ABCF->ab-angle angle

Percentage Accurate: 54.5% → 81.4%

Time: 9.2s

Alternatives: 19

Speedup: 2.5×

Specification

Math

\[\begin{array}{l} \\ 180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (*
  180.0
  (/
   (atan (* (/ 1.0 B) (- (- C A) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))
   (PI))))

Initial Program: 54.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (*
  180.0
  (/
   (atan (* (/ 1.0 B) (- (- C A) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))
   (PI))))

Alternative 1: 81.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;A \leq -6 \cdot 10^{+103}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{0.5 \cdot \mathsf{fma}\left(\frac{C}{A}, B, B\right)}{A}\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(B, A - C\right)}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= A -6e+103)
   (* 180.0 (/ (atan (/ (* 0.5 (fma (/ C A) B B)) A)) (PI)))
   (/ (* (atan (/ (- (- C A) (hypot B (- A C))) B)) 180.0) (PI))))

Derivation

1. Split input into 2 regimes

2. if A < -6e103

   1. Initial program 18.3%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-1 \cdot \frac{\frac{-1}{2} \cdot B + \frac{-1}{2} \cdot \frac{B \cdot C}{A}}{A}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-1 \cdot \left(\frac{-1}{2} \cdot B + \frac{-1}{2} \cdot \frac{B \cdot C}{A}\right)}{A}\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-1 \cdot \left(\frac{-1}{2} \cdot B + \frac{-1}{2} \cdot \frac{B \cdot C}{A}\right)}{A}\right)}}{\mathsf{PI}\left(\right)} \]

      3. distribute-lft-out N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1 \cdot \color{blue}{\left(\frac{-1}{2} \cdot \left(B + \frac{B \cdot C}{A}\right)\right)}}{A}\right)}{\mathsf{PI}\left(\right)} \]

      4. associate-*r* N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{\left(-1 \cdot \frac{-1}{2}\right) \cdot \left(B + \frac{B \cdot C}{A}\right)}}{A}\right)}{\mathsf{PI}\left(\right)} \]

      5. metadata-eval N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{\frac{1}{2}} \cdot \left(B + \frac{B \cdot C}{A}\right)}{A}\right)}{\mathsf{PI}\left(\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{\frac{1}{2} \cdot \left(B + \frac{B \cdot C}{A}\right)}}{A}\right)}{\mathsf{PI}\left(\right)} \]

      7. +-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\frac{1}{2} \cdot \color{blue}{\left(\frac{B \cdot C}{A} + B\right)}}{A}\right)}{\mathsf{PI}\left(\right)} \]

      8. associate-/l* N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\frac{1}{2} \cdot \left(\color{blue}{B \cdot \frac{C}{A}} + B\right)}{A}\right)}{\mathsf{PI}\left(\right)} \]

      9. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\frac{1}{2} \cdot \left(\color{blue}{\frac{C}{A} \cdot B} + B\right)}{A}\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-fma.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\frac{1}{2} \cdot \color{blue}{\mathsf{fma}\left(\frac{C}{A}, B, B\right)}}{A}\right)}{\mathsf{PI}\left(\right)} \]

      11. lower-/.f64 86.9

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{0.5 \cdot \mathsf{fma}\left(\color{blue}{\frac{C}{A}}, B, B\right)}{A}\right)}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 86.9%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{0.5 \cdot \mathsf{fma}\left(\frac{C}{A}, B, B\right)}{A}\right)}}{\mathsf{PI}\left(\right)} \]

   if -6e103 < A

   1. Initial program 58.5%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto 180 \cdot \color{blue}{\frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{180 \cdot \tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{180 \cdot \tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}} \]

   4. Applied rewrites 83.3%

      \[\leadsto \color{blue}{\frac{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(B, A - C\right)}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -6 \cdot 10^{+103}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{0.5 \cdot \mathsf{fma}\left(\frac{C}{A}, B, B\right)}{A}\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(B, A - C\right)}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 60.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {B}^{-1} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\\ \mathbf{if}\;t\_0 \leq -1:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - B}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;t\_0 \leq 10^{-14}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{B}{A} \cdot 0.5\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+307}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{C}{B} + 1\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(1 - \frac{A}{B}\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (let* ((t_0
         (*
          (pow B -1.0)
          (- (- C A) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0)))))))
   (if (<= t_0 -1.0)
     (/ (* (atan (/ (- C B) B)) 180.0) (PI))
     (if (<= t_0 1e-14)
       (* 180.0 (/ (atan (* (/ B A) 0.5)) (PI)))
       (if (<= t_0 5e+307)
         (* 180.0 (/ (atan (+ (/ C B) 1.0)) (PI)))
         (* 180.0 (/ (atan (- 1.0 (/ A B))) (PI))))))))

Derivation

1. Split input into 4 regimes

2. if (*.f64 (/.f64 #s(literal 1 binary64) B) (-.f64 (-.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))) < -1

   1. Initial program 54.0%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto 180 \cdot \color{blue}{\frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{180 \cdot \tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{180 \cdot \tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}} \]

   4. Applied rewrites 87.9%

      \[\leadsto \color{blue}{\frac{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(B, A - C\right)}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)}} \]

   5. Taylor expanded in A around 0

      \[\leadsto \frac{\tan^{-1} \left(\frac{\color{blue}{C - \sqrt{{B}^{2} + {C}^{2}}}}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)} \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{\tan^{-1} \left(\frac{\color{blue}{C - \sqrt{{B}^{2} + {C}^{2}}}}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)} \]

      2. unpow2 N/A

         \[\leadsto \frac{\tan^{-1} \left(\frac{C - \sqrt{\color{blue}{B \cdot B} + {C}^{2}}}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{\tan^{-1} \left(\frac{C - \sqrt{B \cdot B + \color{blue}{C \cdot C}}}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)} \]

      4. lower-hypot.f64 75.1

         \[\leadsto \frac{\tan^{-1} \left(\frac{C - \color{blue}{\mathsf{hypot}\left(B, C\right)}}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)} \]

   7. Applied rewrites 75.1%

      \[\leadsto \frac{\tan^{-1} \left(\frac{\color{blue}{C - \mathsf{hypot}\left(B, C\right)}}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)} \]

   8. Taylor expanded in C around 0

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

   10. Applied rewrites 66.7%

       \[\leadsto \frac{\tan^{-1} \left(\frac{C - \color{blue}{B}}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)} \]

   if -1 < (*.f64 (/.f64 #s(literal 1 binary64) B) (-.f64 (-.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))) < 9.99999999999999999e-15

   1. Initial program 17.0%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{B}{A}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{B}{A} \cdot \frac{1}{2}\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{B}{A} \cdot \frac{1}{2}\right)}}{\mathsf{PI}\left(\right)} \]

      3. lower-/.f64 42.7

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\frac{B}{A}} \cdot 0.5\right)}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 42.7%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{B}{A} \cdot 0.5\right)}}{\mathsf{PI}\left(\right)} \]

   if 9.99999999999999999e-15 < (*.f64 (/.f64 #s(literal 1 binary64) B) (-.f64 (-.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))) < 5e307

   1. Initial program 96.8%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around -inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\left(1 + \frac{C}{B}\right) - \frac{A}{B}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(1 + \left(\frac{C}{B} - \frac{A}{B}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      2. div-sub N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(1 + \color{blue}{\frac{C - A}{B}}\right)}{\mathsf{PI}\left(\right)} \]

      3. +-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C - A}{B} + 1\right)}}{\mathsf{PI}\left(\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C - A}{B} + 1\right)}}{\mathsf{PI}\left(\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\frac{C - A}{B}} + 1\right)}{\mathsf{PI}\left(\right)} \]

      6. lower--.f64 96.6

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{C - A}}{B} + 1\right)}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 96.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C - A}{B} + 1\right)}}{\mathsf{PI}\left(\right)} \]

   6. Taylor expanded in A around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{C}{B} + 1\right)}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 88.0%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{C}{B} + 1\right)}{\mathsf{PI}\left(\right)} \]

   if 5e307 < (*.f64 (/.f64 #s(literal 1 binary64) B) (-.f64 (-.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64))))))

   1. Initial program 43.9%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in C around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-1 \cdot \frac{A + \sqrt{{A}^{2} + {B}^{2}}}{B}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{A + \sqrt{{A}^{2} + {B}^{2}}}{B}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      2. distribute-neg-frac2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{A + \sqrt{{A}^{2} + {B}^{2}}}{\mathsf{neg}\left(B\right)}\right)}}{\mathsf{PI}\left(\right)} \]

      3. mul-1-neg N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{A + \sqrt{{A}^{2} + {B}^{2}}}{\color{blue}{-1 \cdot B}}\right)}{\mathsf{PI}\left(\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{A + \sqrt{{A}^{2} + {B}^{2}}}{-1 \cdot B}\right)}}{\mathsf{PI}\left(\right)} \]

      5. +-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{\sqrt{{A}^{2} + {B}^{2}} + A}}{-1 \cdot B}\right)}{\mathsf{PI}\left(\right)} \]

      6. lower-+.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{\sqrt{{A}^{2} + {B}^{2}} + A}}{-1 \cdot B}\right)}{\mathsf{PI}\left(\right)} \]

      7. +-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\sqrt{\color{blue}{{B}^{2} + {A}^{2}}} + A}{-1 \cdot B}\right)}{\mathsf{PI}\left(\right)} \]

      8. unpow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\sqrt{\color{blue}{B \cdot B} + {A}^{2}} + A}{-1 \cdot B}\right)}{\mathsf{PI}\left(\right)} \]

      9. unpow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\sqrt{B \cdot B + \color{blue}{A \cdot A}} + A}{-1 \cdot B}\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-hypot.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{\mathsf{hypot}\left(B, A\right)} + A}{-1 \cdot B}\right)}{\mathsf{PI}\left(\right)} \]

      11. mul-1-neg N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\mathsf{hypot}\left(B, A\right) + A}{\color{blue}{\mathsf{neg}\left(B\right)}}\right)}{\mathsf{PI}\left(\right)} \]

      12. lower-neg.f64 63.7

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\mathsf{hypot}\left(B, A\right) + A}{\color{blue}{-B}}\right)}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 63.7%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{\mathsf{hypot}\left(B, A\right) + A}{-B}\right)}}{\mathsf{PI}\left(\right)} \]

   6. Taylor expanded in B around -inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(1 + \color{blue}{-1 \cdot \frac{A}{B}}\right)}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 54.7%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(1 - \color{blue}{\frac{A}{B}}\right)}{\mathsf{PI}\left(\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{-1} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \leq -1:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - B}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;{B}^{-1} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \leq 10^{-14}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{B}{A} \cdot 0.5\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;{B}^{-1} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \leq 5 \cdot 10^{+307}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{C}{B} + 1\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(1 - \frac{A}{B}\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 180 \cdot \frac{\tan^{-1} \left({B}^{-1} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}\\ t_1 := \frac{C - A}{B}\\ \mathbf{if}\;t\_0 \leq -40:\\ \;\;\;\;\frac{\tan^{-1} \left(t\_1 - 1\right) \cdot 180}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{\tan^{-1} \left(\mathsf{fma}\left(\frac{B}{C}, -0.5, 0\right)\right) \cdot 180}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(t\_1 + 1\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (let* ((t_0
         (*
          180.0
          (/
           (atan
            (*
             (pow B -1.0)
             (- (- C A) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))
           (PI))))
        (t_1 (/ (- C A) B)))
   (if (<= t_0 -40.0)
     (/ (* (atan (- t_1 1.0)) 180.0) (PI))
     (if (<= t_0 0.0)
       (/ (* (atan (fma (/ B C) -0.5 0.0)) 180.0) (PI))
       (* 180.0 (/ (atan (+ t_1 1.0)) (PI)))))))

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal 180 binary64) (/.f64 (atan.f64 (*.f64 (/.f64 #s(literal 1 binary64) B) (-.f64 (-.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64))))))) (PI.f64))) < -40

   1. Initial program 54.0%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto 180 \cdot \color{blue}{\frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{180 \cdot \tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{180 \cdot \tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}} \]

   4. Applied rewrites 87.9%

      \[\leadsto \color{blue}{\frac{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(B, A - C\right)}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)}} \]

   5. Taylor expanded in B around inf

      \[\leadsto \frac{\tan^{-1} \color{blue}{\left(\frac{C}{B} - \left(1 + \frac{A}{B}\right)\right)} \cdot 180}{\mathsf{PI}\left(\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\tan^{-1} \left(\frac{C}{B} - \color{blue}{\left(\frac{A}{B} + 1\right)}\right) \cdot 180}{\mathsf{PI}\left(\right)} \]

      2. associate--r+ N/A

         \[\leadsto \frac{\tan^{-1} \color{blue}{\left(\left(\frac{C}{B} - \frac{A}{B}\right) - 1\right)} \cdot 180}{\mathsf{PI}\left(\right)} \]

      3. div-sub N/A

         \[\leadsto \frac{\tan^{-1} \left(\color{blue}{\frac{C - A}{B}} - 1\right) \cdot 180}{\mathsf{PI}\left(\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\tan^{-1} \color{blue}{\left(\frac{C - A}{B} - 1\right)} \cdot 180}{\mathsf{PI}\left(\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\tan^{-1} \left(\color{blue}{\frac{C - A}{B}} - 1\right) \cdot 180}{\mathsf{PI}\left(\right)} \]

      6. lower--.f64 76.5

         \[\leadsto \frac{\tan^{-1} \left(\frac{\color{blue}{C - A}}{B} - 1\right) \cdot 180}{\mathsf{PI}\left(\right)} \]

   7. Applied rewrites 76.5%

      \[\leadsto \frac{\tan^{-1} \color{blue}{\left(\frac{C - A}{B} - 1\right)} \cdot 180}{\mathsf{PI}\left(\right)} \]

   if -40 < (*.f64 #s(literal 180 binary64) (/.f64 (atan.f64 (*.f64 (/.f64 #s(literal 1 binary64) B) (-.f64 (-.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64))))))) (PI.f64))) < 0.0

   1. Initial program 17.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto 180 \cdot \color{blue}{\frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{180 \cdot \tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{180 \cdot \tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}} \]

   4. Applied rewrites 17.4%

      \[\leadsto \color{blue}{\frac{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(B, A - C\right)}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)}} \]

   5. Taylor expanded in C around inf

      \[\leadsto \frac{\tan^{-1} \color{blue}{\left(-1 \cdot \frac{A + -1 \cdot A}{B} + \frac{-1}{2} \cdot \frac{B}{C}\right)} \cdot 180}{\mathsf{PI}\left(\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\tan^{-1} \color{blue}{\left(\frac{-1}{2} \cdot \frac{B}{C} + -1 \cdot \frac{A + -1 \cdot A}{B}\right)} \cdot 180}{\mathsf{PI}\left(\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\tan^{-1} \left(\color{blue}{\frac{B}{C} \cdot \frac{-1}{2}} + -1 \cdot \frac{A + -1 \cdot A}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\tan^{-1} \color{blue}{\left(\mathsf{fma}\left(\frac{B}{C}, \frac{-1}{2}, -1 \cdot \frac{A + -1 \cdot A}{B}\right)\right)} \cdot 180}{\mathsf{PI}\left(\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\tan^{-1} \left(\mathsf{fma}\left(\color{blue}{\frac{B}{C}}, \frac{-1}{2}, -1 \cdot \frac{A + -1 \cdot A}{B}\right)\right) \cdot 180}{\mathsf{PI}\left(\right)} \]

      5. mul-1-neg N/A

         \[\leadsto \frac{\tan^{-1} \left(\mathsf{fma}\left(\frac{B}{C}, \frac{-1}{2}, \color{blue}{\mathsf{neg}\left(\frac{A + -1 \cdot A}{B}\right)}\right)\right) \cdot 180}{\mathsf{PI}\left(\right)} \]

      6. distribute-frac-neg N/A

         \[\leadsto \frac{\tan^{-1} \left(\mathsf{fma}\left(\frac{B}{C}, \frac{-1}{2}, \color{blue}{\frac{\mathsf{neg}\left(\left(A + -1 \cdot A\right)\right)}{B}}\right)\right) \cdot 180}{\mathsf{PI}\left(\right)} \]

      7. distribute-rgt1-in N/A

         \[\leadsto \frac{\tan^{-1} \left(\mathsf{fma}\left(\frac{B}{C}, \frac{-1}{2}, \frac{\mathsf{neg}\left(\color{blue}{\left(-1 + 1\right) \cdot A}\right)}{B}\right)\right) \cdot 180}{\mathsf{PI}\left(\right)} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\tan^{-1} \left(\mathsf{fma}\left(\frac{B}{C}, \frac{-1}{2}, \frac{\mathsf{neg}\left(\color{blue}{0} \cdot A\right)}{B}\right)\right) \cdot 180}{\mathsf{PI}\left(\right)} \]

      9. distribute-lft-neg-in N/A

         \[\leadsto \frac{\tan^{-1} \left(\mathsf{fma}\left(\frac{B}{C}, \frac{-1}{2}, \frac{\color{blue}{\left(\mathsf{neg}\left(0\right)\right) \cdot A}}{B}\right)\right) \cdot 180}{\mathsf{PI}\left(\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\tan^{-1} \left(\mathsf{fma}\left(\frac{B}{C}, \frac{-1}{2}, \frac{\color{blue}{0} \cdot A}{B}\right)\right) \cdot 180}{\mathsf{PI}\left(\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{\tan^{-1} \left(\mathsf{fma}\left(\frac{B}{C}, \frac{-1}{2}, \frac{\color{blue}{\left(-1 + 1\right)} \cdot A}{B}\right)\right) \cdot 180}{\mathsf{PI}\left(\right)} \]

      12. distribute-rgt1-in N/A

          \[\leadsto \frac{\tan^{-1} \left(\mathsf{fma}\left(\frac{B}{C}, \frac{-1}{2}, \frac{\color{blue}{A + -1 \cdot A}}{B}\right)\right) \cdot 180}{\mathsf{PI}\left(\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{\tan^{-1} \left(\mathsf{fma}\left(\frac{B}{C}, \frac{-1}{2}, \color{blue}{\frac{A + -1 \cdot A}{B}}\right)\right) \cdot 180}{\mathsf{PI}\left(\right)} \]

      14. distribute-rgt1-in N/A

          \[\leadsto \frac{\tan^{-1} \left(\mathsf{fma}\left(\frac{B}{C}, \frac{-1}{2}, \frac{\color{blue}{\left(-1 + 1\right) \cdot A}}{B}\right)\right) \cdot 180}{\mathsf{PI}\left(\right)} \]

      15. metadata-eval N/A

          \[\leadsto \frac{\tan^{-1} \left(\mathsf{fma}\left(\frac{B}{C}, \frac{-1}{2}, \frac{\color{blue}{0} \cdot A}{B}\right)\right) \cdot 180}{\mathsf{PI}\left(\right)} \]

      16. mul0-lft 62.8

          \[\leadsto \frac{\tan^{-1} \left(\mathsf{fma}\left(\frac{B}{C}, -0.5, \frac{\color{blue}{0}}{B}\right)\right) \cdot 180}{\mathsf{PI}\left(\right)} \]

   7. Applied rewrites 62.8%

      \[\leadsto \frac{\tan^{-1} \color{blue}{\left(\mathsf{fma}\left(\frac{B}{C}, -0.5, \frac{0}{B}\right)\right)} \cdot 180}{\mathsf{PI}\left(\right)} \]

   if 0.0 < (*.f64 #s(literal 180 binary64) (/.f64 (atan.f64 (*.f64 (/.f64 #s(literal 1 binary64) B) (-.f64 (-.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64))))))) (PI.f64)))

   1. Initial program 61.1%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around -inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\left(1 + \frac{C}{B}\right) - \frac{A}{B}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(1 + \left(\frac{C}{B} - \frac{A}{B}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      2. div-sub N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(1 + \color{blue}{\frac{C - A}{B}}\right)}{\mathsf{PI}\left(\right)} \]

      3. +-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C - A}{B} + 1\right)}}{\mathsf{PI}\left(\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C - A}{B} + 1\right)}}{\mathsf{PI}\left(\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\frac{C - A}{B}} + 1\right)}{\mathsf{PI}\left(\right)} \]

      6. lower--.f64 72.9

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{C - A}}{B} + 1\right)}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 72.9%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C - A}{B} + 1\right)}}{\mathsf{PI}\left(\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;180 \cdot \frac{\tan^{-1} \left({B}^{-1} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \leq -40:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - A}{B} - 1\right) \cdot 180}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;180 \cdot \frac{\tan^{-1} \left({B}^{-1} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \leq 0:\\ \;\;\;\;\frac{\tan^{-1} \left(\mathsf{fma}\left(\frac{B}{C}, -0.5, 0\right)\right) \cdot 180}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{C - A}{B} + 1\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 180 \cdot \frac{\tan^{-1} \left({B}^{-1} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}\\ t_1 := \frac{C - A}{B}\\ \mathbf{if}\;t\_0 \leq -40:\\ \;\;\;\;\frac{\tan^{-1} \left(t\_1 - 1\right) \cdot 180}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\mathsf{fma}\left(\frac{B}{C}, -0.5, 0\right)\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(t\_1 + 1\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (let* ((t_0
         (*
          180.0
          (/
           (atan
            (*
             (pow B -1.0)
             (- (- C A) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))
           (PI))))
        (t_1 (/ (- C A) B)))
   (if (<= t_0 -40.0)
     (/ (* (atan (- t_1 1.0)) 180.0) (PI))
     (if (<= t_0 0.0)
       (* 180.0 (/ (atan (fma (/ B C) -0.5 0.0)) (PI)))
       (* 180.0 (/ (atan (+ t_1 1.0)) (PI)))))))

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal 180 binary64) (/.f64 (atan.f64 (*.f64 (/.f64 #s(literal 1 binary64) B) (-.f64 (-.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64))))))) (PI.f64))) < -40

   1. Initial program 54.0%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto 180 \cdot \color{blue}{\frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{180 \cdot \tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{180 \cdot \tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}} \]

   4. Applied rewrites 87.9%

      \[\leadsto \color{blue}{\frac{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(B, A - C\right)}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)}} \]

   5. Taylor expanded in B around inf

      \[\leadsto \frac{\tan^{-1} \color{blue}{\left(\frac{C}{B} - \left(1 + \frac{A}{B}\right)\right)} \cdot 180}{\mathsf{PI}\left(\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\tan^{-1} \left(\frac{C}{B} - \color{blue}{\left(\frac{A}{B} + 1\right)}\right) \cdot 180}{\mathsf{PI}\left(\right)} \]

      2. associate--r+ N/A

         \[\leadsto \frac{\tan^{-1} \color{blue}{\left(\left(\frac{C}{B} - \frac{A}{B}\right) - 1\right)} \cdot 180}{\mathsf{PI}\left(\right)} \]

      3. div-sub N/A

         \[\leadsto \frac{\tan^{-1} \left(\color{blue}{\frac{C - A}{B}} - 1\right) \cdot 180}{\mathsf{PI}\left(\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\tan^{-1} \color{blue}{\left(\frac{C - A}{B} - 1\right)} \cdot 180}{\mathsf{PI}\left(\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\tan^{-1} \left(\color{blue}{\frac{C - A}{B}} - 1\right) \cdot 180}{\mathsf{PI}\left(\right)} \]

      6. lower--.f64 76.5

         \[\leadsto \frac{\tan^{-1} \left(\frac{\color{blue}{C - A}}{B} - 1\right) \cdot 180}{\mathsf{PI}\left(\right)} \]

   7. Applied rewrites 76.5%

      \[\leadsto \frac{\tan^{-1} \color{blue}{\left(\frac{C - A}{B} - 1\right)} \cdot 180}{\mathsf{PI}\left(\right)} \]

   if -40 < (*.f64 #s(literal 180 binary64) (/.f64 (atan.f64 (*.f64 (/.f64 #s(literal 1 binary64) B) (-.f64 (-.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64))))))) (PI.f64))) < 0.0

   1. Initial program 17.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in C around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-1 \cdot \frac{A + -1 \cdot A}{B} + \frac{-1}{2} \cdot \frac{B}{C}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-1}{2} \cdot \frac{B}{C} + -1 \cdot \frac{A + -1 \cdot A}{B}\right)}}{\mathsf{PI}\left(\right)} \]

      2. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\frac{B}{C} \cdot \frac{-1}{2}} + -1 \cdot \frac{A + -1 \cdot A}{B}\right)}{\mathsf{PI}\left(\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{fma}\left(\frac{B}{C}, \frac{-1}{2}, -1 \cdot \frac{A + -1 \cdot A}{B}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{fma}\left(\color{blue}{\frac{B}{C}}, \frac{-1}{2}, -1 \cdot \frac{A + -1 \cdot A}{B}\right)\right)}{\mathsf{PI}\left(\right)} \]

      5. mul-1-neg N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{fma}\left(\frac{B}{C}, \frac{-1}{2}, \color{blue}{\mathsf{neg}\left(\frac{A + -1 \cdot A}{B}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      6. distribute-frac-neg N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{fma}\left(\frac{B}{C}, \frac{-1}{2}, \color{blue}{\frac{\mathsf{neg}\left(\left(A + -1 \cdot A\right)\right)}{B}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      7. distribute-rgt1-in N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{fma}\left(\frac{B}{C}, \frac{-1}{2}, \frac{\mathsf{neg}\left(\color{blue}{\left(-1 + 1\right) \cdot A}\right)}{B}\right)\right)}{\mathsf{PI}\left(\right)} \]

      8. metadata-eval N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{fma}\left(\frac{B}{C}, \frac{-1}{2}, \frac{\mathsf{neg}\left(\color{blue}{0} \cdot A\right)}{B}\right)\right)}{\mathsf{PI}\left(\right)} \]

      9. distribute-lft-neg-in N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{fma}\left(\frac{B}{C}, \frac{-1}{2}, \frac{\color{blue}{\left(\mathsf{neg}\left(0\right)\right) \cdot A}}{B}\right)\right)}{\mathsf{PI}\left(\right)} \]

      10. metadata-eval N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{fma}\left(\frac{B}{C}, \frac{-1}{2}, \frac{\color{blue}{0} \cdot A}{B}\right)\right)}{\mathsf{PI}\left(\right)} \]

      11. metadata-eval N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{fma}\left(\frac{B}{C}, \frac{-1}{2}, \frac{\color{blue}{\left(-1 + 1\right)} \cdot A}{B}\right)\right)}{\mathsf{PI}\left(\right)} \]

      12. distribute-rgt1-in N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{fma}\left(\frac{B}{C}, \frac{-1}{2}, \frac{\color{blue}{A + -1 \cdot A}}{B}\right)\right)}{\mathsf{PI}\left(\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{fma}\left(\frac{B}{C}, \frac{-1}{2}, \color{blue}{\frac{A + -1 \cdot A}{B}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      14. distribute-rgt1-in N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{fma}\left(\frac{B}{C}, \frac{-1}{2}, \frac{\color{blue}{\left(-1 + 1\right) \cdot A}}{B}\right)\right)}{\mathsf{PI}\left(\right)} \]

      15. metadata-eval N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{fma}\left(\frac{B}{C}, \frac{-1}{2}, \frac{\color{blue}{0} \cdot A}{B}\right)\right)}{\mathsf{PI}\left(\right)} \]

      16. mul0-lft 62.8

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{fma}\left(\frac{B}{C}, -0.5, \frac{\color{blue}{0}}{B}\right)\right)}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 62.8%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{fma}\left(\frac{B}{C}, -0.5, \frac{0}{B}\right)\right)}}{\mathsf{PI}\left(\right)} \]

   if 0.0 < (*.f64 #s(literal 180 binary64) (/.f64 (atan.f64 (*.f64 (/.f64 #s(literal 1 binary64) B) (-.f64 (-.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64))))))) (PI.f64)))

   1. Initial program 61.1%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around -inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\left(1 + \frac{C}{B}\right) - \frac{A}{B}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(1 + \left(\frac{C}{B} - \frac{A}{B}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      2. div-sub N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(1 + \color{blue}{\frac{C - A}{B}}\right)}{\mathsf{PI}\left(\right)} \]

      3. +-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C - A}{B} + 1\right)}}{\mathsf{PI}\left(\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C - A}{B} + 1\right)}}{\mathsf{PI}\left(\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\frac{C - A}{B}} + 1\right)}{\mathsf{PI}\left(\right)} \]

      6. lower--.f64 72.9

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{C - A}}{B} + 1\right)}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 72.9%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C - A}{B} + 1\right)}}{\mathsf{PI}\left(\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;180 \cdot \frac{\tan^{-1} \left({B}^{-1} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \leq -40:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - A}{B} - 1\right) \cdot 180}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;180 \cdot \frac{\tan^{-1} \left({B}^{-1} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \leq 0:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\mathsf{fma}\left(\frac{B}{C}, -0.5, 0\right)\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{C - A}{B} + 1\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 180 \cdot \frac{\tan^{-1} \left({B}^{-1} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}\\ t_1 := \frac{C - A}{B}\\ \mathbf{if}\;t\_0 \leq -40:\\ \;\;\;\;\frac{\tan^{-1} \left(t\_1 - 1\right) \cdot 180}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-13}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{B}{A} \cdot 0.5\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(t\_1 + 1\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (let* ((t_0
         (*
          180.0
          (/
           (atan
            (*
             (pow B -1.0)
             (- (- C A) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))
           (PI))))
        (t_1 (/ (- C A) B)))
   (if (<= t_0 -40.0)
     (/ (* (atan (- t_1 1.0)) 180.0) (PI))
     (if (<= t_0 5e-13)
       (* 180.0 (/ (atan (* (/ B A) 0.5)) (PI)))
       (* 180.0 (/ (atan (+ t_1 1.0)) (PI)))))))

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal 180 binary64) (/.f64 (atan.f64 (*.f64 (/.f64 #s(literal 1 binary64) B) (-.f64 (-.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64))))))) (PI.f64))) < -40

   1. Initial program 54.0%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto 180 \cdot \color{blue}{\frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{180 \cdot \tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{180 \cdot \tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}} \]

   4. Applied rewrites 87.9%

      \[\leadsto \color{blue}{\frac{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(B, A - C\right)}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)}} \]

   5. Taylor expanded in B around inf

      \[\leadsto \frac{\tan^{-1} \color{blue}{\left(\frac{C}{B} - \left(1 + \frac{A}{B}\right)\right)} \cdot 180}{\mathsf{PI}\left(\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\tan^{-1} \left(\frac{C}{B} - \color{blue}{\left(\frac{A}{B} + 1\right)}\right) \cdot 180}{\mathsf{PI}\left(\right)} \]

      2. associate--r+ N/A

         \[\leadsto \frac{\tan^{-1} \color{blue}{\left(\left(\frac{C}{B} - \frac{A}{B}\right) - 1\right)} \cdot 180}{\mathsf{PI}\left(\right)} \]

      3. div-sub N/A

         \[\leadsto \frac{\tan^{-1} \left(\color{blue}{\frac{C - A}{B}} - 1\right) \cdot 180}{\mathsf{PI}\left(\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\tan^{-1} \color{blue}{\left(\frac{C - A}{B} - 1\right)} \cdot 180}{\mathsf{PI}\left(\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\tan^{-1} \left(\color{blue}{\frac{C - A}{B}} - 1\right) \cdot 180}{\mathsf{PI}\left(\right)} \]

      6. lower--.f64 76.5

         \[\leadsto \frac{\tan^{-1} \left(\frac{\color{blue}{C - A}}{B} - 1\right) \cdot 180}{\mathsf{PI}\left(\right)} \]

   7. Applied rewrites 76.5%

      \[\leadsto \frac{\tan^{-1} \color{blue}{\left(\frac{C - A}{B} - 1\right)} \cdot 180}{\mathsf{PI}\left(\right)} \]

   if -40 < (*.f64 #s(literal 180 binary64) (/.f64 (atan.f64 (*.f64 (/.f64 #s(literal 1 binary64) B) (-.f64 (-.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64))))))) (PI.f64))) < 4.9999999999999999e-13

   1. Initial program 17.0%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{B}{A}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{B}{A} \cdot \frac{1}{2}\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{B}{A} \cdot \frac{1}{2}\right)}}{\mathsf{PI}\left(\right)} \]

      3. lower-/.f64 42.7

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\frac{B}{A}} \cdot 0.5\right)}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 42.7%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{B}{A} \cdot 0.5\right)}}{\mathsf{PI}\left(\right)} \]

   if 4.9999999999999999e-13 < (*.f64 #s(literal 180 binary64) (/.f64 (atan.f64 (*.f64 (/.f64 #s(literal 1 binary64) B) (-.f64 (-.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64))))))) (PI.f64)))

   1. Initial program 61.7%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around -inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\left(1 + \frac{C}{B}\right) - \frac{A}{B}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(1 + \left(\frac{C}{B} - \frac{A}{B}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      2. div-sub N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(1 + \color{blue}{\frac{C - A}{B}}\right)}{\mathsf{PI}\left(\right)} \]

      3. +-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C - A}{B} + 1\right)}}{\mathsf{PI}\left(\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C - A}{B} + 1\right)}}{\mathsf{PI}\left(\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\frac{C - A}{B}} + 1\right)}{\mathsf{PI}\left(\right)} \]

      6. lower--.f64 73.5

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{C - A}}{B} + 1\right)}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 73.5%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C - A}{B} + 1\right)}}{\mathsf{PI}\left(\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;180 \cdot \frac{\tan^{-1} \left({B}^{-1} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \leq -40:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - A}{B} - 1\right) \cdot 180}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;180 \cdot \frac{\tan^{-1} \left({B}^{-1} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \leq 5 \cdot 10^{-13}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{B}{A} \cdot 0.5\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{C - A}{B} + 1\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 180 \cdot \frac{\tan^{-1} \left({B}^{-1} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}\\ t_1 := \frac{C - A}{B}\\ \mathbf{if}\;t\_0 \leq -40:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(t\_1 - 1\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-13}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{B}{A} \cdot 0.5\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(t\_1 + 1\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (let* ((t_0
         (*
          180.0
          (/
           (atan
            (*
             (pow B -1.0)
             (- (- C A) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))
           (PI))))
        (t_1 (/ (- C A) B)))
   (if (<= t_0 -40.0)
     (* 180.0 (/ (atan (- t_1 1.0)) (PI)))
     (if (<= t_0 5e-13)
       (* 180.0 (/ (atan (* (/ B A) 0.5)) (PI)))
       (* 180.0 (/ (atan (+ t_1 1.0)) (PI)))))))

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal 180 binary64) (/.f64 (atan.f64 (*.f64 (/.f64 #s(literal 1 binary64) B) (-.f64 (-.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64))))))) (PI.f64))) < -40

   1. Initial program 54.0%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C}{B} - \left(1 + \frac{A}{B}\right)\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{C}{B} - \color{blue}{\left(\frac{A}{B} + 1\right)}\right)}{\mathsf{PI}\left(\right)} \]

      2. associate--r+ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\left(\frac{C}{B} - \frac{A}{B}\right) - 1\right)}}{\mathsf{PI}\left(\right)} \]

      3. div-sub N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\frac{C - A}{B}} - 1\right)}{\mathsf{PI}\left(\right)} \]

      4. lower--.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C - A}{B} - 1\right)}}{\mathsf{PI}\left(\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\frac{C - A}{B}} - 1\right)}{\mathsf{PI}\left(\right)} \]

      6. lower--.f64 76.5

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{C - A}}{B} - 1\right)}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 76.5%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C - A}{B} - 1\right)}}{\mathsf{PI}\left(\right)} \]

   if -40 < (*.f64 #s(literal 180 binary64) (/.f64 (atan.f64 (*.f64 (/.f64 #s(literal 1 binary64) B) (-.f64 (-.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64))))))) (PI.f64))) < 4.9999999999999999e-13

   1. Initial program 17.0%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{B}{A}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{B}{A} \cdot \frac{1}{2}\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{B}{A} \cdot \frac{1}{2}\right)}}{\mathsf{PI}\left(\right)} \]

      3. lower-/.f64 42.7

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\frac{B}{A}} \cdot 0.5\right)}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 42.7%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{B}{A} \cdot 0.5\right)}}{\mathsf{PI}\left(\right)} \]

   if 4.9999999999999999e-13 < (*.f64 #s(literal 180 binary64) (/.f64 (atan.f64 (*.f64 (/.f64 #s(literal 1 binary64) B) (-.f64 (-.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64))))))) (PI.f64)))

   1. Initial program 61.7%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around -inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\left(1 + \frac{C}{B}\right) - \frac{A}{B}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(1 + \left(\frac{C}{B} - \frac{A}{B}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      2. div-sub N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(1 + \color{blue}{\frac{C - A}{B}}\right)}{\mathsf{PI}\left(\right)} \]

      3. +-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C - A}{B} + 1\right)}}{\mathsf{PI}\left(\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C - A}{B} + 1\right)}}{\mathsf{PI}\left(\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\frac{C - A}{B}} + 1\right)}{\mathsf{PI}\left(\right)} \]

      6. lower--.f64 73.5

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{C - A}}{B} + 1\right)}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 73.5%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C - A}{B} + 1\right)}}{\mathsf{PI}\left(\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;180 \cdot \frac{\tan^{-1} \left({B}^{-1} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \leq -40:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{C - A}{B} - 1\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;180 \cdot \frac{\tan^{-1} \left({B}^{-1} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \leq 5 \cdot 10^{-13}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{B}{A} \cdot 0.5\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{C - A}{B} + 1\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 180 \cdot \frac{\tan^{-1} \left({B}^{-1} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{if}\;t\_0 \leq -40:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - B}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-13}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{B}{A} \cdot 0.5\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{C - A}{B} + 1\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (let* ((t_0
         (*
          180.0
          (/
           (atan
            (*
             (pow B -1.0)
             (- (- C A) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))
           (PI)))))
   (if (<= t_0 -40.0)
     (/ (* (atan (/ (- C B) B)) 180.0) (PI))
     (if (<= t_0 5e-13)
       (* 180.0 (/ (atan (* (/ B A) 0.5)) (PI)))
       (* 180.0 (/ (atan (+ (/ (- C A) B) 1.0)) (PI)))))))

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal 180 binary64) (/.f64 (atan.f64 (*.f64 (/.f64 #s(literal 1 binary64) B) (-.f64 (-.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64))))))) (PI.f64))) < -40

   1. Initial program 54.0%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto 180 \cdot \color{blue}{\frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{180 \cdot \tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{180 \cdot \tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}} \]

   4. Applied rewrites 87.9%

      \[\leadsto \color{blue}{\frac{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(B, A - C\right)}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)}} \]

   5. Taylor expanded in A around 0

      \[\leadsto \frac{\tan^{-1} \left(\frac{\color{blue}{C - \sqrt{{B}^{2} + {C}^{2}}}}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)} \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{\tan^{-1} \left(\frac{\color{blue}{C - \sqrt{{B}^{2} + {C}^{2}}}}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)} \]

      2. unpow2 N/A

         \[\leadsto \frac{\tan^{-1} \left(\frac{C - \sqrt{\color{blue}{B \cdot B} + {C}^{2}}}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{\tan^{-1} \left(\frac{C - \sqrt{B \cdot B + \color{blue}{C \cdot C}}}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)} \]

      4. lower-hypot.f64 75.1

         \[\leadsto \frac{\tan^{-1} \left(\frac{C - \color{blue}{\mathsf{hypot}\left(B, C\right)}}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)} \]

   7. Applied rewrites 75.1%

      \[\leadsto \frac{\tan^{-1} \left(\frac{\color{blue}{C - \mathsf{hypot}\left(B, C\right)}}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)} \]

   8. Taylor expanded in C around 0

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

   10. Applied rewrites 66.7%

       \[\leadsto \frac{\tan^{-1} \left(\frac{C - \color{blue}{B}}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)} \]

   if -40 < (*.f64 #s(literal 180 binary64) (/.f64 (atan.f64 (*.f64 (/.f64 #s(literal 1 binary64) B) (-.f64 (-.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64))))))) (PI.f64))) < 4.9999999999999999e-13

   1. Initial program 17.0%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{B}{A}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{B}{A} \cdot \frac{1}{2}\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{B}{A} \cdot \frac{1}{2}\right)}}{\mathsf{PI}\left(\right)} \]

      3. lower-/.f64 42.7

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\frac{B}{A}} \cdot 0.5\right)}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 42.7%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{B}{A} \cdot 0.5\right)}}{\mathsf{PI}\left(\right)} \]

   if 4.9999999999999999e-13 < (*.f64 #s(literal 180 binary64) (/.f64 (atan.f64 (*.f64 (/.f64 #s(literal 1 binary64) B) (-.f64 (-.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64))))))) (PI.f64)))

   1. Initial program 61.7%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around -inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\left(1 + \frac{C}{B}\right) - \frac{A}{B}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(1 + \left(\frac{C}{B} - \frac{A}{B}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      2. div-sub N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(1 + \color{blue}{\frac{C - A}{B}}\right)}{\mathsf{PI}\left(\right)} \]

      3. +-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C - A}{B} + 1\right)}}{\mathsf{PI}\left(\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C - A}{B} + 1\right)}}{\mathsf{PI}\left(\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\frac{C - A}{B}} + 1\right)}{\mathsf{PI}\left(\right)} \]

      6. lower--.f64 73.5

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{C - A}}{B} + 1\right)}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 73.5%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C - A}{B} + 1\right)}}{\mathsf{PI}\left(\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;180 \cdot \frac{\tan^{-1} \left({B}^{-1} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \leq -40:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - B}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;180 \cdot \frac{\tan^{-1} \left({B}^{-1} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \leq 5 \cdot 10^{-13}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{B}{A} \cdot 0.5\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{C - A}{B} + 1\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;A \leq -5.5 \cdot 10^{+103}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{0.5 \cdot \mathsf{fma}\left(\frac{C}{A}, B, B\right)}{A}\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;A \leq 1.9 \cdot 10^{-40}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left({B}^{-1} \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{\mathsf{hypot}\left(B, A\right) + A}{-B}\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= A -5.5e+103)
   (* 180.0 (/ (atan (/ (* 0.5 (fma (/ C A) B B)) A)) (PI)))
   (if (<= A 1.9e-40)
     (* 180.0 (/ (atan (* (pow B -1.0) (- C (hypot B C)))) (PI)))
     (* 180.0 (/ (atan (/ (+ (hypot B A) A) (- B))) (PI))))))

Derivation

1. Split input into 3 regimes

2. if A < -5.50000000000000001e103

   1. Initial program 18.3%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-1 \cdot \frac{\frac{-1}{2} \cdot B + \frac{-1}{2} \cdot \frac{B \cdot C}{A}}{A}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-1 \cdot \left(\frac{-1}{2} \cdot B + \frac{-1}{2} \cdot \frac{B \cdot C}{A}\right)}{A}\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-1 \cdot \left(\frac{-1}{2} \cdot B + \frac{-1}{2} \cdot \frac{B \cdot C}{A}\right)}{A}\right)}}{\mathsf{PI}\left(\right)} \]

      3. distribute-lft-out N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1 \cdot \color{blue}{\left(\frac{-1}{2} \cdot \left(B + \frac{B \cdot C}{A}\right)\right)}}{A}\right)}{\mathsf{PI}\left(\right)} \]

      4. associate-*r* N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{\left(-1 \cdot \frac{-1}{2}\right) \cdot \left(B + \frac{B \cdot C}{A}\right)}}{A}\right)}{\mathsf{PI}\left(\right)} \]

      5. metadata-eval N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{\frac{1}{2}} \cdot \left(B + \frac{B \cdot C}{A}\right)}{A}\right)}{\mathsf{PI}\left(\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{\frac{1}{2} \cdot \left(B + \frac{B \cdot C}{A}\right)}}{A}\right)}{\mathsf{PI}\left(\right)} \]

      7. +-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\frac{1}{2} \cdot \color{blue}{\left(\frac{B \cdot C}{A} + B\right)}}{A}\right)}{\mathsf{PI}\left(\right)} \]

      8. associate-/l* N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\frac{1}{2} \cdot \left(\color{blue}{B \cdot \frac{C}{A}} + B\right)}{A}\right)}{\mathsf{PI}\left(\right)} \]

      9. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\frac{1}{2} \cdot \left(\color{blue}{\frac{C}{A} \cdot B} + B\right)}{A}\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-fma.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\frac{1}{2} \cdot \color{blue}{\mathsf{fma}\left(\frac{C}{A}, B, B\right)}}{A}\right)}{\mathsf{PI}\left(\right)} \]

      11. lower-/.f64 86.9

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{0.5 \cdot \mathsf{fma}\left(\color{blue}{\frac{C}{A}}, B, B\right)}{A}\right)}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 86.9%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{0.5 \cdot \mathsf{fma}\left(\frac{C}{A}, B, B\right)}{A}\right)}}{\mathsf{PI}\left(\right)} \]

   if -5.50000000000000001e103 < A < 1.8999999999999999e-40

   1. Initial program 53.0%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \color{blue}{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \color{blue}{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)}{\mathsf{PI}\left(\right)} \]

      2. unpow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(C - \sqrt{\color{blue}{B \cdot B} + {C}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      3. unpow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(C - \sqrt{B \cdot B + \color{blue}{C \cdot C}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      4. lower-hypot.f64 78.9

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(C - \color{blue}{\mathsf{hypot}\left(B, C\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 78.9%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \color{blue}{\left(C - \mathsf{hypot}\left(B, C\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

   if 1.8999999999999999e-40 < A

   1. Initial program 70.5%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in C around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-1 \cdot \frac{A + \sqrt{{A}^{2} + {B}^{2}}}{B}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{A + \sqrt{{A}^{2} + {B}^{2}}}{B}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      2. distribute-neg-frac2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{A + \sqrt{{A}^{2} + {B}^{2}}}{\mathsf{neg}\left(B\right)}\right)}}{\mathsf{PI}\left(\right)} \]

      3. mul-1-neg N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{A + \sqrt{{A}^{2} + {B}^{2}}}{\color{blue}{-1 \cdot B}}\right)}{\mathsf{PI}\left(\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{A + \sqrt{{A}^{2} + {B}^{2}}}{-1 \cdot B}\right)}}{\mathsf{PI}\left(\right)} \]

      5. +-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{\sqrt{{A}^{2} + {B}^{2}} + A}}{-1 \cdot B}\right)}{\mathsf{PI}\left(\right)} \]

      6. lower-+.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{\sqrt{{A}^{2} + {B}^{2}} + A}}{-1 \cdot B}\right)}{\mathsf{PI}\left(\right)} \]

      7. +-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\sqrt{\color{blue}{{B}^{2} + {A}^{2}}} + A}{-1 \cdot B}\right)}{\mathsf{PI}\left(\right)} \]

      8. unpow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\sqrt{\color{blue}{B \cdot B} + {A}^{2}} + A}{-1 \cdot B}\right)}{\mathsf{PI}\left(\right)} \]

      9. unpow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\sqrt{B \cdot B + \color{blue}{A \cdot A}} + A}{-1 \cdot B}\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-hypot.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{\mathsf{hypot}\left(B, A\right)} + A}{-1 \cdot B}\right)}{\mathsf{PI}\left(\right)} \]

      11. mul-1-neg N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\mathsf{hypot}\left(B, A\right) + A}{\color{blue}{\mathsf{neg}\left(B\right)}}\right)}{\mathsf{PI}\left(\right)} \]

      12. lower-neg.f64 83.6

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\mathsf{hypot}\left(B, A\right) + A}{\color{blue}{-B}}\right)}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 83.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{\mathsf{hypot}\left(B, A\right) + A}{-B}\right)}}{\mathsf{PI}\left(\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -5.5 \cdot 10^{+103}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{0.5 \cdot \mathsf{fma}\left(\frac{C}{A}, B, B\right)}{A}\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;A \leq 1.9 \cdot 10^{-40}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left({B}^{-1} \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{\mathsf{hypot}\left(B, A\right) + A}{-B}\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 78.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;A \leq -5.5 \cdot 10^{+103}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{0.5 \cdot \mathsf{fma}\left(\frac{C}{A}, B, B\right)}{A}\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;A \leq 1.9 \cdot 10^{-40}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - \mathsf{hypot}\left(B, C\right)}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{\mathsf{hypot}\left(B, A\right) + A}{-B}\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= A -5.5e+103)
   (* 180.0 (/ (atan (/ (* 0.5 (fma (/ C A) B B)) A)) (PI)))
   (if (<= A 1.9e-40)
     (/ (* (atan (/ (- C (hypot B C)) B)) 180.0) (PI))
     (* 180.0 (/ (atan (/ (+ (hypot B A) A) (- B))) (PI))))))

Derivation

1. Split input into 3 regimes

2. if A < -5.50000000000000001e103

   1. Initial program 18.3%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-1 \cdot \frac{\frac{-1}{2} \cdot B + \frac{-1}{2} \cdot \frac{B \cdot C}{A}}{A}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-1 \cdot \left(\frac{-1}{2} \cdot B + \frac{-1}{2} \cdot \frac{B \cdot C}{A}\right)}{A}\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-1 \cdot \left(\frac{-1}{2} \cdot B + \frac{-1}{2} \cdot \frac{B \cdot C}{A}\right)}{A}\right)}}{\mathsf{PI}\left(\right)} \]

      3. distribute-lft-out N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1 \cdot \color{blue}{\left(\frac{-1}{2} \cdot \left(B + \frac{B \cdot C}{A}\right)\right)}}{A}\right)}{\mathsf{PI}\left(\right)} \]

      4. associate-*r* N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{\left(-1 \cdot \frac{-1}{2}\right) \cdot \left(B + \frac{B \cdot C}{A}\right)}}{A}\right)}{\mathsf{PI}\left(\right)} \]

      5. metadata-eval N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{\frac{1}{2}} \cdot \left(B + \frac{B \cdot C}{A}\right)}{A}\right)}{\mathsf{PI}\left(\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{\frac{1}{2} \cdot \left(B + \frac{B \cdot C}{A}\right)}}{A}\right)}{\mathsf{PI}\left(\right)} \]

      7. +-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\frac{1}{2} \cdot \color{blue}{\left(\frac{B \cdot C}{A} + B\right)}}{A}\right)}{\mathsf{PI}\left(\right)} \]

      8. associate-/l* N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\frac{1}{2} \cdot \left(\color{blue}{B \cdot \frac{C}{A}} + B\right)}{A}\right)}{\mathsf{PI}\left(\right)} \]

      9. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\frac{1}{2} \cdot \left(\color{blue}{\frac{C}{A} \cdot B} + B\right)}{A}\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-fma.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\frac{1}{2} \cdot \color{blue}{\mathsf{fma}\left(\frac{C}{A}, B, B\right)}}{A}\right)}{\mathsf{PI}\left(\right)} \]

      11. lower-/.f64 86.9

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{0.5 \cdot \mathsf{fma}\left(\color{blue}{\frac{C}{A}}, B, B\right)}{A}\right)}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 86.9%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{0.5 \cdot \mathsf{fma}\left(\frac{C}{A}, B, B\right)}{A}\right)}}{\mathsf{PI}\left(\right)} \]

   if -5.50000000000000001e103 < A < 1.8999999999999999e-40

   1. Initial program 53.0%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto 180 \cdot \color{blue}{\frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{180 \cdot \tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{180 \cdot \tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}} \]

   4. Applied rewrites 80.1%

      \[\leadsto \color{blue}{\frac{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(B, A - C\right)}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)}} \]

   5. Taylor expanded in A around 0

      \[\leadsto \frac{\tan^{-1} \left(\frac{\color{blue}{C - \sqrt{{B}^{2} + {C}^{2}}}}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)} \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{\tan^{-1} \left(\frac{\color{blue}{C - \sqrt{{B}^{2} + {C}^{2}}}}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)} \]

      2. unpow2 N/A

         \[\leadsto \frac{\tan^{-1} \left(\frac{C - \sqrt{\color{blue}{B \cdot B} + {C}^{2}}}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{\tan^{-1} \left(\frac{C - \sqrt{B \cdot B + \color{blue}{C \cdot C}}}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)} \]

      4. lower-hypot.f64 78.9

         \[\leadsto \frac{\tan^{-1} \left(\frac{C - \color{blue}{\mathsf{hypot}\left(B, C\right)}}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)} \]

   7. Applied rewrites 78.9%

      \[\leadsto \frac{\tan^{-1} \left(\frac{\color{blue}{C - \mathsf{hypot}\left(B, C\right)}}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)} \]

   if 1.8999999999999999e-40 < A

   1. Initial program 70.5%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in C around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-1 \cdot \frac{A + \sqrt{{A}^{2} + {B}^{2}}}{B}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{A + \sqrt{{A}^{2} + {B}^{2}}}{B}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      2. distribute-neg-frac2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{A + \sqrt{{A}^{2} + {B}^{2}}}{\mathsf{neg}\left(B\right)}\right)}}{\mathsf{PI}\left(\right)} \]

      3. mul-1-neg N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{A + \sqrt{{A}^{2} + {B}^{2}}}{\color{blue}{-1 \cdot B}}\right)}{\mathsf{PI}\left(\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{A + \sqrt{{A}^{2} + {B}^{2}}}{-1 \cdot B}\right)}}{\mathsf{PI}\left(\right)} \]

      5. +-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{\sqrt{{A}^{2} + {B}^{2}} + A}}{-1 \cdot B}\right)}{\mathsf{PI}\left(\right)} \]

      6. lower-+.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{\sqrt{{A}^{2} + {B}^{2}} + A}}{-1 \cdot B}\right)}{\mathsf{PI}\left(\right)} \]

      7. +-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\sqrt{\color{blue}{{B}^{2} + {A}^{2}}} + A}{-1 \cdot B}\right)}{\mathsf{PI}\left(\right)} \]

      8. unpow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\sqrt{\color{blue}{B \cdot B} + {A}^{2}} + A}{-1 \cdot B}\right)}{\mathsf{PI}\left(\right)} \]

      9. unpow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\sqrt{B \cdot B + \color{blue}{A \cdot A}} + A}{-1 \cdot B}\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-hypot.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{\mathsf{hypot}\left(B, A\right)} + A}{-1 \cdot B}\right)}{\mathsf{PI}\left(\right)} \]

      11. mul-1-neg N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\mathsf{hypot}\left(B, A\right) + A}{\color{blue}{\mathsf{neg}\left(B\right)}}\right)}{\mathsf{PI}\left(\right)} \]

      12. lower-neg.f64 83.6

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\mathsf{hypot}\left(B, A\right) + A}{\color{blue}{-B}}\right)}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 83.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{\mathsf{hypot}\left(B, A\right) + A}{-B}\right)}}{\mathsf{PI}\left(\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -5.5 \cdot 10^{+103}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{0.5 \cdot \mathsf{fma}\left(\frac{C}{A}, B, B\right)}{A}\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;A \leq 1.9 \cdot 10^{-40}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - \mathsf{hypot}\left(B, C\right)}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{\mathsf{hypot}\left(B, A\right) + A}{-B}\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 75.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;A \leq -5.5 \cdot 10^{+103}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{0.5 \cdot \mathsf{fma}\left(\frac{C}{A}, B, B\right)}{A}\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;A \leq 2.75 \cdot 10^{+20}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - \mathsf{hypot}\left(B, C\right)}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{C - A}{B} - 1\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= A -5.5e+103)
   (* 180.0 (/ (atan (/ (* 0.5 (fma (/ C A) B B)) A)) (PI)))
   (if (<= A 2.75e+20)
     (/ (* (atan (/ (- C (hypot B C)) B)) 180.0) (PI))
     (* 180.0 (/ (atan (- (/ (- C A) B) 1.0)) (PI))))))

Derivation

1. Split input into 3 regimes

2. if A < -5.50000000000000001e103

   1. Initial program 18.3%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-1 \cdot \frac{\frac{-1}{2} \cdot B + \frac{-1}{2} \cdot \frac{B \cdot C}{A}}{A}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-1 \cdot \left(\frac{-1}{2} \cdot B + \frac{-1}{2} \cdot \frac{B \cdot C}{A}\right)}{A}\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-1 \cdot \left(\frac{-1}{2} \cdot B + \frac{-1}{2} \cdot \frac{B \cdot C}{A}\right)}{A}\right)}}{\mathsf{PI}\left(\right)} \]

      3. distribute-lft-out N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1 \cdot \color{blue}{\left(\frac{-1}{2} \cdot \left(B + \frac{B \cdot C}{A}\right)\right)}}{A}\right)}{\mathsf{PI}\left(\right)} \]

      4. associate-*r* N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{\left(-1 \cdot \frac{-1}{2}\right) \cdot \left(B + \frac{B \cdot C}{A}\right)}}{A}\right)}{\mathsf{PI}\left(\right)} \]

      5. metadata-eval N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{\frac{1}{2}} \cdot \left(B + \frac{B \cdot C}{A}\right)}{A}\right)}{\mathsf{PI}\left(\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{\frac{1}{2} \cdot \left(B + \frac{B \cdot C}{A}\right)}}{A}\right)}{\mathsf{PI}\left(\right)} \]

      7. +-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\frac{1}{2} \cdot \color{blue}{\left(\frac{B \cdot C}{A} + B\right)}}{A}\right)}{\mathsf{PI}\left(\right)} \]

      8. associate-/l* N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\frac{1}{2} \cdot \left(\color{blue}{B \cdot \frac{C}{A}} + B\right)}{A}\right)}{\mathsf{PI}\left(\right)} \]

      9. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\frac{1}{2} \cdot \left(\color{blue}{\frac{C}{A} \cdot B} + B\right)}{A}\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-fma.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\frac{1}{2} \cdot \color{blue}{\mathsf{fma}\left(\frac{C}{A}, B, B\right)}}{A}\right)}{\mathsf{PI}\left(\right)} \]

      11. lower-/.f64 86.9

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{0.5 \cdot \mathsf{fma}\left(\color{blue}{\frac{C}{A}}, B, B\right)}{A}\right)}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 86.9%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{0.5 \cdot \mathsf{fma}\left(\frac{C}{A}, B, B\right)}{A}\right)}}{\mathsf{PI}\left(\right)} \]

   if -5.50000000000000001e103 < A < 2.75e20

   1. Initial program 52.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto 180 \cdot \color{blue}{\frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{180 \cdot \tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{180 \cdot \tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}} \]

   4. Applied rewrites 79.0%

      \[\leadsto \color{blue}{\frac{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(B, A - C\right)}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)}} \]

   5. Taylor expanded in A around 0

      \[\leadsto \frac{\tan^{-1} \left(\frac{\color{blue}{C - \sqrt{{B}^{2} + {C}^{2}}}}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)} \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{\tan^{-1} \left(\frac{\color{blue}{C - \sqrt{{B}^{2} + {C}^{2}}}}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)} \]

      2. unpow2 N/A

         \[\leadsto \frac{\tan^{-1} \left(\frac{C - \sqrt{\color{blue}{B \cdot B} + {C}^{2}}}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{\tan^{-1} \left(\frac{C - \sqrt{B \cdot B + \color{blue}{C \cdot C}}}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)} \]

      4. lower-hypot.f64 77.2

         \[\leadsto \frac{\tan^{-1} \left(\frac{C - \color{blue}{\mathsf{hypot}\left(B, C\right)}}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)} \]

   7. Applied rewrites 77.2%

      \[\leadsto \frac{\tan^{-1} \left(\frac{\color{blue}{C - \mathsf{hypot}\left(B, C\right)}}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)} \]

   if 2.75e20 < A

   1. Initial program 77.2%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C}{B} - \left(1 + \frac{A}{B}\right)\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{C}{B} - \color{blue}{\left(\frac{A}{B} + 1\right)}\right)}{\mathsf{PI}\left(\right)} \]

      2. associate--r+ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\left(\frac{C}{B} - \frac{A}{B}\right) - 1\right)}}{\mathsf{PI}\left(\right)} \]

      3. div-sub N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\frac{C - A}{B}} - 1\right)}{\mathsf{PI}\left(\right)} \]

      4. lower--.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C - A}{B} - 1\right)}}{\mathsf{PI}\left(\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\frac{C - A}{B}} - 1\right)}{\mathsf{PI}\left(\right)} \]

      6. lower--.f64 84.8

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{C - A}}{B} - 1\right)}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 84.8%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -5.5 \cdot 10^{+103}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{0.5 \cdot \mathsf{fma}\left(\frac{C}{A}, B, B\right)}{A}\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;A \leq 2.75 \cdot 10^{+20}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - \mathsf{hypot}\left(B, C\right)}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{C - A}{B} - 1\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 75.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;A \leq -5.5 \cdot 10^{+103}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{0.5 \cdot \mathsf{fma}\left(\frac{C}{A}, B, B\right)}{A}\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;A \leq 2.75 \cdot 10^{+20}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{C - \mathsf{hypot}\left(B, C\right)}{B}\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{C - A}{B} - 1\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= A -5.5e+103)
   (* 180.0 (/ (atan (/ (* 0.5 (fma (/ C A) B B)) A)) (PI)))
   (if (<= A 2.75e+20)
     (* 180.0 (/ (atan (/ (- C (hypot B C)) B)) (PI)))
     (* 180.0 (/ (atan (- (/ (- C A) B) 1.0)) (PI))))))

Derivation

1. Split input into 3 regimes

2. if A < -5.50000000000000001e103

   1. Initial program 18.3%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-1 \cdot \frac{\frac{-1}{2} \cdot B + \frac{-1}{2} \cdot \frac{B \cdot C}{A}}{A}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-1 \cdot \left(\frac{-1}{2} \cdot B + \frac{-1}{2} \cdot \frac{B \cdot C}{A}\right)}{A}\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-1 \cdot \left(\frac{-1}{2} \cdot B + \frac{-1}{2} \cdot \frac{B \cdot C}{A}\right)}{A}\right)}}{\mathsf{PI}\left(\right)} \]

      3. distribute-lft-out N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1 \cdot \color{blue}{\left(\frac{-1}{2} \cdot \left(B + \frac{B \cdot C}{A}\right)\right)}}{A}\right)}{\mathsf{PI}\left(\right)} \]

      4. associate-*r* N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{\left(-1 \cdot \frac{-1}{2}\right) \cdot \left(B + \frac{B \cdot C}{A}\right)}}{A}\right)}{\mathsf{PI}\left(\right)} \]

      5. metadata-eval N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{\frac{1}{2}} \cdot \left(B + \frac{B \cdot C}{A}\right)}{A}\right)}{\mathsf{PI}\left(\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{\frac{1}{2} \cdot \left(B + \frac{B \cdot C}{A}\right)}}{A}\right)}{\mathsf{PI}\left(\right)} \]

      7. +-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\frac{1}{2} \cdot \color{blue}{\left(\frac{B \cdot C}{A} + B\right)}}{A}\right)}{\mathsf{PI}\left(\right)} \]

      8. associate-/l* N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\frac{1}{2} \cdot \left(\color{blue}{B \cdot \frac{C}{A}} + B\right)}{A}\right)}{\mathsf{PI}\left(\right)} \]

      9. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\frac{1}{2} \cdot \left(\color{blue}{\frac{C}{A} \cdot B} + B\right)}{A}\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-fma.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\frac{1}{2} \cdot \color{blue}{\mathsf{fma}\left(\frac{C}{A}, B, B\right)}}{A}\right)}{\mathsf{PI}\left(\right)} \]

      11. lower-/.f64 86.9

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{0.5 \cdot \mathsf{fma}\left(\color{blue}{\frac{C}{A}}, B, B\right)}{A}\right)}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 86.9%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{0.5 \cdot \mathsf{fma}\left(\frac{C}{A}, B, B\right)}{A}\right)}}{\mathsf{PI}\left(\right)} \]

   if -5.50000000000000001e103 < A < 2.75e20

   1. Initial program 52.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C - \sqrt{{B}^{2} + {C}^{2}}}{B}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C - \sqrt{{B}^{2} + {C}^{2}}}{B}\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower--.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{C - \sqrt{{B}^{2} + {C}^{2}}}}{B}\right)}{\mathsf{PI}\left(\right)} \]

      3. unpow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{C - \sqrt{\color{blue}{B \cdot B} + {C}^{2}}}{B}\right)}{\mathsf{PI}\left(\right)} \]

      4. unpow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{C - \sqrt{B \cdot B + \color{blue}{C \cdot C}}}{B}\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-hypot.f64 77.2

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{C - \color{blue}{\mathsf{hypot}\left(B, C\right)}}{B}\right)}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 77.2%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C - \mathsf{hypot}\left(B, C\right)}{B}\right)}}{\mathsf{PI}\left(\right)} \]

   if 2.75e20 < A

   1. Initial program 77.2%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C}{B} - \left(1 + \frac{A}{B}\right)\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{C}{B} - \color{blue}{\left(\frac{A}{B} + 1\right)}\right)}{\mathsf{PI}\left(\right)} \]

      2. associate--r+ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\left(\frac{C}{B} - \frac{A}{B}\right) - 1\right)}}{\mathsf{PI}\left(\right)} \]

      3. div-sub N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\frac{C - A}{B}} - 1\right)}{\mathsf{PI}\left(\right)} \]

      4. lower--.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C - A}{B} - 1\right)}}{\mathsf{PI}\left(\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\frac{C - A}{B}} - 1\right)}{\mathsf{PI}\left(\right)} \]

      6. lower--.f64 84.8

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{C - A}}{B} - 1\right)}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 84.8%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -5.5 \cdot 10^{+103}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{0.5 \cdot \mathsf{fma}\left(\frac{C}{A}, B, B\right)}{A}\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;A \leq 2.75 \cdot 10^{+20}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{C - \mathsf{hypot}\left(B, C\right)}{B}\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{C - A}{B} - 1\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 59.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;A \leq -9.6 \cdot 10^{+54}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{B}{A} \cdot 0.5\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;A \leq -1.12 \cdot 10^{-148}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - B}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;A \leq 5.2 \cdot 10^{-183}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{C}{B} + 1\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{B + A}{-B}\right) \cdot 180}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= A -9.6e+54)
   (* 180.0 (/ (atan (* (/ B A) 0.5)) (PI)))
   (if (<= A -1.12e-148)
     (/ (* (atan (/ (- C B) B)) 180.0) (PI))
     (if (<= A 5.2e-183)
       (* 180.0 (/ (atan (+ (/ C B) 1.0)) (PI)))
       (/ (* (atan (/ (+ B A) (- B))) 180.0) (PI))))))

Derivation

1. Split input into 4 regimes

2. if A < -9.59999999999999993e54

   1. Initial program 19.0%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{B}{A}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{B}{A} \cdot \frac{1}{2}\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{B}{A} \cdot \frac{1}{2}\right)}}{\mathsf{PI}\left(\right)} \]

      3. lower-/.f64 77.0

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\frac{B}{A}} \cdot 0.5\right)}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 77.0%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{B}{A} \cdot 0.5\right)}}{\mathsf{PI}\left(\right)} \]

   if -9.59999999999999993e54 < A < -1.1199999999999999e-148

   1. Initial program 52.7%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto 180 \cdot \color{blue}{\frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{180 \cdot \tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{180 \cdot \tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}} \]

   4. Applied rewrites 80.3%

      \[\leadsto \color{blue}{\frac{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(B, A - C\right)}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)}} \]

   5. Taylor expanded in A around 0

      \[\leadsto \frac{\tan^{-1} \left(\frac{\color{blue}{C - \sqrt{{B}^{2} + {C}^{2}}}}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)} \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{\tan^{-1} \left(\frac{\color{blue}{C - \sqrt{{B}^{2} + {C}^{2}}}}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)} \]

      2. unpow2 N/A

         \[\leadsto \frac{\tan^{-1} \left(\frac{C - \sqrt{\color{blue}{B \cdot B} + {C}^{2}}}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{\tan^{-1} \left(\frac{C - \sqrt{B \cdot B + \color{blue}{C \cdot C}}}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)} \]

      4. lower-hypot.f64 80.4

         \[\leadsto \frac{\tan^{-1} \left(\frac{C - \color{blue}{\mathsf{hypot}\left(B, C\right)}}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)} \]

   7. Applied rewrites 80.4%

      \[\leadsto \frac{\tan^{-1} \left(\frac{\color{blue}{C - \mathsf{hypot}\left(B, C\right)}}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)} \]

   8. Taylor expanded in C around 0

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

   10. Applied rewrites 64.9%

       \[\leadsto \frac{\tan^{-1} \left(\frac{C - \color{blue}{B}}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)} \]

   if -1.1199999999999999e-148 < A < 5.1999999999999998e-183

   1. Initial program 55.3%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around -inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\left(1 + \frac{C}{B}\right) - \frac{A}{B}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(1 + \left(\frac{C}{B} - \frac{A}{B}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      2. div-sub N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(1 + \color{blue}{\frac{C - A}{B}}\right)}{\mathsf{PI}\left(\right)} \]

      3. +-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C - A}{B} + 1\right)}}{\mathsf{PI}\left(\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C - A}{B} + 1\right)}}{\mathsf{PI}\left(\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\frac{C - A}{B}} + 1\right)}{\mathsf{PI}\left(\right)} \]

      6. lower--.f64 52.3

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{C - A}}{B} + 1\right)}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 52.3%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C - A}{B} + 1\right)}}{\mathsf{PI}\left(\right)} \]

   6. Taylor expanded in A around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{C}{B} + 1\right)}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 52.3%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{C}{B} + 1\right)}{\mathsf{PI}\left(\right)} \]

   if 5.1999999999999998e-183 < A

   1. Initial program 65.7%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in C around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-1 \cdot \frac{A + \sqrt{{A}^{2} + {B}^{2}}}{B}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{A + \sqrt{{A}^{2} + {B}^{2}}}{B}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      2. distribute-neg-frac2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{A + \sqrt{{A}^{2} + {B}^{2}}}{\mathsf{neg}\left(B\right)}\right)}}{\mathsf{PI}\left(\right)} \]

      3. mul-1-neg N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{A + \sqrt{{A}^{2} + {B}^{2}}}{\color{blue}{-1 \cdot B}}\right)}{\mathsf{PI}\left(\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{A + \sqrt{{A}^{2} + {B}^{2}}}{-1 \cdot B}\right)}}{\mathsf{PI}\left(\right)} \]

      5. +-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{\sqrt{{A}^{2} + {B}^{2}} + A}}{-1 \cdot B}\right)}{\mathsf{PI}\left(\right)} \]

      6. lower-+.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{\sqrt{{A}^{2} + {B}^{2}} + A}}{-1 \cdot B}\right)}{\mathsf{PI}\left(\right)} \]

      7. +-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\sqrt{\color{blue}{{B}^{2} + {A}^{2}}} + A}{-1 \cdot B}\right)}{\mathsf{PI}\left(\right)} \]

      8. unpow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\sqrt{\color{blue}{B \cdot B} + {A}^{2}} + A}{-1 \cdot B}\right)}{\mathsf{PI}\left(\right)} \]

      9. unpow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\sqrt{B \cdot B + \color{blue}{A \cdot A}} + A}{-1 \cdot B}\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-hypot.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{\mathsf{hypot}\left(B, A\right)} + A}{-1 \cdot B}\right)}{\mathsf{PI}\left(\right)} \]

      11. mul-1-neg N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\mathsf{hypot}\left(B, A\right) + A}{\color{blue}{\mathsf{neg}\left(B\right)}}\right)}{\mathsf{PI}\left(\right)} \]

      12. lower-neg.f64 77.7

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\mathsf{hypot}\left(B, A\right) + A}{\color{blue}{-B}}\right)}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 77.7%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{\mathsf{hypot}\left(B, A\right) + A}{-B}\right)}}{\mathsf{PI}\left(\right)} \]

   6. Taylor expanded in A around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{A + B}{-\color{blue}{B}}\right)}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 66.9%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{B + A}{-\color{blue}{B}}\right)}{\mathsf{PI}\left(\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{180 \cdot \frac{\tan^{-1} \left(\frac{B + A}{-B}\right)}{\mathsf{PI}\left(\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto 180 \cdot \color{blue}{\frac{\tan^{-1} \left(\frac{B + A}{-B}\right)}{\mathsf{PI}\left(\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{180 \cdot \tan^{-1} \left(\frac{B + A}{-B}\right)}{\mathsf{PI}\left(\right)}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{180 \cdot \tan^{-1} \left(\frac{B + A}{-B}\right)}{\mathsf{PI}\left(\right)}} \]

   10. Applied rewrites 66.9%

       \[\leadsto \color{blue}{\frac{\tan^{-1} \left(\frac{B + A}{-B}\right) \cdot 180}{\mathsf{PI}\left(\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -9.6 \cdot 10^{+54}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{B}{A} \cdot 0.5\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;A \leq -1.12 \cdot 10^{-148}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - B}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;A \leq 5.2 \cdot 10^{-183}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{C}{B} + 1\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{B + A}{-B}\right) \cdot 180}{\mathsf{PI}\left(\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 59.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;A \leq -9.6 \cdot 10^{+54}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{B}{A} \cdot 0.5\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;A \leq -1.12 \cdot 10^{-148}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - B}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;A \leq 5.2 \cdot 10^{-183}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{C}{B} + 1\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{B + A}{-B}\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= A -9.6e+54)
   (* 180.0 (/ (atan (* (/ B A) 0.5)) (PI)))
   (if (<= A -1.12e-148)
     (/ (* (atan (/ (- C B) B)) 180.0) (PI))
     (if (<= A 5.2e-183)
       (* 180.0 (/ (atan (+ (/ C B) 1.0)) (PI)))
       (* 180.0 (/ (atan (/ (+ B A) (- B))) (PI)))))))

Derivation

1. Split input into 4 regimes

2. if A < -9.59999999999999993e54

   1. Initial program 19.0%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{B}{A}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{B}{A} \cdot \frac{1}{2}\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{B}{A} \cdot \frac{1}{2}\right)}}{\mathsf{PI}\left(\right)} \]

      3. lower-/.f64 77.0

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\frac{B}{A}} \cdot 0.5\right)}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 77.0%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{B}{A} \cdot 0.5\right)}}{\mathsf{PI}\left(\right)} \]

   if -9.59999999999999993e54 < A < -1.1199999999999999e-148

   1. Initial program 52.7%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto 180 \cdot \color{blue}{\frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{180 \cdot \tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{180 \cdot \tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}} \]

   4. Applied rewrites 80.3%

      \[\leadsto \color{blue}{\frac{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(B, A - C\right)}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)}} \]

   5. Taylor expanded in A around 0

      \[\leadsto \frac{\tan^{-1} \left(\frac{\color{blue}{C - \sqrt{{B}^{2} + {C}^{2}}}}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)} \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{\tan^{-1} \left(\frac{\color{blue}{C - \sqrt{{B}^{2} + {C}^{2}}}}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)} \]

      2. unpow2 N/A

         \[\leadsto \frac{\tan^{-1} \left(\frac{C - \sqrt{\color{blue}{B \cdot B} + {C}^{2}}}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{\tan^{-1} \left(\frac{C - \sqrt{B \cdot B + \color{blue}{C \cdot C}}}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)} \]

      4. lower-hypot.f64 80.4

         \[\leadsto \frac{\tan^{-1} \left(\frac{C - \color{blue}{\mathsf{hypot}\left(B, C\right)}}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)} \]

   7. Applied rewrites 80.4%

      \[\leadsto \frac{\tan^{-1} \left(\frac{\color{blue}{C - \mathsf{hypot}\left(B, C\right)}}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)} \]

   8. Taylor expanded in C around 0

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

   10. Applied rewrites 64.9%

       \[\leadsto \frac{\tan^{-1} \left(\frac{C - \color{blue}{B}}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)} \]

   if -1.1199999999999999e-148 < A < 5.1999999999999998e-183

   1. Initial program 55.3%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around -inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\left(1 + \frac{C}{B}\right) - \frac{A}{B}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(1 + \left(\frac{C}{B} - \frac{A}{B}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      2. div-sub N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(1 + \color{blue}{\frac{C - A}{B}}\right)}{\mathsf{PI}\left(\right)} \]

      3. +-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C - A}{B} + 1\right)}}{\mathsf{PI}\left(\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C - A}{B} + 1\right)}}{\mathsf{PI}\left(\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\frac{C - A}{B}} + 1\right)}{\mathsf{PI}\left(\right)} \]

      6. lower--.f64 52.3

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{C - A}}{B} + 1\right)}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 52.3%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C - A}{B} + 1\right)}}{\mathsf{PI}\left(\right)} \]

   6. Taylor expanded in A around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{C}{B} + 1\right)}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 52.3%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{C}{B} + 1\right)}{\mathsf{PI}\left(\right)} \]

   if 5.1999999999999998e-183 < A

   1. Initial program 65.7%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in C around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-1 \cdot \frac{A + \sqrt{{A}^{2} + {B}^{2}}}{B}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{A + \sqrt{{A}^{2} + {B}^{2}}}{B}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      2. distribute-neg-frac2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{A + \sqrt{{A}^{2} + {B}^{2}}}{\mathsf{neg}\left(B\right)}\right)}}{\mathsf{PI}\left(\right)} \]

      3. mul-1-neg N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{A + \sqrt{{A}^{2} + {B}^{2}}}{\color{blue}{-1 \cdot B}}\right)}{\mathsf{PI}\left(\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{A + \sqrt{{A}^{2} + {B}^{2}}}{-1 \cdot B}\right)}}{\mathsf{PI}\left(\right)} \]

      5. +-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{\sqrt{{A}^{2} + {B}^{2}} + A}}{-1 \cdot B}\right)}{\mathsf{PI}\left(\right)} \]

      6. lower-+.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{\sqrt{{A}^{2} + {B}^{2}} + A}}{-1 \cdot B}\right)}{\mathsf{PI}\left(\right)} \]

      7. +-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\sqrt{\color{blue}{{B}^{2} + {A}^{2}}} + A}{-1 \cdot B}\right)}{\mathsf{PI}\left(\right)} \]

      8. unpow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\sqrt{\color{blue}{B \cdot B} + {A}^{2}} + A}{-1 \cdot B}\right)}{\mathsf{PI}\left(\right)} \]

      9. unpow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\sqrt{B \cdot B + \color{blue}{A \cdot A}} + A}{-1 \cdot B}\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-hypot.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{\mathsf{hypot}\left(B, A\right)} + A}{-1 \cdot B}\right)}{\mathsf{PI}\left(\right)} \]

      11. mul-1-neg N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\mathsf{hypot}\left(B, A\right) + A}{\color{blue}{\mathsf{neg}\left(B\right)}}\right)}{\mathsf{PI}\left(\right)} \]

      12. lower-neg.f64 77.7

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\mathsf{hypot}\left(B, A\right) + A}{\color{blue}{-B}}\right)}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 77.7%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{\mathsf{hypot}\left(B, A\right) + A}{-B}\right)}}{\mathsf{PI}\left(\right)} \]

   6. Taylor expanded in A around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{A + B}{-\color{blue}{B}}\right)}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 66.9%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{B + A}{-\color{blue}{B}}\right)}{\mathsf{PI}\left(\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -9.6 \cdot 10^{+54}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{B}{A} \cdot 0.5\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;A \leq -1.12 \cdot 10^{-148}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - B}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;A \leq 5.2 \cdot 10^{-183}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{C}{B} + 1\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{B + A}{-B}\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 54.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 6 \cdot 10^{-269}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{C}{B} + 1\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;B \leq 1.05 \cdot 10^{-225}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} 0}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - B}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= B 6e-269)
   (* 180.0 (/ (atan (+ (/ C B) 1.0)) (PI)))
   (if (<= B 1.05e-225)
     (* 180.0 (/ (atan 0.0) (PI)))
     (/ (* (atan (/ (- C B) B)) 180.0) (PI)))))

Derivation

1. Split input into 3 regimes

2. if B < 5.9999999999999997e-269

   1. Initial program 59.0%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around -inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\left(1 + \frac{C}{B}\right) - \frac{A}{B}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(1 + \left(\frac{C}{B} - \frac{A}{B}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      2. div-sub N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(1 + \color{blue}{\frac{C - A}{B}}\right)}{\mathsf{PI}\left(\right)} \]

      3. +-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C - A}{B} + 1\right)}}{\mathsf{PI}\left(\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C - A}{B} + 1\right)}}{\mathsf{PI}\left(\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\frac{C - A}{B}} + 1\right)}{\mathsf{PI}\left(\right)} \]

      6. lower--.f64 68.1

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{C - A}}{B} + 1\right)}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 68.1%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C - A}{B} + 1\right)}}{\mathsf{PI}\left(\right)} \]

   6. Taylor expanded in A around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{C}{B} + 1\right)}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 57.9%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{C}{B} + 1\right)}{\mathsf{PI}\left(\right)} \]

   if 5.9999999999999997e-269 < B < 1.05e-225

   1. Initial program 34.1%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in C around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-1 \cdot \frac{A + -1 \cdot A}{B}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{A + -1 \cdot A}{B}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      2. distribute-frac-neg N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{\mathsf{neg}\left(\left(A + -1 \cdot A\right)\right)}{B}\right)}}{\mathsf{PI}\left(\right)} \]

      3. distribute-rgt1-in N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\mathsf{neg}\left(\color{blue}{\left(-1 + 1\right) \cdot A}\right)}{B}\right)}{\mathsf{PI}\left(\right)} \]

      4. metadata-eval N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\mathsf{neg}\left(\color{blue}{0} \cdot A\right)}{B}\right)}{\mathsf{PI}\left(\right)} \]

      5. distribute-lft-neg-in N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{\left(\mathsf{neg}\left(0\right)\right) \cdot A}}{B}\right)}{\mathsf{PI}\left(\right)} \]

      6. metadata-eval N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{0} \cdot A}{B}\right)}{\mathsf{PI}\left(\right)} \]

      7. metadata-eval N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{\left(-1 + 1\right)} \cdot A}{B}\right)}{\mathsf{PI}\left(\right)} \]

      8. distribute-rgt1-in N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{A + -1 \cdot A}}{B}\right)}{\mathsf{PI}\left(\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{A + -1 \cdot A}{B}\right)}}{\mathsf{PI}\left(\right)} \]

      10. distribute-rgt1-in N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{\left(-1 + 1\right) \cdot A}}{B}\right)}{\mathsf{PI}\left(\right)} \]

      11. metadata-eval N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{0} \cdot A}{B}\right)}{\mathsf{PI}\left(\right)} \]

      12. mul0-lft 63.9

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{0}}{B}\right)}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 63.9%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{0}{B}\right)}}{\mathsf{PI}\left(\right)} \]

   if 1.05e-225 < B

   1. Initial program 47.5%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto 180 \cdot \color{blue}{\frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{180 \cdot \tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{180 \cdot \tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}} \]

   4. Applied rewrites 76.5%

      \[\leadsto \color{blue}{\frac{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(B, A - C\right)}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)}} \]

   5. Taylor expanded in A around 0

      \[\leadsto \frac{\tan^{-1} \left(\frac{\color{blue}{C - \sqrt{{B}^{2} + {C}^{2}}}}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)} \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{\tan^{-1} \left(\frac{\color{blue}{C - \sqrt{{B}^{2} + {C}^{2}}}}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)} \]

      2. unpow2 N/A

         \[\leadsto \frac{\tan^{-1} \left(\frac{C - \sqrt{\color{blue}{B \cdot B} + {C}^{2}}}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{\tan^{-1} \left(\frac{C - \sqrt{B \cdot B + \color{blue}{C \cdot C}}}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)} \]

      4. lower-hypot.f64 66.2

         \[\leadsto \frac{\tan^{-1} \left(\frac{C - \color{blue}{\mathsf{hypot}\left(B, C\right)}}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)} \]

   7. Applied rewrites 66.2%

      \[\leadsto \frac{\tan^{-1} \left(\frac{\color{blue}{C - \mathsf{hypot}\left(B, C\right)}}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)} \]

   8. Taylor expanded in C around 0

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

   10. Applied rewrites 59.8%

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

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 6 \cdot 10^{-269}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{C}{B} + 1\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;B \leq 1.05 \cdot 10^{-225}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} 0}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - B}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 51.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 3.8 \cdot 10^{-41}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{C}{B} + 1\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= B 3.8e-41)
   (* 180.0 (/ (atan (+ (/ C B) 1.0)) (PI)))
   (* 180.0 (/ (atan -1.0) (PI)))))

Derivation

1. Split input into 2 regimes

2. if B < 3.79999999999999979e-41

   1. Initial program 56.7%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around -inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\left(1 + \frac{C}{B}\right) - \frac{A}{B}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(1 + \left(\frac{C}{B} - \frac{A}{B}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      2. div-sub N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(1 + \color{blue}{\frac{C - A}{B}}\right)}{\mathsf{PI}\left(\right)} \]

      3. +-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C - A}{B} + 1\right)}}{\mathsf{PI}\left(\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C - A}{B} + 1\right)}}{\mathsf{PI}\left(\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\frac{C - A}{B}} + 1\right)}{\mathsf{PI}\left(\right)} \]

      6. lower--.f64 58.6

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{C - A}}{B} + 1\right)}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 58.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C - A}{B} + 1\right)}}{\mathsf{PI}\left(\right)} \]

   6. Taylor expanded in A around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{C}{B} + 1\right)}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 48.5%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{C}{B} + 1\right)}{\mathsf{PI}\left(\right)} \]

   if 3.79999999999999979e-41 < B

   1. Initial program 41.8%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around inf

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

   5. Applied rewrites 61.4%

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

4. Final simplification 52.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3.8 \cdot 10^{-41}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{C}{B} + 1\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 50.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 1.75 \cdot 10^{-264}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(1 - \frac{A}{B}\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;B \leq 2.75 \cdot 10^{-129}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} 0}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= B 1.75e-264)
   (* 180.0 (/ (atan (- 1.0 (/ A B))) (PI)))
   (if (<= B 2.75e-129)
     (* 180.0 (/ (atan 0.0) (PI)))
     (* 180.0 (/ (atan -1.0) (PI))))))

Derivation

1. Split input into 3 regimes

2. if B < 1.75e-264

   1. Initial program 59.0%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in C around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-1 \cdot \frac{A + \sqrt{{A}^{2} + {B}^{2}}}{B}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{A + \sqrt{{A}^{2} + {B}^{2}}}{B}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      2. distribute-neg-frac2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{A + \sqrt{{A}^{2} + {B}^{2}}}{\mathsf{neg}\left(B\right)}\right)}}{\mathsf{PI}\left(\right)} \]

      3. mul-1-neg N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{A + \sqrt{{A}^{2} + {B}^{2}}}{\color{blue}{-1 \cdot B}}\right)}{\mathsf{PI}\left(\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{A + \sqrt{{A}^{2} + {B}^{2}}}{-1 \cdot B}\right)}}{\mathsf{PI}\left(\right)} \]

      5. +-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{\sqrt{{A}^{2} + {B}^{2}} + A}}{-1 \cdot B}\right)}{\mathsf{PI}\left(\right)} \]

      6. lower-+.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{\sqrt{{A}^{2} + {B}^{2}} + A}}{-1 \cdot B}\right)}{\mathsf{PI}\left(\right)} \]

      7. +-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\sqrt{\color{blue}{{B}^{2} + {A}^{2}}} + A}{-1 \cdot B}\right)}{\mathsf{PI}\left(\right)} \]

      8. unpow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\sqrt{\color{blue}{B \cdot B} + {A}^{2}} + A}{-1 \cdot B}\right)}{\mathsf{PI}\left(\right)} \]

      9. unpow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\sqrt{B \cdot B + \color{blue}{A \cdot A}} + A}{-1 \cdot B}\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-hypot.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{\mathsf{hypot}\left(B, A\right)} + A}{-1 \cdot B}\right)}{\mathsf{PI}\left(\right)} \]

      11. mul-1-neg N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\mathsf{hypot}\left(B, A\right) + A}{\color{blue}{\mathsf{neg}\left(B\right)}}\right)}{\mathsf{PI}\left(\right)} \]

      12. lower-neg.f64 62.1

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\mathsf{hypot}\left(B, A\right) + A}{\color{blue}{-B}}\right)}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 62.1%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{\mathsf{hypot}\left(B, A\right) + A}{-B}\right)}}{\mathsf{PI}\left(\right)} \]

   6. Taylor expanded in B around -inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(1 + \color{blue}{-1 \cdot \frac{A}{B}}\right)}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 56.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(1 - \color{blue}{\frac{A}{B}}\right)}{\mathsf{PI}\left(\right)} \]

   if 1.75e-264 < B < 2.75000000000000012e-129

   1. Initial program 42.6%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in C around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-1 \cdot \frac{A + -1 \cdot A}{B}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{A + -1 \cdot A}{B}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      2. distribute-frac-neg N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{\mathsf{neg}\left(\left(A + -1 \cdot A\right)\right)}{B}\right)}}{\mathsf{PI}\left(\right)} \]

      3. distribute-rgt1-in N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\mathsf{neg}\left(\color{blue}{\left(-1 + 1\right) \cdot A}\right)}{B}\right)}{\mathsf{PI}\left(\right)} \]

      4. metadata-eval N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\mathsf{neg}\left(\color{blue}{0} \cdot A\right)}{B}\right)}{\mathsf{PI}\left(\right)} \]

      5. distribute-lft-neg-in N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{\left(\mathsf{neg}\left(0\right)\right) \cdot A}}{B}\right)}{\mathsf{PI}\left(\right)} \]

      6. metadata-eval N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{0} \cdot A}{B}\right)}{\mathsf{PI}\left(\right)} \]

      7. metadata-eval N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{\left(-1 + 1\right)} \cdot A}{B}\right)}{\mathsf{PI}\left(\right)} \]

      8. distribute-rgt1-in N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{A + -1 \cdot A}}{B}\right)}{\mathsf{PI}\left(\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{A + -1 \cdot A}{B}\right)}}{\mathsf{PI}\left(\right)} \]

      10. distribute-rgt1-in N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{\left(-1 + 1\right) \cdot A}}{B}\right)}{\mathsf{PI}\left(\right)} \]

      11. metadata-eval N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{0} \cdot A}{B}\right)}{\mathsf{PI}\left(\right)} \]

      12. mul0-lft 42.1

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{0}}{B}\right)}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 42.1%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{0}{B}\right)}}{\mathsf{PI}\left(\right)} \]

   if 2.75000000000000012e-129 < B

   1. Initial program 47.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around inf

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

   5. Applied rewrites 52.3%

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

4. Final simplification 52.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.75 \cdot 10^{-264}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(1 - \frac{A}{B}\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;B \leq 2.75 \cdot 10^{-129}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} 0}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 44.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq -4.9 \cdot 10^{-80}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} 1}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;B \leq 2.75 \cdot 10^{-129}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} 0}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= B -4.9e-80)
   (* 180.0 (/ (atan 1.0) (PI)))
   (if (<= B 2.75e-129)
     (* 180.0 (/ (atan 0.0) (PI)))
     (* 180.0 (/ (atan -1.0) (PI))))))

Derivation

1. Split input into 3 regimes

2. if B < -4.8999999999999999e-80

   1. Initial program 53.1%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around -inf

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

   5. Applied rewrites 57.6%

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

   if -4.8999999999999999e-80 < B < 2.75000000000000012e-129

   1. Initial program 56.9%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in C around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-1 \cdot \frac{A + -1 \cdot A}{B}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{A + -1 \cdot A}{B}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      2. distribute-frac-neg N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{\mathsf{neg}\left(\left(A + -1 \cdot A\right)\right)}{B}\right)}}{\mathsf{PI}\left(\right)} \]

      3. distribute-rgt1-in N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\mathsf{neg}\left(\color{blue}{\left(-1 + 1\right) \cdot A}\right)}{B}\right)}{\mathsf{PI}\left(\right)} \]

      4. metadata-eval N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\mathsf{neg}\left(\color{blue}{0} \cdot A\right)}{B}\right)}{\mathsf{PI}\left(\right)} \]

      5. distribute-lft-neg-in N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{\left(\mathsf{neg}\left(0\right)\right) \cdot A}}{B}\right)}{\mathsf{PI}\left(\right)} \]

      6. metadata-eval N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{0} \cdot A}{B}\right)}{\mathsf{PI}\left(\right)} \]

      7. metadata-eval N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{\left(-1 + 1\right)} \cdot A}{B}\right)}{\mathsf{PI}\left(\right)} \]

      8. distribute-rgt1-in N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{A + -1 \cdot A}}{B}\right)}{\mathsf{PI}\left(\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{A + -1 \cdot A}{B}\right)}}{\mathsf{PI}\left(\right)} \]

      10. distribute-rgt1-in N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{\left(-1 + 1\right) \cdot A}}{B}\right)}{\mathsf{PI}\left(\right)} \]

      11. metadata-eval N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{0} \cdot A}{B}\right)}{\mathsf{PI}\left(\right)} \]

      12. mul0-lft 31.2

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{0}}{B}\right)}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 31.2%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{0}{B}\right)}}{\mathsf{PI}\left(\right)} \]

   if 2.75000000000000012e-129 < B

   1. Initial program 47.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around inf

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

   5. Applied rewrites 52.3%

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

4. Final simplification 46.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -4.9 \cdot 10^{-80}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} 1}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;B \leq 2.75 \cdot 10^{-129}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} 0}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 39.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq -5 \cdot 10^{-310}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} 1}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= B -5e-310)
   (* 180.0 (/ (atan 1.0) (PI)))
   (* 180.0 (/ (atan -1.0) (PI)))))

Derivation

1. Split input into 2 regimes

2. if B < -4.999999999999985e-310

   1. Initial program 57.7%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around -inf

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

   5. Applied rewrites 40.0%

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

   if -4.999999999999985e-310 < B

   1. Initial program 47.7%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around inf

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

   5. Applied rewrites 41.5%

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

4. Final simplification 40.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -5 \cdot 10^{-310}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} 1}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 20.4% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ 180 \cdot \frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)} \end{array} \]

FPCore

(FPCore (A B C) :precision binary64 (* 180.0 (/ (atan -1.0) (PI))))

Derivation

1. Initial program 52.1%

   \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
2. Add Preprocessing

3. Taylor expanded in B around inf

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

5. Applied rewrites 24.2%

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

6. Final simplification 24.2%

   \[\leadsto 180 \cdot \frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024350 
(FPCore (A B C)
  :name "ABCF->ab-angle angle"
  :precision binary64
  (* 180.0 (/ (atan (* (/ 1.0 B) (- (- C A) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0)))))) (PI))))