \left(x-scale \cdot \left(x-scale \cdot y-scale\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.9% accurate, 35.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 23.9%

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

3. Taylor expanded in angle around 0

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

   1. associate-*r/ N/A

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

   2. /-lowering-/.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

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

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

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

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

   7. unpow2 N/A

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

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

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

   9. unpow2 N/A

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

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

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

   11. unpow2 N/A

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

   12. associate-*l* N/A

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

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

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

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

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

   15. unpow2 N/A

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

   16. *-lowering-*.f64 57.0

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

5. Simplified 57.0%

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

   1. associate-*r* N/A

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

   2. associate-*r* N/A

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

   3. unswap-sqr N/A

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

   4. times-frac N/A

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

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

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

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

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

   7. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

   13. *-lowering-*.f64 79.6

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

7. Applied egg-rr 79.6%

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

   1. associate-*l* N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

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

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

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

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

   6. *-commutative N/A

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

   7. *-lowering-*.f64 87.0

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

9. Applied egg-rr 87.0%

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

    1. associate-/l* N/A

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

    2. associate-*l* N/A

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

    3. *-commutative N/A

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

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

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

    5. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

    12. *-lowering-*.f64 93.4

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

11. Applied egg-rr 93.4%

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

12. Final simplification 93.4%

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

Alternative 4: 89.8% accurate, 35.9× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 23.9%

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

3. Taylor expanded in angle around 0

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

   1. associate-*r/ N/A

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

   2. /-lowering-/.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

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

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

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

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

   7. unpow2 N/A

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

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

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

   9. unpow2 N/A

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

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

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

   11. unpow2 N/A

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

   12. associate-*l* N/A

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

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

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

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

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

   15. unpow2 N/A

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

   16. *-lowering-*.f64 57.0

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

5. Simplified 57.0%

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

   1. associate-*r* N/A

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

   2. associate-*r* N/A

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

   3. unswap-sqr N/A

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

   4. times-frac N/A

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

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

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

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

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

   7. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

   13. *-lowering-*.f64 79.6

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

7. Applied egg-rr 79.6%

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

   1. frac-times N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

   9. associate-*r* N/A

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

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

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

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

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

   12. *-lowering-*.f64 71.4

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

9. Applied egg-rr 71.4%

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

    1. associate-*l* N/A

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

    2. associate-*l* N/A

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

    3. *-commutative N/A

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

    4. associate-*l* N/A

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

    5. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

    12. *-lowering-*.f64 92.7

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

11. Applied egg-rr 92.7%

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

12. Final simplification 92.7%

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

Alternative 5: 77.9% accurate, 40.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[(a * N[(N[(b * -4.0), $MachinePrecision] * N[(N[(b * a), $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 23.9%

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

3. Taylor expanded in angle around 0

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

   1. associate-*r/ N/A

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

   2. /-lowering-/.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

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

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

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

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

   7. unpow2 N/A

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

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

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

   9. unpow2 N/A

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

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

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

   11. unpow2 N/A

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

   12. associate-*l* N/A

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

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

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

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

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

   15. unpow2 N/A

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

   16. *-lowering-*.f64 57.0

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

5. Simplified 57.0%

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

   1. associate-*r* N/A

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

   2. associate-*r* N/A

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

   3. unswap-sqr N/A

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

   4. times-frac N/A

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

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

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

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

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

   7. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

   13. *-lowering-*.f64 79.6

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

7. Applied egg-rr 79.6%

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

   1. frac-times N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

   9. associate-*r* N/A

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

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

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

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

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

   12. *-lowering-*.f64 71.4

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

9. Applied egg-rr 71.4%

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

    1. associate-*l* N/A

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

    2. associate-*l* N/A

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

    3. associate-/l* N/A

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

    4. *-commutative N/A

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

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

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

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

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

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

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

    8. *-commutative N/A

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

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

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

    10. *-commutative N/A

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

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

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

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

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

    13. *-lowering-*.f64 82.3

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

11. Applied egg-rr 82.3%

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

12. Final simplification 82.3%

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

Reproduce

herbie shell --seed 2024198 
(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))))