raw-angle from scale-rotated-ellipse

Percentage Accurate: 13.7% → 56.0%

Time: 33.9s

Alternatives: 7

Speedup: 22.2×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \mathsf{PI}\left(\right)\\ t_1 := \cos t\_0\\ t_2 := \sin t\_0\\ t_3 := \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot t\_2\right) \cdot t\_1}{x-scale}}{y-scale}\\ t_4 := \frac{\frac{{\left(a \cdot t\_1\right)}^{2} + {\left(b \cdot t\_2\right)}^{2}}{y-scale}}{y-scale}\\ t_5 := \frac{\frac{{\left(a \cdot t\_2\right)}^{2} + {\left(b \cdot t\_1\right)}^{2}}{x-scale}}{x-scale}\\ 180 \cdot \frac{\tan^{-1} \left(\frac{\left(t\_4 - t\_5\right) - \sqrt{{\left(t\_5 - t\_4\right)}^{2} + {t\_3}^{2}}}{t\_3}\right)}{\mathsf{PI}\left(\right)} \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (/ angle 180.0) (PI)))
        (t_1 (cos t_0))
        (t_2 (sin t_0))
        (t_3
         (/
          (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) t_2) t_1) x-scale)
          y-scale))
        (t_4
         (/ (/ (+ (pow (* a t_1) 2.0) (pow (* b t_2) 2.0)) y-scale) y-scale))
        (t_5
         (/ (/ (+ (pow (* a t_2) 2.0) (pow (* b t_1) 2.0)) x-scale) x-scale)))
   (*
    180.0
    (/
     (atan
      (/ (- (- t_4 t_5) (sqrt (+ (pow (- t_5 t_4) 2.0) (pow t_3 2.0)))) t_3))
     (PI)))))

Initial Program: 13.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \mathsf{PI}\left(\right)\\ t_1 := \cos t\_0\\ t_2 := \sin t\_0\\ t_3 := \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot t\_2\right) \cdot t\_1}{x-scale}}{y-scale}\\ t_4 := \frac{\frac{{\left(a \cdot t\_1\right)}^{2} + {\left(b \cdot t\_2\right)}^{2}}{y-scale}}{y-scale}\\ t_5 := \frac{\frac{{\left(a \cdot t\_2\right)}^{2} + {\left(b \cdot t\_1\right)}^{2}}{x-scale}}{x-scale}\\ 180 \cdot \frac{\tan^{-1} \left(\frac{\left(t\_4 - t\_5\right) - \sqrt{{\left(t\_5 - t\_4\right)}^{2} + {t\_3}^{2}}}{t\_3}\right)}{\mathsf{PI}\left(\right)} \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (/ angle 180.0) (PI)))
        (t_1 (cos t_0))
        (t_2 (sin t_0))
        (t_3
         (/
          (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) t_2) t_1) x-scale)
          y-scale))
        (t_4
         (/ (/ (+ (pow (* a t_1) 2.0) (pow (* b t_2) 2.0)) y-scale) y-scale))
        (t_5
         (/ (/ (+ (pow (* a t_2) 2.0) (pow (* b t_1) 2.0)) x-scale) x-scale)))
   (*
    180.0
    (/
     (atan
      (/ (- (- t_4 t_5) (sqrt (+ (pow (- t_5 t_4) 2.0) (pow t_3 2.0)))) t_3))
     (PI)))))

Alternative 1: 56.0% accurate, 12.4× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\\ \mathbf{if}\;b\_m \leq 6.4 \cdot 10^{-52}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \tan t\_0\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{\left(-y-scale\right) \cdot 1}{x-scale \cdot \sin t\_0}\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle (PI)))))
   (if (<= b_m 6.4e-52)
     (* 180.0 (/ (atan (* (/ y-scale x-scale) (tan t_0))) (PI)))
     (* 180.0 (/ (atan (/ (* (- y-scale) 1.0) (* x-scale (sin t_0)))) (PI))))))

Derivation

1. Split input into 2 regimes

2. if b < 6.4000000000000002e-52

   1. Initial program 11.1%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale}}\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x-scale around 0

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

   5. Applied rewrites 32.3%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(2, {\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2}, 2 \cdot {\left(b \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2}\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

   6. Taylor expanded in a around inf

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. quot-tan N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \tan \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-tan.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \tan \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{\mathsf{PI}\left(\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \tan \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{\mathsf{PI}\left(\right)} \]

      7. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \tan \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{\mathsf{PI}\left(\right)} \]

      8. lift-*.f64 52.9

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{\mathsf{PI}\left(\right)} \]

   8. Applied rewrites 52.9%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \color{blue}{\tan \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

   if 6.4000000000000002e-52 < b

   1. Initial program 15.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale}}\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x-scale around 0

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

   5. Applied rewrites 28.8%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(2, {\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2}, 2 \cdot {\left(b \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2}\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

   6. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

   8. Applied rewrites 63.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \color{blue}{\left(\frac{y-scale}{x-scale} \cdot \frac{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}\right)}{\mathsf{PI}\left(\right)} \]

   9. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{1}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 64.2%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{1}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{1}{\color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

       2. lift-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{1}{\sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

       3. lift-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{1}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

       4. lift-sin.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{1}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

       5. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{1}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

       6. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{1}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

       7. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{1}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

       8. frac-times N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \frac{y-scale \cdot 1}{x-scale \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

       9. lower-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \frac{y-scale \cdot 1}{x-scale \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

       10. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \frac{y-scale \cdot 1}{x-scale \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

       11. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \frac{y-scale \cdot 1}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

       12. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \frac{y-scale \cdot 1}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

       13. lift-PI.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \frac{y-scale \cdot 1}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

       14. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \frac{y-scale \cdot 1}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

   13. Applied rewrites 65.4%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \frac{y-scale \cdot 1}{x-scale \cdot \color{blue}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.4 \cdot 10^{-52}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{\left(-y-scale\right) \cdot 1}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 53.6% accurate, 12.8× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 6.4 \cdot 10^{-52}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;b\_m \leq 9.5 \cdot 10^{+89}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{-y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, -0.002777777777777778 \cdot \mathsf{PI}\left(\right) - -0.000925925925925926 \cdot \mathsf{PI}\left(\right), 180 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{angle}\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (if (<= b_m 6.4e-52)
   (*
    180.0
    (/
     (atan
      (* (/ y-scale x-scale) (tan (* 0.005555555555555556 (* angle (PI))))))
     (PI)))
   (if (<= b_m 9.5e+89)
     (*
      180.0
      (/
       (atan
        (*
         (/ (- y-scale) x-scale)
         (/
          (fma
           (* angle angle)
           (- (* -0.002777777777777778 (PI)) (* -0.000925925925925926 (PI)))
           (* 180.0 (/ 1.0 (PI))))
          angle)))
       (PI)))
     (*
      180.0
      (/ (atan (* -180.0 (/ y-scale (* angle (* x-scale (PI)))))) (PI))))))

Derivation

1. Split input into 3 regimes

2. if b < 6.4000000000000002e-52

   1. Initial program 11.1%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale}}\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x-scale around 0

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

   5. Applied rewrites 32.3%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(2, {\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2}, 2 \cdot {\left(b \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2}\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

   6. Taylor expanded in a around inf

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. quot-tan N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \tan \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-tan.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \tan \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{\mathsf{PI}\left(\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \tan \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{\mathsf{PI}\left(\right)} \]

      7. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \tan \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{\mathsf{PI}\left(\right)} \]

      8. lift-*.f64 52.9

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{\mathsf{PI}\left(\right)} \]

   8. Applied rewrites 52.9%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \color{blue}{\tan \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

   if 6.4000000000000002e-52 < b < 9.5000000000000003e89

   1. Initial program 35.0%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale}}\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x-scale around 0

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

   5. Applied rewrites 62.8%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(2, {\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2}, 2 \cdot {\left(b \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2}\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

   6. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

   8. Applied rewrites 62.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \color{blue}{\left(\frac{y-scale}{x-scale} \cdot \frac{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}\right)}{\mathsf{PI}\left(\right)} \]

   9. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{{angle}^{2} \cdot \left(\frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right)\right) + 180 \cdot \frac{1}{\mathsf{PI}\left(\right)}}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{{angle}^{2} \cdot \left(\frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right)\right) + 180 \cdot \frac{1}{\mathsf{PI}\left(\right)}}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]

       2. lower-fma.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left({angle}^{2}, \frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right), 180 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]

       3. unpow2 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, \frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right), 180 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, \frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right), 180 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]

       5. lower--.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, \frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right), 180 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]

       6. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, \frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right), 180 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]

       7. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, \frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right), 180 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]

       8. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, \frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right), 180 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]

       9. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, \frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right), 180 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]

       10. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, \frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right), 180 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]

       11. inv-pow N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, \frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right), 180 \cdot {\mathsf{PI}\left(\right)}^{-1}\right)}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]

       12. lower-pow.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, \frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right), 180 \cdot {\mathsf{PI}\left(\right)}^{-1}\right)}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]

       13. lift-PI.f64 60.3

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, -0.002777777777777778 \cdot \mathsf{PI}\left(\right) - -0.000925925925925926 \cdot \mathsf{PI}\left(\right), 180 \cdot {\mathsf{PI}\left(\right)}^{-1}\right)}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]

   11. Applied rewrites 60.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, -0.002777777777777778 \cdot \mathsf{PI}\left(\right) - -0.000925925925925926 \cdot \mathsf{PI}\left(\right), 180 \cdot {\mathsf{PI}\left(\right)}^{-1}\right)}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]
   12. Step-by-step derivation

       1. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, \frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right), 180 \cdot {\mathsf{PI}\left(\right)}^{-1}\right)}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]

       2. lift-pow.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, \frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right), 180 \cdot {\mathsf{PI}\left(\right)}^{-1}\right)}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]

       3. inv-pow N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, \frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right), 180 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]

       4. lower-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, \frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right), 180 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]

       5. lift-PI.f64 60.3

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, -0.002777777777777778 \cdot \mathsf{PI}\left(\right) - -0.000925925925925926 \cdot \mathsf{PI}\left(\right), 180 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]

   13. Applied rewrites 60.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, -0.002777777777777778 \cdot \mathsf{PI}\left(\right) - -0.000925925925925926 \cdot \mathsf{PI}\left(\right), 180 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]

   if 9.5000000000000003e89 < b

   1. Initial program 4.1%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale}}\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 4.2%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \left(\frac{x-scale}{angle} \cdot \frac{y-scale \cdot \left(2 \cdot \frac{a \cdot a}{y-scale \cdot y-scale} - 2 \cdot \frac{b \cdot b}{x-scale \cdot x-scale}\right)}{\mathsf{PI}\left(\right) \cdot \left(b \cdot b - a \cdot a\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

   6. Taylor expanded in a around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \color{blue}{\frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{\color{blue}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \color{blue}{\left(x-scale \cdot \mathsf{PI}\left(\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)}\right)}{\mathsf{PI}\left(\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      5. lift-PI.f64 59.2

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

   8. Applied rewrites 59.2%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \color{blue}{\frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.4 \cdot 10^{-52}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{+89}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{-y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, -0.002777777777777778 \cdot \mathsf{PI}\left(\right) - -0.000925925925925926 \cdot \mathsf{PI}\left(\right), 180 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{angle}\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 47.9% accurate, 16.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := -0.002777777777777778 \cdot \mathsf{PI}\left(\right) - -0.000925925925925926 \cdot \mathsf{PI}\left(\right)\\ t_1 := \frac{-y-scale}{x-scale}\\ \mathbf{if}\;b\_m \leq 6.2 \cdot 10^{-52}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(t\_1 \cdot \left(angle \cdot t\_0\right)\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;b\_m \leq 9.5 \cdot 10^{+89}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(t\_1 \cdot \frac{\mathsf{fma}\left(angle \cdot angle, t\_0, 180 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{angle}\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (- (* -0.002777777777777778 (PI)) (* -0.000925925925925926 (PI))))
        (t_1 (/ (- y-scale) x-scale)))
   (if (<= b_m 6.2e-52)
     (* 180.0 (/ (atan (* t_1 (* angle t_0))) (PI)))
     (if (<= b_m 9.5e+89)
       (*
        180.0
        (/
         (atan
          (* t_1 (/ (fma (* angle angle) t_0 (* 180.0 (/ 1.0 (PI)))) angle)))
         (PI)))
       (*
        180.0
        (/ (atan (* -180.0 (/ y-scale (* angle (* x-scale (PI)))))) (PI)))))))

Derivation

1. Split input into 3 regimes

2. if b < 6.1999999999999998e-52

   1. Initial program 11.1%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale}}\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x-scale around 0

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

   5. Applied rewrites 32.3%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(2, {\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2}, 2 \cdot {\left(b \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2}\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

   6. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

   8. Applied rewrites 35.9%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \color{blue}{\left(\frac{y-scale}{x-scale} \cdot \frac{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}\right)}{\mathsf{PI}\left(\right)} \]

   9. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{{angle}^{2} \cdot \left(\frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right)\right) + 180 \cdot \frac{1}{\mathsf{PI}\left(\right)}}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{{angle}^{2} \cdot \left(\frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right)\right) + 180 \cdot \frac{1}{\mathsf{PI}\left(\right)}}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]

       2. lower-fma.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left({angle}^{2}, \frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right), 180 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]

       3. unpow2 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, \frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right), 180 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, \frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right), 180 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]

       5. lower--.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, \frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right), 180 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]

       6. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, \frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right), 180 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]

       7. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, \frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right), 180 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]

       8. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, \frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right), 180 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]

       9. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, \frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right), 180 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]

       10. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, \frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right), 180 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]

       11. inv-pow N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, \frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right), 180 \cdot {\mathsf{PI}\left(\right)}^{-1}\right)}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]

       12. lower-pow.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, \frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right), 180 \cdot {\mathsf{PI}\left(\right)}^{-1}\right)}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]

       13. lift-PI.f64 34.8

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, -0.002777777777777778 \cdot \mathsf{PI}\left(\right) - -0.000925925925925926 \cdot \mathsf{PI}\left(\right), 180 \cdot {\mathsf{PI}\left(\right)}^{-1}\right)}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]

   11. Applied rewrites 34.8%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, -0.002777777777777778 \cdot \mathsf{PI}\left(\right) - -0.000925925925925926 \cdot \mathsf{PI}\left(\right), 180 \cdot {\mathsf{PI}\left(\right)}^{-1}\right)}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]

   12. Taylor expanded in angle around inf

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(angle \cdot \left(\frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right)\right)}{\mathsf{PI}\left(\right)} \]
   13. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(angle \cdot \left(\frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}{\mathsf{PI}\left(\right)} \]

       2. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(angle \cdot \left(\frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}{\mathsf{PI}\left(\right)} \]

       3. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(angle \cdot \left(\frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}{\mathsf{PI}\left(\right)} \]

       4. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(angle \cdot \left(\frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}{\mathsf{PI}\left(\right)} \]

       5. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(angle \cdot \left(\frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}{\mathsf{PI}\left(\right)} \]

       6. lift--.f64 41.4

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(angle \cdot \left(-0.002777777777777778 \cdot \mathsf{PI}\left(\right) - -0.000925925925925926 \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}{\mathsf{PI}\left(\right)} \]

   14. Applied rewrites 41.4%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(angle \cdot \left(-0.002777777777777778 \cdot \mathsf{PI}\left(\right) - -0.000925925925925926 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right)\right)}{\mathsf{PI}\left(\right)} \]

   if 6.1999999999999998e-52 < b < 9.5000000000000003e89

   1. Initial program 35.0%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale}}\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x-scale around 0

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

   5. Applied rewrites 62.8%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(2, {\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2}, 2 \cdot {\left(b \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2}\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

   6. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

   8. Applied rewrites 62.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \color{blue}{\left(\frac{y-scale}{x-scale} \cdot \frac{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}\right)}{\mathsf{PI}\left(\right)} \]

   9. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{{angle}^{2} \cdot \left(\frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right)\right) + 180 \cdot \frac{1}{\mathsf{PI}\left(\right)}}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{{angle}^{2} \cdot \left(\frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right)\right) + 180 \cdot \frac{1}{\mathsf{PI}\left(\right)}}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]

       2. lower-fma.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left({angle}^{2}, \frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right), 180 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]

       3. unpow2 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, \frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right), 180 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, \frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right), 180 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]

       5. lower--.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, \frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right), 180 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]

       6. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, \frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right), 180 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]

       7. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, \frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right), 180 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]

       8. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, \frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right), 180 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]

       9. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, \frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right), 180 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]

       10. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, \frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right), 180 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]

       11. inv-pow N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, \frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right), 180 \cdot {\mathsf{PI}\left(\right)}^{-1}\right)}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]

       12. lower-pow.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, \frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right), 180 \cdot {\mathsf{PI}\left(\right)}^{-1}\right)}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]

       13. lift-PI.f64 60.3

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, -0.002777777777777778 \cdot \mathsf{PI}\left(\right) - -0.000925925925925926 \cdot \mathsf{PI}\left(\right), 180 \cdot {\mathsf{PI}\left(\right)}^{-1}\right)}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]

   11. Applied rewrites 60.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, -0.002777777777777778 \cdot \mathsf{PI}\left(\right) - -0.000925925925925926 \cdot \mathsf{PI}\left(\right), 180 \cdot {\mathsf{PI}\left(\right)}^{-1}\right)}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]
   12. Step-by-step derivation

       1. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, \frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right), 180 \cdot {\mathsf{PI}\left(\right)}^{-1}\right)}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]

       2. lift-pow.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, \frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right), 180 \cdot {\mathsf{PI}\left(\right)}^{-1}\right)}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]

       3. inv-pow N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, \frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right), 180 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]

       4. lower-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, \frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right), 180 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]

       5. lift-PI.f64 60.3

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, -0.002777777777777778 \cdot \mathsf{PI}\left(\right) - -0.000925925925925926 \cdot \mathsf{PI}\left(\right), 180 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]

   13. Applied rewrites 60.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, -0.002777777777777778 \cdot \mathsf{PI}\left(\right) - -0.000925925925925926 \cdot \mathsf{PI}\left(\right), 180 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]

   if 9.5000000000000003e89 < b

   1. Initial program 4.1%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale}}\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 4.2%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \left(\frac{x-scale}{angle} \cdot \frac{y-scale \cdot \left(2 \cdot \frac{a \cdot a}{y-scale \cdot y-scale} - 2 \cdot \frac{b \cdot b}{x-scale \cdot x-scale}\right)}{\mathsf{PI}\left(\right) \cdot \left(b \cdot b - a \cdot a\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

   6. Taylor expanded in a around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \color{blue}{\frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{\color{blue}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \color{blue}{\left(x-scale \cdot \mathsf{PI}\left(\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)}\right)}{\mathsf{PI}\left(\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      5. lift-PI.f64 59.2

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

   8. Applied rewrites 59.2%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \color{blue}{\frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 47.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.2 \cdot 10^{-52}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{-y-scale}{x-scale} \cdot \left(angle \cdot \left(-0.002777777777777778 \cdot \mathsf{PI}\left(\right) - -0.000925925925925926 \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{+89}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{-y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, -0.002777777777777778 \cdot \mathsf{PI}\left(\right) - -0.000925925925925926 \cdot \mathsf{PI}\left(\right), 180 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{angle}\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 47.4% accurate, 20.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 7.5 \cdot 10^{+48}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{-y-scale}{x-scale} \cdot \left(angle \cdot \left(-0.002777777777777778 \cdot \mathsf{PI}\left(\right) - -0.000925925925925926 \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (if (<= b_m 7.5e+48)
   (*
    180.0
    (/
     (atan
      (*
       (/ (- y-scale) x-scale)
       (*
        angle
        (- (* -0.002777777777777778 (PI)) (* -0.000925925925925926 (PI))))))
     (PI)))
   (*
    180.0
    (/ (atan (* -180.0 (/ y-scale (* angle (* x-scale (PI)))))) (PI)))))

Derivation

1. Split input into 2 regimes

2. if b < 7.5000000000000006e48

   1. Initial program 14.0%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale}}\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x-scale around 0

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

   5. Applied rewrites 34.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(2, {\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2}, 2 \cdot {\left(b \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2}\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

   6. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

   8. Applied rewrites 37.8%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \color{blue}{\left(\frac{y-scale}{x-scale} \cdot \frac{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}\right)}{\mathsf{PI}\left(\right)} \]

   9. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{{angle}^{2} \cdot \left(\frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right)\right) + 180 \cdot \frac{1}{\mathsf{PI}\left(\right)}}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{{angle}^{2} \cdot \left(\frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right)\right) + 180 \cdot \frac{1}{\mathsf{PI}\left(\right)}}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]

       2. lower-fma.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left({angle}^{2}, \frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right), 180 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]

       3. unpow2 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, \frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right), 180 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, \frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right), 180 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]

       5. lower--.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, \frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right), 180 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]

       6. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, \frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right), 180 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]

       7. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, \frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right), 180 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]

       8. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, \frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right), 180 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]

       9. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, \frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right), 180 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]

       10. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, \frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right), 180 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]

       11. inv-pow N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, \frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right), 180 \cdot {\mathsf{PI}\left(\right)}^{-1}\right)}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]

       12. lower-pow.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, \frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right), 180 \cdot {\mathsf{PI}\left(\right)}^{-1}\right)}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]

       13. lift-PI.f64 36.8

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, -0.002777777777777778 \cdot \mathsf{PI}\left(\right) - -0.000925925925925926 \cdot \mathsf{PI}\left(\right), 180 \cdot {\mathsf{PI}\left(\right)}^{-1}\right)}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]

   11. Applied rewrites 36.8%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, -0.002777777777777778 \cdot \mathsf{PI}\left(\right) - -0.000925925925925926 \cdot \mathsf{PI}\left(\right), 180 \cdot {\mathsf{PI}\left(\right)}^{-1}\right)}{angle}\right)\right)}{\mathsf{PI}\left(\right)} \]

   12. Taylor expanded in angle around inf

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(angle \cdot \left(\frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right)\right)}{\mathsf{PI}\left(\right)} \]
   13. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(angle \cdot \left(\frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}{\mathsf{PI}\left(\right)} \]

       2. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(angle \cdot \left(\frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}{\mathsf{PI}\left(\right)} \]

       3. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(angle \cdot \left(\frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}{\mathsf{PI}\left(\right)} \]

       4. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(angle \cdot \left(\frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}{\mathsf{PI}\left(\right)} \]

       5. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(angle \cdot \left(\frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}{\mathsf{PI}\left(\right)} \]

       6. lift--.f64 40.0

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(angle \cdot \left(-0.002777777777777778 \cdot \mathsf{PI}\left(\right) - -0.000925925925925926 \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}{\mathsf{PI}\left(\right)} \]

   14. Applied rewrites 40.0%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(angle \cdot \left(-0.002777777777777778 \cdot \mathsf{PI}\left(\right) - -0.000925925925925926 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right)\right)}{\mathsf{PI}\left(\right)} \]

   if 7.5000000000000006e48 < b

   1. Initial program 7.1%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale}}\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 7.2%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \left(\frac{x-scale}{angle} \cdot \frac{y-scale \cdot \left(2 \cdot \frac{a \cdot a}{y-scale \cdot y-scale} - 2 \cdot \frac{b \cdot b}{x-scale \cdot x-scale}\right)}{\mathsf{PI}\left(\right) \cdot \left(b \cdot b - a \cdot a\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

   6. Taylor expanded in a around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \color{blue}{\frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{\color{blue}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \color{blue}{\left(x-scale \cdot \mathsf{PI}\left(\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)}\right)}{\mathsf{PI}\left(\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      5. lift-PI.f64 59.8

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

   8. Applied rewrites 59.8%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \color{blue}{\frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.5 \cdot 10^{+48}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{-y-scale}{x-scale} \cdot \left(angle \cdot \left(-0.002777777777777778 \cdot \mathsf{PI}\left(\right) - -0.000925925925925926 \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 39.5% accurate, 21.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ 180 \cdot \frac{\tan^{-1} \left(\frac{-y-scale}{x-scale} \cdot \frac{180}{angle \cdot \mathsf{PI}\left(\right)}\right)}{\mathsf{PI}\left(\right)} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (*
  180.0
  (/ (atan (* (/ (- y-scale) x-scale) (/ 180.0 (* angle (PI))))) (PI))))

Derivation

1. Initial program 12.4%

   \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale}}\right)}{\mathsf{PI}\left(\right)} \]
2. Add Preprocessing

3. Taylor expanded in x-scale around 0

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

5. Applied rewrites 31.2%

   \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(2, {\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2}, 2 \cdot {\left(b \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2}\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

6. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. times-frac N/A

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

   3. lower-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lower-/.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

8. Applied rewrites 44.4%

   \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \color{blue}{\left(\frac{y-scale}{x-scale} \cdot \frac{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}\right)}{\mathsf{PI}\left(\right)} \]

9. Taylor expanded in angle around 0

   \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{180}{angle \cdot \color{blue}{\mathsf{PI}\left(\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]
10. Step-by-step derivation

    1. lower-/.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{180}{angle \cdot \mathsf{PI}\left(\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

    2. lift-*.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{180}{angle \cdot \mathsf{PI}\left(\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

    3. lift-PI.f64 39.2

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{180}{angle \cdot \mathsf{PI}\left(\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

11. Applied rewrites 39.2%

    \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{180}{angle \cdot \color{blue}{\mathsf{PI}\left(\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

12. Final simplification 39.2%

    \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-y-scale}{x-scale} \cdot \frac{180}{angle \cdot \mathsf{PI}\left(\right)}\right)}{\mathsf{PI}\left(\right)} \]
13. Add Preprocessing

Alternative 6: 38.0% accurate, 22.2× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right)}{\mathsf{PI}\left(\right)} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (* 180.0 (/ (atan (* -180.0 (/ y-scale (* angle (* x-scale (PI)))))) (PI))))

Derivation

1. Initial program 12.4%

   \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale}}\right)}{\mathsf{PI}\left(\right)} \]
2. Add Preprocessing

3. Taylor expanded in angle around 0

   \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
4. Step-by-step derivation

5. Applied rewrites 8.6%

   \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \left(\frac{x-scale}{angle} \cdot \frac{y-scale \cdot \left(2 \cdot \frac{a \cdot a}{y-scale \cdot y-scale} - 2 \cdot \frac{b \cdot b}{x-scale \cdot x-scale}\right)}{\mathsf{PI}\left(\right) \cdot \left(b \cdot b - a \cdot a\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

6. Taylor expanded in a around 0

   \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \color{blue}{\frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{\color{blue}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

   2. lower-/.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \color{blue}{\left(x-scale \cdot \mathsf{PI}\left(\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

   3. lower-*.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)}\right)}{\mathsf{PI}\left(\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

   5. lift-PI.f64 38.1

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

8. Applied rewrites 38.1%

   \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \color{blue}{\frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]
9. Add Preprocessing

Alternative 7: 12.2% accurate, 22.2× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{x-scale}{angle \cdot \left(y-scale \cdot \mathsf{PI}\left(\right)\right)}\right)}{\mathsf{PI}\left(\right)} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (* 180.0 (/ (atan (* -180.0 (/ x-scale (* angle (* y-scale (PI)))))) (PI))))

Derivation

1. Initial program 12.4%

   \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale}}\right)}{\mathsf{PI}\left(\right)} \]
2. Add Preprocessing

3. Taylor expanded in angle around 0

   \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
4. Step-by-step derivation

5. Applied rewrites 8.6%

   \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \left(\frac{x-scale}{angle} \cdot \frac{y-scale \cdot \left(2 \cdot \frac{a \cdot a}{y-scale \cdot y-scale} - 2 \cdot \frac{b \cdot b}{x-scale \cdot x-scale}\right)}{\mathsf{PI}\left(\right) \cdot \left(b \cdot b - a \cdot a\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

6. Taylor expanded in a around inf

   \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \color{blue}{\frac{x-scale}{angle \cdot \left(y-scale \cdot \mathsf{PI}\left(\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{x-scale}{\color{blue}{angle \cdot \left(y-scale \cdot \mathsf{PI}\left(\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

   2. lower-/.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{x-scale}{angle \cdot \color{blue}{\left(y-scale \cdot \mathsf{PI}\left(\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

   3. lower-*.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{x-scale}{angle \cdot \left(y-scale \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)}\right)}{\mathsf{PI}\left(\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{x-scale}{angle \cdot \left(y-scale \cdot \mathsf{PI}\left(\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

   5. lift-PI.f64 12.0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{x-scale}{angle \cdot \left(y-scale \cdot \mathsf{PI}\left(\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

8. Applied rewrites 12.0%

   \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \color{blue}{\frac{x-scale}{angle \cdot \left(y-scale \cdot \mathsf{PI}\left(\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025047 
(FPCore (a b angle x-scale y-scale)
  :name "raw-angle from scale-rotated-ellipse"
  :precision binary64
  (* 180.0 (/ (atan (/ (- (- (/ (/ (+ (pow (* a (cos (* (/ angle 180.0) (PI)))) 2.0) (pow (* b (sin (* (/ angle 180.0) (PI)))) 2.0)) y-scale) y-scale) (/ (/ (+ (pow (* a (sin (* (/ angle 180.0) (PI)))) 2.0) (pow (* b (cos (* (/ angle 180.0) (PI)))) 2.0)) x-scale) x-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)))) (/ (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin (* (/ angle 180.0) (PI)))) (cos (* (/ angle 180.0) (PI)))) x-scale) y-scale))) (PI))))