a from scale-rotated-ellipse

Percentage Accurate: 2.9% → 51.6%

Time: 24.5s

Alternatives: 8

Speedup: 484.7×

• 30.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 2.9% 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: 51.6% accurate, 5.9× speedup

Math

\[\begin{array}{l} y-scale_m = \left|y-scale\right| \\ x-scale_m = \left|x-scale\right| \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot angle\\ \mathbf{if}\;y-scale\_m \leq 1.8 \cdot 10^{-113}:\\ \;\;\;\;\left(0.25 \cdot \left(\sqrt{8} \cdot x-scale\_m\right)\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(a \cdot a, {\cos \left(-0.005555555555555556 \cdot t\_0\right)}^{2}, \left(b \cdot b\right) \cdot {\sin \left(t\_0 \cdot 0.005555555555555556\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{hypot}\left(\sin \left(\mathsf{fma}\left(-0.005555555555555556, t\_0, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot b, a \cdot \sin \left(0.005555555555555556 \cdot t\_0\right)\right) \cdot \sqrt{2}\right) \cdot \left(\left(\left(\sqrt{8} \cdot y-scale\_m\right) \cdot x-scale\_m\right) \cdot 0.25\right)}{\left|x-scale\_m\right|}\\ \end{array} \end{array} \]

FPCore

y-scale_m = (fabs.f64 y-scale)
x-scale_m = (fabs.f64 x-scale)
(FPCore (a b angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (* (PI) angle)))
   (if (<= y-scale_m 1.8e-113)
     (*
      (* 0.25 (* (sqrt 8.0) x-scale_m))
      (sqrt
       (*
        2.0
        (fma
         (* a a)
         (pow (cos (* -0.005555555555555556 t_0)) 2.0)
         (* (* b b) (pow (sin (* t_0 0.005555555555555556)) 2.0))))))
     (/
      (*
       (*
        (hypot
         (* (sin (fma -0.005555555555555556 t_0 (/ (PI) 2.0))) b)
         (* a (sin (* 0.005555555555555556 t_0))))
        (sqrt 2.0))
       (* (* (* (sqrt 8.0) y-scale_m) x-scale_m) 0.25))
      (fabs x-scale_m)))))

Derivation

1. Split input into 2 regimes

2. if y-scale < 1.79999999999999987e-113

   1. Initial program 1.7%

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

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. distribute-lft-out N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 24.6%

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

   if 1.79999999999999987e-113 < y-scale

   1. Initial program 1.5%

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

   3. Taylor expanded in y-scale around inf

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

   5. Applied rewrites 24.6%

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

   7. Applied rewrites 42.8%

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

   9. Applied rewrites 42.8%

      \[\leadsto \frac{\left(\mathsf{hypot}\left(\sin \left(\mathsf{fma}\left(-0.005555555555555556, \mathsf{PI}\left(\right) \cdot angle, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot b, a \cdot \sin \left(0.005555555555555556 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)\right) \cdot \sqrt{2}\right) \cdot \left(\left(\left(\sqrt{8} \cdot y-scale\right) \cdot x-scale\right) \cdot 0.25\right)}{\left|x-scale\right|} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 46.2% accurate, 7.3× speedup

Math

\[\begin{array}{l} y-scale_m = \left|y-scale\right| \\ x-scale_m = \left|x-scale\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\\ \mathbf{if}\;y-scale\_m \leq 4.9 \cdot 10^{-77}:\\ \;\;\;\;\left(0.25 \cdot a\right) \cdot \left(\left(x-scale\_m \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{hypot}\left(\cos t\_0 \cdot b, a \cdot \sin t\_0\right) \cdot \sqrt{2}\right) \cdot \left(\left(\left(\sqrt{8} \cdot y-scale\_m\right) \cdot x-scale\_m\right) \cdot 0.25\right)}{\left|x-scale\_m\right|}\\ \end{array} \end{array} \]

FPCore

y-scale_m = (fabs.f64 y-scale)
x-scale_m = (fabs.f64 x-scale)
(FPCore (a b angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* (PI) angle))))
   (if (<= y-scale_m 4.9e-77)
     (* (* 0.25 a) (* (* x-scale_m (sqrt 2.0)) (sqrt 8.0)))
     (/
      (*
       (* (hypot (* (cos t_0) b) (* a (sin t_0))) (sqrt 2.0))
       (* (* (* (sqrt 8.0) y-scale_m) x-scale_m) 0.25))
      (fabs x-scale_m)))))

Derivation

1. Split input into 2 regimes

2. if y-scale < 4.8999999999999997e-77

   1. Initial program 1.6%

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

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. distribute-lft-out N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 8.6%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 9.2%

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

   9. Taylor expanded in angle around 0

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

   11. Applied rewrites 17.2%

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

   if 4.8999999999999997e-77 < y-scale

   1. Initial program 1.5%

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

   3. Taylor expanded in y-scale around inf

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

   5. Applied rewrites 26.5%

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

   7. Applied rewrites 46.0%

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

Alternative 3: 46.2% accurate, 7.3× speedup

Math

\[\begin{array}{l} y-scale_m = \left|y-scale\right| \\ x-scale_m = \left|x-scale\right| \\ \begin{array}{l} t_0 := \left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\\ \mathbf{if}\;y-scale\_m \leq 4.9 \cdot 10^{-77}:\\ \;\;\;\;\left(0.25 \cdot a\right) \cdot \left(\left(x-scale\_m \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\left(0.25 \cdot \left(\sqrt{8} \cdot y-scale\_m\right)\right) \cdot x-scale\_m\right) \cdot \sqrt{2}\right) \cdot \mathsf{hypot}\left(\sin t\_0 \cdot a, \cos t\_0 \cdot b\right)}{\left|x-scale\_m\right|}\\ \end{array} \end{array} \]

FPCore

y-scale_m = (fabs.f64 y-scale)
x-scale_m = (fabs.f64 x-scale)
(FPCore (a b angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (* (* (PI) angle) 0.005555555555555556)))
   (if (<= y-scale_m 4.9e-77)
     (* (* 0.25 a) (* (* x-scale_m (sqrt 2.0)) (sqrt 8.0)))
     (/
      (*
       (* (* (* 0.25 (* (sqrt 8.0) y-scale_m)) x-scale_m) (sqrt 2.0))
       (hypot (* (sin t_0) a) (* (cos t_0) b)))
      (fabs x-scale_m)))))

Derivation

1. Split input into 2 regimes

2. if y-scale < 4.8999999999999997e-77

   1. Initial program 1.6%

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

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. distribute-lft-out N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 8.6%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 9.2%

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

   9. Taylor expanded in angle around 0

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

   11. Applied rewrites 17.2%

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

   if 4.8999999999999997e-77 < y-scale

   1. Initial program 1.5%

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

   3. Taylor expanded in y-scale around inf

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

   5. Applied rewrites 26.5%

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

   7. Applied rewrites 46.0%

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

   9. Applied rewrites 46.0%

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

Alternative 4: 46.2% accurate, 10.2× speedup

Math

\[\begin{array}{l} y-scale_m = \left|y-scale\right| \\ x-scale_m = \left|x-scale\right| \\ \begin{array}{l} \mathbf{if}\;y-scale\_m \leq 4.9 \cdot 10^{-77}:\\ \;\;\;\;\left(0.25 \cdot a\right) \cdot \left(\left(x-scale\_m \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{hypot}\left(1 \cdot b, a \cdot \sin \left(0.005555555555555556 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)\right) \cdot \sqrt{2}\right) \cdot \left(\left(\left(\sqrt{8} \cdot y-scale\_m\right) \cdot x-scale\_m\right) \cdot 0.25\right)}{\left|x-scale\_m\right|}\\ \end{array} \end{array} \]

FPCore

y-scale_m = (fabs.f64 y-scale)
x-scale_m = (fabs.f64 x-scale)
(FPCore (a b angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= y-scale_m 4.9e-77)
   (* (* 0.25 a) (* (* x-scale_m (sqrt 2.0)) (sqrt 8.0)))
   (/
    (*
     (*
      (hypot (* 1.0 b) (* a (sin (* 0.005555555555555556 (* (PI) angle)))))
      (sqrt 2.0))
     (* (* (* (sqrt 8.0) y-scale_m) x-scale_m) 0.25))
    (fabs x-scale_m))))

Derivation

1. Split input into 2 regimes

2. if y-scale < 4.8999999999999997e-77

   1. Initial program 1.6%

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

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. distribute-lft-out N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 8.6%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 9.2%

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

   9. Taylor expanded in angle around 0

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

   11. Applied rewrites 17.2%

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

   if 4.8999999999999997e-77 < y-scale

   1. Initial program 1.5%

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

   3. Taylor expanded in y-scale around inf

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

   5. Applied rewrites 26.5%

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

   7. Applied rewrites 46.0%

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

   8. Taylor expanded in angle around 0

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

   10. Applied rewrites 46.1%

       \[\leadsto \frac{\left(\mathsf{hypot}\left(1 \cdot b, a \cdot \sin \left(0.005555555555555556 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)\right) \cdot \sqrt{2}\right) \cdot \left(\left(\left(\sqrt{8} \cdot y-scale\right) \cdot x-scale\right) \cdot 0.25\right)}{\left|x-scale\right|} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 28.6% accurate, 44.7× speedup

Math

\[\begin{array}{l} y-scale_m = \left|y-scale\right| \\ x-scale_m = \left|x-scale\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 3.25 \cdot 10^{-105}:\\ \;\;\;\;\left(0.25 \cdot a\right) \cdot \left(\left(x-scale\_m \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{\left(b \cdot x-scale\_m\right) \cdot \left(\left(y-scale\_m \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)}{\left|x-scale\_m\right|}\\ \end{array} \end{array} \]

FPCore

y-scale_m = (fabs.f64 y-scale)
x-scale_m = (fabs.f64 x-scale)
(FPCore (a b angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= b 3.25e-105)
   (* (* 0.25 a) (* (* x-scale_m (sqrt 2.0)) (sqrt 8.0)))
   (*
    0.25
    (/
     (* (* b x-scale_m) (* (* y-scale_m (sqrt 2.0)) (sqrt 8.0)))
     (fabs x-scale_m)))))

C

y-scale_m = fabs(y_45_scale);
x-scale_m = fabs(x_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (b <= 3.25e-105) {
		tmp = (0.25 * a) * ((x_45_scale_m * sqrt(2.0)) * sqrt(8.0));
	} else {
		tmp = 0.25 * (((b * x_45_scale_m) * ((y_45_scale_m * sqrt(2.0)) * sqrt(8.0))) / fabs(x_45_scale_m));
	}
	return tmp;
}

Fortran

y-scale_m =     private
x-scale_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, angle, x_45scale_m, y_45scale_m)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale_m
    real(8) :: tmp
    if (b <= 3.25d-105) then
        tmp = (0.25d0 * a) * ((x_45scale_m * sqrt(2.0d0)) * sqrt(8.0d0))
    else
        tmp = 0.25d0 * (((b * x_45scale_m) * ((y_45scale_m * sqrt(2.0d0)) * sqrt(8.0d0))) / abs(x_45scale_m))
    end if
    code = tmp
end function

Java

y-scale_m = Math.abs(y_45_scale);
x-scale_m = Math.abs(x_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (b <= 3.25e-105) {
		tmp = (0.25 * a) * ((x_45_scale_m * Math.sqrt(2.0)) * Math.sqrt(8.0));
	} else {
		tmp = 0.25 * (((b * x_45_scale_m) * ((y_45_scale_m * Math.sqrt(2.0)) * Math.sqrt(8.0))) / Math.abs(x_45_scale_m));
	}
	return tmp;
}

Python

y-scale_m = math.fabs(y_45_scale)
x-scale_m = math.fabs(x_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if b <= 3.25e-105:
		tmp = (0.25 * a) * ((x_45_scale_m * math.sqrt(2.0)) * math.sqrt(8.0))
	else:
		tmp = 0.25 * (((b * x_45_scale_m) * ((y_45_scale_m * math.sqrt(2.0)) * math.sqrt(8.0))) / math.fabs(x_45_scale_m))
	return tmp

Julia

y-scale_m = abs(y_45_scale)
x-scale_m = abs(x_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (b <= 3.25e-105)
		tmp = Float64(Float64(0.25 * a) * Float64(Float64(x_45_scale_m * sqrt(2.0)) * sqrt(8.0)));
	else
		tmp = Float64(0.25 * Float64(Float64(Float64(b * x_45_scale_m) * Float64(Float64(y_45_scale_m * sqrt(2.0)) * sqrt(8.0))) / abs(x_45_scale_m)));
	end
	return tmp
end

MATLAB

y-scale_m = abs(y_45_scale);
x-scale_m = abs(x_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0;
	if (b <= 3.25e-105)
		tmp = (0.25 * a) * ((x_45_scale_m * sqrt(2.0)) * sqrt(8.0));
	else
		tmp = 0.25 * (((b * x_45_scale_m) * ((y_45_scale_m * sqrt(2.0)) * sqrt(8.0))) / abs(x_45_scale_m));
	end
	tmp_2 = tmp;
end

Wolfram

y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[b, 3.25e-105], N[(N[(0.25 * a), $MachinePrecision] * N[(N[(x$45$scale$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(N[(b * x$45$scale$95$m), $MachinePrecision] * N[(N[(y$45$scale$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[x$45$scale$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.25000000000000003e-105

   1. Initial program 1.5%

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

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. distribute-lft-out N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 23.1%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 12.9%

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

   9. Taylor expanded in angle around 0

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

   11. Applied rewrites 17.1%

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

   if 3.25000000000000003e-105 < b

   1. Initial program 1.7%

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

   3. Taylor expanded in y-scale around inf

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

   5. Applied rewrites 15.5%

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

   7. Applied rewrites 30.9%

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

   8. Taylor expanded in angle around 0

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

   10. Applied rewrites 31.9%

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

Alternative 6: 24.6% accurate, 46.1× speedup

Math

\[\begin{array}{l} y-scale_m = \left|y-scale\right| \\ x-scale_m = \left|x-scale\right| \\ \begin{array}{l} \mathbf{if}\;x-scale\_m \leq 2.05 \cdot 10^{+40}:\\ \;\;\;\;b \cdot y-scale\_m\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot \left(\left(\left(\sqrt{8} \cdot y-scale\_m\right) \cdot x-scale\_m\right) \cdot a\right)\right) \cdot \frac{\sqrt{2}}{y-scale\_m}\\ \end{array} \end{array} \]

FPCore

y-scale_m = (fabs.f64 y-scale)
x-scale_m = (fabs.f64 x-scale)
(FPCore (a b angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= x-scale_m 2.05e+40)
   (* b y-scale_m)
   (*
    (* 0.25 (* (* (* (sqrt 8.0) y-scale_m) x-scale_m) a))
    (/ (sqrt 2.0) y-scale_m))))

C

y-scale_m = fabs(y_45_scale);
x-scale_m = fabs(x_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (x_45_scale_m <= 2.05e+40) {
		tmp = b * y_45_scale_m;
	} else {
		tmp = (0.25 * (((sqrt(8.0) * y_45_scale_m) * x_45_scale_m) * a)) * (sqrt(2.0) / y_45_scale_m);
	}
	return tmp;
}

Fortran

y-scale_m =     private
x-scale_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, angle, x_45scale_m, y_45scale_m)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale_m
    real(8) :: tmp
    if (x_45scale_m <= 2.05d+40) then
        tmp = b * y_45scale_m
    else
        tmp = (0.25d0 * (((sqrt(8.0d0) * y_45scale_m) * x_45scale_m) * a)) * (sqrt(2.0d0) / y_45scale_m)
    end if
    code = tmp
end function

Java

y-scale_m = Math.abs(y_45_scale);
x-scale_m = Math.abs(x_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (x_45_scale_m <= 2.05e+40) {
		tmp = b * y_45_scale_m;
	} else {
		tmp = (0.25 * (((Math.sqrt(8.0) * y_45_scale_m) * x_45_scale_m) * a)) * (Math.sqrt(2.0) / y_45_scale_m);
	}
	return tmp;
}

Python

y-scale_m = math.fabs(y_45_scale)
x-scale_m = math.fabs(x_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if x_45_scale_m <= 2.05e+40:
		tmp = b * y_45_scale_m
	else:
		tmp = (0.25 * (((math.sqrt(8.0) * y_45_scale_m) * x_45_scale_m) * a)) * (math.sqrt(2.0) / y_45_scale_m)
	return tmp

Julia

y-scale_m = abs(y_45_scale)
x-scale_m = abs(x_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (x_45_scale_m <= 2.05e+40)
		tmp = Float64(b * y_45_scale_m);
	else
		tmp = Float64(Float64(0.25 * Float64(Float64(Float64(sqrt(8.0) * y_45_scale_m) * x_45_scale_m) * a)) * Float64(sqrt(2.0) / y_45_scale_m));
	end
	return tmp
end

MATLAB

y-scale_m = abs(y_45_scale);
x-scale_m = abs(x_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0;
	if (x_45_scale_m <= 2.05e+40)
		tmp = b * y_45_scale_m;
	else
		tmp = (0.25 * (((sqrt(8.0) * y_45_scale_m) * x_45_scale_m) * a)) * (sqrt(2.0) / y_45_scale_m);
	end
	tmp_2 = tmp;
end

Wolfram

y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[x$45$scale$95$m, 2.05e+40], N[(b * y$45$scale$95$m), $MachinePrecision], N[(N[(0.25 * N[(N[(N[(N[Sqrt[8.0], $MachinePrecision] * y$45$scale$95$m), $MachinePrecision] * x$45$scale$95$m), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 2.0500000000000001e40

   1. Initial program 0.9%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   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 20.6

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

   5. Applied rewrites 20.6%

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

   7. Applied rewrites 20.8%

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

   8. Taylor expanded in b around 0

      \[\leadsto b \cdot \color{blue}{y-scale} \]
   9. Step-by-step derivation

   10. Applied rewrites 20.8%

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

   if 2.0500000000000001e40 < x-scale

   1. Initial program 3.5%

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

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. distribute-lft-out N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 19.9%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 22.3%

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

   9. Taylor expanded in angle around 0

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

   11. Applied rewrites 19.9%

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

Alternative 7: 23.4% accurate, 61.9× speedup

Math

\[\begin{array}{l} y-scale_m = \left|y-scale\right| \\ x-scale_m = \left|x-scale\right| \\ \begin{array}{l} \mathbf{if}\;y-scale\_m \leq 58:\\ \;\;\;\;\left(0.25 \cdot a\right) \cdot \left(\left(x-scale\_m \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot y-scale\_m\\ \end{array} \end{array} \]

FPCore

y-scale_m = (fabs.f64 y-scale)
x-scale_m = (fabs.f64 x-scale)
(FPCore (a b angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= y-scale_m 58.0)
   (* (* 0.25 a) (* (* x-scale_m (sqrt 2.0)) (sqrt 8.0)))
   (* b y-scale_m)))

C

y-scale_m = fabs(y_45_scale);
x-scale_m = fabs(x_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (y_45_scale_m <= 58.0) {
		tmp = (0.25 * a) * ((x_45_scale_m * sqrt(2.0)) * sqrt(8.0));
	} else {
		tmp = b * y_45_scale_m;
	}
	return tmp;
}

Fortran

y-scale_m =     private
x-scale_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, angle, x_45scale_m, y_45scale_m)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale_m
    real(8) :: tmp
    if (y_45scale_m <= 58.0d0) then
        tmp = (0.25d0 * a) * ((x_45scale_m * sqrt(2.0d0)) * sqrt(8.0d0))
    else
        tmp = b * y_45scale_m
    end if
    code = tmp
end function

Java

y-scale_m = Math.abs(y_45_scale);
x-scale_m = Math.abs(x_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (y_45_scale_m <= 58.0) {
		tmp = (0.25 * a) * ((x_45_scale_m * Math.sqrt(2.0)) * Math.sqrt(8.0));
	} else {
		tmp = b * y_45_scale_m;
	}
	return tmp;
}

Python

y-scale_m = math.fabs(y_45_scale)
x-scale_m = math.fabs(x_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if y_45_scale_m <= 58.0:
		tmp = (0.25 * a) * ((x_45_scale_m * math.sqrt(2.0)) * math.sqrt(8.0))
	else:
		tmp = b * y_45_scale_m
	return tmp

Julia

y-scale_m = abs(y_45_scale)
x-scale_m = abs(x_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (y_45_scale_m <= 58.0)
		tmp = Float64(Float64(0.25 * a) * Float64(Float64(x_45_scale_m * sqrt(2.0)) * sqrt(8.0)));
	else
		tmp = Float64(b * y_45_scale_m);
	end
	return tmp
end

MATLAB

y-scale_m = abs(y_45_scale);
x-scale_m = abs(x_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0;
	if (y_45_scale_m <= 58.0)
		tmp = (0.25 * a) * ((x_45_scale_m * sqrt(2.0)) * sqrt(8.0));
	else
		tmp = b * y_45_scale_m;
	end
	tmp_2 = tmp;
end

Wolfram

y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[y$45$scale$95$m, 58.0], N[(N[(0.25 * a), $MachinePrecision] * N[(N[(x$45$scale$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * y$45$scale$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 58

   1. Initial program 1.5%

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

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. distribute-lft-out N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 11.7%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 10.5%

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

   9. Taylor expanded in angle around 0

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

   11. Applied rewrites 18.2%

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

   if 58 < y-scale

   1. Initial program 1.8%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   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 29.7

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

   5. Applied rewrites 29.7%

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

   7. Applied rewrites 29.9%

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

   8. Taylor expanded in b around 0

      \[\leadsto b \cdot \color{blue}{y-scale} \]
   9. Step-by-step derivation

   10. Applied rewrites 29.9%

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

Alternative 8: 17.9% accurate, 484.7× speedup

Math

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

FPCore

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

C

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

Fortran

y-scale_m =     private
x-scale_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, angle, x_45scale_m, y_45scale_m)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale_m
    code = b * y_45scale_m
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 1.6%

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

3. Taylor expanded in angle around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-sqrt.f64 N/A

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

   8. lower-sqrt.f64 18.2

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

5. Applied rewrites 18.2%

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

7. Applied rewrites 18.3%

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

8. Taylor expanded in b around 0

   \[\leadsto b \cdot \color{blue}{y-scale} \]
9. Step-by-step derivation

10. Applied rewrites 18.3%

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

Reproduce

herbie shell --seed 2024360 
(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))))