Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 24.5% → 93.7%

Time: 16.4s

Alternatives: 10

Speedup: 40.5×

• Could not converge on a ground truth

  1. a = 9.391604065296596e-134
  2. b = 8.730431492359815e-134
  3. angle = 6.86349992282063e+90
  4. x-scale = 2.596029502085003e+70
  5. y-scale = 1.408465565573225e+63

• 33.1% 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: 24.5% 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: 93.7% accurate, 33.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 28.2%

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

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

   2. *-commutative N/A

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

   3. times-frac N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. unpow2 N/A

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

   10. times-frac N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f64 60.7

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

5. Applied rewrites 60.7%

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

7. Applied rewrites 84.3%

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

9. Applied rewrites 88.6%

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

11. Applied rewrites 96.6%

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

Alternative 2: 84.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \mathsf{PI}\left(\right)\\ t_1 := \sin t\_0\\ t_2 := \cos t\_0\\ t_3 := \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot t\_1\right) \cdot t\_2}{x-scale}}{y-scale}\\ \mathbf{if}\;t\_3 \cdot t\_3 - \left(4 \cdot \frac{\frac{{\left(a \cdot t\_1\right)}^{2} + {\left(b \cdot t\_2\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot t\_2\right)}^{2} + {\left(b \cdot t\_1\right)}^{2}}{y-scale}}{y-scale} \leq 5 \cdot 10^{+260}:\\ \;\;\;\;\left(\frac{-4 \cdot a}{y-scale \cdot x-scale} \cdot \frac{a}{y-scale \cdot x-scale}\right) \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot b\right) \cdot \left(\frac{a \cdot b}{x-scale} \cdot \frac{-4}{\left(y-scale \cdot x-scale\right) \cdot y-scale}\right)\\ \end{array} \end{array} \]

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)))
   (if (<=
        (-
         (* 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)))
        5e+260)
     (*
      (* (/ (* -4.0 a) (* y-scale x-scale)) (/ a (* y-scale x-scale)))
      (* b b))
     (*
      (* a b)
      (* (/ (* a b) x-scale) (/ -4.0 (* (* y-scale x-scale) y-scale)))))))

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))) < 4.9999999999999996e260

   1. Initial program 68.8%

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

   3. Taylor expanded in 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 72.7%

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

   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 80.3%

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

   10. Applied rewrites 90.7%

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

   if 4.9999999999999996e260 < (-.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-*r/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 48.2

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

   5. Applied rewrites 48.2%

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

   7. Applied rewrites 80.8%

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

   9. Applied rewrites 86.0%

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

   11. Applied rewrites 84.9%

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

Alternative 3: 84.9% accurate, 29.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y-scale \cdot x-scale\right) \cdot y-scale\\ t_1 := \left(-4 \cdot a\right) \cdot b\\ t_2 := \frac{a \cdot b}{x-scale}\\ \mathbf{if}\;x-scale \leq 4.2 \cdot 10^{-158}:\\ \;\;\;\;\left(a \cdot b\right) \cdot \left(t\_2 \cdot \frac{-4}{t\_0}\right)\\ \mathbf{elif}\;x-scale \leq 2.6 \cdot 10^{+179}:\\ \;\;\;\;\frac{t\_1}{\left(x-scale \cdot x-scale\right) \cdot y-scale} \cdot \frac{a \cdot b}{y-scale}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{t\_0} \cdot t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (* y-scale x-scale) y-scale))
        (t_1 (* (* -4.0 a) b))
        (t_2 (/ (* a b) x-scale)))
   (if (<= x-scale 4.2e-158)
     (* (* a b) (* t_2 (/ -4.0 t_0)))
     (if (<= x-scale 2.6e+179)
       (* (/ t_1 (* (* x-scale x-scale) y-scale)) (/ (* a b) y-scale))
       (* (/ t_1 t_0) t_2)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (y_45_scale * x_45_scale) * y_45_scale;
	double t_1 = (-4.0 * a) * b;
	double t_2 = (a * b) / x_45_scale;
	double tmp;
	if (x_45_scale <= 4.2e-158) {
		tmp = (a * b) * (t_2 * (-4.0 / t_0));
	} else if (x_45_scale <= 2.6e+179) {
		tmp = (t_1 / ((x_45_scale * x_45_scale) * y_45_scale)) * ((a * b) / y_45_scale);
	} else {
		tmp = (t_1 / t_0) * t_2;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (y_45scale * x_45scale) * y_45scale
    t_1 = ((-4.0d0) * a) * b
    t_2 = (a * b) / x_45scale
    if (x_45scale <= 4.2d-158) then
        tmp = (a * b) * (t_2 * ((-4.0d0) / t_0))
    else if (x_45scale <= 2.6d+179) then
        tmp = (t_1 / ((x_45scale * x_45scale) * y_45scale)) * ((a * b) / y_45scale)
    else
        tmp = (t_1 / t_0) * t_2
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (y_45_scale * x_45_scale) * y_45_scale;
	double t_1 = (-4.0 * a) * b;
	double t_2 = (a * b) / x_45_scale;
	double tmp;
	if (x_45_scale <= 4.2e-158) {
		tmp = (a * b) * (t_2 * (-4.0 / t_0));
	} else if (x_45_scale <= 2.6e+179) {
		tmp = (t_1 / ((x_45_scale * x_45_scale) * y_45_scale)) * ((a * b) / y_45_scale);
	} else {
		tmp = (t_1 / t_0) * t_2;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (y_45_scale * x_45_scale) * y_45_scale
	t_1 = (-4.0 * a) * b
	t_2 = (a * b) / x_45_scale
	tmp = 0
	if x_45_scale <= 4.2e-158:
		tmp = (a * b) * (t_2 * (-4.0 / t_0))
	elif x_45_scale <= 2.6e+179:
		tmp = (t_1 / ((x_45_scale * x_45_scale) * y_45_scale)) * ((a * b) / y_45_scale)
	else:
		tmp = (t_1 / t_0) * t_2
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(y_45_scale * x_45_scale) * y_45_scale)
	t_1 = Float64(Float64(-4.0 * a) * b)
	t_2 = Float64(Float64(a * b) / x_45_scale)
	tmp = 0.0
	if (x_45_scale <= 4.2e-158)
		tmp = Float64(Float64(a * b) * Float64(t_2 * Float64(-4.0 / t_0)));
	elseif (x_45_scale <= 2.6e+179)
		tmp = Float64(Float64(t_1 / Float64(Float64(x_45_scale * x_45_scale) * y_45_scale)) * Float64(Float64(a * b) / y_45_scale));
	else
		tmp = Float64(Float64(t_1 / t_0) * t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (y_45_scale * x_45_scale) * y_45_scale;
	t_1 = (-4.0 * a) * b;
	t_2 = (a * b) / x_45_scale;
	tmp = 0.0;
	if (x_45_scale <= 4.2e-158)
		tmp = (a * b) * (t_2 * (-4.0 / t_0));
	elseif (x_45_scale <= 2.6e+179)
		tmp = (t_1 / ((x_45_scale * x_45_scale) * y_45_scale)) * ((a * b) / y_45_scale);
	else
		tmp = (t_1 / t_0) * t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-4.0 * a), $MachinePrecision] * b), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a * b), $MachinePrecision] / x$45$scale), $MachinePrecision]}, If[LessEqual[x$45$scale, 4.2e-158], N[(N[(a * b), $MachinePrecision] * N[(t$95$2 * N[(-4.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$45$scale, 2.6e+179], N[(N[(t$95$1 / N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(a * b), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 / t$95$0), $MachinePrecision] * t$95$2), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x-scale < 4.19999999999999983e-158

   1. Initial program 21.6%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 57.4

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

   5. Applied rewrites 57.4%

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

   7. Applied rewrites 84.4%

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

   9. Applied rewrites 88.2%

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

   11. Applied rewrites 88.2%

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

   if 4.19999999999999983e-158 < x-scale < 2.6000000000000002e179

   1. Initial program 35.0%

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

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 69.7

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

   5. Applied rewrites 69.7%

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

   7. Applied rewrites 86.6%

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

   9. Applied rewrites 90.6%

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

   11. Applied rewrites 93.5%

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

   if 2.6000000000000002e179 < x-scale

   1. Initial program 47.1%

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

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 52.6

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

   5. Applied rewrites 52.6%

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

   7. Applied rewrites 76.3%

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

   9. Applied rewrites 84.4%

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

   11. Applied rewrites 81.1%

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

Alternative 4: 84.8% accurate, 29.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot b}{x-scale}\\ t_1 := \frac{-4}{\left(y-scale \cdot x-scale\right) \cdot y-scale}\\ \mathbf{if}\;x-scale \leq 4.2 \cdot 10^{-158}:\\ \;\;\;\;\left(a \cdot b\right) \cdot \left(t\_0 \cdot t\_1\right)\\ \mathbf{elif}\;x-scale \leq 2.6 \cdot 10^{+179}:\\ \;\;\;\;\frac{\left(-4 \cdot a\right) \cdot b}{\left(x-scale \cdot x-scale\right) \cdot y-scale} \cdot \frac{a \cdot b}{y-scale}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\left(a \cdot b\right) \cdot t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ (* a b) x-scale))
        (t_1 (/ -4.0 (* (* y-scale x-scale) y-scale))))
   (if (<= x-scale 4.2e-158)
     (* (* a b) (* t_0 t_1))
     (if (<= x-scale 2.6e+179)
       (*
        (/ (* (* -4.0 a) b) (* (* x-scale x-scale) y-scale))
        (/ (* a b) y-scale))
       (* t_0 (* (* a b) t_1))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (a * b) / x_45_scale;
	double t_1 = -4.0 / ((y_45_scale * x_45_scale) * y_45_scale);
	double tmp;
	if (x_45_scale <= 4.2e-158) {
		tmp = (a * b) * (t_0 * t_1);
	} else if (x_45_scale <= 2.6e+179) {
		tmp = (((-4.0 * a) * b) / ((x_45_scale * x_45_scale) * y_45_scale)) * ((a * b) / y_45_scale);
	} else {
		tmp = t_0 * ((a * b) * t_1);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (a * b) / x_45scale
    t_1 = (-4.0d0) / ((y_45scale * x_45scale) * y_45scale)
    if (x_45scale <= 4.2d-158) then
        tmp = (a * b) * (t_0 * t_1)
    else if (x_45scale <= 2.6d+179) then
        tmp = ((((-4.0d0) * a) * b) / ((x_45scale * x_45scale) * y_45scale)) * ((a * b) / y_45scale)
    else
        tmp = t_0 * ((a * b) * t_1)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (a * b) / x_45_scale;
	double t_1 = -4.0 / ((y_45_scale * x_45_scale) * y_45_scale);
	double tmp;
	if (x_45_scale <= 4.2e-158) {
		tmp = (a * b) * (t_0 * t_1);
	} else if (x_45_scale <= 2.6e+179) {
		tmp = (((-4.0 * a) * b) / ((x_45_scale * x_45_scale) * y_45_scale)) * ((a * b) / y_45_scale);
	} else {
		tmp = t_0 * ((a * b) * t_1);
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (a * b) / x_45_scale
	t_1 = -4.0 / ((y_45_scale * x_45_scale) * y_45_scale)
	tmp = 0
	if x_45_scale <= 4.2e-158:
		tmp = (a * b) * (t_0 * t_1)
	elif x_45_scale <= 2.6e+179:
		tmp = (((-4.0 * a) * b) / ((x_45_scale * x_45_scale) * y_45_scale)) * ((a * b) / y_45_scale)
	else:
		tmp = t_0 * ((a * b) * t_1)
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(a * b) / x_45_scale)
	t_1 = Float64(-4.0 / Float64(Float64(y_45_scale * x_45_scale) * y_45_scale))
	tmp = 0.0
	if (x_45_scale <= 4.2e-158)
		tmp = Float64(Float64(a * b) * Float64(t_0 * t_1));
	elseif (x_45_scale <= 2.6e+179)
		tmp = Float64(Float64(Float64(Float64(-4.0 * a) * b) / Float64(Float64(x_45_scale * x_45_scale) * y_45_scale)) * Float64(Float64(a * b) / y_45_scale));
	else
		tmp = Float64(t_0 * Float64(Float64(a * b) * t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (a * b) / x_45_scale;
	t_1 = -4.0 / ((y_45_scale * x_45_scale) * y_45_scale);
	tmp = 0.0;
	if (x_45_scale <= 4.2e-158)
		tmp = (a * b) * (t_0 * t_1);
	elseif (x_45_scale <= 2.6e+179)
		tmp = (((-4.0 * a) * b) / ((x_45_scale * x_45_scale) * y_45_scale)) * ((a * b) / y_45_scale);
	else
		tmp = t_0 * ((a * b) * t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(a * b), $MachinePrecision] / x$45$scale), $MachinePrecision]}, Block[{t$95$1 = N[(-4.0 / N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$45$scale, 4.2e-158], N[(N[(a * b), $MachinePrecision] * N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$45$scale, 2.6e+179], N[(N[(N[(N[(-4.0 * a), $MachinePrecision] * b), $MachinePrecision] / N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(a * b), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(a * b), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x-scale < 4.19999999999999983e-158

   1. Initial program 21.6%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 57.4

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

   5. Applied rewrites 57.4%

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

   7. Applied rewrites 84.4%

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

   9. Applied rewrites 88.2%

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

   11. Applied rewrites 88.2%

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

   if 4.19999999999999983e-158 < x-scale < 2.6000000000000002e179

   1. Initial program 35.0%

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

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 69.7

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

   5. Applied rewrites 69.7%

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

   7. Applied rewrites 86.6%

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

   9. Applied rewrites 90.6%

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

   11. Applied rewrites 93.5%

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

   if 2.6000000000000002e179 < x-scale

   1. Initial program 47.1%

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

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 52.6

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

   5. Applied rewrites 52.6%

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

   7. Applied rewrites 76.3%

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

   9. Applied rewrites 84.4%

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

   11. Applied rewrites 81.1%

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

Alternative 5: 82.5% accurate, 32.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y-scale \leq 4.3 \cdot 10^{+139}:\\ \;\;\;\;\frac{a \cdot b}{x-scale} \cdot \left(\left(a \cdot b\right) \cdot \frac{-4}{\left(y-scale \cdot x-scale\right) \cdot y-scale}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-4 \cdot a}{y-scale \cdot x-scale} \cdot \frac{a}{y-scale \cdot x-scale}\right) \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= y-scale 4.3e+139)
   (* (/ (* a b) x-scale) (* (* a b) (/ -4.0 (* (* y-scale x-scale) y-scale))))
   (*
    (* (/ (* -4.0 a) (* y-scale x-scale)) (/ a (* y-scale x-scale)))
    (* b b))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (y_45_scale <= 4.3e+139) {
		tmp = ((a * b) / x_45_scale) * ((a * b) * (-4.0 / ((y_45_scale * x_45_scale) * y_45_scale)));
	} else {
		tmp = (((-4.0 * a) / (y_45_scale * x_45_scale)) * (a / (y_45_scale * x_45_scale))) * (b * b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: tmp
    if (y_45scale <= 4.3d+139) then
        tmp = ((a * b) / x_45scale) * ((a * b) * ((-4.0d0) / ((y_45scale * x_45scale) * y_45scale)))
    else
        tmp = ((((-4.0d0) * a) / (y_45scale * x_45scale)) * (a / (y_45scale * x_45scale))) * (b * b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (y_45_scale <= 4.3e+139) {
		tmp = ((a * b) / x_45_scale) * ((a * b) * (-4.0 / ((y_45_scale * x_45_scale) * y_45_scale)));
	} else {
		tmp = (((-4.0 * a) / (y_45_scale * x_45_scale)) * (a / (y_45_scale * x_45_scale))) * (b * b);
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if y_45_scale <= 4.3e+139:
		tmp = ((a * b) / x_45_scale) * ((a * b) * (-4.0 / ((y_45_scale * x_45_scale) * y_45_scale)))
	else:
		tmp = (((-4.0 * a) / (y_45_scale * x_45_scale)) * (a / (y_45_scale * x_45_scale))) * (b * b)
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (y_45_scale <= 4.3e+139)
		tmp = Float64(Float64(Float64(a * b) / x_45_scale) * Float64(Float64(a * b) * Float64(-4.0 / Float64(Float64(y_45_scale * x_45_scale) * y_45_scale))));
	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 * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (y_45_scale <= 4.3e+139)
		tmp = ((a * b) / x_45_scale) * ((a * b) * (-4.0 / ((y_45_scale * x_45_scale) * y_45_scale)));
	else
		tmp = (((-4.0 * a) / (y_45_scale * x_45_scale)) * (a / (y_45_scale * x_45_scale))) * (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[y$45$scale, 4.3e+139], N[(N[(N[(a * b), $MachinePrecision] / x$45$scale), $MachinePrecision] * N[(N[(a * b), $MachinePrecision] * N[(-4.0 / N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 4.2999999999999998e139

   1. Initial program 25.0%

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

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 59.9

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

   5. Applied rewrites 59.9%

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

   7. Applied rewrites 84.2%

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

   9. Applied rewrites 87.8%

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

   11. Applied rewrites 84.0%

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

   if 4.2999999999999998e139 < y-scale

   1. Initial program 50.5%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} \]
   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 71.9%

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

   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 78.1%

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

   10. Applied rewrites 90.7%

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

Alternative 6: 75.3% accurate, 32.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;angle \leq 1.05 \cdot 10^{+60}:\\ \;\;\;\;\left(\frac{-4 \cdot a}{y-scale \cdot x-scale} \cdot \frac{a}{y-scale \cdot x-scale}\right) \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(-4 \cdot a\right) \cdot b, a \cdot b, 0\right)}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= angle 1.05e+60)
   (* (* (/ (* -4.0 a) (* y-scale x-scale)) (/ a (* y-scale x-scale))) (* b b))
   (/
    (fma (* (* -4.0 a) b) (* a b) 0.0)
    (* (* y-scale x-scale) (* y-scale x-scale)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (angle <= 1.05e+60) {
		tmp = (((-4.0 * a) / (y_45_scale * x_45_scale)) * (a / (y_45_scale * x_45_scale))) * (b * b);
	} else {
		tmp = fma(((-4.0 * a) * b), (a * b), 0.0) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale));
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (angle <= 1.05e+60)
		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 * b));
	else
		tmp = Float64(fma(Float64(Float64(-4.0 * a) * b), Float64(a * b), 0.0) / Float64(Float64(y_45_scale * x_45_scale) * Float64(y_45_scale * x_45_scale)));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[angle, 1.05e+60], 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 * b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-4.0 * a), $MachinePrecision] * b), $MachinePrecision] * N[(a * b), $MachinePrecision] + 0.0), $MachinePrecision] / N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 1.0500000000000001e60

   1. Initial program 28.9%

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

   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 50.8%

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

   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 64.7%

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

   10. Applied rewrites 80.6%

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

   if 1.0500000000000001e60 < angle

   1. Initial program 24.7%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} \]
   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 61.8%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 84.3%

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

   9. Taylor expanded in angle around 0

      \[\leadsto -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + \frac{{a}^{2} \cdot \left({angle}^{2} \cdot \left({b}^{2} \cdot \left(\frac{-1}{4050} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{4050} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)}{\color{blue}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 34.1%

       \[\leadsto \frac{\mathsf{fma}\left(\left(a \cdot angle\right) \cdot \left(a \cdot angle\right), \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 0\right) \cdot \left(b \cdot b\right), -4 \cdot \left(\left(a \cdot b\right) \cdot \left(a \cdot b\right)\right)\right)}{\left(y-scale \cdot x-scale\right) \cdot \color{blue}{\left(y-scale \cdot x-scale\right)}} \]
   12. Step-by-step derivation

   13. Applied rewrites 84.4%

       \[\leadsto \frac{\mathsf{fma}\left(\left(-4 \cdot a\right) \cdot b, a \cdot b, 0\right)}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 77.1% accurate, 39.7× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\left(-4 \cdot a\right) \cdot b, a \cdot b, 0\right)}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (/
  (fma (* (* -4.0 a) b) (* a b) 0.0)
  (* (* y-scale x-scale) (* y-scale x-scale))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return fma(((-4.0 * a) * b), (a * b), 0.0) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale));
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return Float64(fma(Float64(Float64(-4.0 * a) * b), Float64(a * b), 0.0) / Float64(Float64(y_45_scale * x_45_scale) * Float64(y_45_scale * x_45_scale)))
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[(N[(N[(N[(-4.0 * a), $MachinePrecision] * b), $MachinePrecision] * N[(a * b), $MachinePrecision] + 0.0), $MachinePrecision] / N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 28.2%

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

3. Taylor expanded in 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 52.7%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 79.3%

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

9. Taylor expanded in angle around 0

   \[\leadsto -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + \frac{{a}^{2} \cdot \left({angle}^{2} \cdot \left({b}^{2} \cdot \left(\frac{-1}{4050} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{4050} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)}{\color{blue}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
10. Step-by-step derivation

11. Applied rewrites 42.4%

    \[\leadsto \frac{\mathsf{fma}\left(\left(a \cdot angle\right) \cdot \left(a \cdot angle\right), \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 0\right) \cdot \left(b \cdot b\right), -4 \cdot \left(\left(a \cdot b\right) \cdot \left(a \cdot b\right)\right)\right)}{\left(y-scale \cdot x-scale\right) \cdot \color{blue}{\left(y-scale \cdot x-scale\right)}} \]
12. Step-by-step derivation

13. Applied rewrites 80.0%

    \[\leadsto \frac{\mathsf{fma}\left(\left(-4 \cdot a\right) \cdot b, a \cdot b, 0\right)}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \]
14. Add Preprocessing

Alternative 8: 66.4% accurate, 39.7× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(-4 \cdot a, \left(b \cdot b\right) \cdot a, 0\right)}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (/
  (fma (* -4.0 a) (* (* b b) a) 0.0)
  (* (* y-scale x-scale) (* y-scale x-scale))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return fma((-4.0 * a), ((b * b) * a), 0.0) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale));
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return Float64(fma(Float64(-4.0 * a), Float64(Float64(b * b) * a), 0.0) / Float64(Float64(y_45_scale * x_45_scale) * Float64(y_45_scale * x_45_scale)))
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[(N[(N[(-4.0 * a), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] * a), $MachinePrecision] + 0.0), $MachinePrecision] / N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 28.2%

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

3. Taylor expanded in 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 52.7%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 79.3%

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

9. Taylor expanded in angle around 0

   \[\leadsto -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + \frac{{a}^{2} \cdot \left({angle}^{2} \cdot \left({b}^{2} \cdot \left(\frac{-1}{4050} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{4050} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)}{\color{blue}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
10. Step-by-step derivation

11. Applied rewrites 42.4%

    \[\leadsto \frac{\mathsf{fma}\left(\left(a \cdot angle\right) \cdot \left(a \cdot angle\right), \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 0\right) \cdot \left(b \cdot b\right), -4 \cdot \left(\left(a \cdot b\right) \cdot \left(a \cdot b\right)\right)\right)}{\left(y-scale \cdot x-scale\right) \cdot \color{blue}{\left(y-scale \cdot x-scale\right)}} \]
12. Step-by-step derivation

13. Applied rewrites 72.3%

    \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot a, \left(b \cdot b\right) \cdot a, 0\right)}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \]
14. Add Preprocessing

Alternative 9: 60.7% accurate, 40.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 28.2%

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

3. Taylor expanded in 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 52.7%

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

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 65.5%

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

Alternative 10: 60.0% accurate, 40.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 28.2%

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

3. Taylor expanded in 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 52.7%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 79.3%

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

9. Taylor expanded in angle around 0

   \[\leadsto -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + \frac{{a}^{2} \cdot \left({angle}^{2} \cdot \left({b}^{2} \cdot \left(\frac{-1}{4050} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{4050} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)}{\color{blue}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
10. Step-by-step derivation

11. Applied rewrites 42.4%

    \[\leadsto \frac{\mathsf{fma}\left(\left(a \cdot angle\right) \cdot \left(a \cdot angle\right), \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 0\right) \cdot \left(b \cdot b\right), -4 \cdot \left(\left(a \cdot b\right) \cdot \left(a \cdot b\right)\right)\right)}{\left(y-scale \cdot x-scale\right) \cdot \color{blue}{\left(y-scale \cdot x-scale\right)}} \]

12. Taylor expanded in a around 0

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

14. Applied rewrites 65.2%

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

Reproduce

herbie shell --seed 2024332 
(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))))