Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 25.2% → 89.9%

Time: 24.9s

Alternatives: 5

Speedup: 40.5×

• 33.1% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. a = 5.720280650863164e-65
  2. b = 7.045183375892393e+291
  3. angle = 2.84759400892301e+295
  4. x-scale = -4.1401576775888e+21
  5. y-scale = 2.7378003224177702e-281

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: 25.2% 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: 89.9% accurate, 35.9× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 27.5%

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

3. Taylor expanded in angle around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. associate-*l* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 56.0

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

5. Applied rewrites 56.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/l* N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*r* N/A

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

   11. associate-*l* N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 63.4

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

7. Applied rewrites 63.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. times-frac N/A

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

   4. associate-*r/ N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*r* N/A

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

   9. lift-*.f64 N/A

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

   10. times-frac N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-/.f64 82.9

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

9. Applied rewrites 82.9%

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

    1. lift-*.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-/.f64 N/A

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

    4. associate-/r* N/A

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

    5. lift-*.f64 N/A

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

    6. lift-/.f64 N/A

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

    7. associate-*r* N/A

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

    8. associate-*r* N/A

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

    9. *-commutative N/A

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

11. Applied rewrites 90.8%

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

Alternative 2: 89.2% accurate, 35.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 27.5%

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

3. Taylor expanded in angle around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. associate-*l* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 56.0

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

5. Applied rewrites 56.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/l* N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*r* N/A

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

   11. associate-*l* N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 63.4

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

7. Applied rewrites 63.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. times-frac N/A

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

   4. associate-*r/ N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*r* N/A

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

   9. lift-*.f64 N/A

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

   10. times-frac N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-/.f64 82.9

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

9. Applied rewrites 82.9%

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

    1. lift-*.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. associate-/r* N/A

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

    4. lift-*.f64 N/A

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

    5. lift-/.f64 N/A

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

    6. associate-*r* N/A

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

    7. lift-/.f64 N/A

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

    8. associate-*r/ N/A

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

    9. lower-/.f64 N/A

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

    10. *-commutative N/A

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

    11. lower-*.f64 N/A

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

    12. lift-/.f64 N/A

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

    13. associate-*r/ N/A

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

    14. lower-/.f64 N/A

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

    15. *-commutative N/A

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

    16. lower-*.f64 89.0

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

11. Applied rewrites 89.0%

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

Alternative 3: 85.6% accurate, 35.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 27.5%

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

3. Taylor expanded in angle around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. associate-*l* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 56.0

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

5. Applied rewrites 56.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/l* N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*r* N/A

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

   11. associate-*l* N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 63.4

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

7. Applied rewrites 63.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. times-frac N/A

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

   4. associate-*r/ N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*r* N/A

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

   9. lift-*.f64 N/A

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

   10. times-frac N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-/.f64 82.9

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

9. Applied rewrites 82.9%

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

    1. lift-*.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. lift-/.f64 N/A

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

    4. lift-/.f64 N/A

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

    5. lift-*.f64 82.9

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

    6. lift-/.f64 N/A

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

    7. lift-/.f64 N/A

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

    8. associate-/r* N/A

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

    9. lift-*.f64 N/A

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

    10. lift-/.f64 84.0

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

11. Applied rewrites 84.0%

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

Alternative 4: 78.1% accurate, 40.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 27.5%

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

3. Taylor expanded in angle around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. associate-*l* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 56.0

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

5. Applied rewrites 56.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*r* N/A

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

   8. times-frac N/A

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

   9. lower-*.f64 N/A

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

7. Applied rewrites 68.3%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. frac-times N/A

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

9. Applied rewrites 73.9%

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

    1. associate-*r* N/A

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

    2. lift-*.f64 N/A

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

    3. lower-*.f64 81.9

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

11. Applied rewrites 81.9%

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

12. Final simplification 81.9%

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

Alternative 5: 70.3% accurate, 40.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 27.5%

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

3. Taylor expanded in angle around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. associate-*l* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 56.0

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

5. Applied rewrites 56.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*r* N/A

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

   8. times-frac N/A

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

   9. lower-*.f64 N/A

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

7. Applied rewrites 68.3%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. frac-times N/A

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

9. Applied rewrites 73.9%

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

Reproduce

herbie shell --seed 2024219 
(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))))