raw-angle from scale-rotated-ellipse

Percentage Accurate: 13.3% → 49.0%

Time: 28.0s

Alternatives: 8

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: 13.3% 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: 49.0% accurate, 3.3× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot angle\\ t_1 := \frac{\mathsf{PI}\left(\right)}{2}\\ t_2 := t\_0 \cdot -0.005555555555555556\\ t_3 := \sqrt{\mathsf{PI}\left(\right)}\\ \mathbf{if}\;b\_m \leq 1.02 \cdot 10^{-41}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{\left(\left(-2 \cdot \frac{b\_m}{x-scale \cdot x-scale}\right) \cdot \frac{y-scale}{b\_m - a\_m}\right) \cdot x-scale}{t\_0} \cdot 90\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{\frac{\left(-2 \cdot y-scale\right) \cdot {\left(\mathsf{fma}\left(\sin t\_2, \cos t\_1, \cos t\_2 \cdot \sin t\_1\right)\right)}^{2}}{x-scale}}{\sin \left(t\_0 \cdot 0.005555555555555556\right) \cdot \cos \left(-0.005555555555555556 \cdot \left(\left(t\_3 \cdot t\_3\right) \cdot angle\right)\right)} \cdot 0.5\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (PI) angle))
        (t_1 (/ (PI) 2.0))
        (t_2 (* t_0 -0.005555555555555556))
        (t_3 (sqrt (PI))))
   (if (<= b_m 1.02e-41)
     (*
      180.0
      (/
       (atan
        (*
         (/
          (*
           (* (* -2.0 (/ b_m (* x-scale x-scale))) (/ y-scale (- b_m a_m)))
           x-scale)
          t_0)
         90.0))
       (PI)))
     (*
      180.0
      (/
       (atan
        (*
         (/
          (/
           (*
            (* -2.0 y-scale)
            (pow (fma (sin t_2) (cos t_1) (* (cos t_2) (sin t_1))) 2.0))
           x-scale)
          (*
           (sin (* t_0 0.005555555555555556))
           (cos (* -0.005555555555555556 (* (* t_3 t_3) angle)))))
         0.5))
       (PI))))))

Derivation

1. Split input into 2 regimes

2. if b < 1.02e-41

   1. Initial program 7.7%

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 8.4%

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

   7. Applied rewrites 23.2%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 39.9%

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

   if 1.02e-41 < b

   1. Initial program 11.8%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 33.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{\left(x-scale \cdot y-scale\right) \cdot \left(\left(\frac{{\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}}{y-scale \cdot y-scale} - \sqrt{\mathsf{fma}\left({\cos \left(-0.005555555555555556 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)}^{2} \cdot \frac{{\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}, 4, {\left(\frac{{\cos \left(-0.005555555555555556 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)}^{2}}{x-scale \cdot x-scale} - \frac{{\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}}{y-scale \cdot y-scale}\right)}^{2}\right)}\right) - \frac{{\cos \left(-0.005555555555555556 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)}^{2}}{x-scale \cdot x-scale}\right)}{\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(-0.005555555555555556 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)} \cdot 0.5\right)}}{\mathsf{PI}\left(\right)} \]

   6. Taylor expanded in x-scale around 0

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

   8. Applied rewrites 63.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\frac{\left(-2 \cdot y-scale\right) \cdot {\cos \left(-0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(-0.005555555555555556 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)} \cdot 0.5\right)}{\mathsf{PI}\left(\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 63.4%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\frac{\left(-2 \cdot y-scale\right) \cdot {\left(\mathsf{fma}\left(\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot -0.005555555555555556\right), \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right), \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot -0.005555555555555556\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}^{2}}{x-scale}}{\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(-0.005555555555555556 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)} \cdot 0.5\right)}{\mathsf{PI}\left(\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 61.7%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\frac{\left(-2 \cdot y-scale\right) \cdot {\left(\mathsf{fma}\left(\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot -0.005555555555555556\right), \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right), \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot -0.005555555555555556\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}^{2}}{x-scale}}{\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(-0.005555555555555556 \cdot \left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot angle\right)\right)} \cdot 0.5\right)}{\mathsf{PI}\left(\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 48.6% accurate, 5.3× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot angle\\ \mathbf{if}\;b\_m \leq 4 \cdot 10^{-42}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{\left(\left(-2 \cdot \frac{b\_m}{x-scale \cdot x-scale}\right) \cdot \frac{y-scale}{b\_m - a\_m}\right) \cdot x-scale}{t\_0} \cdot 90\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{\frac{\left(-2 \cdot y-scale\right) \cdot {\cos \left(-0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{\sin \left(t\_0 \cdot 0.005555555555555556\right) \cdot \cos \left(-0.005555555555555556 \cdot t\_0\right)} \cdot 0.5\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (PI) angle)))
   (if (<= b_m 4e-42)
     (*
      180.0
      (/
       (atan
        (*
         (/
          (*
           (* (* -2.0 (/ b_m (* x-scale x-scale))) (/ y-scale (- b_m a_m)))
           x-scale)
          t_0)
         90.0))
       (PI)))
     (*
      180.0
      (/
       (atan
        (*
         (/
          (/
           (*
            (* -2.0 y-scale)
            (pow (cos (* -0.005555555555555556 (* angle (PI)))) 2.0))
           x-scale)
          (*
           (sin (* t_0 0.005555555555555556))
           (cos (* -0.005555555555555556 t_0))))
         0.5))
       (PI))))))

Derivation

1. Split input into 2 regimes

2. if b < 4.00000000000000015e-42

   1. Initial program 7.7%

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 8.4%

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

   7. Applied rewrites 23.2%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 39.9%

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

   if 4.00000000000000015e-42 < b

   1. Initial program 11.8%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 33.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{\left(x-scale \cdot y-scale\right) \cdot \left(\left(\frac{{\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}}{y-scale \cdot y-scale} - \sqrt{\mathsf{fma}\left({\cos \left(-0.005555555555555556 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)}^{2} \cdot \frac{{\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}, 4, {\left(\frac{{\cos \left(-0.005555555555555556 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)}^{2}}{x-scale \cdot x-scale} - \frac{{\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}}{y-scale \cdot y-scale}\right)}^{2}\right)}\right) - \frac{{\cos \left(-0.005555555555555556 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)}^{2}}{x-scale \cdot x-scale}\right)}{\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(-0.005555555555555556 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)} \cdot 0.5\right)}}{\mathsf{PI}\left(\right)} \]

   6. Taylor expanded in x-scale around 0

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

   8. Applied rewrites 63.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\frac{\left(-2 \cdot y-scale\right) \cdot {\cos \left(-0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(-0.005555555555555556 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)} \cdot 0.5\right)}{\mathsf{PI}\left(\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 48.4% accurate, 8.4× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot angle\\ \mathbf{if}\;b\_m \leq 4 \cdot 10^{-42}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{\left(\left(-2 \cdot \frac{b\_m}{x-scale \cdot x-scale}\right) \cdot \frac{y-scale}{b\_m - a\_m}\right) \cdot x-scale}{t\_0} \cdot 90\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{-2 \cdot \frac{y-scale}{x-scale}}{\sin \left(t\_0 \cdot 0.005555555555555556\right) \cdot \cos \left(-0.005555555555555556 \cdot t\_0\right)} \cdot 0.5\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (PI) angle)))
   (if (<= b_m 4e-42)
     (*
      180.0
      (/
       (atan
        (*
         (/
          (*
           (* (* -2.0 (/ b_m (* x-scale x-scale))) (/ y-scale (- b_m a_m)))
           x-scale)
          t_0)
         90.0))
       (PI)))
     (*
      180.0
      (/
       (atan
        (*
         (/
          (* -2.0 (/ y-scale x-scale))
          (*
           (sin (* t_0 0.005555555555555556))
           (cos (* -0.005555555555555556 t_0))))
         0.5))
       (PI))))))

Derivation

1. Split input into 2 regimes

2. if b < 4.00000000000000015e-42

   1. Initial program 7.7%

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 8.4%

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

   7. Applied rewrites 23.2%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 39.9%

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

   if 4.00000000000000015e-42 < b

   1. Initial program 11.8%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 33.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{\left(x-scale \cdot y-scale\right) \cdot \left(\left(\frac{{\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}}{y-scale \cdot y-scale} - \sqrt{\mathsf{fma}\left({\cos \left(-0.005555555555555556 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)}^{2} \cdot \frac{{\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}, 4, {\left(\frac{{\cos \left(-0.005555555555555556 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)}^{2}}{x-scale \cdot x-scale} - \frac{{\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}}{y-scale \cdot y-scale}\right)}^{2}\right)}\right) - \frac{{\cos \left(-0.005555555555555556 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)}^{2}}{x-scale \cdot x-scale}\right)}{\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(-0.005555555555555556 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)} \cdot 0.5\right)}}{\mathsf{PI}\left(\right)} \]

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 62.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-2 \cdot \frac{y-scale}{x-scale}}{\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(-0.005555555555555556 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)} \cdot 0.5\right)}{\mathsf{PI}\left(\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 47.0% accurate, 16.8× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 3.9 \cdot 10^{+155}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{\left(\left(-2 \cdot \frac{b\_m}{x-scale \cdot x-scale}\right) \cdot \frac{y-scale}{b\_m - a\_m}\right) \cdot x-scale}{\mathsf{PI}\left(\right) \cdot angle} \cdot 90\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle x-scale y-scale)
 :precision binary64
 (if (<= b_m 3.9e+155)
   (*
    180.0
    (/
     (atan
      (*
       (/
        (*
         (* (* -2.0 (/ b_m (* x-scale x-scale))) (/ y-scale (- b_m a_m)))
         x-scale)
        (* (PI) angle))
       90.0))
     (PI)))
   (*
    180.0
    (/ (atan (* -180.0 (/ y-scale (* angle (* x-scale (PI)))))) (PI)))))

Derivation

1. Split input into 2 regimes

2. if b < 3.8999999999999998e155

   1. Initial program 10.8%

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 11.8%

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

   7. Applied rewrites 25.3%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 41.0%

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

   if 3.8999999999999998e155 < b

   1. Initial program 0.0%

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

   3. Taylor expanded in 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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 0.0%

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

   7. Applied rewrites 38.2%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 50.8%

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

Alternative 5: 41.0% accurate, 18.7× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;a\_m \leq 5.5 \cdot 10^{-91}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{\left(-2 \cdot \frac{y-scale}{x-scale \cdot x-scale}\right) \cdot x-scale}{\mathsf{PI}\left(\right) \cdot angle} \cdot 90\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle x-scale y-scale)
 :precision binary64
 (if (<= a_m 5.5e-91)
   (* 180.0 (/ (atan (* -180.0 (/ y-scale (* angle (* x-scale (PI)))))) (PI)))
   (*
    180.0
    (/
     (atan
      (*
       (/ (* (* -2.0 (/ y-scale (* x-scale x-scale))) x-scale) (* (PI) angle))
       90.0))
     (PI)))))

Derivation

1. Split input into 2 regimes

2. if a < 5.49999999999999965e-91

   1. Initial program 11.3%

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 11.1%

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

   7. Applied rewrites 32.3%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 44.4%

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

   if 5.49999999999999965e-91 < a

   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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 6.8%

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

   7. Applied rewrites 17.4%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 32.2%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-2 \cdot \frac{y-scale}{x-scale \cdot x-scale}\right) \cdot x-scale}{\mathsf{PI}\left(\right) \cdot angle} \cdot 90\right)}{\mathsf{PI}\left(\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 39.1% accurate, 20.6× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ 180 \cdot \frac{\tan^{-1} \left(\frac{-2 \cdot \frac{y-scale}{x-scale}}{\mathsf{PI}\left(\right) \cdot angle} \cdot 90\right)}{\mathsf{PI}\left(\right)} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle x-scale y-scale)
 :precision binary64
 (*
  180.0
  (/ (atan (* (/ (* -2.0 (/ y-scale x-scale)) (* (PI) angle)) 90.0)) (PI))))

Derivation

1. Initial program 9.0%

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

3. Taylor expanded in 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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 9.8%

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

7. Applied rewrites 27.5%

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

8. Taylor expanded in a around 0

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

10. Applied rewrites 42.0%

    \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-2 \cdot \frac{y-scale}{x-scale}}{\mathsf{PI}\left(\right) \cdot angle} \cdot 90\right)}{\mathsf{PI}\left(\right)} \]
11. Add Preprocessing

Alternative 7: 37.3% accurate, 22.2× speedup

Math

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

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m 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 9.0%

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

3. Taylor expanded in 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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 9.8%

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

7. Applied rewrites 27.5%

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

8. Taylor expanded in a around 0

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

10. Applied rewrites 39.0%

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

Alternative 8: 12.2% accurate, 22.2× speedup

Math

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

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m 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 9.0%

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

3. Taylor expanded in 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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 9.8%

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

7. Applied rewrites 27.5%

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

8. Taylor expanded in a around inf

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

10. Applied rewrites 13.0%

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

Reproduce

herbie shell --seed 2024360 
(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))))