a from scale-rotated-ellipse

Percentage Accurate: 2.8% → 62.1%

Time: 30.3s

Alternatives: 8

Speedup: 484.7×

• 21.5% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. a = -3.8341936916213894e+173
  2. b = -9.892866783139104e-144
  3. angle = -3.5956156938990484e+108
  4. x-scale = 4.250205968268575e+75
  5. y-scale = -9.16474631752597e-90

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \mathsf{PI}\left(\right)\\ t_1 := \sin t\_0\\ t_2 := \cos t\_0\\ t_3 := \frac{\frac{{\left(a \cdot t\_2\right)}^{2} + {\left(b \cdot t\_1\right)}^{2}}{y-scale}}{y-scale}\\ t_4 := \frac{\frac{{\left(a \cdot t\_1\right)}^{2} + {\left(b \cdot t\_2\right)}^{2}}{x-scale}}{x-scale}\\ t_5 := \left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\\ t_6 := \frac{4 \cdot t\_5}{{\left(x-scale \cdot y-scale\right)}^{2}}\\ \frac{-\sqrt{\left(\left(2 \cdot t\_6\right) \cdot t\_5\right) \cdot \left(\left(t\_4 + t\_3\right) + \sqrt{{\left(t\_4 - t\_3\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot t\_1\right) \cdot t\_2}{x-scale}}{y-scale}\right)}^{2}}\right)}}{t\_6} \end{array} \end{array} \]

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
         (/ (/ (+ (pow (* a t_2) 2.0) (pow (* b t_1) 2.0)) y-scale) y-scale))
        (t_4
         (/ (/ (+ (pow (* a t_1) 2.0) (pow (* b t_2) 2.0)) x-scale) x-scale))
        (t_5 (* (* b a) (* b (- a))))
        (t_6 (/ (* 4.0 t_5) (pow (* x-scale y-scale) 2.0))))
   (/
    (-
     (sqrt
      (*
       (* (* 2.0 t_6) t_5)
       (+
        (+ t_4 t_3)
        (sqrt
         (+
          (pow (- t_4 t_3) 2.0)
          (pow
           (/
            (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) t_1) t_2) x-scale)
            y-scale)
           2.0)))))))
    t_6)))

Initial Program: 2.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \mathsf{PI}\left(\right)\\ t_1 := \sin t\_0\\ t_2 := \cos t\_0\\ t_3 := \frac{\frac{{\left(a \cdot t\_2\right)}^{2} + {\left(b \cdot t\_1\right)}^{2}}{y-scale}}{y-scale}\\ t_4 := \frac{\frac{{\left(a \cdot t\_1\right)}^{2} + {\left(b \cdot t\_2\right)}^{2}}{x-scale}}{x-scale}\\ t_5 := \left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\\ t_6 := \frac{4 \cdot t\_5}{{\left(x-scale \cdot y-scale\right)}^{2}}\\ \frac{-\sqrt{\left(\left(2 \cdot t\_6\right) \cdot t\_5\right) \cdot \left(\left(t\_4 + t\_3\right) + \sqrt{{\left(t\_4 - t\_3\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot t\_1\right) \cdot t\_2}{x-scale}}{y-scale}\right)}^{2}}\right)}}{t\_6} \end{array} \end{array} \]

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
         (/ (/ (+ (pow (* a t_2) 2.0) (pow (* b t_1) 2.0)) y-scale) y-scale))
        (t_4
         (/ (/ (+ (pow (* a t_1) 2.0) (pow (* b t_2) 2.0)) x-scale) x-scale))
        (t_5 (* (* b a) (* b (- a))))
        (t_6 (/ (* 4.0 t_5) (pow (* x-scale y-scale) 2.0))))
   (/
    (-
     (sqrt
      (*
       (* (* 2.0 t_6) t_5)
       (+
        (+ t_4 t_3)
        (sqrt
         (+
          (pow (- t_4 t_3) 2.0)
          (pow
           (/
            (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) t_1) t_2) x-scale)
            y-scale)
           2.0)))))))
    t_6)))

Alternative 1: 62.1% accurate, 7.7× speedup

Math

\[\begin{array}{l} y-scale_m = \left|y-scale\right| \\ x-scale_m = \left|x-scale\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\\ t_1 := \sin t\_0\\ \mathbf{if}\;x-scale\_m \leq 6.8 \cdot 10^{+48}:\\ \;\;\;\;\left(0.25 \cdot \left(\left(y-scale\_m \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)\right) \cdot \mathsf{hypot}\left(a \cdot t\_1, b \cdot \cos t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x-scale\_m \cdot 0.25\right) \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right) \cdot \mathsf{hypot}\left(a, t\_1 \cdot b\right)\\ \end{array} \end{array} \]

FPCore

y-scale_m = (fabs.f64 y-scale)
x-scale_m = (fabs.f64 x-scale)
(FPCore (a b angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle (PI)))) (t_1 (sin t_0)))
   (if (<= x-scale_m 6.8e+48)
     (*
      (* 0.25 (* (* y-scale_m (sqrt 2.0)) (sqrt 8.0)))
      (hypot (* a t_1) (* b (cos t_0))))
     (*
      (* (* x-scale_m 0.25) (* (sqrt 2.0) (sqrt 8.0)))
      (hypot a (* t_1 b))))))

Derivation

1. Split input into 2 regimes

2. if x-scale < 6.8000000000000006e48

   1. Initial program 3.4%

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

   3. Taylor expanded in y-scale around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

   5. Applied rewrites 9.2%

      \[\leadsto \color{blue}{\left(0.25 \cdot \left(\left(x-scale \cdot y-scale\right) \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(a \cdot a, \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale \cdot x-scale}, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \left(b \cdot b\right)}{x-scale \cdot x-scale}\right)}} \]

   6. Taylor expanded in y-scale around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

   8. Applied rewrites 9.2%

      \[\leadsto \color{blue}{\left(0.25 \cdot \left(\left(x-scale \cdot y-scale\right) \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right) \cdot \sqrt{\mathsf{fma}\left(b \cdot b, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale \cdot x-scale}, \frac{\left(a \cdot a\right) \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale \cdot x-scale}\right)}} \]

   9. Taylor expanded in x-scale around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(\left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right) \cdot \sqrt{{a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. associate-*r* N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-sqrt.f64 N/A

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

       8. lower-sqrt.f64 N/A

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

   11. Applied rewrites 25.9%

       \[\leadsto \color{blue}{\left(0.25 \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)\right) \cdot \mathsf{hypot}\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), b \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

   if 6.8000000000000006e48 < x-scale

   1. Initial program 3.5%

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

   3. Taylor expanded in y-scale around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

   5. Applied rewrites 56.4%

      \[\leadsto \color{blue}{\left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left({\sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, b \cdot b, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)}} \]

   7. Applied rewrites 64.7%

      \[\leadsto \left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \color{blue}{\mathsf{fma}\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot b, \sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot b, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot \left(a \cdot a\right)\right)}} \]

   8. Taylor expanded in angle around 0

      \[\leadsto \left(\frac{1}{4} \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot b, \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot b, \color{blue}{1} \cdot \left(a \cdot a\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 64.7%

       \[\leadsto \left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot b, \sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot b, \color{blue}{1} \cdot \left(a \cdot a\right)\right)} \]

   11. Taylor expanded in x-scale around 0

       \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(\left(x-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right) \cdot \sqrt{{b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {a}^{2}}\right)} \]
   12. Step-by-step derivation

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-sqrt.f64 N/A

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

       8. lower-sqrt.f64 N/A

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

       9. +-commutative N/A

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

       10. unpow2 N/A

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

       11. unpow2 N/A

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

       12. unpow2 N/A

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

       13. unswap-sqr N/A

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

       14. lower-hypot.f64 N/A

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

   13. Applied rewrites 74.2%

       \[\leadsto \color{blue}{\left(\left(0.25 \cdot x-scale\right) \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right) \cdot \mathsf{hypot}\left(a, b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq 6.8 \cdot 10^{+48}:\\ \;\;\;\;\left(0.25 \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)\right) \cdot \mathsf{hypot}\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), b \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x-scale \cdot 0.25\right) \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right) \cdot \mathsf{hypot}\left(a, \sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 46.9% accurate, 11.1× speedup

Math

\[\begin{array}{l} y-scale_m = \left|y-scale\right| \\ x-scale_m = \left|x-scale\right| \\ \begin{array}{l} \mathbf{if}\;x-scale\_m \leq 3.3 \cdot 10^{-56}:\\ \;\;\;\;y-scale\_m \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x-scale\_m \cdot 0.25\right) \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right) \cdot \mathsf{hypot}\left(a, \sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot b\right)\\ \end{array} \end{array} \]

FPCore

y-scale_m = (fabs.f64 y-scale)
x-scale_m = (fabs.f64 x-scale)
(FPCore (a b angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= x-scale_m 3.3e-56)
   (* y-scale_m b)
   (*
    (* (* x-scale_m 0.25) (* (sqrt 2.0) (sqrt 8.0)))
    (hypot a (* (sin (* 0.005555555555555556 (* angle (PI)))) b)))))

Derivation

1. Split input into 2 regimes

2. if x-scale < 3.29999999999999984e-56

   1. Initial program 3.2%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot b\right) \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot b\right) \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot b\right)} \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \color{blue}{\left(\left(y-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \color{blue}{\left(\left(y-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \left(\color{blue}{\left(y-scale \cdot \sqrt{2}\right)} \cdot \sqrt{8}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \left(\left(y-scale \cdot \color{blue}{\sqrt{2}}\right) \cdot \sqrt{8}\right) \]

      8. lower-sqrt.f64 18.3

         \[\leadsto \left(0.25 \cdot b\right) \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \color{blue}{\sqrt{8}}\right) \]

   5. Applied rewrites 18.3%

      \[\leadsto \color{blue}{\left(0.25 \cdot b\right) \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot b\right)} \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \left(\left(y-scale \cdot \color{blue}{\sqrt{2}}\right) \cdot \sqrt{8}\right) \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \color{blue}{\sqrt{8}}\right) \]

      4. associate-*l* N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \color{blue}{\left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \left(y-scale \cdot \left(\color{blue}{\sqrt{2}} \cdot \sqrt{8}\right)\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \color{blue}{\sqrt{8}}\right)\right) \]

      7. sqrt-unprod N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \left(y-scale \cdot \color{blue}{\sqrt{2 \cdot 8}}\right) \]

      8. metadata-eval N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \left(y-scale \cdot \sqrt{\color{blue}{16}}\right) \]

      9. metadata-eval N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 18.3

          \[\leadsto \color{blue}{\left(\left(0.25 \cdot b\right) \cdot y-scale\right)} \cdot 4 \]

   7. Applied rewrites 18.3%

      \[\leadsto \color{blue}{\left(\left(0.25 \cdot b\right) \cdot y-scale\right) \cdot 4} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. metadata-eval N/A

         \[\leadsto \left(\left(\frac{1}{4} \cdot b\right) \cdot y-scale\right) \cdot \color{blue}{\sqrt{16}} \]

      4. metadata-eval N/A

         \[\leadsto \left(\left(\frac{1}{4} \cdot b\right) \cdot y-scale\right) \cdot \sqrt{\color{blue}{2 \cdot 8}} \]

      5. sqrt-unprod N/A

         \[\leadsto \left(\left(\frac{1}{4} \cdot b\right) \cdot y-scale\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{8}\right)} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{1}{4} \cdot b\right) \cdot y-scale\right) \cdot \left(\color{blue}{\sqrt{2}} \cdot \sqrt{8}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{1}{4} \cdot b\right) \cdot y-scale\right) \cdot \left(\sqrt{2} \cdot \color{blue}{\sqrt{8}}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{4} \cdot b\right) \cdot y-scale\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{8}\right)} \]

      9. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \sqrt{8}\right) \cdot \left(\left(\frac{1}{4} \cdot b\right) \cdot y-scale\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{2} \cdot \sqrt{8}\right) \cdot \color{blue}{\left(\left(\frac{1}{4} \cdot b\right) \cdot y-scale\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{2} \cdot \sqrt{8}\right) \cdot \left(\color{blue}{\left(\frac{1}{4} \cdot b\right)} \cdot y-scale\right) \]

      12. associate-*l* N/A

          \[\leadsto \left(\sqrt{2} \cdot \sqrt{8}\right) \cdot \color{blue}{\left(\frac{1}{4} \cdot \left(b \cdot y-scale\right)\right)} \]

      13. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(\sqrt{2} \cdot \sqrt{8}\right) \cdot \frac{1}{4}\right) \cdot \left(b \cdot y-scale\right)} \]

      14. lift-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left(\sqrt{2} \cdot \sqrt{8}\right)} \cdot \frac{1}{4}\right) \cdot \left(b \cdot y-scale\right) \]

      15. lift-sqrt.f64 N/A

          \[\leadsto \left(\left(\color{blue}{\sqrt{2}} \cdot \sqrt{8}\right) \cdot \frac{1}{4}\right) \cdot \left(b \cdot y-scale\right) \]

      16. lift-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{2} \cdot \color{blue}{\sqrt{8}}\right) \cdot \frac{1}{4}\right) \cdot \left(b \cdot y-scale\right) \]

      17. sqrt-unprod N/A

          \[\leadsto \left(\color{blue}{\sqrt{2 \cdot 8}} \cdot \frac{1}{4}\right) \cdot \left(b \cdot y-scale\right) \]

      18. metadata-eval N/A

          \[\leadsto \left(\sqrt{\color{blue}{16}} \cdot \frac{1}{4}\right) \cdot \left(b \cdot y-scale\right) \]

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. *-lft-identity N/A

          \[\leadsto \color{blue}{b \cdot y-scale} \]

      22. *-commutative N/A

          \[\leadsto \color{blue}{y-scale \cdot b} \]

      23. lower-*.f64 18.3

          \[\leadsto \color{blue}{y-scale \cdot b} \]

   9. Applied rewrites 18.3%

      \[\leadsto \color{blue}{y-scale \cdot b} \]

   if 3.29999999999999984e-56 < x-scale

   1. Initial program 3.8%

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

   3. Taylor expanded in y-scale around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

   5. Applied rewrites 46.8%

      \[\leadsto \color{blue}{\left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left({\sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, b \cdot b, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)}} \]

   7. Applied rewrites 51.5%

      \[\leadsto \left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \color{blue}{\mathsf{fma}\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot b, \sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot b, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot \left(a \cdot a\right)\right)}} \]

   8. Taylor expanded in angle around 0

      \[\leadsto \left(\frac{1}{4} \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot b, \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot b, \color{blue}{1} \cdot \left(a \cdot a\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 51.5%

       \[\leadsto \left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot b, \sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot b, \color{blue}{1} \cdot \left(a \cdot a\right)\right)} \]

   11. Taylor expanded in x-scale around 0

       \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(\left(x-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right) \cdot \sqrt{{b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {a}^{2}}\right)} \]
   12. Step-by-step derivation

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-sqrt.f64 N/A

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

       8. lower-sqrt.f64 N/A

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

       9. +-commutative N/A

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

       10. unpow2 N/A

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

       11. unpow2 N/A

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

       12. unpow2 N/A

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

       13. unswap-sqr N/A

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

       14. lower-hypot.f64 N/A

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

   13. Applied rewrites 60.3%

       \[\leadsto \color{blue}{\left(\left(0.25 \cdot x-scale\right) \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right) \cdot \mathsf{hypot}\left(a, b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq 3.3 \cdot 10^{-56}:\\ \;\;\;\;y-scale \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x-scale \cdot 0.25\right) \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right) \cdot \mathsf{hypot}\left(a, \sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 29.1% accurate, 14.5× speedup

Math

\[\begin{array}{l} y-scale_m = \left|y-scale\right| \\ x-scale_m = \left|x-scale\right| \\ \begin{array}{l} t_0 := \sqrt{2} \cdot \sqrt{8}\\ t_1 := \mathsf{fma}\left(3.858024691358025 \cdot 10^{-6}, \frac{\left(angle \cdot angle\right) \cdot \left(\left(b \cdot b\right) \cdot \left(t\_0 \cdot \left(x-scale\_m \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}{a}, 0.25 \cdot \left(t\_0 \cdot \left(x-scale\_m \cdot a\right)\right)\right)\\ t_2 := 0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\\ \mathbf{if}\;x-scale\_m \leq 1.65 \cdot 10^{-98}:\\ \;\;\;\;y-scale\_m \cdot b\\ \mathbf{elif}\;x-scale\_m \leq 7.6 \cdot 10^{+39}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x-scale\_m \leq 3.6 \cdot 10^{+154}:\\ \;\;\;\;\left(0.25 \cdot \left(x-scale\_m \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(t\_2 \cdot b, \sin t\_2 \cdot b, a \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

y-scale_m = (fabs.f64 y-scale)
x-scale_m = (fabs.f64 x-scale)
(FPCore (a b angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (* (sqrt 2.0) (sqrt 8.0)))
        (t_1
         (fma
          3.858024691358025e-6
          (/
           (* (* angle angle) (* (* b b) (* t_0 (* x-scale_m (* (PI) (PI))))))
           a)
          (* 0.25 (* t_0 (* x-scale_m a)))))
        (t_2 (* 0.005555555555555556 (* angle (PI)))))
   (if (<= x-scale_m 1.65e-98)
     (* y-scale_m b)
     (if (<= x-scale_m 7.6e+39)
       t_1
       (if (<= x-scale_m 3.6e+154)
         (*
          (* 0.25 (* x-scale_m (sqrt 8.0)))
          (sqrt (* 2.0 (fma (* t_2 b) (* (sin t_2) b) (* a a)))))
         t_1)))))

Derivation

1. Split input into 3 regimes

2. if x-scale < 1.6500000000000001e-98

   1. Initial program 2.8%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot b\right) \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot b\right) \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot b\right)} \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \color{blue}{\left(\left(y-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \color{blue}{\left(\left(y-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \left(\color{blue}{\left(y-scale \cdot \sqrt{2}\right)} \cdot \sqrt{8}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \left(\left(y-scale \cdot \color{blue}{\sqrt{2}}\right) \cdot \sqrt{8}\right) \]

      8. lower-sqrt.f64 18.8

         \[\leadsto \left(0.25 \cdot b\right) \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \color{blue}{\sqrt{8}}\right) \]

   5. Applied rewrites 18.8%

      \[\leadsto \color{blue}{\left(0.25 \cdot b\right) \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot b\right)} \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \left(\left(y-scale \cdot \color{blue}{\sqrt{2}}\right) \cdot \sqrt{8}\right) \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \color{blue}{\sqrt{8}}\right) \]

      4. associate-*l* N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \color{blue}{\left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \left(y-scale \cdot \left(\color{blue}{\sqrt{2}} \cdot \sqrt{8}\right)\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \color{blue}{\sqrt{8}}\right)\right) \]

      7. sqrt-unprod N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \left(y-scale \cdot \color{blue}{\sqrt{2 \cdot 8}}\right) \]

      8. metadata-eval N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \left(y-scale \cdot \sqrt{\color{blue}{16}}\right) \]

      9. metadata-eval N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 18.9

          \[\leadsto \color{blue}{\left(\left(0.25 \cdot b\right) \cdot y-scale\right)} \cdot 4 \]

   7. Applied rewrites 18.9%

      \[\leadsto \color{blue}{\left(\left(0.25 \cdot b\right) \cdot y-scale\right) \cdot 4} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. metadata-eval N/A

         \[\leadsto \left(\left(\frac{1}{4} \cdot b\right) \cdot y-scale\right) \cdot \color{blue}{\sqrt{16}} \]

      4. metadata-eval N/A

         \[\leadsto \left(\left(\frac{1}{4} \cdot b\right) \cdot y-scale\right) \cdot \sqrt{\color{blue}{2 \cdot 8}} \]

      5. sqrt-unprod N/A

         \[\leadsto \left(\left(\frac{1}{4} \cdot b\right) \cdot y-scale\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{8}\right)} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{1}{4} \cdot b\right) \cdot y-scale\right) \cdot \left(\color{blue}{\sqrt{2}} \cdot \sqrt{8}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{1}{4} \cdot b\right) \cdot y-scale\right) \cdot \left(\sqrt{2} \cdot \color{blue}{\sqrt{8}}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{4} \cdot b\right) \cdot y-scale\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{8}\right)} \]

      9. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \sqrt{8}\right) \cdot \left(\left(\frac{1}{4} \cdot b\right) \cdot y-scale\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{2} \cdot \sqrt{8}\right) \cdot \color{blue}{\left(\left(\frac{1}{4} \cdot b\right) \cdot y-scale\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{2} \cdot \sqrt{8}\right) \cdot \left(\color{blue}{\left(\frac{1}{4} \cdot b\right)} \cdot y-scale\right) \]

      12. associate-*l* N/A

          \[\leadsto \left(\sqrt{2} \cdot \sqrt{8}\right) \cdot \color{blue}{\left(\frac{1}{4} \cdot \left(b \cdot y-scale\right)\right)} \]

      13. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(\sqrt{2} \cdot \sqrt{8}\right) \cdot \frac{1}{4}\right) \cdot \left(b \cdot y-scale\right)} \]

      14. lift-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left(\sqrt{2} \cdot \sqrt{8}\right)} \cdot \frac{1}{4}\right) \cdot \left(b \cdot y-scale\right) \]

      15. lift-sqrt.f64 N/A

          \[\leadsto \left(\left(\color{blue}{\sqrt{2}} \cdot \sqrt{8}\right) \cdot \frac{1}{4}\right) \cdot \left(b \cdot y-scale\right) \]

      16. lift-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{2} \cdot \color{blue}{\sqrt{8}}\right) \cdot \frac{1}{4}\right) \cdot \left(b \cdot y-scale\right) \]

      17. sqrt-unprod N/A

          \[\leadsto \left(\color{blue}{\sqrt{2 \cdot 8}} \cdot \frac{1}{4}\right) \cdot \left(b \cdot y-scale\right) \]

      18. metadata-eval N/A

          \[\leadsto \left(\sqrt{\color{blue}{16}} \cdot \frac{1}{4}\right) \cdot \left(b \cdot y-scale\right) \]

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. *-lft-identity N/A

          \[\leadsto \color{blue}{b \cdot y-scale} \]

      22. *-commutative N/A

          \[\leadsto \color{blue}{y-scale \cdot b} \]

      23. lower-*.f64 18.9

          \[\leadsto \color{blue}{y-scale \cdot b} \]

   9. Applied rewrites 18.9%

      \[\leadsto \color{blue}{y-scale \cdot b} \]

   if 1.6500000000000001e-98 < x-scale < 7.5999999999999996e39 or 3.6000000000000001e154 < x-scale

   1. Initial program 3.3%

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

   3. Taylor expanded in y-scale around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

   5. Applied rewrites 47.0%

      \[\leadsto \color{blue}{\left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left({\sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, b \cdot b, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)}} \]

   7. Applied rewrites 45.9%

      \[\leadsto \left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \color{blue}{\mathsf{fma}\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot b, \sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot b, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot \left(a \cdot a\right)\right)}} \]

   8. Taylor expanded in angle around 0

      \[\leadsto \left(\frac{1}{4} \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot b, \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot b, \color{blue}{1} \cdot \left(a \cdot a\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 45.9%

       \[\leadsto \left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot b, \sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot b, \color{blue}{1} \cdot \left(a \cdot a\right)\right)} \]

   11. Taylor expanded in angle around 0

       \[\leadsto \color{blue}{\frac{1}{259200} \cdot \frac{{angle}^{2} \cdot \left({b}^{2} \cdot \left(x-scale \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)\right)}{a} + \frac{1}{4} \cdot \left(a \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]
   12. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{259200}, \frac{{angle}^{2} \cdot \left({b}^{2} \cdot \left(x-scale \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)\right)}{a}, \frac{1}{4} \cdot \left(a \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)\right)} \]

   13. Applied rewrites 33.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(3.858024691358025 \cdot 10^{-6}, \frac{\left(angle \cdot angle\right) \cdot \left(\left(b \cdot b\right) \cdot \left(\left(x-scale \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)}{a}, 0.25 \cdot \left(\left(a \cdot x-scale\right) \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]

   if 7.5999999999999996e39 < x-scale < 3.6000000000000001e154

   1. Initial program 7.8%

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

   3. Taylor expanded in y-scale around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

   5. Applied rewrites 47.4%

      \[\leadsto \color{blue}{\left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left({\sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, b \cdot b, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)}} \]

   7. Applied rewrites 58.6%

      \[\leadsto \left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \color{blue}{\mathsf{fma}\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot b, \sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot b, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot \left(a \cdot a\right)\right)}} \]

   8. Taylor expanded in angle around 0

      \[\leadsto \left(\frac{1}{4} \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot b, \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot b, \color{blue}{1} \cdot \left(a \cdot a\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 58.6%

       \[\leadsto \left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot b, \sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot b, \color{blue}{1} \cdot \left(a \cdot a\right)\right)} \]

   11. Taylor expanded in angle around 0

       \[\leadsto \left(\frac{1}{4} \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(\color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot b, \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot b, 1 \cdot \left(a \cdot a\right)\right)} \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-PI.f64 57.3

          \[\leadsto \left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(\left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot b, \sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot b, 1 \cdot \left(a \cdot a\right)\right)} \]

   13. Applied rewrites 57.3%

       \[\leadsto \left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(\color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot b, \sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot b, 1 \cdot \left(a \cdot a\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 26.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq 1.65 \cdot 10^{-98}:\\ \;\;\;\;y-scale \cdot b\\ \mathbf{elif}\;x-scale \leq 7.6 \cdot 10^{+39}:\\ \;\;\;\;\mathsf{fma}\left(3.858024691358025 \cdot 10^{-6}, \frac{\left(angle \cdot angle\right) \cdot \left(\left(b \cdot b\right) \cdot \left(\left(\sqrt{2} \cdot \sqrt{8}\right) \cdot \left(x-scale \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}{a}, 0.25 \cdot \left(\left(\sqrt{2} \cdot \sqrt{8}\right) \cdot \left(x-scale \cdot a\right)\right)\right)\\ \mathbf{elif}\;x-scale \leq 3.6 \cdot 10^{+154}:\\ \;\;\;\;\left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(\left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot b, \sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot b, a \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.858024691358025 \cdot 10^{-6}, \frac{\left(angle \cdot angle\right) \cdot \left(\left(b \cdot b\right) \cdot \left(\left(\sqrt{2} \cdot \sqrt{8}\right) \cdot \left(x-scale \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}{a}, 0.25 \cdot \left(\left(\sqrt{2} \cdot \sqrt{8}\right) \cdot \left(x-scale \cdot a\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 25.1% accurate, 22.4× speedup

Math

\[\begin{array}{l} y-scale_m = \left|y-scale\right| \\ x-scale_m = \left|x-scale\right| \\ \begin{array}{l} t_0 := \sqrt{2} \cdot \sqrt{8}\\ \mathbf{if}\;x-scale\_m \leq 3.3 \cdot 10^{-56}:\\ \;\;\;\;y-scale\_m \cdot b\\ \mathbf{elif}\;x-scale\_m \leq 5.5 \cdot 10^{+162}:\\ \;\;\;\;\left(0.25 \cdot \left(\left(x-scale\_m \cdot a\right) \cdot \left(y-scale\_m \cdot \sqrt{8}\right)\right)\right) \cdot \frac{\sqrt{2}}{y-scale\_m}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.858024691358025 \cdot 10^{-6}, \frac{\left(angle \cdot angle\right) \cdot \left(\left(b \cdot b\right) \cdot \left(t\_0 \cdot \left(x-scale\_m \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}{a}, 0.25 \cdot \left(t\_0 \cdot \left(x-scale\_m \cdot a\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

y-scale_m = (fabs.f64 y-scale)
x-scale_m = (fabs.f64 x-scale)
(FPCore (a b angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (* (sqrt 2.0) (sqrt 8.0))))
   (if (<= x-scale_m 3.3e-56)
     (* y-scale_m b)
     (if (<= x-scale_m 5.5e+162)
       (*
        (* 0.25 (* (* x-scale_m a) (* y-scale_m (sqrt 8.0))))
        (/ (sqrt 2.0) y-scale_m))
       (fma
        3.858024691358025e-6
        (/
         (* (* angle angle) (* (* b b) (* t_0 (* x-scale_m (* (PI) (PI))))))
         a)
        (* 0.25 (* t_0 (* x-scale_m a))))))))

Derivation

1. Split input into 3 regimes

2. if x-scale < 3.29999999999999984e-56

   1. Initial program 3.2%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot b\right) \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot b\right) \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot b\right)} \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \color{blue}{\left(\left(y-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \color{blue}{\left(\left(y-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \left(\color{blue}{\left(y-scale \cdot \sqrt{2}\right)} \cdot \sqrt{8}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \left(\left(y-scale \cdot \color{blue}{\sqrt{2}}\right) \cdot \sqrt{8}\right) \]

      8. lower-sqrt.f64 18.3

         \[\leadsto \left(0.25 \cdot b\right) \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \color{blue}{\sqrt{8}}\right) \]

   5. Applied rewrites 18.3%

      \[\leadsto \color{blue}{\left(0.25 \cdot b\right) \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot b\right)} \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \left(\left(y-scale \cdot \color{blue}{\sqrt{2}}\right) \cdot \sqrt{8}\right) \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \color{blue}{\sqrt{8}}\right) \]

      4. associate-*l* N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \color{blue}{\left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \left(y-scale \cdot \left(\color{blue}{\sqrt{2}} \cdot \sqrt{8}\right)\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \color{blue}{\sqrt{8}}\right)\right) \]

      7. sqrt-unprod N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \left(y-scale \cdot \color{blue}{\sqrt{2 \cdot 8}}\right) \]

      8. metadata-eval N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \left(y-scale \cdot \sqrt{\color{blue}{16}}\right) \]

      9. metadata-eval N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 18.3

          \[\leadsto \color{blue}{\left(\left(0.25 \cdot b\right) \cdot y-scale\right)} \cdot 4 \]

   7. Applied rewrites 18.3%

      \[\leadsto \color{blue}{\left(\left(0.25 \cdot b\right) \cdot y-scale\right) \cdot 4} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. metadata-eval N/A

         \[\leadsto \left(\left(\frac{1}{4} \cdot b\right) \cdot y-scale\right) \cdot \color{blue}{\sqrt{16}} \]

      4. metadata-eval N/A

         \[\leadsto \left(\left(\frac{1}{4} \cdot b\right) \cdot y-scale\right) \cdot \sqrt{\color{blue}{2 \cdot 8}} \]

      5. sqrt-unprod N/A

         \[\leadsto \left(\left(\frac{1}{4} \cdot b\right) \cdot y-scale\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{8}\right)} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{1}{4} \cdot b\right) \cdot y-scale\right) \cdot \left(\color{blue}{\sqrt{2}} \cdot \sqrt{8}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{1}{4} \cdot b\right) \cdot y-scale\right) \cdot \left(\sqrt{2} \cdot \color{blue}{\sqrt{8}}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{4} \cdot b\right) \cdot y-scale\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{8}\right)} \]

      9. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \sqrt{8}\right) \cdot \left(\left(\frac{1}{4} \cdot b\right) \cdot y-scale\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{2} \cdot \sqrt{8}\right) \cdot \color{blue}{\left(\left(\frac{1}{4} \cdot b\right) \cdot y-scale\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{2} \cdot \sqrt{8}\right) \cdot \left(\color{blue}{\left(\frac{1}{4} \cdot b\right)} \cdot y-scale\right) \]

      12. associate-*l* N/A

          \[\leadsto \left(\sqrt{2} \cdot \sqrt{8}\right) \cdot \color{blue}{\left(\frac{1}{4} \cdot \left(b \cdot y-scale\right)\right)} \]

      13. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(\sqrt{2} \cdot \sqrt{8}\right) \cdot \frac{1}{4}\right) \cdot \left(b \cdot y-scale\right)} \]

      14. lift-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left(\sqrt{2} \cdot \sqrt{8}\right)} \cdot \frac{1}{4}\right) \cdot \left(b \cdot y-scale\right) \]

      15. lift-sqrt.f64 N/A

          \[\leadsto \left(\left(\color{blue}{\sqrt{2}} \cdot \sqrt{8}\right) \cdot \frac{1}{4}\right) \cdot \left(b \cdot y-scale\right) \]

      16. lift-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{2} \cdot \color{blue}{\sqrt{8}}\right) \cdot \frac{1}{4}\right) \cdot \left(b \cdot y-scale\right) \]

      17. sqrt-unprod N/A

          \[\leadsto \left(\color{blue}{\sqrt{2 \cdot 8}} \cdot \frac{1}{4}\right) \cdot \left(b \cdot y-scale\right) \]

      18. metadata-eval N/A

          \[\leadsto \left(\sqrt{\color{blue}{16}} \cdot \frac{1}{4}\right) \cdot \left(b \cdot y-scale\right) \]

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. *-lft-identity N/A

          \[\leadsto \color{blue}{b \cdot y-scale} \]

      22. *-commutative N/A

          \[\leadsto \color{blue}{y-scale \cdot b} \]

      23. lower-*.f64 18.3

          \[\leadsto \color{blue}{y-scale \cdot b} \]

   9. Applied rewrites 18.3%

      \[\leadsto \color{blue}{y-scale \cdot b} \]

   if 3.29999999999999984e-56 < x-scale < 5.49999999999999966e162

   1. Initial program 6.1%

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

   3. Taylor expanded in a around inf

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

   5. Applied rewrites 12.9%

      \[\leadsto \color{blue}{\left(0.25 \cdot \left(\left(a \cdot x-scale\right) \cdot \left(y-scale \cdot \sqrt{8}\right)\right)\right) \cdot \sqrt{\sqrt{\mathsf{fma}\left(4, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{\left(x-scale \cdot x-scale\right) \cdot \left(y-scale \cdot y-scale\right)}, {\left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale \cdot x-scale} - \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale \cdot y-scale}\right)}^{2}\right)} + \left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale \cdot x-scale} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale \cdot y-scale}\right)}} \]

   6. Taylor expanded in angle around 0

      \[\leadsto \left(\frac{1}{4} \cdot \left(\left(a \cdot x-scale\right) \cdot \left(y-scale \cdot \sqrt{8}\right)\right)\right) \cdot \color{blue}{\frac{\sqrt{2}}{y-scale}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left(\left(a \cdot x-scale\right) \cdot \left(y-scale \cdot \sqrt{8}\right)\right)\right) \cdot \color{blue}{\frac{\sqrt{2}}{y-scale}} \]

      2. lower-sqrt.f64 28.7

         \[\leadsto \left(0.25 \cdot \left(\left(a \cdot x-scale\right) \cdot \left(y-scale \cdot \sqrt{8}\right)\right)\right) \cdot \frac{\color{blue}{\sqrt{2}}}{y-scale} \]

   8. Applied rewrites 28.7%

      \[\leadsto \left(0.25 \cdot \left(\left(a \cdot x-scale\right) \cdot \left(y-scale \cdot \sqrt{8}\right)\right)\right) \cdot \color{blue}{\frac{\sqrt{2}}{y-scale}} \]

   if 5.49999999999999966e162 < x-scale

   1. Initial program 0.1%

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

   3. Taylor expanded in y-scale around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

   5. Applied rewrites 61.8%

      \[\leadsto \color{blue}{\left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left({\sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, b \cdot b, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)}} \]

   7. Applied rewrites 67.8%

      \[\leadsto \left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \color{blue}{\mathsf{fma}\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot b, \sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot b, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot \left(a \cdot a\right)\right)}} \]

   8. Taylor expanded in angle around 0

      \[\leadsto \left(\frac{1}{4} \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot b, \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot b, \color{blue}{1} \cdot \left(a \cdot a\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 67.9%

       \[\leadsto \left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot b, \sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot b, \color{blue}{1} \cdot \left(a \cdot a\right)\right)} \]

   11. Taylor expanded in angle around 0

       \[\leadsto \color{blue}{\frac{1}{259200} \cdot \frac{{angle}^{2} \cdot \left({b}^{2} \cdot \left(x-scale \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)\right)}{a} + \frac{1}{4} \cdot \left(a \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]
   12. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{259200}, \frac{{angle}^{2} \cdot \left({b}^{2} \cdot \left(x-scale \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)\right)}{a}, \frac{1}{4} \cdot \left(a \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)\right)} \]

   13. Applied rewrites 36.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(3.858024691358025 \cdot 10^{-6}, \frac{\left(angle \cdot angle\right) \cdot \left(\left(b \cdot b\right) \cdot \left(\left(x-scale \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)}{a}, 0.25 \cdot \left(\left(a \cdot x-scale\right) \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 22.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq 3.3 \cdot 10^{-56}:\\ \;\;\;\;y-scale \cdot b\\ \mathbf{elif}\;x-scale \leq 5.5 \cdot 10^{+162}:\\ \;\;\;\;\left(0.25 \cdot \left(\left(x-scale \cdot a\right) \cdot \left(y-scale \cdot \sqrt{8}\right)\right)\right) \cdot \frac{\sqrt{2}}{y-scale}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.858024691358025 \cdot 10^{-6}, \frac{\left(angle \cdot angle\right) \cdot \left(\left(b \cdot b\right) \cdot \left(\left(\sqrt{2} \cdot \sqrt{8}\right) \cdot \left(x-scale \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}{a}, 0.25 \cdot \left(\left(\sqrt{2} \cdot \sqrt{8}\right) \cdot \left(x-scale \cdot a\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 29.5% accurate, 32.7× speedup

Math

\[\begin{array}{l} y-scale_m = \left|y-scale\right| \\ x-scale_m = \left|x-scale\right| \\ \begin{array}{l} t_0 := 0.25 \cdot \left(x-scale\_m \cdot \sqrt{8}\right)\\ \mathbf{if}\;b \leq 7.8 \cdot 10^{-132}:\\ \;\;\;\;t\_0 \cdot \left(\sqrt{2} \cdot a\right)\\ \mathbf{elif}\;b \leq 4.8 \cdot 10^{+106}:\\ \;\;\;\;t\_0 \cdot \sqrt{2 \cdot \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}, \left(b \cdot b\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), a \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;y-scale\_m \cdot b\\ \end{array} \end{array} \]

FPCore

y-scale_m = (fabs.f64 y-scale)
x-scale_m = (fabs.f64 x-scale)
(FPCore (a b angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (* 0.25 (* x-scale_m (sqrt 8.0)))))
   (if (<= b 7.8e-132)
     (* t_0 (* (sqrt 2.0) a))
     (if (<= b 4.8e+106)
       (*
        t_0
        (sqrt
         (*
          2.0
          (fma
           (* (* angle angle) 3.08641975308642e-5)
           (* (* b b) (* (PI) (PI)))
           (* a a)))))
       (* y-scale_m b)))))

Derivation

1. Split input into 3 regimes

2. if b < 7.79999999999999964e-132

   1. Initial program 3.0%

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

   3. Taylor expanded in y-scale around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

   5. Applied rewrites 21.7%

      \[\leadsto \color{blue}{\left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left({\sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, b \cdot b, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)}} \]

   6. Taylor expanded in angle around 0

      \[\leadsto \left(\frac{1}{4} \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\left(a \cdot \sqrt{2}\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\left(a \cdot \sqrt{2}\right)} \]

      2. lower-sqrt.f64 24.6

         \[\leadsto \left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \left(a \cdot \color{blue}{\sqrt{2}}\right) \]

   8. Applied rewrites 24.6%

      \[\leadsto \left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\left(a \cdot \sqrt{2}\right)} \]

   if 7.79999999999999964e-132 < b < 4.8000000000000001e106

   1. Initial program 5.5%

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

   3. Taylor expanded in y-scale around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

   5. Applied rewrites 27.4%

      \[\leadsto \color{blue}{\left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left({\sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, b \cdot b, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)}} \]

   7. Applied rewrites 28.2%

      \[\leadsto \left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \color{blue}{\mathsf{fma}\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot b, \sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot b, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot \left(a \cdot a\right)\right)}} \]

   8. Taylor expanded in angle around 0

      \[\leadsto \left(\frac{1}{4} \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot b, \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot b, \color{blue}{1} \cdot \left(a \cdot a\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 28.2%

       \[\leadsto \left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot b, \sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot b, \color{blue}{1} \cdot \left(a \cdot a\right)\right)} \]

   11. Taylor expanded in angle around 0

       \[\leadsto \left(\frac{1}{4} \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \color{blue}{\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}\right)}} \]
   12. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{1}{4} \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left(\color{blue}{\left(\frac{1}{32400} \cdot {angle}^{2}\right) \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)} + {a}^{2}\right)} \]

       2. lower-fma.f64 N/A

          \[\leadsto \left(\frac{1}{4} \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{32400} \cdot {angle}^{2}, {b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}, {a}^{2}\right)}} \]

       3. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{4} \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(\color{blue}{\frac{1}{32400} \cdot {angle}^{2}}, {b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}, {a}^{2}\right)} \]

       4. unpow2 N/A

          \[\leadsto \left(\frac{1}{4} \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(\frac{1}{32400} \cdot \color{blue}{\left(angle \cdot angle\right)}, {b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}, {a}^{2}\right)} \]

       5. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{4} \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(\frac{1}{32400} \cdot \color{blue}{\left(angle \cdot angle\right)}, {b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}, {a}^{2}\right)} \]

       6. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{4} \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(\frac{1}{32400} \cdot \left(angle \cdot angle\right), \color{blue}{{b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}}, {a}^{2}\right)} \]

       7. unpow2 N/A

          \[\leadsto \left(\frac{1}{4} \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(\frac{1}{32400} \cdot \left(angle \cdot angle\right), \color{blue}{\left(b \cdot b\right)} \cdot {\mathsf{PI}\left(\right)}^{2}, {a}^{2}\right)} \]

       8. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{4} \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(\frac{1}{32400} \cdot \left(angle \cdot angle\right), \color{blue}{\left(b \cdot b\right)} \cdot {\mathsf{PI}\left(\right)}^{2}, {a}^{2}\right)} \]

       9. unpow2 N/A

          \[\leadsto \left(\frac{1}{4} \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(\frac{1}{32400} \cdot \left(angle \cdot angle\right), \left(b \cdot b\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, {a}^{2}\right)} \]

       10. lower-*.f64 N/A

           \[\leadsto \left(\frac{1}{4} \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(\frac{1}{32400} \cdot \left(angle \cdot angle\right), \left(b \cdot b\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, {a}^{2}\right)} \]

       11. lower-PI.f64 N/A

           \[\leadsto \left(\frac{1}{4} \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(\frac{1}{32400} \cdot \left(angle \cdot angle\right), \left(b \cdot b\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right), {a}^{2}\right)} \]

       12. lower-PI.f64 N/A

           \[\leadsto \left(\frac{1}{4} \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(\frac{1}{32400} \cdot \left(angle \cdot angle\right), \left(b \cdot b\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right), {a}^{2}\right)} \]

       13. unpow2 N/A

           \[\leadsto \left(\frac{1}{4} \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(\frac{1}{32400} \cdot \left(angle \cdot angle\right), \left(b \cdot b\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \color{blue}{a \cdot a}\right)} \]

       14. lower-*.f64 26.0

           \[\leadsto \left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot angle\right), \left(b \cdot b\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \color{blue}{a \cdot a}\right)} \]

   13. Applied rewrites 26.0%

       \[\leadsto \left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \color{blue}{\mathsf{fma}\left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot angle\right), \left(b \cdot b\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), a \cdot a\right)}} \]

   if 4.8000000000000001e106 < b

   1. Initial program 3.1%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot b\right) \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot b\right) \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot b\right)} \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \color{blue}{\left(\left(y-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \color{blue}{\left(\left(y-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \left(\color{blue}{\left(y-scale \cdot \sqrt{2}\right)} \cdot \sqrt{8}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \left(\left(y-scale \cdot \color{blue}{\sqrt{2}}\right) \cdot \sqrt{8}\right) \]

      8. lower-sqrt.f64 41.6

         \[\leadsto \left(0.25 \cdot b\right) \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \color{blue}{\sqrt{8}}\right) \]

   5. Applied rewrites 41.6%

      \[\leadsto \color{blue}{\left(0.25 \cdot b\right) \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot b\right)} \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \left(\left(y-scale \cdot \color{blue}{\sqrt{2}}\right) \cdot \sqrt{8}\right) \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \color{blue}{\sqrt{8}}\right) \]

      4. associate-*l* N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \color{blue}{\left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \left(y-scale \cdot \left(\color{blue}{\sqrt{2}} \cdot \sqrt{8}\right)\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \color{blue}{\sqrt{8}}\right)\right) \]

      7. sqrt-unprod N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \left(y-scale \cdot \color{blue}{\sqrt{2 \cdot 8}}\right) \]

      8. metadata-eval N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \left(y-scale \cdot \sqrt{\color{blue}{16}}\right) \]

      9. metadata-eval N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 41.7

          \[\leadsto \color{blue}{\left(\left(0.25 \cdot b\right) \cdot y-scale\right)} \cdot 4 \]

   7. Applied rewrites 41.7%

      \[\leadsto \color{blue}{\left(\left(0.25 \cdot b\right) \cdot y-scale\right) \cdot 4} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. metadata-eval N/A

         \[\leadsto \left(\left(\frac{1}{4} \cdot b\right) \cdot y-scale\right) \cdot \color{blue}{\sqrt{16}} \]

      4. metadata-eval N/A

         \[\leadsto \left(\left(\frac{1}{4} \cdot b\right) \cdot y-scale\right) \cdot \sqrt{\color{blue}{2 \cdot 8}} \]

      5. sqrt-unprod N/A

         \[\leadsto \left(\left(\frac{1}{4} \cdot b\right) \cdot y-scale\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{8}\right)} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{1}{4} \cdot b\right) \cdot y-scale\right) \cdot \left(\color{blue}{\sqrt{2}} \cdot \sqrt{8}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{1}{4} \cdot b\right) \cdot y-scale\right) \cdot \left(\sqrt{2} \cdot \color{blue}{\sqrt{8}}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{4} \cdot b\right) \cdot y-scale\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{8}\right)} \]

      9. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \sqrt{8}\right) \cdot \left(\left(\frac{1}{4} \cdot b\right) \cdot y-scale\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{2} \cdot \sqrt{8}\right) \cdot \color{blue}{\left(\left(\frac{1}{4} \cdot b\right) \cdot y-scale\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{2} \cdot \sqrt{8}\right) \cdot \left(\color{blue}{\left(\frac{1}{4} \cdot b\right)} \cdot y-scale\right) \]

      12. associate-*l* N/A

          \[\leadsto \left(\sqrt{2} \cdot \sqrt{8}\right) \cdot \color{blue}{\left(\frac{1}{4} \cdot \left(b \cdot y-scale\right)\right)} \]

      13. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(\sqrt{2} \cdot \sqrt{8}\right) \cdot \frac{1}{4}\right) \cdot \left(b \cdot y-scale\right)} \]

      14. lift-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left(\sqrt{2} \cdot \sqrt{8}\right)} \cdot \frac{1}{4}\right) \cdot \left(b \cdot y-scale\right) \]

      15. lift-sqrt.f64 N/A

          \[\leadsto \left(\left(\color{blue}{\sqrt{2}} \cdot \sqrt{8}\right) \cdot \frac{1}{4}\right) \cdot \left(b \cdot y-scale\right) \]

      16. lift-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{2} \cdot \color{blue}{\sqrt{8}}\right) \cdot \frac{1}{4}\right) \cdot \left(b \cdot y-scale\right) \]

      17. sqrt-unprod N/A

          \[\leadsto \left(\color{blue}{\sqrt{2 \cdot 8}} \cdot \frac{1}{4}\right) \cdot \left(b \cdot y-scale\right) \]

      18. metadata-eval N/A

          \[\leadsto \left(\sqrt{\color{blue}{16}} \cdot \frac{1}{4}\right) \cdot \left(b \cdot y-scale\right) \]

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. *-lft-identity N/A

          \[\leadsto \color{blue}{b \cdot y-scale} \]

      22. *-commutative N/A

          \[\leadsto \color{blue}{y-scale \cdot b} \]

      23. lower-*.f64 41.7

          \[\leadsto \color{blue}{y-scale \cdot b} \]

   9. Applied rewrites 41.7%

      \[\leadsto \color{blue}{y-scale \cdot b} \]
3. Recombined 3 regimes into one program.

4. Final simplification 27.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.8 \cdot 10^{-132}:\\ \;\;\;\;\left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \left(\sqrt{2} \cdot a\right)\\ \mathbf{elif}\;b \leq 4.8 \cdot 10^{+106}:\\ \;\;\;\;\left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}, \left(b \cdot b\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), a \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;y-scale \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 24.1% accurate, 61.9× speedup

Math

\[\begin{array}{l} y-scale_m = \left|y-scale\right| \\ x-scale_m = \left|x-scale\right| \\ \begin{array}{l} \mathbf{if}\;x-scale\_m \leq 3.6 \cdot 10^{-56}:\\ \;\;\;\;y-scale\_m \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot \left(x-scale\_m \cdot \sqrt{8}\right)\right) \cdot \left(\sqrt{2} \cdot a\right)\\ \end{array} \end{array} \]

FPCore

y-scale_m = (fabs.f64 y-scale)
x-scale_m = (fabs.f64 x-scale)
(FPCore (a b angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= x-scale_m 3.6e-56)
   (* y-scale_m b)
   (* (* 0.25 (* x-scale_m (sqrt 8.0))) (* (sqrt 2.0) a))))

C

y-scale_m = fabs(y_45_scale);
x-scale_m = fabs(x_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (x_45_scale_m <= 3.6e-56) {
		tmp = y_45_scale_m * b;
	} else {
		tmp = (0.25 * (x_45_scale_m * sqrt(8.0))) * (sqrt(2.0) * a);
	}
	return tmp;
}

Fortran

y-scale_m = abs(y_45scale)
x-scale_m = abs(x_45scale)
real(8) function code(a, b, angle, x_45scale_m, y_45scale_m)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale_m
    real(8) :: tmp
    if (x_45scale_m <= 3.6d-56) then
        tmp = y_45scale_m * b
    else
        tmp = (0.25d0 * (x_45scale_m * sqrt(8.0d0))) * (sqrt(2.0d0) * a)
    end if
    code = tmp
end function

Java

y-scale_m = Math.abs(y_45_scale);
x-scale_m = Math.abs(x_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (x_45_scale_m <= 3.6e-56) {
		tmp = y_45_scale_m * b;
	} else {
		tmp = (0.25 * (x_45_scale_m * Math.sqrt(8.0))) * (Math.sqrt(2.0) * a);
	}
	return tmp;
}

Python

y-scale_m = math.fabs(y_45_scale)
x-scale_m = math.fabs(x_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if x_45_scale_m <= 3.6e-56:
		tmp = y_45_scale_m * b
	else:
		tmp = (0.25 * (x_45_scale_m * math.sqrt(8.0))) * (math.sqrt(2.0) * a)
	return tmp

Julia

y-scale_m = abs(y_45_scale)
x-scale_m = abs(x_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (x_45_scale_m <= 3.6e-56)
		tmp = Float64(y_45_scale_m * b);
	else
		tmp = Float64(Float64(0.25 * Float64(x_45_scale_m * sqrt(8.0))) * Float64(sqrt(2.0) * a));
	end
	return tmp
end

MATLAB

y-scale_m = abs(y_45_scale);
x-scale_m = abs(x_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0;
	if (x_45_scale_m <= 3.6e-56)
		tmp = y_45_scale_m * b;
	else
		tmp = (0.25 * (x_45_scale_m * sqrt(8.0))) * (sqrt(2.0) * a);
	end
	tmp_2 = tmp;
end

Wolfram

y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[x$45$scale$95$m, 3.6e-56], N[(y$45$scale$95$m * b), $MachinePrecision], N[(N[(0.25 * N[(x$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 3.59999999999999978e-56

   1. Initial program 3.2%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot b\right) \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot b\right) \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot b\right)} \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \color{blue}{\left(\left(y-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \color{blue}{\left(\left(y-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \left(\color{blue}{\left(y-scale \cdot \sqrt{2}\right)} \cdot \sqrt{8}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \left(\left(y-scale \cdot \color{blue}{\sqrt{2}}\right) \cdot \sqrt{8}\right) \]

      8. lower-sqrt.f64 18.3

         \[\leadsto \left(0.25 \cdot b\right) \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \color{blue}{\sqrt{8}}\right) \]

   5. Applied rewrites 18.3%

      \[\leadsto \color{blue}{\left(0.25 \cdot b\right) \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot b\right)} \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \left(\left(y-scale \cdot \color{blue}{\sqrt{2}}\right) \cdot \sqrt{8}\right) \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \color{blue}{\sqrt{8}}\right) \]

      4. associate-*l* N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \color{blue}{\left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \left(y-scale \cdot \left(\color{blue}{\sqrt{2}} \cdot \sqrt{8}\right)\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \color{blue}{\sqrt{8}}\right)\right) \]

      7. sqrt-unprod N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \left(y-scale \cdot \color{blue}{\sqrt{2 \cdot 8}}\right) \]

      8. metadata-eval N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \left(y-scale \cdot \sqrt{\color{blue}{16}}\right) \]

      9. metadata-eval N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 18.3

          \[\leadsto \color{blue}{\left(\left(0.25 \cdot b\right) \cdot y-scale\right)} \cdot 4 \]

   7. Applied rewrites 18.3%

      \[\leadsto \color{blue}{\left(\left(0.25 \cdot b\right) \cdot y-scale\right) \cdot 4} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. metadata-eval N/A

         \[\leadsto \left(\left(\frac{1}{4} \cdot b\right) \cdot y-scale\right) \cdot \color{blue}{\sqrt{16}} \]

      4. metadata-eval N/A

         \[\leadsto \left(\left(\frac{1}{4} \cdot b\right) \cdot y-scale\right) \cdot \sqrt{\color{blue}{2 \cdot 8}} \]

      5. sqrt-unprod N/A

         \[\leadsto \left(\left(\frac{1}{4} \cdot b\right) \cdot y-scale\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{8}\right)} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{1}{4} \cdot b\right) \cdot y-scale\right) \cdot \left(\color{blue}{\sqrt{2}} \cdot \sqrt{8}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{1}{4} \cdot b\right) \cdot y-scale\right) \cdot \left(\sqrt{2} \cdot \color{blue}{\sqrt{8}}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{4} \cdot b\right) \cdot y-scale\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{8}\right)} \]

      9. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \sqrt{8}\right) \cdot \left(\left(\frac{1}{4} \cdot b\right) \cdot y-scale\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{2} \cdot \sqrt{8}\right) \cdot \color{blue}{\left(\left(\frac{1}{4} \cdot b\right) \cdot y-scale\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{2} \cdot \sqrt{8}\right) \cdot \left(\color{blue}{\left(\frac{1}{4} \cdot b\right)} \cdot y-scale\right) \]

      12. associate-*l* N/A

          \[\leadsto \left(\sqrt{2} \cdot \sqrt{8}\right) \cdot \color{blue}{\left(\frac{1}{4} \cdot \left(b \cdot y-scale\right)\right)} \]

      13. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(\sqrt{2} \cdot \sqrt{8}\right) \cdot \frac{1}{4}\right) \cdot \left(b \cdot y-scale\right)} \]

      14. lift-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left(\sqrt{2} \cdot \sqrt{8}\right)} \cdot \frac{1}{4}\right) \cdot \left(b \cdot y-scale\right) \]

      15. lift-sqrt.f64 N/A

          \[\leadsto \left(\left(\color{blue}{\sqrt{2}} \cdot \sqrt{8}\right) \cdot \frac{1}{4}\right) \cdot \left(b \cdot y-scale\right) \]

      16. lift-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{2} \cdot \color{blue}{\sqrt{8}}\right) \cdot \frac{1}{4}\right) \cdot \left(b \cdot y-scale\right) \]

      17. sqrt-unprod N/A

          \[\leadsto \left(\color{blue}{\sqrt{2 \cdot 8}} \cdot \frac{1}{4}\right) \cdot \left(b \cdot y-scale\right) \]

      18. metadata-eval N/A

          \[\leadsto \left(\sqrt{\color{blue}{16}} \cdot \frac{1}{4}\right) \cdot \left(b \cdot y-scale\right) \]

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. *-lft-identity N/A

          \[\leadsto \color{blue}{b \cdot y-scale} \]

      22. *-commutative N/A

          \[\leadsto \color{blue}{y-scale \cdot b} \]

      23. lower-*.f64 18.3

          \[\leadsto \color{blue}{y-scale \cdot b} \]

   9. Applied rewrites 18.3%

      \[\leadsto \color{blue}{y-scale \cdot b} \]

   if 3.59999999999999978e-56 < x-scale

   1. Initial program 3.8%

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

   3. Taylor expanded in y-scale around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

   5. Applied rewrites 46.8%

      \[\leadsto \color{blue}{\left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left({\sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, b \cdot b, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)}} \]

   6. Taylor expanded in angle around 0

      \[\leadsto \left(\frac{1}{4} \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\left(a \cdot \sqrt{2}\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\left(a \cdot \sqrt{2}\right)} \]

      2. lower-sqrt.f64 25.5

         \[\leadsto \left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \left(a \cdot \color{blue}{\sqrt{2}}\right) \]

   8. Applied rewrites 25.5%

      \[\leadsto \left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\left(a \cdot \sqrt{2}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 20.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq 3.6 \cdot 10^{-56}:\\ \;\;\;\;y-scale \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \left(\sqrt{2} \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 24.1% accurate, 61.9× speedup

Math

\[\begin{array}{l} y-scale_m = \left|y-scale\right| \\ x-scale_m = \left|x-scale\right| \\ \begin{array}{l} \mathbf{if}\;x-scale\_m \leq 3.6 \cdot 10^{-56}:\\ \;\;\;\;y-scale\_m \cdot b\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(\sqrt{2} \cdot \sqrt{8}\right) \cdot \left(x-scale\_m \cdot a\right)\right)\\ \end{array} \end{array} \]

FPCore

y-scale_m = (fabs.f64 y-scale)
x-scale_m = (fabs.f64 x-scale)
(FPCore (a b angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= x-scale_m 3.6e-56)
   (* y-scale_m b)
   (* 0.25 (* (* (sqrt 2.0) (sqrt 8.0)) (* x-scale_m a)))))

C

y-scale_m = fabs(y_45_scale);
x-scale_m = fabs(x_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (x_45_scale_m <= 3.6e-56) {
		tmp = y_45_scale_m * b;
	} else {
		tmp = 0.25 * ((sqrt(2.0) * sqrt(8.0)) * (x_45_scale_m * a));
	}
	return tmp;
}

Fortran

y-scale_m = abs(y_45scale)
x-scale_m = abs(x_45scale)
real(8) function code(a, b, angle, x_45scale_m, y_45scale_m)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale_m
    real(8) :: tmp
    if (x_45scale_m <= 3.6d-56) then
        tmp = y_45scale_m * b
    else
        tmp = 0.25d0 * ((sqrt(2.0d0) * sqrt(8.0d0)) * (x_45scale_m * a))
    end if
    code = tmp
end function

Java

y-scale_m = Math.abs(y_45_scale);
x-scale_m = Math.abs(x_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (x_45_scale_m <= 3.6e-56) {
		tmp = y_45_scale_m * b;
	} else {
		tmp = 0.25 * ((Math.sqrt(2.0) * Math.sqrt(8.0)) * (x_45_scale_m * a));
	}
	return tmp;
}

Python

y-scale_m = math.fabs(y_45_scale)
x-scale_m = math.fabs(x_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if x_45_scale_m <= 3.6e-56:
		tmp = y_45_scale_m * b
	else:
		tmp = 0.25 * ((math.sqrt(2.0) * math.sqrt(8.0)) * (x_45_scale_m * a))
	return tmp

Julia

y-scale_m = abs(y_45_scale)
x-scale_m = abs(x_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (x_45_scale_m <= 3.6e-56)
		tmp = Float64(y_45_scale_m * b);
	else
		tmp = Float64(0.25 * Float64(Float64(sqrt(2.0) * sqrt(8.0)) * Float64(x_45_scale_m * a)));
	end
	return tmp
end

MATLAB

y-scale_m = abs(y_45_scale);
x-scale_m = abs(x_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0;
	if (x_45_scale_m <= 3.6e-56)
		tmp = y_45_scale_m * b;
	else
		tmp = 0.25 * ((sqrt(2.0) * sqrt(8.0)) * (x_45_scale_m * a));
	end
	tmp_2 = tmp;
end

Wolfram

y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[x$45$scale$95$m, 3.6e-56], N[(y$45$scale$95$m * b), $MachinePrecision], N[(0.25 * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision] * N[(x$45$scale$95$m * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 3.59999999999999978e-56

   1. Initial program 3.2%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot b\right) \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot b\right) \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot b\right)} \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \color{blue}{\left(\left(y-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \color{blue}{\left(\left(y-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \left(\color{blue}{\left(y-scale \cdot \sqrt{2}\right)} \cdot \sqrt{8}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \left(\left(y-scale \cdot \color{blue}{\sqrt{2}}\right) \cdot \sqrt{8}\right) \]

      8. lower-sqrt.f64 18.3

         \[\leadsto \left(0.25 \cdot b\right) \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \color{blue}{\sqrt{8}}\right) \]

   5. Applied rewrites 18.3%

      \[\leadsto \color{blue}{\left(0.25 \cdot b\right) \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot b\right)} \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \left(\left(y-scale \cdot \color{blue}{\sqrt{2}}\right) \cdot \sqrt{8}\right) \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \color{blue}{\sqrt{8}}\right) \]

      4. associate-*l* N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \color{blue}{\left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \left(y-scale \cdot \left(\color{blue}{\sqrt{2}} \cdot \sqrt{8}\right)\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \color{blue}{\sqrt{8}}\right)\right) \]

      7. sqrt-unprod N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \left(y-scale \cdot \color{blue}{\sqrt{2 \cdot 8}}\right) \]

      8. metadata-eval N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \left(y-scale \cdot \sqrt{\color{blue}{16}}\right) \]

      9. metadata-eval N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 18.3

          \[\leadsto \color{blue}{\left(\left(0.25 \cdot b\right) \cdot y-scale\right)} \cdot 4 \]

   7. Applied rewrites 18.3%

      \[\leadsto \color{blue}{\left(\left(0.25 \cdot b\right) \cdot y-scale\right) \cdot 4} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. metadata-eval N/A

         \[\leadsto \left(\left(\frac{1}{4} \cdot b\right) \cdot y-scale\right) \cdot \color{blue}{\sqrt{16}} \]

      4. metadata-eval N/A

         \[\leadsto \left(\left(\frac{1}{4} \cdot b\right) \cdot y-scale\right) \cdot \sqrt{\color{blue}{2 \cdot 8}} \]

      5. sqrt-unprod N/A

         \[\leadsto \left(\left(\frac{1}{4} \cdot b\right) \cdot y-scale\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{8}\right)} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{1}{4} \cdot b\right) \cdot y-scale\right) \cdot \left(\color{blue}{\sqrt{2}} \cdot \sqrt{8}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{1}{4} \cdot b\right) \cdot y-scale\right) \cdot \left(\sqrt{2} \cdot \color{blue}{\sqrt{8}}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{4} \cdot b\right) \cdot y-scale\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{8}\right)} \]

      9. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \sqrt{8}\right) \cdot \left(\left(\frac{1}{4} \cdot b\right) \cdot y-scale\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{2} \cdot \sqrt{8}\right) \cdot \color{blue}{\left(\left(\frac{1}{4} \cdot b\right) \cdot y-scale\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{2} \cdot \sqrt{8}\right) \cdot \left(\color{blue}{\left(\frac{1}{4} \cdot b\right)} \cdot y-scale\right) \]

      12. associate-*l* N/A

          \[\leadsto \left(\sqrt{2} \cdot \sqrt{8}\right) \cdot \color{blue}{\left(\frac{1}{4} \cdot \left(b \cdot y-scale\right)\right)} \]

      13. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(\sqrt{2} \cdot \sqrt{8}\right) \cdot \frac{1}{4}\right) \cdot \left(b \cdot y-scale\right)} \]

      14. lift-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left(\sqrt{2} \cdot \sqrt{8}\right)} \cdot \frac{1}{4}\right) \cdot \left(b \cdot y-scale\right) \]

      15. lift-sqrt.f64 N/A

          \[\leadsto \left(\left(\color{blue}{\sqrt{2}} \cdot \sqrt{8}\right) \cdot \frac{1}{4}\right) \cdot \left(b \cdot y-scale\right) \]

      16. lift-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{2} \cdot \color{blue}{\sqrt{8}}\right) \cdot \frac{1}{4}\right) \cdot \left(b \cdot y-scale\right) \]

      17. sqrt-unprod N/A

          \[\leadsto \left(\color{blue}{\sqrt{2 \cdot 8}} \cdot \frac{1}{4}\right) \cdot \left(b \cdot y-scale\right) \]

      18. metadata-eval N/A

          \[\leadsto \left(\sqrt{\color{blue}{16}} \cdot \frac{1}{4}\right) \cdot \left(b \cdot y-scale\right) \]

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. *-lft-identity N/A

          \[\leadsto \color{blue}{b \cdot y-scale} \]

      22. *-commutative N/A

          \[\leadsto \color{blue}{y-scale \cdot b} \]

      23. lower-*.f64 18.3

          \[\leadsto \color{blue}{y-scale \cdot b} \]

   9. Applied rewrites 18.3%

      \[\leadsto \color{blue}{y-scale \cdot b} \]

   if 3.59999999999999978e-56 < x-scale

   1. Initial program 3.8%

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

   3. Taylor expanded in y-scale around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

   5. Applied rewrites 46.8%

      \[\leadsto \color{blue}{\left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left({\sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, b \cdot b, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)}} \]

   6. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(a \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(a \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{1}{4} \cdot \color{blue}{\left(\left(a \cdot x-scale\right) \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \color{blue}{\left(\left(a \cdot x-scale\right) \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \left(\color{blue}{\left(a \cdot x-scale\right)} \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \left(\left(a \cdot x-scale\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{8}\right)}\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \left(\left(a \cdot x-scale\right) \cdot \left(\color{blue}{\sqrt{2}} \cdot \sqrt{8}\right)\right) \]

      7. lower-sqrt.f64 25.5

         \[\leadsto 0.25 \cdot \left(\left(a \cdot x-scale\right) \cdot \left(\sqrt{2} \cdot \color{blue}{\sqrt{8}}\right)\right) \]

   8. Applied rewrites 25.5%

      \[\leadsto \color{blue}{0.25 \cdot \left(\left(a \cdot x-scale\right) \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 20.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq 3.6 \cdot 10^{-56}:\\ \;\;\;\;y-scale \cdot b\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(\sqrt{2} \cdot \sqrt{8}\right) \cdot \left(x-scale \cdot a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 19.5% accurate, 484.7× speedup

Math

\[\begin{array}{l} y-scale_m = \left|y-scale\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale\_m \cdot b \end{array} \]

FPCore

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

C

y-scale_m = fabs(y_45_scale);
x-scale_m = fabs(x_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	return y_45_scale_m * b;
}

Fortran

y-scale_m = abs(y_45scale)
x-scale_m = abs(x_45scale)
real(8) function code(a, b, angle, x_45scale_m, y_45scale_m)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale_m
    code = y_45scale_m * b
end function

Java

y-scale_m = Math.abs(y_45_scale);
x-scale_m = Math.abs(x_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	return y_45_scale_m * b;
}

Python

y-scale_m = math.fabs(y_45_scale)
x-scale_m = math.fabs(x_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale_m):
	return y_45_scale_m * b

Julia

y-scale_m = abs(y_45_scale)
x-scale_m = abs(x_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale_m)
	return Float64(y_45_scale_m * b)
end

MATLAB

y-scale_m = abs(y_45_scale);
x-scale_m = abs(x_45_scale);
function tmp = code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = y_45_scale_m * b;
end

Wolfram

y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := N[(y$45$scale$95$m * b), $MachinePrecision]

Derivation

1. Initial program 3.4%

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

3. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot b\right) \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot b\right) \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)} \]

   3. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot b\right)} \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \color{blue}{\left(\left(y-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)} \]

   5. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \color{blue}{\left(\left(y-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \left(\color{blue}{\left(y-scale \cdot \sqrt{2}\right)} \cdot \sqrt{8}\right) \]

   7. lower-sqrt.f64 N/A

      \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \left(\left(y-scale \cdot \color{blue}{\sqrt{2}}\right) \cdot \sqrt{8}\right) \]

   8. lower-sqrt.f64 16.0

      \[\leadsto \left(0.25 \cdot b\right) \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \color{blue}{\sqrt{8}}\right) \]

5. Applied rewrites 16.0%

   \[\leadsto \color{blue}{\left(0.25 \cdot b\right) \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot b\right)} \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right) \]

   2. lift-sqrt.f64 N/A

      \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \left(\left(y-scale \cdot \color{blue}{\sqrt{2}}\right) \cdot \sqrt{8}\right) \]

   3. lift-sqrt.f64 N/A

      \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \color{blue}{\sqrt{8}}\right) \]

   4. associate-*l* N/A

      \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \color{blue}{\left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)} \]

   5. lift-sqrt.f64 N/A

      \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \left(y-scale \cdot \left(\color{blue}{\sqrt{2}} \cdot \sqrt{8}\right)\right) \]

   6. lift-sqrt.f64 N/A

      \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \color{blue}{\sqrt{8}}\right)\right) \]

   7. sqrt-unprod N/A

      \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \left(y-scale \cdot \color{blue}{\sqrt{2 \cdot 8}}\right) \]

   8. metadata-eval N/A

      \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \left(y-scale \cdot \sqrt{\color{blue}{16}}\right) \]

   9. metadata-eval N/A

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

   10. associate-*r* N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 16.1

       \[\leadsto \color{blue}{\left(\left(0.25 \cdot b\right) \cdot y-scale\right)} \cdot 4 \]

7. Applied rewrites 16.1%

   \[\leadsto \color{blue}{\left(\left(0.25 \cdot b\right) \cdot y-scale\right) \cdot 4} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. metadata-eval N/A

      \[\leadsto \left(\left(\frac{1}{4} \cdot b\right) \cdot y-scale\right) \cdot \color{blue}{\sqrt{16}} \]

   4. metadata-eval N/A

      \[\leadsto \left(\left(\frac{1}{4} \cdot b\right) \cdot y-scale\right) \cdot \sqrt{\color{blue}{2 \cdot 8}} \]

   5. sqrt-unprod N/A

      \[\leadsto \left(\left(\frac{1}{4} \cdot b\right) \cdot y-scale\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{8}\right)} \]

   6. lift-sqrt.f64 N/A

      \[\leadsto \left(\left(\frac{1}{4} \cdot b\right) \cdot y-scale\right) \cdot \left(\color{blue}{\sqrt{2}} \cdot \sqrt{8}\right) \]

   7. lift-sqrt.f64 N/A

      \[\leadsto \left(\left(\frac{1}{4} \cdot b\right) \cdot y-scale\right) \cdot \left(\sqrt{2} \cdot \color{blue}{\sqrt{8}}\right) \]

   8. lift-*.f64 N/A

      \[\leadsto \left(\left(\frac{1}{4} \cdot b\right) \cdot y-scale\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{8}\right)} \]

   9. *-commutative N/A

      \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \sqrt{8}\right) \cdot \left(\left(\frac{1}{4} \cdot b\right) \cdot y-scale\right)} \]

   10. lift-*.f64 N/A

       \[\leadsto \left(\sqrt{2} \cdot \sqrt{8}\right) \cdot \color{blue}{\left(\left(\frac{1}{4} \cdot b\right) \cdot y-scale\right)} \]

   11. lift-*.f64 N/A

       \[\leadsto \left(\sqrt{2} \cdot \sqrt{8}\right) \cdot \left(\color{blue}{\left(\frac{1}{4} \cdot b\right)} \cdot y-scale\right) \]

   12. associate-*l* N/A

       \[\leadsto \left(\sqrt{2} \cdot \sqrt{8}\right) \cdot \color{blue}{\left(\frac{1}{4} \cdot \left(b \cdot y-scale\right)\right)} \]

   13. associate-*r* N/A

       \[\leadsto \color{blue}{\left(\left(\sqrt{2} \cdot \sqrt{8}\right) \cdot \frac{1}{4}\right) \cdot \left(b \cdot y-scale\right)} \]

   14. lift-*.f64 N/A

       \[\leadsto \left(\color{blue}{\left(\sqrt{2} \cdot \sqrt{8}\right)} \cdot \frac{1}{4}\right) \cdot \left(b \cdot y-scale\right) \]

   15. lift-sqrt.f64 N/A

       \[\leadsto \left(\left(\color{blue}{\sqrt{2}} \cdot \sqrt{8}\right) \cdot \frac{1}{4}\right) \cdot \left(b \cdot y-scale\right) \]

   16. lift-sqrt.f64 N/A

       \[\leadsto \left(\left(\sqrt{2} \cdot \color{blue}{\sqrt{8}}\right) \cdot \frac{1}{4}\right) \cdot \left(b \cdot y-scale\right) \]

   17. sqrt-unprod N/A

       \[\leadsto \left(\color{blue}{\sqrt{2 \cdot 8}} \cdot \frac{1}{4}\right) \cdot \left(b \cdot y-scale\right) \]

   18. metadata-eval N/A

       \[\leadsto \left(\sqrt{\color{blue}{16}} \cdot \frac{1}{4}\right) \cdot \left(b \cdot y-scale\right) \]

   19. metadata-eval N/A

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

   20. metadata-eval N/A

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

   21. *-lft-identity N/A

       \[\leadsto \color{blue}{b \cdot y-scale} \]

   22. *-commutative N/A

       \[\leadsto \color{blue}{y-scale \cdot b} \]

   23. lower-*.f64 16.1

       \[\leadsto \color{blue}{y-scale \cdot b} \]

9. Applied rewrites 16.1%

   \[\leadsto \color{blue}{y-scale \cdot b} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024219 
(FPCore (a b angle x-scale y-scale)
  :name "a from scale-rotated-ellipse"
  :precision binary64
  (/ (- (sqrt (* (* (* 2.0 (/ (* 4.0 (* (* b a) (* b (- a)))) (pow (* x-scale y-scale) 2.0))) (* (* b a) (* b (- a)))) (+ (+ (/ (/ (+ (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)) (sqrt (+ (pow (- (/ (/ (+ (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)) 2.0) (pow (/ (/ (* (* (* 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))))))) (/ (* 4.0 (* (* b a) (* b (- a)))) (pow (* x-scale y-scale) 2.0))))