Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 25.0% → 94.8%

Time: 22.1s

Alternatives: 12

Speedup: 86.6×

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

Specification

Math

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

FPCore

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

Initial Program: 25.0% 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: 94.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \mathsf{PI}\left(\right)\\ t_1 := \sin t\_0\\ t_2 := \cos t\_0\\ t_3 := \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot t\_1\right) \cdot t\_2}{x-scale}}{y-scale}\\ t_4 := \frac{a \cdot b}{y-scale \cdot x-scale}\\ \mathbf{if}\;t\_3 \cdot t\_3 - \left(4 \cdot \frac{\frac{{\left(a \cdot t\_1\right)}^{2} + {\left(b \cdot t\_2\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot t\_2\right)}^{2} + {\left(b \cdot t\_1\right)}^{2}}{y-scale}}{y-scale} \leq 10^{+35}:\\ \;\;\;\;\left(\frac{a}{y-scale} \cdot \left(\frac{b}{x-scale} \cdot -4\right)\right) \cdot \left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_4 \cdot t\_4\right) \cdot -4\\ \end{array} \end{array} \]

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_4 (/ (* a b) (* y-scale x-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)))
        1e+35)
     (*
      (* (/ a y-scale) (* (/ b x-scale) -4.0))
      (* (/ a x-scale) (/ b y-scale)))
     (* (* t_4 t_4) -4.0))))

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) x-scale) y-scale) (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) x-scale) y-scale)) (*.f64 (*.f64 #s(literal 4 binary64) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) x-scale) x-scale)) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) y-scale) y-scale))) < 9.9999999999999997e34

   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 angle around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 69.7

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

   5. Applied rewrites 69.7%

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

   7. Applied rewrites 73.8%

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

   9. Applied rewrites 93.2%

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

   11. Applied rewrites 97.9%

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

   if 9.9999999999999997e34 < (-.f64 (*.f64 (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) x-scale) y-scale) (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) x-scale) y-scale)) (*.f64 (*.f64 #s(literal 4 binary64) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) x-scale) x-scale)) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) y-scale) y-scale)))

   1. Initial program 0.0%

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

   3. Taylor expanded in angle around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 45.8

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

   5. Applied rewrites 45.8%

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

   7. Applied rewrites 79.1%

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

   9. Applied rewrites 96.5%

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

Alternative 2: 79.7% accurate, 29.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.1 \cdot 10^{-99} \lor \neg \left(b \leq 6.2 \cdot 10^{+62}\right):\\ \;\;\;\;\frac{\left(b \cdot a\right) \cdot \left(b \cdot a\right)}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-4 \cdot a}{y-scale \cdot x-scale} \cdot \frac{a}{y-scale \cdot x-scale}\right) \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (or (<= b 2.1e-99) (not (<= b 6.2e+62)))
   (* (/ (* (* b a) (* b a)) (* (* x-scale y-scale) (* x-scale y-scale))) -4.0)
   (*
    (* (/ (* -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 ((b <= 2.1e-99) || !(b <= 6.2e+62)) {
		tmp = (((b * a) * (b * a)) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale))) * -4.0;
	} 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 ((b <= 2.1d-99) .or. (.not. (b <= 6.2d+62))) then
        tmp = (((b * a) * (b * a)) / ((x_45scale * y_45scale) * (x_45scale * y_45scale))) * (-4.0d0)
    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 ((b <= 2.1e-99) || !(b <= 6.2e+62)) {
		tmp = (((b * a) * (b * a)) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale))) * -4.0;
	} 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 (b <= 2.1e-99) or not (b <= 6.2e+62):
		tmp = (((b * a) * (b * a)) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale))) * -4.0
	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 ((b <= 2.1e-99) || !(b <= 6.2e+62))
		tmp = Float64(Float64(Float64(Float64(b * a) * Float64(b * a)) / Float64(Float64(x_45_scale * y_45_scale) * Float64(x_45_scale * y_45_scale))) * -4.0);
	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 ((b <= 2.1e-99) || ~((b <= 6.2e+62)))
		tmp = (((b * a) * (b * a)) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale))) * -4.0;
	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[Or[LessEqual[b, 2.1e-99], N[Not[LessEqual[b, 6.2e+62]], $MachinePrecision]], N[(N[(N[(N[(b * a), $MachinePrecision] * N[(b * a), $MachinePrecision]), $MachinePrecision] / N[(N[(x$45$scale * y$45$scale), $MachinePrecision] * N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -4.0), $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 b < 2.09999999999999984e-99 or 6.20000000000000029e62 < b

   1. Initial program 25.7%

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

   3. Taylor expanded in angle around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 55.0

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

   5. Applied rewrites 55.0%

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

   7. Applied rewrites 78.2%

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

   9. Applied rewrites 95.0%

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

   11. Applied rewrites 78.2%

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

   if 2.09999999999999984e-99 < b < 6.20000000000000029e62

   1. Initial program 30.5%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} \]
   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 49.3%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 68.6%

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

   10. Applied rewrites 92.1%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.1 \cdot 10^{-99} \lor \neg \left(b \leq 6.2 \cdot 10^{+62}\right):\\ \;\;\;\;\frac{\left(b \cdot a\right) \cdot \left(b \cdot a\right)}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-4 \cdot a}{y-scale \cdot x-scale} \cdot \frac{a}{y-scale \cdot x-scale}\right) \cdot \left(b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.7% accurate, 29.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x-scale \leq 5.8 \cdot 10^{-152}:\\ \;\;\;\;\frac{\left(b \cdot a\right) \cdot \left(b \cdot a\right)}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)} \cdot -4\\ \mathbf{elif}\;x-scale \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(\frac{a \cdot b}{y-scale} \cdot \frac{a \cdot b}{\left(x-scale \cdot x-scale\right) \cdot y-scale}\right) \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-4 \cdot a}{y-scale \cdot x-scale} \cdot \frac{a}{y-scale \cdot x-scale}\right) \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= x-scale 5.8e-152)
   (* (/ (* (* b a) (* b a)) (* (* x-scale y-scale) (* x-scale y-scale))) -4.0)
   (if (<= x-scale 1.35e+154)
     (*
      (* (/ (* a b) y-scale) (/ (* a b) (* (* x-scale x-scale) y-scale)))
      -4.0)
     (*
      (* (/ (* -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 (x_45_scale <= 5.8e-152) {
		tmp = (((b * a) * (b * a)) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale))) * -4.0;
	} else if (x_45_scale <= 1.35e+154) {
		tmp = (((a * b) / y_45_scale) * ((a * b) / ((x_45_scale * x_45_scale) * y_45_scale))) * -4.0;
	} 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 (x_45scale <= 5.8d-152) then
        tmp = (((b * a) * (b * a)) / ((x_45scale * y_45scale) * (x_45scale * y_45scale))) * (-4.0d0)
    else if (x_45scale <= 1.35d+154) then
        tmp = (((a * b) / y_45scale) * ((a * b) / ((x_45scale * x_45scale) * y_45scale))) * (-4.0d0)
    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 (x_45_scale <= 5.8e-152) {
		tmp = (((b * a) * (b * a)) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale))) * -4.0;
	} else if (x_45_scale <= 1.35e+154) {
		tmp = (((a * b) / y_45_scale) * ((a * b) / ((x_45_scale * x_45_scale) * y_45_scale))) * -4.0;
	} 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 x_45_scale <= 5.8e-152:
		tmp = (((b * a) * (b * a)) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale))) * -4.0
	elif x_45_scale <= 1.35e+154:
		tmp = (((a * b) / y_45_scale) * ((a * b) / ((x_45_scale * x_45_scale) * y_45_scale))) * -4.0
	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 (x_45_scale <= 5.8e-152)
		tmp = Float64(Float64(Float64(Float64(b * a) * Float64(b * a)) / Float64(Float64(x_45_scale * y_45_scale) * Float64(x_45_scale * y_45_scale))) * -4.0);
	elseif (x_45_scale <= 1.35e+154)
		tmp = Float64(Float64(Float64(Float64(a * b) / y_45_scale) * Float64(Float64(a * b) / Float64(Float64(x_45_scale * x_45_scale) * y_45_scale))) * -4.0);
	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 (x_45_scale <= 5.8e-152)
		tmp = (((b * a) * (b * a)) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale))) * -4.0;
	elseif (x_45_scale <= 1.35e+154)
		tmp = (((a * b) / y_45_scale) * ((a * b) / ((x_45_scale * x_45_scale) * y_45_scale))) * -4.0;
	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[x$45$scale, 5.8e-152], N[(N[(N[(N[(b * a), $MachinePrecision] * N[(b * a), $MachinePrecision]), $MachinePrecision] / N[(N[(x$45$scale * y$45$scale), $MachinePrecision] * N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision], If[LessEqual[x$45$scale, 1.35e+154], N[(N[(N[(N[(a * b), $MachinePrecision] / y$45$scale), $MachinePrecision] * N[(N[(a * b), $MachinePrecision] / N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -4.0), $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 3 regimes

2. if x-scale < 5.8000000000000003e-152

   1. Initial program 22.9%

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

   3. Taylor expanded in angle around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 52.5

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

   5. Applied rewrites 52.5%

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

   7. Applied rewrites 76.5%

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

   9. Applied rewrites 94.0%

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

   11. Applied rewrites 76.5%

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

   if 5.8000000000000003e-152 < x-scale < 1.35000000000000003e154

   1. Initial program 27.1%

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

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 72.3

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

   5. Applied rewrites 72.3%

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

   7. Applied rewrites 75.7%

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

   9. Applied rewrites 88.6%

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

   if 1.35000000000000003e154 < x-scale

   1. Initial program 41.4%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} \]
   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 41.3%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 62.1%

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

   10. Applied rewrites 76.5%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq 5.8 \cdot 10^{-152}:\\ \;\;\;\;\frac{\left(b \cdot a\right) \cdot \left(b \cdot a\right)}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)} \cdot -4\\ \mathbf{elif}\;x-scale \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(\frac{a \cdot b}{y-scale} \cdot \frac{a \cdot b}{\left(x-scale \cdot x-scale\right) \cdot y-scale}\right) \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-4 \cdot a}{y-scale \cdot x-scale} \cdot \frac{a}{y-scale \cdot x-scale}\right) \cdot \left(b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.0% accurate, 29.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b}{y-scale \cdot x-scale}\\ \mathbf{if}\;a \leq 1.28 \cdot 10^{-160}:\\ \;\;\;\;\left(\frac{-4 \cdot a}{y-scale \cdot x-scale} \cdot \frac{a}{y-scale \cdot x-scale}\right) \cdot \left(b \cdot b\right)\\ \mathbf{elif}\;a \leq 4.2 \cdot 10^{+157}:\\ \;\;\;\;\left(-4 \cdot \left(a \cdot a\right)\right) \cdot \left(t\_0 \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(b \cdot a\right) \cdot \left(b \cdot a\right)}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)} \cdot -4\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ b (* y-scale x-scale))))
   (if (<= a 1.28e-160)
     (*
      (* (/ (* -4.0 a) (* y-scale x-scale)) (/ a (* y-scale x-scale)))
      (* b b))
     (if (<= a 4.2e+157)
       (* (* -4.0 (* a a)) (* t_0 t_0))
       (*
        (/ (* (* b a) (* b a)) (* (* x-scale y-scale) (* x-scale y-scale)))
        -4.0)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = b / (y_45_scale * x_45_scale);
	double tmp;
	if (a <= 1.28e-160) {
		tmp = (((-4.0 * a) / (y_45_scale * x_45_scale)) * (a / (y_45_scale * x_45_scale))) * (b * b);
	} else if (a <= 4.2e+157) {
		tmp = (-4.0 * (a * a)) * (t_0 * t_0);
	} else {
		tmp = (((b * a) * (b * a)) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale))) * -4.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: tmp
    t_0 = b / (y_45scale * x_45scale)
    if (a <= 1.28d-160) then
        tmp = ((((-4.0d0) * a) / (y_45scale * x_45scale)) * (a / (y_45scale * x_45scale))) * (b * b)
    else if (a <= 4.2d+157) then
        tmp = ((-4.0d0) * (a * a)) * (t_0 * t_0)
    else
        tmp = (((b * a) * (b * a)) / ((x_45scale * y_45scale) * (x_45scale * y_45scale))) * (-4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = b / (y_45_scale * x_45_scale);
	double tmp;
	if (a <= 1.28e-160) {
		tmp = (((-4.0 * a) / (y_45_scale * x_45_scale)) * (a / (y_45_scale * x_45_scale))) * (b * b);
	} else if (a <= 4.2e+157) {
		tmp = (-4.0 * (a * a)) * (t_0 * t_0);
	} else {
		tmp = (((b * a) * (b * a)) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale))) * -4.0;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = b / (y_45_scale * x_45_scale)
	tmp = 0
	if a <= 1.28e-160:
		tmp = (((-4.0 * a) / (y_45_scale * x_45_scale)) * (a / (y_45_scale * x_45_scale))) * (b * b)
	elif a <= 4.2e+157:
		tmp = (-4.0 * (a * a)) * (t_0 * t_0)
	else:
		tmp = (((b * a) * (b * a)) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale))) * -4.0
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(b / Float64(y_45_scale * x_45_scale))
	tmp = 0.0
	if (a <= 1.28e-160)
		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));
	elseif (a <= 4.2e+157)
		tmp = Float64(Float64(-4.0 * Float64(a * a)) * Float64(t_0 * t_0));
	else
		tmp = Float64(Float64(Float64(Float64(b * a) * Float64(b * a)) / Float64(Float64(x_45_scale * y_45_scale) * Float64(x_45_scale * y_45_scale))) * -4.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = b / (y_45_scale * x_45_scale);
	tmp = 0.0;
	if (a <= 1.28e-160)
		tmp = (((-4.0 * a) / (y_45_scale * x_45_scale)) * (a / (y_45_scale * x_45_scale))) * (b * b);
	elseif (a <= 4.2e+157)
		tmp = (-4.0 * (a * a)) * (t_0 * t_0);
	else
		tmp = (((b * a) * (b * a)) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale))) * -4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(b / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 1.28e-160], 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], If[LessEqual[a, 4.2e+157], N[(N[(-4.0 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(b * a), $MachinePrecision] * N[(b * a), $MachinePrecision]), $MachinePrecision] / N[(N[(x$45$scale * y$45$scale), $MachinePrecision] * N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < 1.27999999999999994e-160

   1. Initial program 27.9%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} \]
   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 41.9%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 55.3%

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

   10. Applied rewrites 72.5%

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

   if 1.27999999999999994e-160 < a < 4.2e157

   1. Initial program 33.0%

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

   3. Taylor expanded in angle around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 71.0

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

   5. Applied rewrites 71.0%

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

   7. Applied rewrites 95.3%

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

   if 4.2e157 < a

   1. Initial program 0.0%

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

   3. Taylor expanded in angle around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 23.6

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

   5. Applied rewrites 23.6%

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

   7. Applied rewrites 69.4%

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

   9. Applied rewrites 92.0%

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

   11. Applied rewrites 69.4%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.28 \cdot 10^{-160}:\\ \;\;\;\;\left(\frac{-4 \cdot a}{y-scale \cdot x-scale} \cdot \frac{a}{y-scale \cdot x-scale}\right) \cdot \left(b \cdot b\right)\\ \mathbf{elif}\;a \leq 4.2 \cdot 10^{+157}:\\ \;\;\;\;\left(-4 \cdot \left(a \cdot a\right)\right) \cdot \left(\frac{b}{y-scale \cdot x-scale} \cdot \frac{b}{y-scale \cdot x-scale}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(b \cdot a\right) \cdot \left(b \cdot a\right)}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)} \cdot -4\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 50.1% accurate, 35.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4 \cdot 10^{-159}:\\ \;\;\;\;\frac{4}{x-scale} \cdot \frac{0}{x-scale}\\ \mathbf{else}:\\ \;\;\;\;\left(-4 \cdot \left(a \cdot a\right)\right) \cdot \frac{b \cdot b}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= b 4e-159)
   (* (/ 4.0 x-scale) (/ 0.0 x-scale))
   (*
    (* -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) {
	double tmp;
	if (b <= 4e-159) {
		tmp = (4.0 / x_45_scale) * (0.0 / x_45_scale);
	} else {
		tmp = (-4.0 * (a * a)) * ((b * b) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale)));
	}
	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 (b <= 4d-159) then
        tmp = (4.0d0 / x_45scale) * (0.0d0 / x_45scale)
    else
        tmp = ((-4.0d0) * (a * a)) * ((b * b) / ((y_45scale * x_45scale) * (y_45scale * x_45scale)))
    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 (b <= 4e-159) {
		tmp = (4.0 / x_45_scale) * (0.0 / x_45_scale);
	} else {
		tmp = (-4.0 * (a * a)) * ((b * b) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale)));
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if b <= 4e-159:
		tmp = (4.0 / x_45_scale) * (0.0 / x_45_scale)
	else:
		tmp = (-4.0 * (a * a)) * ((b * b) / ((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 (b <= 4e-159)
		tmp = Float64(Float64(4.0 / x_45_scale) * Float64(0.0 / x_45_scale));
	else
		tmp = Float64(Float64(-4.0 * Float64(a * a)) * Float64(Float64(b * b) / Float64(Float64(y_45_scale * x_45_scale) * Float64(y_45_scale * x_45_scale))));
	end
	return tmp
end

MATLAB

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

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[b, 4e-159], N[(N[(4.0 / x$45$scale), $MachinePrecision] * N[(0.0 / x$45$scale), $MachinePrecision]), $MachinePrecision], N[(N[(-4.0 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] / N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.99999999999999995e-159

   1. Initial program 29.2%

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

   3. Taylor expanded in angle around inf

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

   5. Applied rewrites 24.2%

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

   6. Taylor expanded in b around 0

      \[\leadsto \frac{4}{x-scale} \cdot \frac{\frac{b \cdot \left({\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \left({\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \left(-2 \cdot {a}^{3} + 2 \cdot {a}^{3}\right)\right)\right)}{{y-scale}^{2}}}{x-scale} \]

   8. Applied rewrites 33.6%

      \[\leadsto \frac{4}{x-scale} \cdot \frac{b \cdot \frac{\left(0 \cdot {\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}\right) \cdot {\cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}}{y-scale \cdot y-scale}}{x-scale} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{4}{x-scale} \cdot \frac{0}{x-scale} \]
   10. Step-by-step derivation

   11. Applied rewrites 37.7%

       \[\leadsto \frac{4}{x-scale} \cdot \frac{0}{x-scale} \]

   if 3.99999999999999995e-159 < b

   1. Initial program 20.8%

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

   3. Taylor expanded in angle around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 53.9

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

   5. Applied rewrites 53.9%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 64.2%

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

Alternative 6: 50.1% accurate, 35.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4 \cdot 10^{-159}:\\ \;\;\;\;\frac{4}{x-scale} \cdot \frac{0}{x-scale}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-4 \cdot a\right) \cdot a}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= b 4e-159)
   (* (/ 4.0 x-scale) (/ 0.0 x-scale))
   (*
    (/ (* (* -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) {
	double tmp;
	if (b <= 4e-159) {
		tmp = (4.0 / x_45_scale) * (0.0 / x_45_scale);
	} else {
		tmp = (((-4.0 * a) * a) / ((y_45_scale * x_45_scale) * (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 (b <= 4d-159) then
        tmp = (4.0d0 / x_45scale) * (0.0d0 / x_45scale)
    else
        tmp = ((((-4.0d0) * a) * a) / ((y_45scale * x_45scale) * (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 (b <= 4e-159) {
		tmp = (4.0 / x_45_scale) * (0.0 / x_45_scale);
	} else {
		tmp = (((-4.0 * a) * a) / ((y_45_scale * x_45_scale) * (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 b <= 4e-159:
		tmp = (4.0 / x_45_scale) * (0.0 / x_45_scale)
	else:
		tmp = (((-4.0 * a) * a) / ((y_45_scale * x_45_scale) * (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 (b <= 4e-159)
		tmp = Float64(Float64(4.0 / x_45_scale) * Float64(0.0 / x_45_scale));
	else
		tmp = Float64(Float64(Float64(Float64(-4.0 * a) * a) / Float64(Float64(y_45_scale * x_45_scale) * 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 (b <= 4e-159)
		tmp = (4.0 / x_45_scale) * (0.0 / x_45_scale);
	else
		tmp = (((-4.0 * a) * a) / ((y_45_scale * x_45_scale) * (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[b, 4e-159], N[(N[(4.0 / x$45$scale), $MachinePrecision] * N[(0.0 / x$45$scale), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-4.0 * a), $MachinePrecision] * a), $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. Split input into 2 regimes

2. if b < 3.99999999999999995e-159

   1. Initial program 29.2%

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

   3. Taylor expanded in angle around inf

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

   5. Applied rewrites 24.2%

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

   6. Taylor expanded in b around 0

      \[\leadsto \frac{4}{x-scale} \cdot \frac{\frac{b \cdot \left({\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \left({\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \left(-2 \cdot {a}^{3} + 2 \cdot {a}^{3}\right)\right)\right)}{{y-scale}^{2}}}{x-scale} \]

   8. Applied rewrites 33.6%

      \[\leadsto \frac{4}{x-scale} \cdot \frac{b \cdot \frac{\left(0 \cdot {\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}\right) \cdot {\cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}}{y-scale \cdot y-scale}}{x-scale} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{4}{x-scale} \cdot \frac{0}{x-scale} \]
   10. Step-by-step derivation

   11. Applied rewrites 37.7%

       \[\leadsto \frac{4}{x-scale} \cdot \frac{0}{x-scale} \]

   if 3.99999999999999995e-159 < b

   1. Initial program 20.8%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} \]
   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 47.1%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 63.3%

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

   10. Applied rewrites 63.5%

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

Alternative 7: 50.1% accurate, 35.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4 \cdot 10^{-159}:\\ \;\;\;\;\frac{4}{x-scale} \cdot \frac{0}{x-scale}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4 \cdot \left(a \cdot a\right)}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= b 4e-159)
   (* (/ 4.0 x-scale) (/ 0.0 x-scale))
   (*
    (/ (* -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) {
	double tmp;
	if (b <= 4e-159) {
		tmp = (4.0 / x_45_scale) * (0.0 / x_45_scale);
	} else {
		tmp = ((-4.0 * (a * a)) / ((y_45_scale * x_45_scale) * (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 (b <= 4d-159) then
        tmp = (4.0d0 / x_45scale) * (0.0d0 / x_45scale)
    else
        tmp = (((-4.0d0) * (a * a)) / ((y_45scale * x_45scale) * (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 (b <= 4e-159) {
		tmp = (4.0 / x_45_scale) * (0.0 / x_45_scale);
	} else {
		tmp = ((-4.0 * (a * a)) / ((y_45_scale * x_45_scale) * (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 b <= 4e-159:
		tmp = (4.0 / x_45_scale) * (0.0 / x_45_scale)
	else:
		tmp = ((-4.0 * (a * a)) / ((y_45_scale * x_45_scale) * (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 (b <= 4e-159)
		tmp = Float64(Float64(4.0 / x_45_scale) * Float64(0.0 / x_45_scale));
	else
		tmp = 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
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (b <= 4e-159)
		tmp = (4.0 / x_45_scale) * (0.0 / x_45_scale);
	else
		tmp = ((-4.0 * (a * a)) / ((y_45_scale * x_45_scale) * (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[b, 4e-159], N[(N[(4.0 / x$45$scale), $MachinePrecision] * N[(0.0 / x$45$scale), $MachinePrecision]), $MachinePrecision], 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. Split input into 2 regimes

2. if b < 3.99999999999999995e-159

   1. Initial program 29.2%

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

   3. Taylor expanded in angle around inf

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

   5. Applied rewrites 24.2%

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

   6. Taylor expanded in b around 0

      \[\leadsto \frac{4}{x-scale} \cdot \frac{\frac{b \cdot \left({\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \left({\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \left(-2 \cdot {a}^{3} + 2 \cdot {a}^{3}\right)\right)\right)}{{y-scale}^{2}}}{x-scale} \]

   8. Applied rewrites 33.6%

      \[\leadsto \frac{4}{x-scale} \cdot \frac{b \cdot \frac{\left(0 \cdot {\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}\right) \cdot {\cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}}{y-scale \cdot y-scale}}{x-scale} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{4}{x-scale} \cdot \frac{0}{x-scale} \]
   10. Step-by-step derivation

   11. Applied rewrites 37.7%

       \[\leadsto \frac{4}{x-scale} \cdot \frac{0}{x-scale} \]

   if 3.99999999999999995e-159 < b

   1. Initial program 20.8%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} \]
   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 47.1%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 63.3%

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

Alternative 8: 93.5% accurate, 35.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 26.3%

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

3. Taylor expanded in angle around 0

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

   1. associate-/l* N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. *-commutative N/A

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

   9. times-frac N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 55.0

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

5. Applied rewrites 55.0%

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

7. Applied rewrites 77.1%

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

9. Applied rewrites 95.3%

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

Alternative 9: 92.0% accurate, 35.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 26.3%

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

3. Taylor expanded in angle around 0

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

   1. associate-/l* N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. *-commutative N/A

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

   9. times-frac N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 55.0

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

5. Applied rewrites 55.0%

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

7. Applied rewrites 77.1%

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

9. Applied rewrites 95.3%

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

11. Applied rewrites 92.8%

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

Alternative 10: 77.8% accurate, 40.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 26.3%

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

3. Taylor expanded in angle around 0

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

   1. associate-/l* N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. *-commutative N/A

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

   9. times-frac N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 55.0

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

5. Applied rewrites 55.0%

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

7. Applied rewrites 77.1%

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

9. Applied rewrites 95.3%

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

11. Applied rewrites 77.1%

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

12. Final simplification 77.1%

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

Alternative 11: 35.7% accurate, 68.0× speedup

Math

\[\begin{array}{l} \\ \frac{4}{x-scale} \cdot \frac{0}{x-scale} \end{array} \]

FPCore

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

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return (4.0 / x_45_scale) * (0.0 / 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 / x_45scale) * (0.0d0 / 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 / x_45_scale) * (0.0 / x_45_scale);
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	return (4.0 / x_45_scale) * (0.0 / x_45_scale)

Julia

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

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = (4.0 / x_45_scale) * (0.0 / x_45_scale);
end

Wolfram

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

Derivation

1. Initial program 26.3%

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

3. Taylor expanded in angle around inf

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

5. Applied rewrites 22.2%

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

6. Taylor expanded in b around 0

   \[\leadsto \frac{4}{x-scale} \cdot \frac{\frac{b \cdot \left({\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \left({\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \left(-2 \cdot {a}^{3} + 2 \cdot {a}^{3}\right)\right)\right)}{{y-scale}^{2}}}{x-scale} \]

8. Applied rewrites 29.9%

   \[\leadsto \frac{4}{x-scale} \cdot \frac{b \cdot \frac{\left(0 \cdot {\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}\right) \cdot {\cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}}{y-scale \cdot y-scale}}{x-scale} \]

9. Taylor expanded in b around 0

   \[\leadsto \frac{4}{x-scale} \cdot \frac{0}{x-scale} \]
10. Step-by-step derivation

11. Applied rewrites 32.8%

    \[\leadsto \frac{4}{x-scale} \cdot \frac{0}{x-scale} \]
12. Add Preprocessing

Alternative 12: 32.6% accurate, 86.6× speedup

Math

\[\begin{array}{l} \\ \frac{0 \cdot 4}{x-scale \cdot x-scale} \end{array} \]

FPCore

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

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return (0.0 * 4.0) / (x_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 = (0.0d0 * 4.0d0) / (x_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 (0.0 * 4.0) / (x_45_scale * x_45_scale);
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	return (0.0 * 4.0) / (x_45_scale * x_45_scale)

Julia

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

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = (0.0 * 4.0) / (x_45_scale * x_45_scale);
end

Wolfram

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

Derivation

1. Initial program 26.3%

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

3. Taylor expanded in angle around inf

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

5. Applied rewrites 22.2%

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

6. Taylor expanded in b around 0

   \[\leadsto \frac{4}{x-scale} \cdot \frac{\frac{b \cdot \left({\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \left({\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \left(-2 \cdot {a}^{3} + 2 \cdot {a}^{3}\right)\right)\right)}{{y-scale}^{2}}}{x-scale} \]

8. Applied rewrites 29.9%

   \[\leadsto \frac{4}{x-scale} \cdot \frac{b \cdot \frac{\left(0 \cdot {\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}\right) \cdot {\cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}}{y-scale \cdot y-scale}}{x-scale} \]

9. Taylor expanded in b around 0

   \[\leadsto \frac{4}{x-scale} \cdot \frac{0}{x-scale} \]
10. Step-by-step derivation

11. Applied rewrites 32.8%

    \[\leadsto \frac{4}{x-scale} \cdot \frac{0}{x-scale} \]

13. Applied rewrites 29.4%

    \[\leadsto \frac{0 \cdot 4}{\color{blue}{x-scale \cdot x-scale}} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024303 
(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))))