Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 24.7% → 90.4%

Time: 21.3s

Alternatives: 9

Speedup: 40.5×

• 33.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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)))))

Initial Program: 24.7% accurate, 1.0× speedup

Math

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

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)))))

Alternative 1: 90.4% accurate, 10.6× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \frac{b\_m \cdot a}{x-scale}\\ \mathbf{if}\;b\_m \leq 6.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\left(\frac{t\_0}{y-scale} \cdot t\_0\right) \cdot 4}{-y-scale}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\frac{x-scale}{b\_m}}{a} \cdot \left(\frac{y-scale}{a} \cdot \frac{x-scale}{b\_m}\right)\right)}^{-1} \cdot 4}{-y-scale}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ (* b_m a) x-scale)))
   (if (<= b_m 6.5e-12)
     (/ (* (* (/ t_0 y-scale) t_0) 4.0) (- y-scale))
     (/
      (*
       (pow (* (/ (/ x-scale b_m) a) (* (/ y-scale a) (/ x-scale b_m))) -1.0)
       4.0)
      (- y-scale)))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (b_m * a) / x_45_scale;
	double tmp;
	if (b_m <= 6.5e-12) {
		tmp = (((t_0 / y_45_scale) * t_0) * 4.0) / -y_45_scale;
	} else {
		tmp = (pow((((x_45_scale / b_m) / a) * ((y_45_scale / a) * (x_45_scale / b_m))), -1.0) * 4.0) / -y_45_scale;
	}
	return tmp;
}

Fortran

b_m = abs(b)
real(8) function code(a, b_m, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    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_m * a) / x_45scale
    if (b_m <= 6.5d-12) then
        tmp = (((t_0 / y_45scale) * t_0) * 4.0d0) / -y_45scale
    else
        tmp = (((((x_45scale / b_m) / a) * ((y_45scale / a) * (x_45scale / b_m))) ** (-1.0d0)) * 4.0d0) / -y_45scale
    end if
    code = tmp
end function

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (b_m * a) / x_45_scale;
	double tmp;
	if (b_m <= 6.5e-12) {
		tmp = (((t_0 / y_45_scale) * t_0) * 4.0) / -y_45_scale;
	} else {
		tmp = (Math.pow((((x_45_scale / b_m) / a) * ((y_45_scale / a) * (x_45_scale / b_m))), -1.0) * 4.0) / -y_45_scale;
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	t_0 = (b_m * a) / x_45_scale
	tmp = 0
	if b_m <= 6.5e-12:
		tmp = (((t_0 / y_45_scale) * t_0) * 4.0) / -y_45_scale
	else:
		tmp = (math.pow((((x_45_scale / b_m) / a) * ((y_45_scale / a) * (x_45_scale / b_m))), -1.0) * 4.0) / -y_45_scale
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(b_m * a) / x_45_scale)
	tmp = 0.0
	if (b_m <= 6.5e-12)
		tmp = Float64(Float64(Float64(Float64(t_0 / y_45_scale) * t_0) * 4.0) / Float64(-y_45_scale));
	else
		tmp = Float64(Float64((Float64(Float64(Float64(x_45_scale / b_m) / a) * Float64(Float64(y_45_scale / a) * Float64(x_45_scale / b_m))) ^ -1.0) * 4.0) / Float64(-y_45_scale));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = (b_m * a) / x_45_scale;
	tmp = 0.0;
	if (b_m <= 6.5e-12)
		tmp = (((t_0 / y_45_scale) * t_0) * 4.0) / -y_45_scale;
	else
		tmp = (((((x_45_scale / b_m) / a) * ((y_45_scale / a) * (x_45_scale / b_m))) ^ -1.0) * 4.0) / -y_45_scale;
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(b$95$m * a), $MachinePrecision] / x$45$scale), $MachinePrecision]}, If[LessEqual[b$95$m, 6.5e-12], N[(N[(N[(N[(t$95$0 / y$45$scale), $MachinePrecision] * t$95$0), $MachinePrecision] * 4.0), $MachinePrecision] / (-y$45$scale)), $MachinePrecision], N[(N[(N[Power[N[(N[(N[(x$45$scale / b$95$m), $MachinePrecision] / a), $MachinePrecision] * N[(N[(y$45$scale / a), $MachinePrecision] * N[(x$45$scale / b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] * 4.0), $MachinePrecision] / (-y$45$scale)), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 6.5000000000000002e-12

   1. Initial program 30.9%

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

   4. Applied rewrites 32.3%

      \[\leadsto \color{blue}{\frac{\left(-\frac{{\left(\frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}{x-scale}\right)}^{2}}{y-scale}\right) - \frac{{\left(\sin \left(\mathsf{PI}\left(\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot b\right)}^{2} + {\left(\cos \left(\mathsf{PI}\left(\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot a\right)}^{2}}{-y-scale} \cdot \left(\frac{4}{x-scale} \cdot \frac{{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot b\right)}^{2} + {\left(\sin \left(\mathsf{PI}\left(\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot a\right)}^{2}}{x-scale}\right)}{-y-scale}} \]

   5. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. unswap-sqr N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 68.1

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

   7. Applied rewrites 68.1%

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

   9. Applied rewrites 92.3%

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

   11. Applied rewrites 93.3%

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

   if 6.5000000000000002e-12 < b

   1. Initial program 5.8%

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

   4. Applied rewrites 9.0%

      \[\leadsto \color{blue}{\frac{\left(-\frac{{\left(\frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}{x-scale}\right)}^{2}}{y-scale}\right) - \frac{{\left(\sin \left(\mathsf{PI}\left(\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot b\right)}^{2} + {\left(\cos \left(\mathsf{PI}\left(\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot a\right)}^{2}}{-y-scale} \cdot \left(\frac{4}{x-scale} \cdot \frac{{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot b\right)}^{2} + {\left(\sin \left(\mathsf{PI}\left(\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot a\right)}^{2}}{x-scale}\right)}{-y-scale}} \]

   5. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. unswap-sqr N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 69.6

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

   7. Applied rewrites 69.6%

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

   9. Applied rewrites 83.7%

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

   11. Applied rewrites 97.0%

       \[\leadsto \frac{\frac{1}{\frac{\frac{x-scale}{b}}{a} \cdot \left(\frac{y-scale}{a} \cdot \frac{x-scale}{b}\right)} \cdot 4}{-y-scale} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\left(\frac{\frac{b \cdot a}{x-scale}}{y-scale} \cdot \frac{b \cdot a}{x-scale}\right) \cdot 4}{-y-scale}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\frac{x-scale}{b}}{a} \cdot \left(\frac{y-scale}{a} \cdot \frac{x-scale}{b}\right)\right)}^{-1} \cdot 4}{-y-scale}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \mathsf{PI}\left(\right)\\ t_1 := \sin t\_0\\ t_2 := \cos t\_0\\ t_3 := \frac{\frac{\left(\left(2 \cdot \left({b\_m}^{2} - {a}^{2}\right)\right) \cdot t\_1\right) \cdot t\_2}{x-scale}}{y-scale}\\ \mathbf{if}\;t\_3 \cdot t\_3 - \left(4 \cdot \frac{\frac{{\left(a \cdot t\_1\right)}^{2} + {\left(b\_m \cdot t\_2\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot t\_2\right)}^{2} + {\left(b\_m \cdot t\_1\right)}^{2}}{y-scale}}{y-scale} \leq 10^{+120}:\\ \;\;\;\;\left(\frac{-4 \cdot a}{y-scale \cdot x-scale} \cdot \frac{b\_m \cdot b\_m}{y-scale \cdot x-scale}\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{-4 \cdot \left(\left(b\_m \cdot a\right) \cdot b\_m\right)}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \cdot a\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m 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_m 2.0) (pow a 2.0))) t_1) t_2) x-scale)
          y-scale)))
   (if (<=
        (-
         (* t_3 t_3)
         (*
          (*
           4.0
           (/
            (/ (+ (pow (* a t_1) 2.0) (pow (* b_m t_2) 2.0)) x-scale)
            x-scale))
          (/
           (/ (+ (pow (* a t_2) 2.0) (pow (* b_m t_1) 2.0)) y-scale)
           y-scale)))
        1e+120)
     (*
      (*
       (/ (* -4.0 a) (* y-scale x-scale))
       (/ (* b_m b_m) (* y-scale x-scale)))
      a)
     (*
      (/
       (* -4.0 (* (* b_m a) b_m))
       (* (* y-scale x-scale) (* y-scale x-scale)))
      a))))

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) x-scale) y-scale) (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) x-scale) y-scale)) (*.f64 (*.f64 #s(literal 4 binary64) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) x-scale) x-scale)) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) y-scale) y-scale))) < 9.9999999999999998e119

   1. Initial program 68.2%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} \]
   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. unpow2 N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 68.8

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

   5. Applied rewrites 68.8%

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

   7. Applied rewrites 78.2%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 77.7%

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

   12. Applied rewrites 91.6%

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

   if 9.9999999999999998e119 < (-.f64 (*.f64 (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) x-scale) y-scale) (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) x-scale) y-scale)) (*.f64 (*.f64 #s(literal 4 binary64) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) x-scale) x-scale)) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) y-scale) y-scale)))

   1. Initial program 0.0%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} \]
   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. unpow2 N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 54.3

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

   5. Applied rewrites 54.3%

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

   7. Applied rewrites 77.8%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 62.4%

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

   12. Applied rewrites 73.7%

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

Alternative 3: 89.3% accurate, 26.1× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;y-scale \leq 2.7 \cdot 10^{+212}:\\ \;\;\;\;\frac{\left(\frac{a \cdot b\_m}{x-scale} \cdot \frac{a \cdot b\_m}{y-scale \cdot x-scale}\right) \cdot 4}{-y-scale}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(b\_m \cdot \frac{\left(\frac{b\_m}{y-scale} \cdot a\right) \cdot \frac{a}{x-scale}}{x-scale}\right) \cdot 4}{-y-scale}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (if (<= y-scale 2.7e+212)
   (/
    (* (* (/ (* a b_m) x-scale) (/ (* a b_m) (* y-scale x-scale))) 4.0)
    (- y-scale))
   (/
    (* (* b_m (/ (* (* (/ b_m y-scale) a) (/ a x-scale)) x-scale)) 4.0)
    (- y-scale))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (y_45_scale <= 2.7e+212) {
		tmp = ((((a * b_m) / x_45_scale) * ((a * b_m) / (y_45_scale * x_45_scale))) * 4.0) / -y_45_scale;
	} else {
		tmp = ((b_m * ((((b_m / y_45_scale) * a) * (a / x_45_scale)) / x_45_scale)) * 4.0) / -y_45_scale;
	}
	return tmp;
}

Fortran

b_m = abs(b)
real(8) function code(a, b_m, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: tmp
    if (y_45scale <= 2.7d+212) then
        tmp = ((((a * b_m) / x_45scale) * ((a * b_m) / (y_45scale * x_45scale))) * 4.0d0) / -y_45scale
    else
        tmp = ((b_m * ((((b_m / y_45scale) * a) * (a / x_45scale)) / x_45scale)) * 4.0d0) / -y_45scale
    end if
    code = tmp
end function

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (y_45_scale <= 2.7e+212) {
		tmp = ((((a * b_m) / x_45_scale) * ((a * b_m) / (y_45_scale * x_45_scale))) * 4.0) / -y_45_scale;
	} else {
		tmp = ((b_m * ((((b_m / y_45_scale) * a) * (a / x_45_scale)) / x_45_scale)) * 4.0) / -y_45_scale;
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	tmp = 0
	if y_45_scale <= 2.7e+212:
		tmp = ((((a * b_m) / x_45_scale) * ((a * b_m) / (y_45_scale * x_45_scale))) * 4.0) / -y_45_scale
	else:
		tmp = ((b_m * ((((b_m / y_45_scale) * a) * (a / x_45_scale)) / x_45_scale)) * 4.0) / -y_45_scale
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (y_45_scale <= 2.7e+212)
		tmp = Float64(Float64(Float64(Float64(Float64(a * b_m) / x_45_scale) * Float64(Float64(a * b_m) / Float64(y_45_scale * x_45_scale))) * 4.0) / Float64(-y_45_scale));
	else
		tmp = Float64(Float64(Float64(b_m * Float64(Float64(Float64(Float64(b_m / y_45_scale) * a) * Float64(a / x_45_scale)) / x_45_scale)) * 4.0) / Float64(-y_45_scale));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (y_45_scale <= 2.7e+212)
		tmp = ((((a * b_m) / x_45_scale) * ((a * b_m) / (y_45_scale * x_45_scale))) * 4.0) / -y_45_scale;
	else
		tmp = ((b_m * ((((b_m / y_45_scale) * a) * (a / x_45_scale)) / x_45_scale)) * 4.0) / -y_45_scale;
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[y$45$scale, 2.7e+212], N[(N[(N[(N[(N[(a * b$95$m), $MachinePrecision] / x$45$scale), $MachinePrecision] * N[(N[(a * b$95$m), $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision] / (-y$45$scale)), $MachinePrecision], N[(N[(N[(b$95$m * N[(N[(N[(N[(b$95$m / y$45$scale), $MachinePrecision] * a), $MachinePrecision] * N[(a / x$45$scale), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision] / (-y$45$scale)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 2.7e212

   1. Initial program 22.8%

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

   4. Applied rewrites 24.7%

      \[\leadsto \color{blue}{\frac{\left(-\frac{{\left(\frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}{x-scale}\right)}^{2}}{y-scale}\right) - \frac{{\left(\sin \left(\mathsf{PI}\left(\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot b\right)}^{2} + {\left(\cos \left(\mathsf{PI}\left(\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot a\right)}^{2}}{-y-scale} \cdot \left(\frac{4}{x-scale} \cdot \frac{{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot b\right)}^{2} + {\left(\sin \left(\mathsf{PI}\left(\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot a\right)}^{2}}{x-scale}\right)}{-y-scale}} \]

   5. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. unswap-sqr N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 67.9

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

   7. Applied rewrites 67.9%

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

   9. Applied rewrites 90.3%

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

   if 2.7e212 < y-scale

   1. Initial program 53.2%

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

   4. Applied rewrites 53.0%

      \[\leadsto \color{blue}{\frac{\left(-\frac{{\left(\frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}{x-scale}\right)}^{2}}{y-scale}\right) - \frac{{\left(\sin \left(\mathsf{PI}\left(\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot b\right)}^{2} + {\left(\cos \left(\mathsf{PI}\left(\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot a\right)}^{2}}{-y-scale} \cdot \left(\frac{4}{x-scale} \cdot \frac{{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot b\right)}^{2} + {\left(\sin \left(\mathsf{PI}\left(\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot a\right)}^{2}}{x-scale}\right)}{-y-scale}} \]

   5. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. unswap-sqr N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 76.9

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

   7. Applied rewrites 76.9%

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

   9. Applied rewrites 77.0%

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

   11. Applied rewrites 88.7%

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

Alternative 4: 91.3% accurate, 28.4× speedup

Math

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

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ (* b_m a) x-scale)))
   (/ (* (* (/ t_0 y-scale) t_0) 4.0) (- y-scale))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 24.8%

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

4. Applied rewrites 26.6%

   \[\leadsto \color{blue}{\frac{\left(-\frac{{\left(\frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}{x-scale}\right)}^{2}}{y-scale}\right) - \frac{{\left(\sin \left(\mathsf{PI}\left(\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot b\right)}^{2} + {\left(\cos \left(\mathsf{PI}\left(\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot a\right)}^{2}}{-y-scale} \cdot \left(\frac{4}{x-scale} \cdot \frac{{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot b\right)}^{2} + {\left(\sin \left(\mathsf{PI}\left(\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot a\right)}^{2}}{x-scale}\right)}{-y-scale}} \]

5. Taylor expanded in angle around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. unpow2 N/A

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

   5. unpow2 N/A

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

   6. unswap-sqr N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 68.5

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

7. Applied rewrites 68.5%

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

9. Applied rewrites 90.2%

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

11. Applied rewrites 91.5%

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

Alternative 5: 87.1% accurate, 29.3× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \frac{b\_m}{y-scale \cdot x-scale}\\ \mathbf{if}\;b\_m \leq 1.94 \cdot 10^{-145} \lor \neg \left(b\_m \leq 3.7 \cdot 10^{+112}\right):\\ \;\;\;\;\left(\left(t\_0 \cdot t\_0\right) \cdot \left(-4 \cdot a\right)\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-4 \cdot a}{y-scale \cdot x-scale} \cdot \frac{a}{y-scale \cdot x-scale}\right) \cdot \left(b\_m \cdot b\_m\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ b_m (* y-scale x-scale))))
   (if (or (<= b_m 1.94e-145) (not (<= b_m 3.7e+112)))
     (* (* (* t_0 t_0) (* -4.0 a)) a)
     (*
      (* (/ (* -4.0 a) (* y-scale x-scale)) (/ a (* y-scale x-scale)))
      (* b_m b_m)))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = b_m / (y_45_scale * x_45_scale);
	double tmp;
	if ((b_m <= 1.94e-145) || !(b_m <= 3.7e+112)) {
		tmp = ((t_0 * t_0) * (-4.0 * a)) * a;
	} else {
		tmp = (((-4.0 * a) / (y_45_scale * x_45_scale)) * (a / (y_45_scale * x_45_scale))) * (b_m * b_m);
	}
	return tmp;
}

Fortran

b_m = abs(b)
real(8) function code(a, b_m, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    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_m / (y_45scale * x_45scale)
    if ((b_m <= 1.94d-145) .or. (.not. (b_m <= 3.7d+112))) then
        tmp = ((t_0 * t_0) * ((-4.0d0) * a)) * a
    else
        tmp = ((((-4.0d0) * a) / (y_45scale * x_45scale)) * (a / (y_45scale * x_45scale))) * (b_m * b_m)
    end if
    code = tmp
end function

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = b_m / (y_45_scale * x_45_scale);
	double tmp;
	if ((b_m <= 1.94e-145) || !(b_m <= 3.7e+112)) {
		tmp = ((t_0 * t_0) * (-4.0 * a)) * a;
	} else {
		tmp = (((-4.0 * a) / (y_45_scale * x_45_scale)) * (a / (y_45_scale * x_45_scale))) * (b_m * b_m);
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	t_0 = b_m / (y_45_scale * x_45_scale)
	tmp = 0
	if (b_m <= 1.94e-145) or not (b_m <= 3.7e+112):
		tmp = ((t_0 * t_0) * (-4.0 * a)) * a
	else:
		tmp = (((-4.0 * a) / (y_45_scale * x_45_scale)) * (a / (y_45_scale * x_45_scale))) * (b_m * b_m)
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(b_m / Float64(y_45_scale * x_45_scale))
	tmp = 0.0
	if ((b_m <= 1.94e-145) || !(b_m <= 3.7e+112))
		tmp = Float64(Float64(Float64(t_0 * t_0) * Float64(-4.0 * a)) * a);
	else
		tmp = Float64(Float64(Float64(Float64(-4.0 * a) / Float64(y_45_scale * x_45_scale)) * Float64(a / Float64(y_45_scale * x_45_scale))) * Float64(b_m * b_m));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = b_m / (y_45_scale * x_45_scale);
	tmp = 0.0;
	if ((b_m <= 1.94e-145) || ~((b_m <= 3.7e+112)))
		tmp = ((t_0 * t_0) * (-4.0 * a)) * a;
	else
		tmp = (((-4.0 * a) / (y_45_scale * x_45_scale)) * (a / (y_45_scale * x_45_scale))) * (b_m * b_m);
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(b$95$m / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[b$95$m, 1.94e-145], N[Not[LessEqual[b$95$m, 3.7e+112]], $MachinePrecision]], N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] * N[(-4.0 * a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], N[(N[(N[(N[(-4.0 * a), $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(a / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.9400000000000001e-145 or 3.70000000000000004e112 < b

   1. Initial program 24.8%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} \]
   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. unpow2 N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 60.0

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

   5. Applied rewrites 60.0%

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

   7. Applied rewrites 79.2%

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

   9. Applied rewrites 87.7%

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

   if 1.9400000000000001e-145 < b < 3.70000000000000004e112

   1. Initial program 24.6%

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

   3. Taylor expanded in b around 0

      \[\leadsto \color{blue}{{b}^{2} \cdot \left(-8 \cdot \frac{{a}^{2} \cdot \left({\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}} - 4 \cdot \left(\frac{{a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{x-scale}^{2} \cdot {y-scale}^{2}} + \frac{{a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)\right)} \]

   5. Applied rewrites 48.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-4, \left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right) \cdot \left(\frac{{\sin \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)}^{4}}{y-scale \cdot y-scale} + \frac{{\cos \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)}^{4}}{y-scale \cdot y-scale}\right), \left(-8 \cdot \left(a \cdot a\right)\right) \cdot \left(\frac{{\sin \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)}^{2}}{\left(\left(y-scale \cdot y-scale\right) \cdot x-scale\right) \cdot x-scale} \cdot {\cos \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)}^{2}\right)\right) \cdot \left(b \cdot b\right)} \]

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 68.4%

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

   10. Applied rewrites 93.3%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.94 \cdot 10^{-145} \lor \neg \left(b \leq 3.7 \cdot 10^{+112}\right):\\ \;\;\;\;\left(\left(\frac{b}{y-scale \cdot x-scale} \cdot \frac{b}{y-scale \cdot x-scale}\right) \cdot \left(-4 \cdot a\right)\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-4 \cdot a}{y-scale \cdot x-scale} \cdot \frac{a}{y-scale \cdot x-scale}\right) \cdot \left(b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.9% accurate, 29.3× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 1.94 \cdot 10^{-145} \lor \neg \left(b\_m \leq 5.8 \cdot 10^{+112}\right):\\ \;\;\;\;\left(\left(b\_m \cdot \frac{\frac{b\_m}{y-scale \cdot x-scale}}{y-scale \cdot x-scale}\right) \cdot \left(-4 \cdot a\right)\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-4 \cdot a}{y-scale \cdot x-scale} \cdot \frac{a}{y-scale \cdot x-scale}\right) \cdot \left(b\_m \cdot b\_m\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (if (or (<= b_m 1.94e-145) (not (<= b_m 5.8e+112)))
   (*
    (* (* b_m (/ (/ b_m (* y-scale x-scale)) (* y-scale x-scale))) (* -4.0 a))
    a)
   (*
    (* (/ (* -4.0 a) (* y-scale x-scale)) (/ a (* y-scale x-scale)))
    (* b_m b_m))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if ((b_m <= 1.94e-145) || !(b_m <= 5.8e+112)) {
		tmp = ((b_m * ((b_m / (y_45_scale * x_45_scale)) / (y_45_scale * x_45_scale))) * (-4.0 * a)) * a;
	} else {
		tmp = (((-4.0 * a) / (y_45_scale * x_45_scale)) * (a / (y_45_scale * x_45_scale))) * (b_m * b_m);
	}
	return tmp;
}

Fortran

b_m = abs(b)
real(8) function code(a, b_m, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: tmp
    if ((b_m <= 1.94d-145) .or. (.not. (b_m <= 5.8d+112))) then
        tmp = ((b_m * ((b_m / (y_45scale * x_45scale)) / (y_45scale * x_45scale))) * ((-4.0d0) * a)) * a
    else
        tmp = ((((-4.0d0) * a) / (y_45scale * x_45scale)) * (a / (y_45scale * x_45scale))) * (b_m * b_m)
    end if
    code = tmp
end function

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if ((b_m <= 1.94e-145) || !(b_m <= 5.8e+112)) {
		tmp = ((b_m * ((b_m / (y_45_scale * x_45_scale)) / (y_45_scale * x_45_scale))) * (-4.0 * a)) * a;
	} else {
		tmp = (((-4.0 * a) / (y_45_scale * x_45_scale)) * (a / (y_45_scale * x_45_scale))) * (b_m * b_m);
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	tmp = 0
	if (b_m <= 1.94e-145) or not (b_m <= 5.8e+112):
		tmp = ((b_m * ((b_m / (y_45_scale * x_45_scale)) / (y_45_scale * x_45_scale))) * (-4.0 * a)) * a
	else:
		tmp = (((-4.0 * a) / (y_45_scale * x_45_scale)) * (a / (y_45_scale * x_45_scale))) * (b_m * b_m)
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if ((b_m <= 1.94e-145) || !(b_m <= 5.8e+112))
		tmp = Float64(Float64(Float64(b_m * Float64(Float64(b_m / Float64(y_45_scale * x_45_scale)) / Float64(y_45_scale * x_45_scale))) * Float64(-4.0 * a)) * a);
	else
		tmp = Float64(Float64(Float64(Float64(-4.0 * a) / Float64(y_45_scale * x_45_scale)) * Float64(a / Float64(y_45_scale * x_45_scale))) * Float64(b_m * b_m));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if ((b_m <= 1.94e-145) || ~((b_m <= 5.8e+112)))
		tmp = ((b_m * ((b_m / (y_45_scale * x_45_scale)) / (y_45_scale * x_45_scale))) * (-4.0 * a)) * a;
	else
		tmp = (((-4.0 * a) / (y_45_scale * x_45_scale)) * (a / (y_45_scale * x_45_scale))) * (b_m * b_m);
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := If[Or[LessEqual[b$95$m, 1.94e-145], N[Not[LessEqual[b$95$m, 5.8e+112]], $MachinePrecision]], N[(N[(N[(b$95$m * N[(N[(b$95$m / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-4.0 * a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], N[(N[(N[(N[(-4.0 * a), $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(a / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.9400000000000001e-145 or 5.8000000000000004e112 < b

   1. Initial program 24.8%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} \]
   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. unpow2 N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 60.0

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

   5. Applied rewrites 60.0%

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

   7. Applied rewrites 79.2%

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

   9. Applied rewrites 85.1%

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

   if 1.9400000000000001e-145 < b < 5.8000000000000004e112

   1. Initial program 24.6%

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

   3. Taylor expanded in b around 0

      \[\leadsto \color{blue}{{b}^{2} \cdot \left(-8 \cdot \frac{{a}^{2} \cdot \left({\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}} - 4 \cdot \left(\frac{{a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{x-scale}^{2} \cdot {y-scale}^{2}} + \frac{{a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)\right)} \]

   5. Applied rewrites 48.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-4, \left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right) \cdot \left(\frac{{\sin \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)}^{4}}{y-scale \cdot y-scale} + \frac{{\cos \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)}^{4}}{y-scale \cdot y-scale}\right), \left(-8 \cdot \left(a \cdot a\right)\right) \cdot \left(\frac{{\sin \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)}^{2}}{\left(\left(y-scale \cdot y-scale\right) \cdot x-scale\right) \cdot x-scale} \cdot {\cos \left(\left(\mathsf{PI}\left(\right) \cdot 0.005555555555555556\right) \cdot angle\right)}^{2}\right)\right) \cdot \left(b \cdot b\right)} \]

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 68.4%

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

   10. Applied rewrites 93.3%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.94 \cdot 10^{-145} \lor \neg \left(b \leq 5.8 \cdot 10^{+112}\right):\\ \;\;\;\;\left(\left(b \cdot \frac{\frac{b}{y-scale \cdot x-scale}}{y-scale \cdot x-scale}\right) \cdot \left(-4 \cdot a\right)\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-4 \cdot a}{y-scale \cdot x-scale} \cdot \frac{a}{y-scale \cdot x-scale}\right) \cdot \left(b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.4% accurate, 31.2× speedup

Math

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

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (/
  (* (* (/ (* a b_m) x-scale) (/ (* a b_m) (* y-scale x-scale))) 4.0)
  (- y-scale)))

C

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

Fortran

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

Java

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

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	return ((((a * b_m) / x_45_scale) * ((a * b_m) / (y_45_scale * x_45_scale))) * 4.0) / -y_45_scale

Julia

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

MATLAB

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

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := N[(N[(N[(N[(N[(a * b$95$m), $MachinePrecision] / x$45$scale), $MachinePrecision] * N[(N[(a * b$95$m), $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision] / (-y$45$scale)), $MachinePrecision]

Derivation

1. Initial program 24.8%

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

4. Applied rewrites 26.6%

   \[\leadsto \color{blue}{\frac{\left(-\frac{{\left(\frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}{x-scale}\right)}^{2}}{y-scale}\right) - \frac{{\left(\sin \left(\mathsf{PI}\left(\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot b\right)}^{2} + {\left(\cos \left(\mathsf{PI}\left(\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot a\right)}^{2}}{-y-scale} \cdot \left(\frac{4}{x-scale} \cdot \frac{{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot b\right)}^{2} + {\left(\sin \left(\mathsf{PI}\left(\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot a\right)}^{2}}{x-scale}\right)}{-y-scale}} \]

5. Taylor expanded in angle around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. unpow2 N/A

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

   5. unpow2 N/A

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

   6. unswap-sqr N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 68.5

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

7. Applied rewrites 68.5%

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

9. Applied rewrites 90.2%

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

Alternative 8: 75.9% accurate, 40.5× speedup

Math

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

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (*
  (/ (* -4.0 (* (* b_m a) b_m)) (* (* y-scale x-scale) (* y-scale x-scale)))
  a))

C

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

Fortran

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

Java

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

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	return ((-4.0 * ((b_m * a) * b_m)) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * a

Julia

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

MATLAB

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

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := N[(N[(N[(-4.0 * N[(N[(b$95$m * a), $MachinePrecision] * b$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]

Derivation

1. Initial program 24.8%

   \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} \]
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. unpow2 N/A

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

   8. *-commutative N/A

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

   9. times-frac N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 59.6

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

5. Applied rewrites 59.6%

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

7. Applied rewrites 77.9%

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

8. Taylor expanded in a around 0

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

10. Applied rewrites 67.9%

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

12. Applied rewrites 75.1%

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

Alternative 9: 69.4% accurate, 40.5× speedup

Math

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

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (*
  (/ (* -4.0 (* a (* b_m b_m))) (* (* y-scale x-scale) (* y-scale x-scale)))
  a))

C

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

Fortran

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

Java

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

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	return ((-4.0 * (a * (b_m * b_m))) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * a

Julia

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

MATLAB

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

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := N[(N[(N[(-4.0 * N[(a * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]

Derivation

1. Initial program 24.8%

   \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} \]
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. unpow2 N/A

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

   8. *-commutative N/A

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

   9. times-frac N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 59.6

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

5. Applied rewrites 59.6%

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

7. Applied rewrites 77.9%

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

8. Taylor expanded in a around 0

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

10. Applied rewrites 67.9%

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

Reproduce

herbie shell --seed 2024308 
(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))))