Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 24.6% → 93.5%

Time: 33.4s

Alternatives: 9

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 = 1.3615917866522844e+307
  2. b = -3.824814947932823e+174
  3. angle = -1.9560804376891005e+195
  4. x-scale = 1.2195172002082184e-165
  5. y-scale = -7.199770577778049e-305

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 24.6% 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: 93.5% accurate, 35.9× speedup

Math

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

FPCore

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

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (b / (y_45_scale * x_45_scale)) * a;
	return -4.0 * (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 = (b / (y_45scale * x_45scale)) * a
    code = (-4.0d0) * (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 = (b / (y_45_scale * x_45_scale)) * a;
	return -4.0 * (t_0 * t_0);
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 27.4%

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

3. Taylor expanded in angle around 0

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

   1. associate-/l* N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. unpow2 N/A

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

   12. associate-*r* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 51.1

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

5. Applied rewrites 51.1%

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

6. Taylor expanded in angle around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. unpow2 N/A

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

   12. unpow2 N/A

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

   13. unswap-sqr N/A

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

   14. lower-*.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-*.f64 57.5

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

8. Applied rewrites 57.5%

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

10. Applied rewrites 93.4%

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

11. Final simplification 93.4%

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

Alternative 2: 77.4% accurate, 29.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b}{y-scale \cdot x-scale}\\ \mathbf{if}\;a \leq 3.8 \cdot 10^{-164}:\\ \;\;\;\;\left(\frac{a \cdot b}{\left(y-scale \cdot y-scale\right) \cdot x-scale} \cdot \frac{-4 \cdot a}{x-scale}\right) \cdot b\\ \mathbf{elif}\;a \leq 6.8 \cdot 10^{+153}:\\ \;\;\;\;\left(t\_0 \cdot t\_0\right) \cdot \left(\left(a \cdot a\right) \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(y-scale \cdot x-scale\right) \cdot y-scale\right) \cdot x-scale} \cdot \left(\left(\left(-4 \cdot b\right) \cdot a\right) \cdot \left(a \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ b (* y-scale x-scale))))
   (if (<= a 3.8e-164)
     (*
      (* (/ (* a b) (* (* y-scale y-scale) x-scale)) (/ (* -4.0 a) x-scale))
      b)
     (if (<= a 6.8e+153)
       (* (* t_0 t_0) (* (* a a) -4.0))
       (*
        (/ 1.0 (* (* (* y-scale x-scale) y-scale) x-scale))
        (* (* (* -4.0 b) a) (* a b)))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = b / (y_45_scale * x_45_scale);
	double tmp;
	if (a <= 3.8e-164) {
		tmp = (((a * b) / ((y_45_scale * y_45_scale) * x_45_scale)) * ((-4.0 * a) / x_45_scale)) * b;
	} else if (a <= 6.8e+153) {
		tmp = (t_0 * t_0) * ((a * a) * -4.0);
	} else {
		tmp = (1.0 / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale)) * (((-4.0 * b) * a) * (a * b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: tmp
    t_0 = b / (y_45scale * x_45scale)
    if (a <= 3.8d-164) then
        tmp = (((a * b) / ((y_45scale * y_45scale) * x_45scale)) * (((-4.0d0) * a) / x_45scale)) * b
    else if (a <= 6.8d+153) then
        tmp = (t_0 * t_0) * ((a * a) * (-4.0d0))
    else
        tmp = (1.0d0 / (((y_45scale * x_45scale) * y_45scale) * x_45scale)) * ((((-4.0d0) * b) * a) * (a * b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = b / (y_45_scale * x_45_scale);
	double tmp;
	if (a <= 3.8e-164) {
		tmp = (((a * b) / ((y_45_scale * y_45_scale) * x_45_scale)) * ((-4.0 * a) / x_45_scale)) * b;
	} else if (a <= 6.8e+153) {
		tmp = (t_0 * t_0) * ((a * a) * -4.0);
	} else {
		tmp = (1.0 / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale)) * (((-4.0 * b) * a) * (a * b));
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = b / (y_45_scale * x_45_scale)
	tmp = 0
	if a <= 3.8e-164:
		tmp = (((a * b) / ((y_45_scale * y_45_scale) * x_45_scale)) * ((-4.0 * a) / x_45_scale)) * b
	elif a <= 6.8e+153:
		tmp = (t_0 * t_0) * ((a * a) * -4.0)
	else:
		tmp = (1.0 / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale)) * (((-4.0 * b) * a) * (a * b))
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(b / Float64(y_45_scale * x_45_scale))
	tmp = 0.0
	if (a <= 3.8e-164)
		tmp = Float64(Float64(Float64(Float64(a * b) / Float64(Float64(y_45_scale * y_45_scale) * x_45_scale)) * Float64(Float64(-4.0 * a) / x_45_scale)) * b);
	elseif (a <= 6.8e+153)
		tmp = Float64(Float64(t_0 * t_0) * Float64(Float64(a * a) * -4.0));
	else
		tmp = Float64(Float64(1.0 / Float64(Float64(Float64(y_45_scale * x_45_scale) * y_45_scale) * x_45_scale)) * Float64(Float64(Float64(-4.0 * b) * a) * Float64(a * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = b / (y_45_scale * x_45_scale);
	tmp = 0.0;
	if (a <= 3.8e-164)
		tmp = (((a * b) / ((y_45_scale * y_45_scale) * x_45_scale)) * ((-4.0 * a) / x_45_scale)) * b;
	elseif (a <= 6.8e+153)
		tmp = (t_0 * t_0) * ((a * a) * -4.0);
	else
		tmp = (1.0 / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale)) * (((-4.0 * b) * a) * (a * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(b / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 3.8e-164], N[(N[(N[(N[(a * b), $MachinePrecision] / N[(N[(y$45$scale * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(-4.0 * a), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[a, 6.8e+153], N[(N[(t$95$0 * t$95$0), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(-4.0 * b), $MachinePrecision] * a), $MachinePrecision] * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < 3.79999999999999989e-164

   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-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 49.8

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

   5. Applied rewrites 49.8%

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

   7. Applied rewrites 71.5%

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

   9. Applied rewrites 62.4%

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

   11. Applied rewrites 76.4%

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

   if 3.79999999999999989e-164 < a < 6.7999999999999995e153

   1. Initial program 25.3%

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

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 65.9

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

   5. Applied rewrites 65.9%

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

   7. Applied rewrites 92.8%

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

   if 6.7999999999999995e153 < a

   1. Initial program 0.0%

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

   3. Taylor expanded in angle around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 29.2

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

   5. Applied rewrites 29.2%

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

   7. Applied rewrites 32.8%

      \[\leadsto \left(\left(\left(\left(a \cdot a\right) \cdot -4\right) \cdot b\right) \cdot b\right) \cdot \color{blue}{\frac{1}{\left(\left(y-scale \cdot y-scale\right) \cdot x-scale\right) \cdot x-scale}} \]
   8. Step-by-step derivation

   9. Applied rewrites 32.9%

      \[\leadsto \left(\left(\left(\left(a \cdot a\right) \cdot -4\right) \cdot b\right) \cdot b\right) \cdot \frac{1}{\left(\left(x-scale \cdot y-scale\right) \cdot y-scale\right) \cdot x-scale} \]
   10. Step-by-step derivation

   11. Applied rewrites 64.8%

       \[\leadsto \left(\left(\left(-4 \cdot b\right) \cdot a\right) \cdot \left(a \cdot b\right)\right) \cdot \frac{\color{blue}{1}}{\left(\left(x-scale \cdot y-scale\right) \cdot y-scale\right) \cdot x-scale} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3.8 \cdot 10^{-164}:\\ \;\;\;\;\left(\frac{a \cdot b}{\left(y-scale \cdot y-scale\right) \cdot x-scale} \cdot \frac{-4 \cdot a}{x-scale}\right) \cdot b\\ \mathbf{elif}\;a \leq 6.8 \cdot 10^{+153}:\\ \;\;\;\;\left(\frac{b}{y-scale \cdot x-scale} \cdot \frac{b}{y-scale \cdot x-scale}\right) \cdot \left(\left(a \cdot a\right) \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(y-scale \cdot x-scale\right) \cdot y-scale\right) \cdot x-scale} \cdot \left(\left(\left(-4 \cdot b\right) \cdot a\right) \cdot \left(a \cdot b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.8% accurate, 29.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(y-scale \cdot x-scale\right) \cdot y-scale\right) \cdot x-scale\\ \mathbf{if}\;y-scale \leq 2.5 \cdot 10^{-98}:\\ \;\;\;\;\left(\left(\frac{b}{t\_0} \cdot b\right) \cdot \left(-4 \cdot a\right)\right) \cdot a\\ \mathbf{elif}\;y-scale \leq 6.8 \cdot 10^{+146}:\\ \;\;\;\;\left(\frac{a \cdot b}{\left(y-scale \cdot y-scale\right) \cdot x-scale} \cdot \frac{-4 \cdot a}{x-scale}\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_0} \cdot \left(\left(\left(-4 \cdot b\right) \cdot a\right) \cdot \left(a \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (* (* y-scale x-scale) y-scale) x-scale)))
   (if (<= y-scale 2.5e-98)
     (* (* (* (/ b t_0) b) (* -4.0 a)) a)
     (if (<= y-scale 6.8e+146)
       (*
        (* (/ (* a b) (* (* y-scale y-scale) x-scale)) (/ (* -4.0 a) x-scale))
        b)
       (* (/ 1.0 t_0) (* (* (* -4.0 b) a) (* a b)))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = ((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale;
	double tmp;
	if (y_45_scale <= 2.5e-98) {
		tmp = (((b / t_0) * b) * (-4.0 * a)) * a;
	} else if (y_45_scale <= 6.8e+146) {
		tmp = (((a * b) / ((y_45_scale * y_45_scale) * x_45_scale)) * ((-4.0 * a) / x_45_scale)) * b;
	} else {
		tmp = (1.0 / t_0) * (((-4.0 * b) * a) * (a * b));
	}
	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 = ((y_45scale * x_45scale) * y_45scale) * x_45scale
    if (y_45scale <= 2.5d-98) then
        tmp = (((b / t_0) * b) * ((-4.0d0) * a)) * a
    else if (y_45scale <= 6.8d+146) then
        tmp = (((a * b) / ((y_45scale * y_45scale) * x_45scale)) * (((-4.0d0) * a) / x_45scale)) * b
    else
        tmp = (1.0d0 / t_0) * ((((-4.0d0) * b) * a) * (a * b))
    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 = ((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale;
	double tmp;
	if (y_45_scale <= 2.5e-98) {
		tmp = (((b / t_0) * b) * (-4.0 * a)) * a;
	} else if (y_45_scale <= 6.8e+146) {
		tmp = (((a * b) / ((y_45_scale * y_45_scale) * x_45_scale)) * ((-4.0 * a) / x_45_scale)) * b;
	} else {
		tmp = (1.0 / t_0) * (((-4.0 * b) * a) * (a * b));
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = ((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale
	tmp = 0
	if y_45_scale <= 2.5e-98:
		tmp = (((b / t_0) * b) * (-4.0 * a)) * a
	elif y_45_scale <= 6.8e+146:
		tmp = (((a * b) / ((y_45_scale * y_45_scale) * x_45_scale)) * ((-4.0 * a) / x_45_scale)) * b
	else:
		tmp = (1.0 / t_0) * (((-4.0 * b) * a) * (a * b))
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(Float64(y_45_scale * x_45_scale) * y_45_scale) * x_45_scale)
	tmp = 0.0
	if (y_45_scale <= 2.5e-98)
		tmp = Float64(Float64(Float64(Float64(b / t_0) * b) * Float64(-4.0 * a)) * a);
	elseif (y_45_scale <= 6.8e+146)
		tmp = Float64(Float64(Float64(Float64(a * b) / Float64(Float64(y_45_scale * y_45_scale) * x_45_scale)) * Float64(Float64(-4.0 * a) / x_45_scale)) * b);
	else
		tmp = Float64(Float64(1.0 / t_0) * Float64(Float64(Float64(-4.0 * b) * a) * Float64(a * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = ((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale;
	tmp = 0.0;
	if (y_45_scale <= 2.5e-98)
		tmp = (((b / t_0) * b) * (-4.0 * a)) * a;
	elseif (y_45_scale <= 6.8e+146)
		tmp = (((a * b) / ((y_45_scale * y_45_scale) * x_45_scale)) * ((-4.0 * a) / x_45_scale)) * b;
	else
		tmp = (1.0 / t_0) * (((-4.0 * b) * a) * (a * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision]}, If[LessEqual[y$45$scale, 2.5e-98], N[(N[(N[(N[(b / t$95$0), $MachinePrecision] * b), $MachinePrecision] * N[(-4.0 * a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[y$45$scale, 6.8e+146], N[(N[(N[(N[(a * b), $MachinePrecision] / N[(N[(y$45$scale * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(-4.0 * a), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], N[(N[(1.0 / t$95$0), $MachinePrecision] * N[(N[(N[(-4.0 * b), $MachinePrecision] * a), $MachinePrecision] * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y-scale < 2.50000000000000009e-98

   1. Initial program 24.1%

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

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 47.1

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

   5. Applied rewrites 47.1%

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

   7. Applied rewrites 65.3%

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

   9. Applied rewrites 74.5%

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

   if 2.50000000000000009e-98 < y-scale < 6.79999999999999981e146

   1. Initial program 31.1%

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

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 68.5

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

   5. Applied rewrites 68.5%

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

   7. Applied rewrites 79.5%

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

   9. Applied rewrites 74.5%

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

   11. Applied rewrites 93.3%

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

   if 6.79999999999999981e146 < y-scale

   1. Initial program 36.4%

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

   3. Taylor expanded in angle around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 37.9

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

   5. Applied rewrites 37.9%

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

   7. Applied rewrites 53.0%

      \[\leadsto \left(\left(\left(\left(a \cdot a\right) \cdot -4\right) \cdot b\right) \cdot b\right) \cdot \color{blue}{\frac{1}{\left(\left(y-scale \cdot y-scale\right) \cdot x-scale\right) \cdot x-scale}} \]
   8. Step-by-step derivation

   9. Applied rewrites 68.2%

      \[\leadsto \left(\left(\left(\left(a \cdot a\right) \cdot -4\right) \cdot b\right) \cdot b\right) \cdot \frac{1}{\left(\left(x-scale \cdot y-scale\right) \cdot y-scale\right) \cdot x-scale} \]
   10. Step-by-step derivation

   11. Applied rewrites 73.3%

       \[\leadsto \left(\left(\left(-4 \cdot b\right) \cdot a\right) \cdot \left(a \cdot b\right)\right) \cdot \frac{\color{blue}{1}}{\left(\left(x-scale \cdot y-scale\right) \cdot y-scale\right) \cdot x-scale} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 2.5 \cdot 10^{-98}:\\ \;\;\;\;\left(\left(\frac{b}{\left(\left(y-scale \cdot x-scale\right) \cdot y-scale\right) \cdot x-scale} \cdot b\right) \cdot \left(-4 \cdot a\right)\right) \cdot a\\ \mathbf{elif}\;y-scale \leq 6.8 \cdot 10^{+146}:\\ \;\;\;\;\left(\frac{a \cdot b}{\left(y-scale \cdot y-scale\right) \cdot x-scale} \cdot \frac{-4 \cdot a}{x-scale}\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(y-scale \cdot x-scale\right) \cdot y-scale\right) \cdot x-scale} \cdot \left(\left(\left(-4 \cdot b\right) \cdot a\right) \cdot \left(a \cdot b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.9% accurate, 29.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left(\left(y-scale \cdot x-scale\right) \cdot y-scale\right) \cdot x-scale} \cdot \left(\left(\left(-4 \cdot b\right) \cdot a\right) \cdot \left(a \cdot b\right)\right)\\ \mathbf{if}\;a \leq 5.5 \cdot 10^{-146}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 5 \cdot 10^{+151}:\\ \;\;\;\;\frac{\left(\left(-4 \cdot a\right) \cdot a\right) \cdot b}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \cdot b\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0
         (*
          (/ 1.0 (* (* (* y-scale x-scale) y-scale) x-scale))
          (* (* (* -4.0 b) a) (* a b)))))
   (if (<= a 5.5e-146)
     t_0
     (if (<= a 5e+151)
       (*
        (/ (* (* (* -4.0 a) a) b) (* (* y-scale x-scale) (* y-scale x-scale)))
        b)
       t_0))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (1.0 / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale)) * (((-4.0 * b) * a) * (a * b));
	double tmp;
	if (a <= 5.5e-146) {
		tmp = t_0;
	} else if (a <= 5e+151) {
		tmp = ((((-4.0 * a) * a) * b) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * b;
	} 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 = (1.0d0 / (((y_45scale * x_45scale) * y_45scale) * x_45scale)) * ((((-4.0d0) * b) * a) * (a * b))
    if (a <= 5.5d-146) then
        tmp = t_0
    else if (a <= 5d+151) then
        tmp = (((((-4.0d0) * a) * a) * b) / ((y_45scale * x_45scale) * (y_45scale * x_45scale))) * b
    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 = (1.0 / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale)) * (((-4.0 * b) * a) * (a * b));
	double tmp;
	if (a <= 5.5e-146) {
		tmp = t_0;
	} else if (a <= 5e+151) {
		tmp = ((((-4.0 * a) * a) * b) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * b;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (1.0 / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale)) * (((-4.0 * b) * a) * (a * b))
	tmp = 0
	if a <= 5.5e-146:
		tmp = t_0
	elif a <= 5e+151:
		tmp = ((((-4.0 * a) * a) * b) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * b
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(1.0 / Float64(Float64(Float64(y_45_scale * x_45_scale) * y_45_scale) * x_45_scale)) * Float64(Float64(Float64(-4.0 * b) * a) * Float64(a * b)))
	tmp = 0.0
	if (a <= 5.5e-146)
		tmp = t_0;
	elseif (a <= 5e+151)
		tmp = Float64(Float64(Float64(Float64(Float64(-4.0 * a) * a) * b) / Float64(Float64(y_45_scale * x_45_scale) * Float64(y_45_scale * x_45_scale))) * b);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (1.0 / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale)) * (((-4.0 * b) * a) * (a * b));
	tmp = 0.0;
	if (a <= 5.5e-146)
		tmp = t_0;
	elseif (a <= 5e+151)
		tmp = ((((-4.0 * a) * a) * b) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * b;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(1.0 / N[(N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(-4.0 * b), $MachinePrecision] * a), $MachinePrecision] * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 5.5e-146], t$95$0, If[LessEqual[a, 5e+151], N[(N[(N[(N[(N[(-4.0 * a), $MachinePrecision] * a), $MachinePrecision] * b), $MachinePrecision] / N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < 5.49999999999999998e-146 or 5.0000000000000002e151 < a

   1. Initial program 27.7%

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

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 46.6

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

   5. Applied rewrites 46.6%

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

   7. Applied rewrites 54.4%

      \[\leadsto \left(\left(\left(\left(a \cdot a\right) \cdot -4\right) \cdot b\right) \cdot b\right) \cdot \color{blue}{\frac{1}{\left(\left(y-scale \cdot y-scale\right) \cdot x-scale\right) \cdot x-scale}} \]
   8. Step-by-step derivation

   9. Applied rewrites 58.7%

      \[\leadsto \left(\left(\left(\left(a \cdot a\right) \cdot -4\right) \cdot b\right) \cdot b\right) \cdot \frac{1}{\left(\left(x-scale \cdot y-scale\right) \cdot y-scale\right) \cdot x-scale} \]
   10. Step-by-step derivation

   11. Applied rewrites 75.9%

       \[\leadsto \left(\left(\left(-4 \cdot b\right) \cdot a\right) \cdot \left(a \cdot b\right)\right) \cdot \frac{\color{blue}{1}}{\left(\left(x-scale \cdot y-scale\right) \cdot y-scale\right) \cdot x-scale} \]

   if 5.49999999999999998e-146 < a < 5.0000000000000002e151

   1. Initial program 26.1%

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

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 66.5

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

   5. Applied rewrites 66.5%

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

   7. Applied rewrites 94.0%

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

   9. Applied rewrites 71.6%

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

   11. Applied rewrites 88.5%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 5.5 \cdot 10^{-146}:\\ \;\;\;\;\frac{1}{\left(\left(y-scale \cdot x-scale\right) \cdot y-scale\right) \cdot x-scale} \cdot \left(\left(\left(-4 \cdot b\right) \cdot a\right) \cdot \left(a \cdot b\right)\right)\\ \mathbf{elif}\;a \leq 5 \cdot 10^{+151}:\\ \;\;\;\;\frac{\left(\left(-4 \cdot a\right) \cdot a\right) \cdot b}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \cdot b\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(y-scale \cdot x-scale\right) \cdot y-scale\right) \cdot x-scale} \cdot \left(\left(\left(-4 \cdot b\right) \cdot a\right) \cdot \left(a \cdot b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.6% accurate, 32.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(\frac{b}{\left(\left(y-scale \cdot x-scale\right) \cdot y-scale\right) \cdot x-scale} \cdot b\right) \cdot \left(-4 \cdot a\right)\right) \cdot a\\ \mathbf{if}\;y-scale \leq 2.5 \cdot 10^{-98}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y-scale \leq 4.15 \cdot 10^{+152}:\\ \;\;\;\;\left(\frac{a \cdot b}{\left(\left(y-scale \cdot y-scale\right) \cdot x-scale\right) \cdot x-scale} \cdot \left(-4 \cdot a\right)\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0
         (*
          (*
           (* (/ b (* (* (* y-scale x-scale) y-scale) x-scale)) b)
           (* -4.0 a))
          a)))
   (if (<= y-scale 2.5e-98)
     t_0
     (if (<= y-scale 4.15e+152)
       (*
        (* (/ (* a b) (* (* (* y-scale y-scale) x-scale) x-scale)) (* -4.0 a))
        b)
       t_0))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (((b / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale)) * b) * (-4.0 * a)) * a;
	double tmp;
	if (y_45_scale <= 2.5e-98) {
		tmp = t_0;
	} else if (y_45_scale <= 4.15e+152) {
		tmp = (((a * b) / (((y_45_scale * y_45_scale) * x_45_scale) * x_45_scale)) * (-4.0 * a)) * b;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((b / (((y_45scale * x_45scale) * y_45scale) * x_45scale)) * b) * ((-4.0d0) * a)) * a
    if (y_45scale <= 2.5d-98) then
        tmp = t_0
    else if (y_45scale <= 4.15d+152) then
        tmp = (((a * b) / (((y_45scale * y_45scale) * x_45scale) * x_45scale)) * ((-4.0d0) * a)) * b
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (((b / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale)) * b) * (-4.0 * a)) * a;
	double tmp;
	if (y_45_scale <= 2.5e-98) {
		tmp = t_0;
	} else if (y_45_scale <= 4.15e+152) {
		tmp = (((a * b) / (((y_45_scale * y_45_scale) * x_45_scale) * x_45_scale)) * (-4.0 * a)) * b;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (((b / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale)) * b) * (-4.0 * a)) * a
	tmp = 0
	if y_45_scale <= 2.5e-98:
		tmp = t_0
	elif y_45_scale <= 4.15e+152:
		tmp = (((a * b) / (((y_45_scale * y_45_scale) * x_45_scale) * x_45_scale)) * (-4.0 * a)) * b
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(Float64(Float64(b / Float64(Float64(Float64(y_45_scale * x_45_scale) * y_45_scale) * x_45_scale)) * b) * Float64(-4.0 * a)) * a)
	tmp = 0.0
	if (y_45_scale <= 2.5e-98)
		tmp = t_0;
	elseif (y_45_scale <= 4.15e+152)
		tmp = Float64(Float64(Float64(Float64(a * b) / Float64(Float64(Float64(y_45_scale * y_45_scale) * x_45_scale) * x_45_scale)) * Float64(-4.0 * a)) * b);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (((b / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale)) * b) * (-4.0 * a)) * a;
	tmp = 0.0;
	if (y_45_scale <= 2.5e-98)
		tmp = t_0;
	elseif (y_45_scale <= 4.15e+152)
		tmp = (((a * b) / (((y_45_scale * y_45_scale) * x_45_scale) * x_45_scale)) * (-4.0 * a)) * b;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(N[(N[(b / N[(N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * N[(-4.0 * a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[y$45$scale, 2.5e-98], t$95$0, If[LessEqual[y$45$scale, 4.15e+152], N[(N[(N[(N[(a * b), $MachinePrecision] / N[(N[(N[(y$45$scale * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(-4.0 * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 2.50000000000000009e-98 or 4.1500000000000001e152 < y-scale

   1. Initial program 26.0%

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

   3. Taylor expanded in angle around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 45.4

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

   5. Applied rewrites 45.4%

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

   7. Applied rewrites 64.0%

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

   9. Applied rewrites 73.9%

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

   if 2.50000000000000009e-98 < y-scale < 4.1500000000000001e152

   1. Initial program 31.4%

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

   3. Taylor expanded in angle around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 67.9

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

   5. Applied rewrites 67.9%

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

   7. Applied rewrites 80.1%

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

   9. Applied rewrites 75.2%

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

   11. Applied rewrites 88.8%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 2.5 \cdot 10^{-98}:\\ \;\;\;\;\left(\left(\frac{b}{\left(\left(y-scale \cdot x-scale\right) \cdot y-scale\right) \cdot x-scale} \cdot b\right) \cdot \left(-4 \cdot a\right)\right) \cdot a\\ \mathbf{elif}\;y-scale \leq 4.15 \cdot 10^{+152}:\\ \;\;\;\;\left(\frac{a \cdot b}{\left(\left(y-scale \cdot y-scale\right) \cdot x-scale\right) \cdot x-scale} \cdot \left(-4 \cdot a\right)\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{b}{\left(\left(y-scale \cdot x-scale\right) \cdot y-scale\right) \cdot x-scale} \cdot b\right) \cdot \left(-4 \cdot a\right)\right) \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.2% accurate, 32.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{a \cdot b}{\left(\left(y-scale \cdot y-scale\right) \cdot x-scale\right) \cdot x-scale} \cdot \left(-4 \cdot a\right)\right) \cdot b\\ \mathbf{if}\;a \leq 3.8 \cdot 10^{-164}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 4 \cdot 10^{+159}:\\ \;\;\;\;\frac{\left(\left(-4 \cdot a\right) \cdot a\right) \cdot b}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \cdot b\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0
         (*
          (*
           (/ (* a b) (* (* (* y-scale y-scale) x-scale) x-scale))
           (* -4.0 a))
          b)))
   (if (<= a 3.8e-164)
     t_0
     (if (<= a 4e+159)
       (*
        (/ (* (* (* -4.0 a) a) b) (* (* y-scale x-scale) (* y-scale x-scale)))
        b)
       t_0))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (((a * b) / (((y_45_scale * y_45_scale) * x_45_scale) * x_45_scale)) * (-4.0 * a)) * b;
	double tmp;
	if (a <= 3.8e-164) {
		tmp = t_0;
	} else if (a <= 4e+159) {
		tmp = ((((-4.0 * a) * a) * b) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * b;
	} 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) / (((y_45scale * y_45scale) * x_45scale) * x_45scale)) * ((-4.0d0) * a)) * b
    if (a <= 3.8d-164) then
        tmp = t_0
    else if (a <= 4d+159) then
        tmp = (((((-4.0d0) * a) * a) * b) / ((y_45scale * x_45scale) * (y_45scale * x_45scale))) * b
    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) / (((y_45_scale * y_45_scale) * x_45_scale) * x_45_scale)) * (-4.0 * a)) * b;
	double tmp;
	if (a <= 3.8e-164) {
		tmp = t_0;
	} else if (a <= 4e+159) {
		tmp = ((((-4.0 * a) * a) * b) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * b;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (((a * b) / (((y_45_scale * y_45_scale) * x_45_scale) * x_45_scale)) * (-4.0 * a)) * b
	tmp = 0
	if a <= 3.8e-164:
		tmp = t_0
	elif a <= 4e+159:
		tmp = ((((-4.0 * a) * a) * b) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * b
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(Float64(Float64(a * b) / Float64(Float64(Float64(y_45_scale * y_45_scale) * x_45_scale) * x_45_scale)) * Float64(-4.0 * a)) * b)
	tmp = 0.0
	if (a <= 3.8e-164)
		tmp = t_0;
	elseif (a <= 4e+159)
		tmp = Float64(Float64(Float64(Float64(Float64(-4.0 * a) * a) * b) / Float64(Float64(y_45_scale * x_45_scale) * Float64(y_45_scale * x_45_scale))) * b);
	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) / (((y_45_scale * y_45_scale) * x_45_scale) * x_45_scale)) * (-4.0 * a)) * b;
	tmp = 0.0;
	if (a <= 3.8e-164)
		tmp = t_0;
	elseif (a <= 4e+159)
		tmp = ((((-4.0 * a) * a) * b) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * b;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(N[(N[(a * b), $MachinePrecision] / N[(N[(N[(y$45$scale * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(-4.0 * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[a, 3.8e-164], t$95$0, If[LessEqual[a, 4e+159], N[(N[(N[(N[(N[(-4.0 * a), $MachinePrecision] * a), $MachinePrecision] * b), $MachinePrecision] / N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < 3.79999999999999989e-164 or 3.9999999999999997e159 < a

   1. Initial program 28.0%

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

   3. Taylor expanded in angle around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 46.5

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

   5. Applied rewrites 46.5%

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

   7. Applied rewrites 66.5%

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

   9. Applied rewrites 57.7%

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

   11. Applied rewrites 71.8%

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

   if 3.79999999999999989e-164 < a < 3.9999999999999997e159

   1. Initial program 25.3%

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

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 65.9

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

   5. Applied rewrites 65.9%

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

   7. Applied rewrites 92.8%

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

   9. Applied rewrites 70.9%

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

   11. Applied rewrites 87.7%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3.8 \cdot 10^{-164}:\\ \;\;\;\;\left(\frac{a \cdot b}{\left(\left(y-scale \cdot y-scale\right) \cdot x-scale\right) \cdot x-scale} \cdot \left(-4 \cdot a\right)\right) \cdot b\\ \mathbf{elif}\;a \leq 4 \cdot 10^{+159}:\\ \;\;\;\;\frac{\left(\left(-4 \cdot a\right) \cdot a\right) \cdot b}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{a \cdot b}{\left(\left(y-scale \cdot y-scale\right) \cdot x-scale\right) \cdot x-scale} \cdot \left(-4 \cdot a\right)\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.8% accurate, 35.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 27.4%

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

3. Taylor expanded in angle around 0

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

   1. associate-/l* N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. unpow2 N/A

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

   12. associate-*r* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 51.1

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

5. Applied rewrites 51.1%

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

6. Taylor expanded in angle around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. unpow2 N/A

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

   12. unpow2 N/A

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

   13. unswap-sqr N/A

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

   14. lower-*.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-*.f64 57.5

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

8. Applied rewrites 57.5%

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

10. Applied rewrites 93.4%

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

12. Applied rewrites 92.6%

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

13. Final simplification 92.6%

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

Alternative 8: 69.8% accurate, 40.5× speedup

Math

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 27.4%

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

3. Taylor expanded in angle around 0

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

   1. associate-/l* N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. unpow2 N/A

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

   12. associate-*r* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 51.1

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

5. Applied rewrites 51.1%

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

7. Applied rewrites 72.7%

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

9. Applied rewrites 60.8%

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

11. Applied rewrites 72.0%

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

12. Final simplification 72.0%

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

Alternative 9: 65.8% accurate, 40.5× speedup

Math

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

FPCore

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

C

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

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) * (-4.0d0)) / (((x_45scale * x_45scale) * y_45scale) * y_45scale)) * b
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) * -4.0) / (((x_45_scale * x_45_scale) * y_45_scale) * y_45_scale)) * b;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 27.4%

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

3. Taylor expanded in angle around 0

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

   1. associate-/l* N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. unpow2 N/A

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

   12. associate-*r* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 51.1

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

5. Applied rewrites 51.1%

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

7. Applied rewrites 72.7%

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

9. Applied rewrites 60.8%

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

10. Taylor expanded in b around 0

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

12. Applied rewrites 66.6%

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

13. Final simplification 66.6%

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

Reproduce

herbie shell --seed 2024240 
(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))))