a from scale-rotated-ellipse

Percentage Accurate: 2.8% → 57.0%

Time: 31.3s

Alternatives: 8

Speedup: 484.7×

• 21.9% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. a = 3.077835033545672e+218
  2. b = 1.1513845481017635e-249
  3. angle = 1.4299495121745184e+161
  4. x-scale = -2.5547910482430886e-119
  5. y-scale = -2.2045607516368646e+265

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: 57.0% accurate, 7.7× speedup

Math

\[\begin{array}{l} y-scale_m = \left|y-scale\right| \\ x-scale_m = \left|x-scale\right| \\ b_m = \left|b\right| \\ \begin{array}{l} t_0 := angle \cdot \left(0.005555555555555556 \cdot \mathsf{PI}\left(\right)\right)\\ \mathbf{if}\;y-scale\_m \leq 1.8 \cdot 10^{+35}:\\ \;\;\;\;\left(0.25 \cdot \left(\left(x-scale\_m \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)\right) \cdot \mathsf{hypot}\left(b\_m \cdot \sin t\_0, a \cdot \cos t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot \left(b\_m \cdot \left(x-scale\_m \cdot \left(y-scale\_m \cdot \sqrt{8}\right)\right)\right)\right) \cdot \frac{\sqrt{2}}{x-scale\_m}\\ \end{array} \end{array} \]

FPCore

y-scale_m = (fabs.f64 y-scale)
x-scale_m = (fabs.f64 x-scale)
b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (* angle (* 0.005555555555555556 (PI)))))
   (if (<= y-scale_m 1.8e+35)
     (*
      (* 0.25 (* (* x-scale_m (sqrt 2.0)) (sqrt 8.0)))
      (hypot (* b_m (sin t_0)) (* a (cos t_0))))
     (*
      (* 0.25 (* b_m (* x-scale_m (* y-scale_m (sqrt 8.0)))))
      (/ (sqrt 2.0) x-scale_m)))))

Derivation

1. Split input into 2 regimes

2. if y-scale < 1.8e35

   1. Initial program 2.4%

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

   3. Taylor expanded in y-scale around 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 20.5%

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

   6. Taylor expanded in x-scale around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-fma.f64 N/A

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

   8. Applied rewrites 20.4%

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

      1. add-cbrt-cube N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left(\left(x-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)\right) \cdot \sqrt{\mathsf{fma}\left(a \cdot a, {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, \left(b \cdot b\right) \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}}\right)\right)}^{2}\right)} \]

      2. pow1/3 N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left(\left(x-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)\right) \cdot \sqrt{\mathsf{fma}\left(a \cdot a, {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, \left(b \cdot b\right) \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{{\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)}^{\frac{1}{3}}}\right)\right)}^{2}\right)} \]

      3. add-sqr-sqrt N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left(\left(x-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)\right) \cdot \sqrt{\mathsf{fma}\left(a \cdot a, {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, \left(b \cdot b\right) \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot {\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)}^{\frac{1}{3}}\right)\right)}^{2}\right)} \]

      4. associate-*r* N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left(\left(x-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)\right) \cdot \sqrt{\mathsf{fma}\left(a \cdot a, {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, \left(b \cdot b\right) \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot {\color{blue}{\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}}^{\frac{1}{3}}\right)\right)}^{2}\right)} \]

      5. unpow-prod-down N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left(\left(x-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)\right) \cdot \sqrt{\mathsf{fma}\left(a \cdot a, {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, \left(b \cdot b\right) \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\left({\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}^{\frac{1}{3}} \cdot {\left(\sqrt{\mathsf{PI}\left(\right)}\right)}^{\frac{1}{3}}\right)}\right)\right)}^{2}\right)} \]

      6. pow1/3 N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left(\left(x-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)\right) \cdot \sqrt{\mathsf{fma}\left(a \cdot a, {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, \left(b \cdot b\right) \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \left({\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}^{\frac{1}{3}} \cdot \color{blue}{\sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}}\right)\right)\right)}^{2}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left(\left(x-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)\right) \cdot \sqrt{\mathsf{fma}\left(a \cdot a, {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, \left(b \cdot b\right) \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\left({\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}^{\frac{1}{3}} \cdot \sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)}\right)\right)}^{2}\right)} \]

      8. lower-pow.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left(\left(x-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)\right) \cdot \sqrt{\mathsf{fma}\left(a \cdot a, {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, \left(b \cdot b\right) \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \left(\color{blue}{{\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}^{\frac{1}{3}}} \cdot \sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)\right)\right)}^{2}\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left(\left(x-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)\right) \cdot \sqrt{\mathsf{fma}\left(a \cdot a, {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, \left(b \cdot b\right) \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \left({\color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}}^{\frac{1}{3}} \cdot \sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)\right)\right)}^{2}\right)} \]

      10. lift-PI.f64 N/A

          \[\leadsto \left(\frac{1}{4} \cdot \left(\left(x-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)\right) \cdot \sqrt{\mathsf{fma}\left(a \cdot a, {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, \left(b \cdot b\right) \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \left({\left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}^{\frac{1}{3}} \cdot \sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)\right)\right)}^{2}\right)} \]

      11. lift-PI.f64 N/A

          \[\leadsto \left(\frac{1}{4} \cdot \left(\left(x-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)\right) \cdot \sqrt{\mathsf{fma}\left(a \cdot a, {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, \left(b \cdot b\right) \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \left({\left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}^{\frac{1}{3}} \cdot \sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)\right)\right)}^{2}\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{4} \cdot \left(\left(x-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)\right) \cdot \sqrt{\mathsf{fma}\left(a \cdot a, {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, \left(b \cdot b\right) \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \left({\left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}^{\frac{1}{3}} \cdot \sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)\right)\right)}^{2}\right)} \]

      13. lift-PI.f64 N/A

          \[\leadsto \left(\frac{1}{4} \cdot \left(\left(x-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)\right) \cdot \sqrt{\mathsf{fma}\left(a \cdot a, {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, \left(b \cdot b\right) \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \left({\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{\mathsf{PI}\left(\right)}}\right)}^{\frac{1}{3}} \cdot \sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)\right)\right)}^{2}\right)} \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \left(\frac{1}{4} \cdot \left(\left(x-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)\right) \cdot \sqrt{\mathsf{fma}\left(a \cdot a, {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, \left(b \cdot b\right) \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \left({\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right)}}\right)}^{\frac{1}{3}} \cdot \sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)\right)\right)}^{2}\right)} \]

      15. lower-cbrt.f64 N/A

          \[\leadsto \left(\frac{1}{4} \cdot \left(\left(x-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)\right) \cdot \sqrt{\mathsf{fma}\left(a \cdot a, {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, \left(b \cdot b\right) \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \left({\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}^{\frac{1}{3}} \cdot \color{blue}{\sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}}\right)\right)\right)}^{2}\right)} \]

      16. lift-PI.f64 N/A

          \[\leadsto \left(\frac{1}{4} \cdot \left(\left(x-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)\right) \cdot \sqrt{\mathsf{fma}\left(a \cdot a, {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, \left(b \cdot b\right) \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \left({\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}^{\frac{1}{3}} \cdot \sqrt[3]{\sqrt{\color{blue}{\mathsf{PI}\left(\right)}}}\right)\right)\right)}^{2}\right)} \]

      17. lower-sqrt.f64 20.5

          \[\leadsto \left(0.25 \cdot \left(\left(x-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)\right) \cdot \sqrt{\mathsf{fma}\left(a \cdot a, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, \left(b \cdot b\right) \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \left({\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}^{0.3333333333333333} \cdot \sqrt[3]{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}}\right)\right)\right)}^{2}\right)} \]

   10. Applied rewrites 20.5%

       \[\leadsto \left(0.25 \cdot \left(\left(x-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)\right) \cdot \sqrt{\mathsf{fma}\left(a \cdot a, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, \left(b \cdot b\right) \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\left({\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}^{0.3333333333333333} \cdot \sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)}\right)\right)}^{2}\right)} \]

   12. Applied rewrites 24.5%

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

   if 1.8e35 < y-scale

   1. Initial program 4.1%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 18.0%

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

   6. Taylor expanded in angle around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 36.7

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

   8. Applied rewrites 36.7%

      \[\leadsto \left(0.25 \cdot \left(b \cdot \left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right)\right)\right) \cdot \color{blue}{\frac{\sqrt{2}}{x-scale}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 42.5% accurate, 10.6× speedup

Math

\[\begin{array}{l} y-scale_m = \left|y-scale\right| \\ x-scale_m = \left|x-scale\right| \\ b_m = \left|b\right| \\ \begin{array}{l} t_0 := \cos \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot 0.011111111111111112\right)\\ \mathbf{if}\;a \leq 1.35 \cdot 10^{+40}:\\ \;\;\;\;b\_m \cdot \left(0.25 \cdot \left(y-scale\_m \cdot 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x-scale\_m \cdot \sqrt{\mathsf{fma}\left(b\_m, b\_m \cdot \mathsf{fma}\left(t\_0, -0.5, 0.5\right), \left(a \cdot a\right) \cdot \mathsf{fma}\left(0.5, t\_0, 0.5\right)\right)}\\ \end{array} \end{array} \]

FPCore

y-scale_m = (fabs.f64 y-scale)
x-scale_m = (fabs.f64 x-scale)
b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (cos (* (* angle (PI)) 0.011111111111111112))))
   (if (<= a 1.35e+40)
     (* b_m (* 0.25 (* y-scale_m 4.0)))
     (*
      x-scale_m
      (sqrt
       (fma b_m (* b_m (fma t_0 -0.5 0.5)) (* (* a a) (fma 0.5 t_0 0.5))))))))

Derivation

1. Split input into 2 regimes

2. if a < 1.35000000000000005e40

   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 27.4

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

   5. Applied rewrites 27.4%

      \[\leadsto \color{blue}{\left(0.25 \cdot b\right) \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)} \]
   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-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.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. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 27.4

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lift-sqrt.f64 N/A

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

      15. lift-sqrt.f64 N/A

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

      16. sqrt-unprod N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. lower-*.f64 27.6

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

   7. Applied rewrites 27.6%

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

   if 1.35000000000000005e40 < a

   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 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 35.5%

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

   6. Taylor expanded in x-scale around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-fma.f64 N/A

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

   8. Applied rewrites 35.5%

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

   10. Applied rewrites 36.0%

       \[\leadsto \color{blue}{\left(x-scale \cdot 1\right) \cdot \sqrt{\mathsf{fma}\left(b, b \cdot \mathsf{fma}\left(\cos \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot 0.011111111111111112\right), -0.5, 0.5\right), \left(a \cdot a\right) \cdot \mathsf{fma}\left(0.5, \cos \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot 0.011111111111111112\right), 0.5\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 29.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.35 \cdot 10^{+40}:\\ \;\;\;\;b \cdot \left(0.25 \cdot \left(y-scale \cdot 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x-scale \cdot \sqrt{\mathsf{fma}\left(b, b \cdot \mathsf{fma}\left(\cos \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot 0.011111111111111112\right), -0.5, 0.5\right), \left(a \cdot a\right) \cdot \mathsf{fma}\left(0.5, \cos \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot 0.011111111111111112\right), 0.5\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 42.7% accurate, 32.7× speedup

Math

\[\begin{array}{l} y-scale_m = \left|y-scale\right| \\ x-scale_m = \left|x-scale\right| \\ b_m = \left|b\right| \\ \begin{array}{l} t_0 := 0.25 \cdot \left(x-scale\_m \cdot \sqrt{8}\right)\\ \mathbf{if}\;a \leq 1.35 \cdot 10^{+40}:\\ \;\;\;\;b\_m \cdot \left(0.25 \cdot \left(y-scale\_m \cdot 4\right)\right)\\ \mathbf{elif}\;a \leq 5.8 \cdot 10^{+133}:\\ \;\;\;\;t\_0 \cdot \sqrt{2 \cdot \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, \left(angle \cdot angle\right) \cdot \left(\left(b\_m \cdot b\_m\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right), a \cdot a\right)}\\ \mathbf{elif}\;a \leq 1.35 \cdot 10^{+244}:\\ \;\;\;\;x-scale\_m \cdot a\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \sqrt{2 \cdot \left(a \cdot a\right)}\\ \end{array} \end{array} \]

FPCore

y-scale_m = (fabs.f64 y-scale)
x-scale_m = (fabs.f64 x-scale)
b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (* 0.25 (* x-scale_m (sqrt 8.0)))))
   (if (<= a 1.35e+40)
     (* b_m (* 0.25 (* y-scale_m 4.0)))
     (if (<= a 5.8e+133)
       (*
        t_0
        (sqrt
         (*
          2.0
          (fma
           3.08641975308642e-5
           (* (* angle angle) (* (* b_m b_m) (* (PI) (PI))))
           (* a a)))))
       (if (<= a 1.35e+244) (* x-scale_m a) (* t_0 (sqrt (* 2.0 (* a a)))))))))

Derivation

1. Split input into 4 regimes

2. if a < 1.35000000000000005e40

   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 27.4

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

   5. Applied rewrites 27.4%

      \[\leadsto \color{blue}{\left(0.25 \cdot b\right) \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)} \]
   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-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.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. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 27.4

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lift-sqrt.f64 N/A

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

      15. lift-sqrt.f64 N/A

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

      16. sqrt-unprod N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. lower-*.f64 27.6

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

   7. Applied rewrites 27.6%

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

   if 1.35000000000000005e40 < a < 5.8000000000000002e133

   1. Initial program 6.2%

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

   3. Taylor expanded in 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 37.0%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 37.1%

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

   9. Taylor expanded in angle around 0

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

       1. lower-fma.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. lower-PI.f64 N/A

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

       11. lower-PI.f64 N/A

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

       12. unpow2 N/A

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

       13. lower-*.f64 37.2

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

   11. Applied rewrites 37.2%

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

   if 5.8000000000000002e133 < a < 1.34999999999999999e244

   1. Initial program 0.0%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   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 21.3%

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

   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 39.0

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

   8. Applied rewrites 39.0%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. sqrt-unprod N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. *-lft-identity 39.3

          \[\leadsto \color{blue}{a \cdot x-scale} \]

      15. lift-*.f64 N/A

          \[\leadsto \color{blue}{a \cdot x-scale} \]

      16. *-commutative N/A

          \[\leadsto \color{blue}{x-scale \cdot a} \]

      17. lower-*.f64 39.3

          \[\leadsto \color{blue}{x-scale \cdot a} \]

   10. Applied rewrites 39.3%

       \[\leadsto \color{blue}{x-scale \cdot a} \]

   if 1.34999999999999999e244 < a

   1. Initial program 0.0%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   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 55.9%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 55.9%

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

   9. Taylor expanded in angle around 0

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

       1. unpow2 N/A

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

       2. lower-*.f64 67.1

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

   11. Applied rewrites 67.1%

       \[\leadsto \left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \color{blue}{\left(a \cdot a\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 30.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.35 \cdot 10^{+40}:\\ \;\;\;\;b \cdot \left(0.25 \cdot \left(y-scale \cdot 4\right)\right)\\ \mathbf{elif}\;a \leq 5.8 \cdot 10^{+133}:\\ \;\;\;\;\left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, \left(angle \cdot angle\right) \cdot \left(\left(b \cdot b\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right), a \cdot a\right)}\\ \mathbf{elif}\;a \leq 1.35 \cdot 10^{+244}:\\ \;\;\;\;x-scale \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left(a \cdot a\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 42.4% accurate, 46.1× speedup

Math

\[\begin{array}{l} y-scale_m = \left|y-scale\right| \\ x-scale_m = \left|x-scale\right| \\ b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 1.65 \cdot 10^{-52}:\\ \;\;\;\;x-scale\_m \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{x-scale\_m} \cdot \left(\left(x-scale\_m \cdot b\_m\right) \cdot \left(y-scale\_m \cdot \left(0.25 \cdot \sqrt{8}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

y-scale_m = (fabs.f64 y-scale)
x-scale_m = (fabs.f64 x-scale)
b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= b_m 1.65e-52)
   (* x-scale_m a)
   (*
    (/ (sqrt 2.0) x-scale_m)
    (* (* x-scale_m b_m) (* y-scale_m (* 0.25 (sqrt 8.0)))))))

C

y-scale_m = fabs(y_45_scale);
x-scale_m = fabs(x_45_scale);
b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (b_m <= 1.65e-52) {
		tmp = x_45_scale_m * a;
	} else {
		tmp = (sqrt(2.0) / x_45_scale_m) * ((x_45_scale_m * b_m) * (y_45_scale_m * (0.25 * sqrt(8.0))));
	}
	return tmp;
}

Fortran

y-scale_m = abs(y_45scale)
x-scale_m = abs(x_45scale)
b_m = abs(b)
real(8) function code(a, b_m, angle, x_45scale_m, y_45scale_m)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale_m
    real(8) :: tmp
    if (b_m <= 1.65d-52) then
        tmp = x_45scale_m * a
    else
        tmp = (sqrt(2.0d0) / x_45scale_m) * ((x_45scale_m * b_m) * (y_45scale_m * (0.25d0 * sqrt(8.0d0))))
    end if
    code = tmp
end function

Java

y-scale_m = Math.abs(y_45_scale);
x-scale_m = Math.abs(x_45_scale);
b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (b_m <= 1.65e-52) {
		tmp = x_45_scale_m * a;
	} else {
		tmp = (Math.sqrt(2.0) / x_45_scale_m) * ((x_45_scale_m * b_m) * (y_45_scale_m * (0.25 * Math.sqrt(8.0))));
	}
	return tmp;
}

Python

y-scale_m = math.fabs(y_45_scale)
x-scale_m = math.fabs(x_45_scale)
b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if b_m <= 1.65e-52:
		tmp = x_45_scale_m * a
	else:
		tmp = (math.sqrt(2.0) / x_45_scale_m) * ((x_45_scale_m * b_m) * (y_45_scale_m * (0.25 * math.sqrt(8.0))))
	return tmp

Julia

y-scale_m = abs(y_45_scale)
x-scale_m = abs(x_45_scale)
b_m = abs(b)
function code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (b_m <= 1.65e-52)
		tmp = Float64(x_45_scale_m * a);
	else
		tmp = Float64(Float64(sqrt(2.0) / x_45_scale_m) * Float64(Float64(x_45_scale_m * b_m) * Float64(y_45_scale_m * Float64(0.25 * sqrt(8.0)))));
	end
	return tmp
end

MATLAB

y-scale_m = abs(y_45_scale);
x-scale_m = abs(x_45_scale);
b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0;
	if (b_m <= 1.65e-52)
		tmp = x_45_scale_m * a;
	else
		tmp = (sqrt(2.0) / x_45_scale_m) * ((x_45_scale_m * b_m) * (y_45_scale_m * (0.25 * sqrt(8.0))));
	end
	tmp_2 = tmp;
end

Wolfram

y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[b$95$m, 1.65e-52], N[(x$45$scale$95$m * a), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / x$45$scale$95$m), $MachinePrecision] * N[(N[(x$45$scale$95$m * b$95$m), $MachinePrecision] * N[(y$45$scale$95$m * N[(0.25 * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.64999999999999998e-52

   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 20.7%

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

   6. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(a \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]
   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 21.3

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

   8. Applied rewrites 21.3%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. sqrt-unprod N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. *-lft-identity 21.5

          \[\leadsto \color{blue}{a \cdot x-scale} \]

      15. lift-*.f64 N/A

          \[\leadsto \color{blue}{a \cdot x-scale} \]

      16. *-commutative N/A

          \[\leadsto \color{blue}{x-scale \cdot a} \]

      17. lower-*.f64 21.5

          \[\leadsto \color{blue}{x-scale \cdot a} \]

   10. Applied rewrites 21.5%

       \[\leadsto \color{blue}{x-scale \cdot a} \]

   if 1.64999999999999998e-52 < b

   1. Initial program 2.3%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 16.0%

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

   7. Applied rewrites 16.9%

      \[\leadsto \color{blue}{\left(\left(b \cdot x-scale\right) \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot 0.25\right)\right)\right) \cdot \sqrt{\sqrt{\mathsf{fma}\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right), \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{y-scale \cdot \left(y-scale \cdot \left(x-scale \cdot x-scale\right)\right)} \cdot 4, {\left(\frac{0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{x-scale \cdot x-scale} - \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{y-scale \cdot y-scale}\right)}^{2}\right)} + \left(\frac{0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{x-scale \cdot x-scale} + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{y-scale \cdot y-scale}\right)}} \]

   8. Taylor expanded in angle around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 32.1

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

   10. Applied rewrites 32.1%

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

4. Final simplification 24.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.65 \cdot 10^{-52}:\\ \;\;\;\;x-scale \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{x-scale} \cdot \left(\left(x-scale \cdot b\right) \cdot \left(y-scale \cdot \left(0.25 \cdot \sqrt{8}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 42.5% accurate, 50.1× speedup

Math

\[\begin{array}{l} y-scale_m = \left|y-scale\right| \\ x-scale_m = \left|x-scale\right| \\ b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 5.8 \cdot 10^{+40}:\\ \;\;\;\;b\_m \cdot \left(0.25 \cdot \left(y-scale\_m \cdot 4\right)\right)\\ \mathbf{elif}\;a \leq 1.35 \cdot 10^{+244}:\\ \;\;\;\;x-scale\_m \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot \left(x-scale\_m \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left(a \cdot a\right)}\\ \end{array} \end{array} \]

FPCore

y-scale_m = (fabs.f64 y-scale)
x-scale_m = (fabs.f64 x-scale)
b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= a 5.8e+40)
   (* b_m (* 0.25 (* y-scale_m 4.0)))
   (if (<= a 1.35e+244)
     (* x-scale_m a)
     (* (* 0.25 (* x-scale_m (sqrt 8.0))) (sqrt (* 2.0 (* a a)))))))

C

y-scale_m = fabs(y_45_scale);
x-scale_m = fabs(x_45_scale);
b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (a <= 5.8e+40) {
		tmp = b_m * (0.25 * (y_45_scale_m * 4.0));
	} else if (a <= 1.35e+244) {
		tmp = x_45_scale_m * a;
	} else {
		tmp = (0.25 * (x_45_scale_m * sqrt(8.0))) * sqrt((2.0 * (a * a)));
	}
	return tmp;
}

Fortran

y-scale_m = abs(y_45scale)
x-scale_m = abs(x_45scale)
b_m = abs(b)
real(8) function code(a, b_m, angle, x_45scale_m, y_45scale_m)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale_m
    real(8) :: tmp
    if (a <= 5.8d+40) then
        tmp = b_m * (0.25d0 * (y_45scale_m * 4.0d0))
    else if (a <= 1.35d+244) then
        tmp = x_45scale_m * a
    else
        tmp = (0.25d0 * (x_45scale_m * sqrt(8.0d0))) * sqrt((2.0d0 * (a * a)))
    end if
    code = tmp
end function

Java

y-scale_m = Math.abs(y_45_scale);
x-scale_m = Math.abs(x_45_scale);
b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (a <= 5.8e+40) {
		tmp = b_m * (0.25 * (y_45_scale_m * 4.0));
	} else if (a <= 1.35e+244) {
		tmp = x_45_scale_m * a;
	} else {
		tmp = (0.25 * (x_45_scale_m * Math.sqrt(8.0))) * Math.sqrt((2.0 * (a * a)));
	}
	return tmp;
}

Python

y-scale_m = math.fabs(y_45_scale)
x-scale_m = math.fabs(x_45_scale)
b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if a <= 5.8e+40:
		tmp = b_m * (0.25 * (y_45_scale_m * 4.0))
	elif a <= 1.35e+244:
		tmp = x_45_scale_m * a
	else:
		tmp = (0.25 * (x_45_scale_m * math.sqrt(8.0))) * math.sqrt((2.0 * (a * a)))
	return tmp

Julia

y-scale_m = abs(y_45_scale)
x-scale_m = abs(x_45_scale)
b_m = abs(b)
function code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (a <= 5.8e+40)
		tmp = Float64(b_m * Float64(0.25 * Float64(y_45_scale_m * 4.0)));
	elseif (a <= 1.35e+244)
		tmp = Float64(x_45_scale_m * a);
	else
		tmp = Float64(Float64(0.25 * Float64(x_45_scale_m * sqrt(8.0))) * sqrt(Float64(2.0 * Float64(a * a))));
	end
	return tmp
end

MATLAB

y-scale_m = abs(y_45_scale);
x-scale_m = abs(x_45_scale);
b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0;
	if (a <= 5.8e+40)
		tmp = b_m * (0.25 * (y_45_scale_m * 4.0));
	elseif (a <= 1.35e+244)
		tmp = x_45_scale_m * a;
	else
		tmp = (0.25 * (x_45_scale_m * sqrt(8.0))) * sqrt((2.0 * (a * 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]
b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[a, 5.8e+40], N[(b$95$m * N[(0.25 * N[(y$45$scale$95$m * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.35e+244], N[(x$45$scale$95$m * a), $MachinePrecision], N[(N[(0.25 * N[(x$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(2.0 * N[(a * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < 5.80000000000000035e40

   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 27.4

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

   5. Applied rewrites 27.4%

      \[\leadsto \color{blue}{\left(0.25 \cdot b\right) \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)} \]
   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-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.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. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 27.4

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lift-sqrt.f64 N/A

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

      15. lift-sqrt.f64 N/A

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

      16. sqrt-unprod N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. lower-*.f64 27.6

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

   7. Applied rewrites 27.6%

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

   if 5.80000000000000035e40 < a < 1.34999999999999999e244

   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 30.3%

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

   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 35.4

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

   8. Applied rewrites 35.4%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. sqrt-unprod N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. *-lft-identity 35.7

          \[\leadsto \color{blue}{a \cdot x-scale} \]

      15. lift-*.f64 N/A

          \[\leadsto \color{blue}{a \cdot x-scale} \]

      16. *-commutative N/A

          \[\leadsto \color{blue}{x-scale \cdot a} \]

      17. lower-*.f64 35.7

          \[\leadsto \color{blue}{x-scale \cdot a} \]

   10. Applied rewrites 35.7%

       \[\leadsto \color{blue}{x-scale \cdot a} \]

   if 1.34999999999999999e244 < a

   1. Initial program 0.0%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   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 55.9%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 55.9%

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

   9. Taylor expanded in angle around 0

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

       1. unpow2 N/A

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

       2. lower-*.f64 67.1

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

   11. Applied rewrites 67.1%

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

4. Final simplification 30.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 5.8 \cdot 10^{+40}:\\ \;\;\;\;b \cdot \left(0.25 \cdot \left(y-scale \cdot 4\right)\right)\\ \mathbf{elif}\;a \leq 1.35 \cdot 10^{+244}:\\ \;\;\;\;x-scale \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left(a \cdot a\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 42.8% accurate, 132.1× speedup

Math

\[\begin{array}{l} y-scale_m = \left|y-scale\right| \\ x-scale_m = \left|x-scale\right| \\ b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 5.8 \cdot 10^{+40}:\\ \;\;\;\;b\_m \cdot \left(0.25 \cdot \left(y-scale\_m \cdot 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x-scale\_m \cdot a\\ \end{array} \end{array} \]

FPCore

y-scale_m = (fabs.f64 y-scale)
x-scale_m = (fabs.f64 x-scale)
b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= a 5.8e+40) (* b_m (* 0.25 (* y-scale_m 4.0))) (* x-scale_m a)))

C

y-scale_m = fabs(y_45_scale);
x-scale_m = fabs(x_45_scale);
b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (a <= 5.8e+40) {
		tmp = b_m * (0.25 * (y_45_scale_m * 4.0));
	} else {
		tmp = x_45_scale_m * a;
	}
	return tmp;
}

Fortran

y-scale_m = abs(y_45scale)
x-scale_m = abs(x_45scale)
b_m = abs(b)
real(8) function code(a, b_m, angle, x_45scale_m, y_45scale_m)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale_m
    real(8) :: tmp
    if (a <= 5.8d+40) then
        tmp = b_m * (0.25d0 * (y_45scale_m * 4.0d0))
    else
        tmp = 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);
b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (a <= 5.8e+40) {
		tmp = b_m * (0.25 * (y_45_scale_m * 4.0));
	} else {
		tmp = x_45_scale_m * a;
	}
	return tmp;
}

Python

y-scale_m = math.fabs(y_45_scale)
x-scale_m = math.fabs(x_45_scale)
b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if a <= 5.8e+40:
		tmp = b_m * (0.25 * (y_45_scale_m * 4.0))
	else:
		tmp = x_45_scale_m * a
	return tmp

Julia

y-scale_m = abs(y_45_scale)
x-scale_m = abs(x_45_scale)
b_m = abs(b)
function code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (a <= 5.8e+40)
		tmp = Float64(b_m * Float64(0.25 * Float64(y_45_scale_m * 4.0)));
	else
		tmp = 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);
b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0;
	if (a <= 5.8e+40)
		tmp = b_m * (0.25 * (y_45_scale_m * 4.0));
	else
		tmp = 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]
b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[a, 5.8e+40], N[(b$95$m * N[(0.25 * N[(y$45$scale$95$m * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$45$scale$95$m * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 5.80000000000000035e40

   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 27.4

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

   5. Applied rewrites 27.4%

      \[\leadsto \color{blue}{\left(0.25 \cdot b\right) \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)} \]
   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-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.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. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 27.4

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lift-sqrt.f64 N/A

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

      15. lift-sqrt.f64 N/A

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

      16. sqrt-unprod N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. lower-*.f64 27.6

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

   7. Applied rewrites 27.6%

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

   if 5.80000000000000035e40 < a

   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 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 35.5%

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

   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 37.6

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

   8. Applied rewrites 37.6%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. sqrt-unprod N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. *-lft-identity 37.8

          \[\leadsto \color{blue}{a \cdot x-scale} \]

      15. lift-*.f64 N/A

          \[\leadsto \color{blue}{a \cdot x-scale} \]

      16. *-commutative N/A

          \[\leadsto \color{blue}{x-scale \cdot a} \]

      17. lower-*.f64 37.8

          \[\leadsto \color{blue}{x-scale \cdot a} \]

   10. Applied rewrites 37.8%

       \[\leadsto \color{blue}{x-scale \cdot a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 29.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 5.8 \cdot 10^{+40}:\\ \;\;\;\;b \cdot \left(0.25 \cdot \left(y-scale \cdot 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x-scale \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 42.8% accurate, 242.0× speedup

Math

\[\begin{array}{l} y-scale_m = \left|y-scale\right| \\ x-scale_m = \left|x-scale\right| \\ b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 5.8 \cdot 10^{+40}:\\ \;\;\;\;y-scale\_m \cdot b\_m\\ \mathbf{else}:\\ \;\;\;\;x-scale\_m \cdot a\\ \end{array} \end{array} \]

FPCore

y-scale_m = (fabs.f64 y-scale)
x-scale_m = (fabs.f64 x-scale)
b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= a 5.8e+40) (* y-scale_m b_m) (* x-scale_m a)))

C

y-scale_m = fabs(y_45_scale);
x-scale_m = fabs(x_45_scale);
b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (a <= 5.8e+40) {
		tmp = y_45_scale_m * b_m;
	} else {
		tmp = x_45_scale_m * a;
	}
	return tmp;
}

Fortran

y-scale_m = abs(y_45scale)
x-scale_m = abs(x_45scale)
b_m = abs(b)
real(8) function code(a, b_m, angle, x_45scale_m, y_45scale_m)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale_m
    real(8) :: tmp
    if (a <= 5.8d+40) then
        tmp = y_45scale_m * b_m
    else
        tmp = 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);
b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (a <= 5.8e+40) {
		tmp = y_45_scale_m * b_m;
	} else {
		tmp = x_45_scale_m * a;
	}
	return tmp;
}

Python

y-scale_m = math.fabs(y_45_scale)
x-scale_m = math.fabs(x_45_scale)
b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if a <= 5.8e+40:
		tmp = y_45_scale_m * b_m
	else:
		tmp = x_45_scale_m * a
	return tmp

Julia

y-scale_m = abs(y_45_scale)
x-scale_m = abs(x_45_scale)
b_m = abs(b)
function code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (a <= 5.8e+40)
		tmp = Float64(y_45_scale_m * b_m);
	else
		tmp = 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);
b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0;
	if (a <= 5.8e+40)
		tmp = y_45_scale_m * b_m;
	else
		tmp = 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]
b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[a, 5.8e+40], N[(y$45$scale$95$m * b$95$m), $MachinePrecision], N[(x$45$scale$95$m * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 5.80000000000000035e40

   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 27.4

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

   5. Applied rewrites 27.4%

      \[\leadsto \color{blue}{\left(0.25 \cdot b\right) \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)} \]
   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-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.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. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 27.4

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lift-sqrt.f64 N/A

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

      15. lift-sqrt.f64 N/A

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

      16. sqrt-unprod N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. lower-*.f64 27.6

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

   7. Applied rewrites 27.6%

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

      1. associate-*l* N/A

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

      2. metadata-eval N/A

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

      3. *-rgt-identity 27.6

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

   9. Applied rewrites 27.6%

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

   if 5.80000000000000035e40 < a

   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 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 35.5%

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

   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 37.6

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

   8. Applied rewrites 37.6%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. sqrt-unprod N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. *-lft-identity 37.8

          \[\leadsto \color{blue}{a \cdot x-scale} \]

      15. lift-*.f64 N/A

          \[\leadsto \color{blue}{a \cdot x-scale} \]

      16. *-commutative N/A

          \[\leadsto \color{blue}{x-scale \cdot a} \]

      17. lower-*.f64 37.8

          \[\leadsto \color{blue}{x-scale \cdot a} \]

   10. Applied rewrites 37.8%

       \[\leadsto \color{blue}{x-scale \cdot a} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 19.4% accurate, 484.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 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 19.4%

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

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 18.7

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

8. Applied rewrites 18.7%

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

   1. lift-*.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lift-*.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. lift-sqrt.f64 N/A

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

   10. sqrt-unprod N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. *-lft-identity 18.9

       \[\leadsto \color{blue}{a \cdot x-scale} \]

   15. lift-*.f64 N/A

       \[\leadsto \color{blue}{a \cdot x-scale} \]

   16. *-commutative N/A

       \[\leadsto \color{blue}{x-scale \cdot a} \]

   17. lower-*.f64 18.9

       \[\leadsto \color{blue}{x-scale \cdot a} \]

10. Applied rewrites 18.9%

    \[\leadsto \color{blue}{x-scale \cdot a} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024216 
(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))))