raw-angle from scale-rotated-ellipse

Percentage Accurate: 13.8% → 57.3%

Time: 42.0s

Alternatives: 14

Speedup: 21.5×

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: 13.8% 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: 57.3% accurate, 4.6× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot angle\\ t_1 := t\_0 \cdot 0.005555555555555556\\ \mathbf{if}\;b\_m \leq 4.1 \cdot 10^{+39}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{\sin t\_1}{\cos t\_1} \cdot \frac{y-scale}{x-scale}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{PI}\left(\right)}{\tan^{-1} \left(\frac{\frac{{\cos \left(\left(\sqrt[3]{{\mathsf{PI}\left(\right)}^{3}} \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}}{x-scale} \cdot \left(-2 \cdot y-scale\right)}{\sin \left(0.011111111111111112 \cdot t\_0\right) \cdot 0.5} \cdot 0.5\right) \cdot 180}}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (PI) angle)) (t_1 (* t_0 0.005555555555555556)))
   (if (<= b_m 4.1e+39)
     (* (/ (atan (* (/ (sin t_1) (cos t_1)) (/ y-scale x-scale))) (PI)) 180.0)
     (/
      1.0
      (/
       (PI)
       (*
        (atan
         (*
          (/
           (*
            (/
             (pow
              (cos (* (* (cbrt (pow (PI) 3.0)) angle) 0.005555555555555556))
              2.0)
             x-scale)
            (* -2.0 y-scale))
           (* (sin (* 0.011111111111111112 t_0)) 0.5))
          0.5))
        180.0))))))

Derivation

1. Split input into 2 regimes

2. if b < 4.10000000000000004e39

   1. Initial program 15.2%

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

   3. Taylor expanded in y-scale around inf

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

   5. Applied rewrites 18.4%

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

   7. Applied rewrites 18.4%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 49.0%

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

   if 4.10000000000000004e39 < b

   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 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 24.3%

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

   6. Taylor expanded in x-scale around 0

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

   8. Applied rewrites 60.0%

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

   10. Applied rewrites 60.1%

       \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{PI}\left(\right)}{180 \cdot \tan^{-1} \left(0.5 \cdot \frac{\left(-2 \cdot y-scale\right) \cdot \frac{{\cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}}{x-scale}}{\left(\sin 0 + \sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.011111111111111112\right)\right) \cdot 0.5}\right)}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 60.4%

       \[\leadsto \frac{1}{\frac{\mathsf{PI}\left(\right)}{180 \cdot \tan^{-1} \left(0.5 \cdot \frac{\left(-2 \cdot y-scale\right) \cdot \frac{{\cos \left(\left(\sqrt[3]{{\mathsf{PI}\left(\right)}^{3}} \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}}{x-scale}}{\left(\sin 0 + \sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.011111111111111112\right)\right) \cdot 0.5}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.1 \cdot 10^{+39}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}{\cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)} \cdot \frac{y-scale}{x-scale}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{PI}\left(\right)}{\tan^{-1} \left(\frac{\frac{{\cos \left(\left(\sqrt[3]{{\mathsf{PI}\left(\right)}^{3}} \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}}{x-scale} \cdot \left(-2 \cdot y-scale\right)}{\sin \left(0.011111111111111112 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \cdot 0.5} \cdot 0.5\right) \cdot 180}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 57.3% accurate, 4.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\\ t_1 := \cos t\_0\\ t_2 := \sin t\_0\\ \mathbf{if}\;b\_m \leq 4.1 \cdot 10^{+39}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{t\_2}{t\_1} \cdot \frac{y-scale}{x-scale}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{\frac{{\cos \left(\left(\sqrt[3]{{\mathsf{PI}\left(\right)}^{3}} \cdot angle\right) \cdot 0.005555555555555556\right)}^{2} \cdot y-scale}{x-scale} \cdot -2}{t\_2 \cdot t\_1} \cdot 0.5\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (* (PI) angle) 0.005555555555555556))
        (t_1 (cos t_0))
        (t_2 (sin t_0)))
   (if (<= b_m 4.1e+39)
     (* (/ (atan (* (/ t_2 t_1) (/ y-scale x-scale))) (PI)) 180.0)
     (*
      (/
       (atan
        (*
         (/
          (*
           (/
            (*
             (pow
              (cos (* (* (cbrt (pow (PI) 3.0)) angle) 0.005555555555555556))
              2.0)
             y-scale)
            x-scale)
           -2.0)
          (* t_2 t_1))
         0.5))
       (PI))
      180.0))))

Derivation

1. Split input into 2 regimes

2. if b < 4.10000000000000004e39

   1. Initial program 15.2%

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

   3. Taylor expanded in y-scale around inf

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

   5. Applied rewrites 18.4%

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

   7. Applied rewrites 18.4%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 49.0%

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

   if 4.10000000000000004e39 < b

   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 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 24.3%

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

   6. Taylor expanded in x-scale around 0

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

   8. Applied rewrites 60.0%

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

   10. Applied rewrites 60.3%

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

4. Final simplification 51.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.1 \cdot 10^{+39}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}{\cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)} \cdot \frac{y-scale}{x-scale}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{\frac{{\cos \left(\left(\sqrt[3]{{\mathsf{PI}\left(\right)}^{3}} \cdot angle\right) \cdot 0.005555555555555556\right)}^{2} \cdot y-scale}{x-scale} \cdot -2}{\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)} \cdot 0.5\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 57.3% accurate, 6.1× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot angle\\ t_1 := \sqrt{\mathsf{PI}\left(\right)}\\ t_2 := t\_0 \cdot 0.005555555555555556\\ \mathbf{if}\;b\_m \leq 4.1 \cdot 10^{+39}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{\sin t\_2}{\cos t\_2} \cdot \frac{y-scale}{x-scale}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{PI}\left(\right)}{\tan^{-1} \left(\frac{\frac{{\cos \left(\left(\left(t\_1 \cdot t\_1\right) \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}}{x-scale} \cdot \left(-2 \cdot y-scale\right)}{\sin \left(0.011111111111111112 \cdot t\_0\right) \cdot 0.5} \cdot 0.5\right) \cdot 180}}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (PI) angle))
        (t_1 (sqrt (PI)))
        (t_2 (* t_0 0.005555555555555556)))
   (if (<= b_m 4.1e+39)
     (* (/ (atan (* (/ (sin t_2) (cos t_2)) (/ y-scale x-scale))) (PI)) 180.0)
     (/
      1.0
      (/
       (PI)
       (*
        (atan
         (*
          (/
           (*
            (/
             (pow (cos (* (* (* t_1 t_1) angle) 0.005555555555555556)) 2.0)
             x-scale)
            (* -2.0 y-scale))
           (* (sin (* 0.011111111111111112 t_0)) 0.5))
          0.5))
        180.0))))))

Derivation

1. Split input into 2 regimes

2. if b < 4.10000000000000004e39

   1. Initial program 15.2%

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

   3. Taylor expanded in y-scale around inf

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

   5. Applied rewrites 18.4%

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

   7. Applied rewrites 18.4%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 49.0%

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

   if 4.10000000000000004e39 < b

   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 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 24.3%

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

   6. Taylor expanded in x-scale around 0

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

   8. Applied rewrites 60.0%

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

   10. Applied rewrites 60.1%

       \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{PI}\left(\right)}{180 \cdot \tan^{-1} \left(0.5 \cdot \frac{\left(-2 \cdot y-scale\right) \cdot \frac{{\cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}}{x-scale}}{\left(\sin 0 + \sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.011111111111111112\right)\right) \cdot 0.5}\right)}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 60.1%

       \[\leadsto \frac{1}{\frac{\mathsf{PI}\left(\right)}{180 \cdot \tan^{-1} \left(0.5 \cdot \frac{\left(-2 \cdot y-scale\right) \cdot \frac{{\cos \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}}{x-scale}}{\left(\sin 0 + \sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.011111111111111112\right)\right) \cdot 0.5}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.1 \cdot 10^{+39}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}{\cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)} \cdot \frac{y-scale}{x-scale}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{PI}\left(\right)}{\tan^{-1} \left(\frac{\frac{{\cos \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}}{x-scale} \cdot \left(-2 \cdot y-scale\right)}{\sin \left(0.011111111111111112 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \cdot 0.5} \cdot 0.5\right) \cdot 180}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 57.3% accurate, 6.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot angle\\ t_1 := t\_0 \cdot 0.005555555555555556\\ t_2 := \cos t\_1\\ \mathbf{if}\;b\_m \leq 4.1 \cdot 10^{+39}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{\sin t\_1}{t\_2} \cdot \frac{y-scale}{x-scale}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{PI}\left(\right)}{\tan^{-1} \left(\left(\frac{{t\_2}^{2}}{\sin \left(0.011111111111111112 \cdot t\_0\right)} \cdot \frac{y-scale}{x-scale}\right) \cdot -2\right) \cdot 180}}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (PI) angle))
        (t_1 (* t_0 0.005555555555555556))
        (t_2 (cos t_1)))
   (if (<= b_m 4.1e+39)
     (* (/ (atan (* (/ (sin t_1) t_2) (/ y-scale x-scale))) (PI)) 180.0)
     (/
      1.0
      (/
       (PI)
       (*
        (atan
         (*
          (*
           (/ (pow t_2 2.0) (sin (* 0.011111111111111112 t_0)))
           (/ y-scale x-scale))
          -2.0))
        180.0))))))

Derivation

1. Split input into 2 regimes

2. if b < 4.10000000000000004e39

   1. Initial program 15.2%

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

   3. Taylor expanded in y-scale around inf

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

   5. Applied rewrites 18.4%

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

   7. Applied rewrites 18.4%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 49.0%

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

   if 4.10000000000000004e39 < b

   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 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 24.3%

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

   6. Taylor expanded in x-scale around 0

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

   8. Applied rewrites 60.0%

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

   10. Applied rewrites 60.1%

       \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{PI}\left(\right)}{180 \cdot \tan^{-1} \left(0.5 \cdot \frac{\left(-2 \cdot y-scale\right) \cdot \frac{{\cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}}{x-scale}}{\left(\sin 0 + \sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.011111111111111112\right)\right) \cdot 0.5}\right)}}} \]

   11. Taylor expanded in x-scale around 0

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

   13. Applied rewrites 60.1%

       \[\leadsto \frac{1}{\frac{\mathsf{PI}\left(\right)}{180 \cdot \tan^{-1} \left(-2 \cdot \color{blue}{\left(\frac{y-scale}{x-scale} \cdot \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{\sin \left(0.011111111111111112 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.1 \cdot 10^{+39}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}{\cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)} \cdot \frac{y-scale}{x-scale}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{PI}\left(\right)}{\tan^{-1} \left(\left(\frac{{\cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}}{\sin \left(0.011111111111111112 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)} \cdot \frac{y-scale}{x-scale}\right) \cdot -2\right) \cdot 180}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 57.3% accurate, 8.4× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\\ t_1 := \cos t\_0\\ t_2 := \sin t\_0\\ \mathbf{if}\;b\_m \leq 4.1 \cdot 10^{+39}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{t\_2}{t\_1} \cdot \frac{y-scale}{x-scale}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\left(\left(\frac{t\_1}{t\_2} \cdot \frac{y-scale}{x-scale}\right) \cdot 2\right) \cdot -0.5\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (* (PI) angle) 0.005555555555555556))
        (t_1 (cos t_0))
        (t_2 (sin t_0)))
   (if (<= b_m 4.1e+39)
     (* (/ (atan (* (/ t_2 t_1) (/ y-scale x-scale))) (PI)) 180.0)
     (*
      (/ (atan (* (* (* (/ t_1 t_2) (/ y-scale x-scale)) 2.0) -0.5)) (PI))
      180.0))))

Derivation

1. Split input into 2 regimes

2. if b < 4.10000000000000004e39

   1. Initial program 15.2%

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

   3. Taylor expanded in y-scale around inf

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

   5. Applied rewrites 18.4%

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

   7. Applied rewrites 18.4%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 49.0%

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

   if 4.10000000000000004e39 < b

   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 y-scale around inf

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

   5. Applied rewrites 18.0%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 60.0%

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

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.1 \cdot 10^{+39}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}{\cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)} \cdot \frac{y-scale}{x-scale}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\left(\left(\frac{\cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}{\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)} \cdot \frac{y-scale}{x-scale}\right) \cdot 2\right) \cdot -0.5\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 57.0% accurate, 8.6× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot angle\\ t_1 := t\_0 \cdot 0.005555555555555556\\ \mathbf{if}\;b\_m \leq 4.1 \cdot 10^{+39}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{\sin t\_1}{\cos t\_1} \cdot \frac{y-scale}{x-scale}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{PI}\left(\right)}{\tan^{-1} \left(\frac{-2 \cdot \frac{y-scale}{x-scale}}{\sin \left(0.011111111111111112 \cdot t\_0\right) \cdot 0.5} \cdot 0.5\right) \cdot 180}}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (PI) angle)) (t_1 (* t_0 0.005555555555555556)))
   (if (<= b_m 4.1e+39)
     (* (/ (atan (* (/ (sin t_1) (cos t_1)) (/ y-scale x-scale))) (PI)) 180.0)
     (/
      1.0
      (/
       (PI)
       (*
        (atan
         (*
          (/
           (* -2.0 (/ y-scale x-scale))
           (* (sin (* 0.011111111111111112 t_0)) 0.5))
          0.5))
        180.0))))))

Derivation

1. Split input into 2 regimes

2. if b < 4.10000000000000004e39

   1. Initial program 15.2%

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

   3. Taylor expanded in y-scale around inf

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

   5. Applied rewrites 18.4%

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

   7. Applied rewrites 18.4%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 49.0%

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

   if 4.10000000000000004e39 < b

   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 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 24.3%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 58.9%

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

   10. Applied rewrites 59.0%

       \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{PI}\left(\right)}{180 \cdot \tan^{-1} \left(\frac{\frac{y-scale}{x-scale} \cdot -2}{\sin \left(0.011111111111111112 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot 0.5} \cdot 0.5\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.1 \cdot 10^{+39}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}{\cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)} \cdot \frac{y-scale}{x-scale}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{PI}\left(\right)}{\tan^{-1} \left(\frac{-2 \cdot \frac{y-scale}{x-scale}}{\sin \left(0.011111111111111112 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \cdot 0.5} \cdot 0.5\right) \cdot 180}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 44.8% accurate, 11.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \frac{1}{\frac{\mathsf{PI}\left(\right)}{\tan^{-1} \left(\frac{-2 \cdot \frac{y-scale}{x-scale}}{\sin \left(0.011111111111111112 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \cdot 0.5} \cdot 0.5\right) \cdot 180}} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (/
  1.0
  (/
   (PI)
   (*
    (atan
     (*
      (/
       (* -2.0 (/ y-scale x-scale))
       (* (sin (* 0.011111111111111112 (* (PI) angle))) 0.5))
      0.5))
    180.0))))

Derivation

1. Initial program 14.3%

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

3. Taylor expanded in 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 26.8%

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

6. Taylor expanded in angle around 0

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

8. Applied rewrites 46.6%

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

10. Applied rewrites 46.6%

    \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{PI}\left(\right)}{180 \cdot \tan^{-1} \left(\frac{\frac{y-scale}{x-scale} \cdot -2}{\sin \left(0.011111111111111112 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot 0.5} \cdot 0.5\right)}}} \]

11. Final simplification 46.6%

    \[\leadsto \frac{1}{\frac{\mathsf{PI}\left(\right)}{\tan^{-1} \left(\frac{-2 \cdot \frac{y-scale}{x-scale}}{\sin \left(0.011111111111111112 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \cdot 0.5} \cdot 0.5\right) \cdot 180}} \]
12. Add Preprocessing

Alternative 8: 40.6% accurate, 11.8× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 1.7 \cdot 10^{-217}:\\ \;\;\;\;\frac{\tan^{-1} \left(90 \cdot \left(\frac{y-scale}{\left(\mathsf{PI}\left(\right) \cdot x-scale\right) \cdot angle} \cdot -2\right)\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{-2 \cdot \frac{y-scale}{x-scale}}{1 \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)} \cdot 0.5\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (if (<= a 1.7e-217)
   (*
    (/ (atan (* 90.0 (* (/ y-scale (* (* (PI) x-scale) angle)) -2.0))) (PI))
    180.0)
   (*
    (/
     (atan
      (*
       (/
        (* -2.0 (/ y-scale x-scale))
        (* 1.0 (sin (* (* (PI) angle) 0.005555555555555556))))
       0.5))
     (PI))
    180.0)))

Derivation

1. Split input into 2 regimes

2. if a < 1.70000000000000008e-217

   1. Initial program 12.6%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale}}\right)}{\mathsf{PI}\left(\right)} \]
   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 7.6%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 14.5%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 40.9%

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

   if 1.70000000000000008e-217 < a

   1. Initial program 16.5%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale}}\right)}{\mathsf{PI}\left(\right)} \]
   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 29.5%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 46.3%

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

   9. Taylor expanded in angle around 0

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

   11. Applied rewrites 42.9%

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

4. Final simplification 41.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.7 \cdot 10^{-217}:\\ \;\;\;\;\frac{\tan^{-1} \left(90 \cdot \left(\frac{y-scale}{\left(\mathsf{PI}\left(\right) \cdot x-scale\right) \cdot angle} \cdot -2\right)\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{-2 \cdot \frac{y-scale}{x-scale}}{1 \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)} \cdot 0.5\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 44.8% accurate, 11.8× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \frac{180}{\frac{\mathsf{PI}\left(\right)}{\tan^{-1} \left(\frac{-2 \cdot \frac{y-scale}{x-scale}}{\sin \left(0.011111111111111112 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \cdot 0.5} \cdot 0.5\right)}} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (/
  180.0
  (/
   (PI)
   (atan
    (*
     (/
      (* -2.0 (/ y-scale x-scale))
      (* (sin (* 0.011111111111111112 (* (PI) angle))) 0.5))
     0.5)))))

Derivation

1. Initial program 14.3%

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

3. Taylor expanded in 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 26.8%

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

6. Taylor expanded in angle around 0

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

8. Applied rewrites 46.6%

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

10. Applied rewrites 46.6%

    \[\leadsto \color{blue}{\frac{180}{\frac{\mathsf{PI}\left(\right)}{\tan^{-1} \left(\frac{\frac{y-scale}{x-scale} \cdot -2}{\sin \left(0.011111111111111112 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot 0.5} \cdot 0.5\right)}}} \]

11. Final simplification 46.6%

    \[\leadsto \frac{180}{\frac{\mathsf{PI}\left(\right)}{\tan^{-1} \left(\frac{-2 \cdot \frac{y-scale}{x-scale}}{\sin \left(0.011111111111111112 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \cdot 0.5} \cdot 0.5\right)}} \]
12. Add Preprocessing

Alternative 10: 44.8% accurate, 12.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \frac{\tan^{-1} \left(\frac{-2 \cdot \frac{y-scale}{x-scale}}{\sin \left(0.011111111111111112 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \cdot 0.5} \cdot 0.5\right)}{\mathsf{PI}\left(\right)} \cdot 180 \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (*
  (/
   (atan
    (*
     (/
      (* -2.0 (/ y-scale x-scale))
      (* (sin (* 0.011111111111111112 (* (PI) angle))) 0.5))
     0.5))
   (PI))
  180.0))

Derivation

1. Initial program 14.3%

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

3. Taylor expanded in 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 26.8%

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

6. Taylor expanded in angle around 0

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

8. Applied rewrites 46.6%

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

10. Applied rewrites 46.6%

    \[\leadsto \color{blue}{\frac{\tan^{-1} \left(\frac{\frac{y-scale}{x-scale} \cdot -2}{\sin \left(0.011111111111111112 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot 0.5} \cdot 0.5\right)}{\mathsf{PI}\left(\right)} \cdot 180} \]

11. Final simplification 46.6%

    \[\leadsto \frac{\tan^{-1} \left(\frac{-2 \cdot \frac{y-scale}{x-scale}}{\sin \left(0.011111111111111112 \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \cdot 0.5} \cdot 0.5\right)}{\mathsf{PI}\left(\right)} \cdot 180 \]
12. Add Preprocessing

Alternative 11: 38.3% accurate, 17.6× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 5.6 \cdot 10^{+82}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{\left(\left(b\_m \cdot b\_m\right) \cdot y-scale\right) \cdot -2}{\left(\left(\left(a + b\_m\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(b\_m - a\right)\right) \cdot \left(x-scale \cdot angle\right)} \cdot 90\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{PI}\left(\right)}{\tan^{-1} \left(-180 \cdot \frac{y-scale}{\left(\mathsf{PI}\left(\right) \cdot x-scale\right) \cdot angle}\right) \cdot 180}}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (if (<= b_m 5.6e+82)
   (*
    (/
     (atan
      (*
       (/
        (* (* (* b_m b_m) y-scale) -2.0)
        (* (* (* (+ a b_m) (PI)) (- b_m a)) (* x-scale angle)))
       90.0))
     (PI))
    180.0)
   (/
    1.0
    (/
     (PI)
     (* (atan (* -180.0 (/ y-scale (* (* (PI) x-scale) angle)))) 180.0)))))

Derivation

1. Split input into 2 regimes

2. if b < 5.6000000000000001e82

   1. Initial program 16.3%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale}}\right)}{\mathsf{PI}\left(\right)} \]
   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 9.4%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 11.8%

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

   9. Taylor expanded in x-scale around 0

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

   11. Applied rewrites 26.9%

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

   if 5.6000000000000001e82 < b

   1. Initial program 6.2%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale}}\right)}{\mathsf{PI}\left(\right)} \]
   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 26.3%

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

   6. Taylor expanded in x-scale around 0

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

   8. Applied rewrites 57.3%

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

   10. Applied rewrites 57.4%

       \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{PI}\left(\right)}{180 \cdot \tan^{-1} \left(0.5 \cdot \frac{\left(-2 \cdot y-scale\right) \cdot \frac{{\cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}}{x-scale}}{\left(\sin 0 + \sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.011111111111111112\right)\right) \cdot 0.5}\right)}}} \]

   11. Taylor expanded in angle around 0

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

   13. Applied rewrites 50.7%

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

4. Final simplification 31.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.6 \cdot 10^{+82}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{\left(\left(b \cdot b\right) \cdot y-scale\right) \cdot -2}{\left(\left(\left(a + b\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(b - a\right)\right) \cdot \left(x-scale \cdot angle\right)} \cdot 90\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{PI}\left(\right)}{\tan^{-1} \left(-180 \cdot \frac{y-scale}{\left(\mathsf{PI}\left(\right) \cdot x-scale\right) \cdot angle}\right) \cdot 180}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 38.3% accurate, 18.1× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 5.6 \cdot 10^{+82}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{-180 \cdot \left(\left(b\_m \cdot b\_m\right) \cdot y-scale\right)}{\left(\left(\left(a + b\_m\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(b\_m - a\right)\right) \cdot \left(x-scale \cdot angle\right)}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{PI}\left(\right)}{\tan^{-1} \left(-180 \cdot \frac{y-scale}{\left(\mathsf{PI}\left(\right) \cdot x-scale\right) \cdot angle}\right) \cdot 180}}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (if (<= b_m 5.6e+82)
   (*
    (/
     (atan
      (/
       (* -180.0 (* (* b_m b_m) y-scale))
       (* (* (* (+ a b_m) (PI)) (- b_m a)) (* x-scale angle))))
     (PI))
    180.0)
   (/
    1.0
    (/
     (PI)
     (* (atan (* -180.0 (/ y-scale (* (* (PI) x-scale) angle)))) 180.0)))))

Derivation

1. Split input into 2 regimes

2. if b < 5.6000000000000001e82

   1. Initial program 16.3%

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

   3. Taylor expanded in y-scale around inf

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

   5. Applied rewrites 19.8%

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

   7. Applied rewrites 19.8%

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

   8. Taylor expanded in angle around 0

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

   10. Applied rewrites 26.9%

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

   if 5.6000000000000001e82 < b

   1. Initial program 6.2%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale}}\right)}{\mathsf{PI}\left(\right)} \]
   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 26.3%

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

   6. Taylor expanded in x-scale around 0

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

   8. Applied rewrites 57.3%

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

   10. Applied rewrites 57.4%

       \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{PI}\left(\right)}{180 \cdot \tan^{-1} \left(0.5 \cdot \frac{\left(-2 \cdot y-scale\right) \cdot \frac{{\cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}}{x-scale}}{\left(\sin 0 + \sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.011111111111111112\right)\right) \cdot 0.5}\right)}}} \]

   11. Taylor expanded in angle around 0

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

   13. Applied rewrites 50.7%

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

4. Final simplification 31.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.6 \cdot 10^{+82}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{-180 \cdot \left(\left(b \cdot b\right) \cdot y-scale\right)}{\left(\left(\left(a + b\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(b - a\right)\right) \cdot \left(x-scale \cdot angle\right)}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{PI}\left(\right)}{\tan^{-1} \left(-180 \cdot \frac{y-scale}{\left(\mathsf{PI}\left(\right) \cdot x-scale\right) \cdot angle}\right) \cdot 180}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 37.1% accurate, 21.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 14.3%

   \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale}}\right)}{\mathsf{PI}\left(\right)} \]
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 8.8%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 14.0%

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

9. Taylor expanded in b around inf

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

11. Applied rewrites 36.7%

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

12. Final simplification 36.7%

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

Alternative 14: 11.6% accurate, 21.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 14.3%

   \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale}}\right)}{\mathsf{PI}\left(\right)} \]
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 8.8%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 14.0%

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

9. Taylor expanded in b around 0

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

11. Applied rewrites 13.9%

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

12. Final simplification 13.9%

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

Reproduce

herbie shell --seed 2024276 
(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))))