ABCF->ab-angle angle

Percentage Accurate: 53.4% → 81.2%

Time: 7.9s

Alternatives: 11

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: 53.4% 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: 81.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;A \leq -3.2 \cdot 10^{+107}:\\ \;\;\;\;\frac{\tan^{-1} \left(0.5 \cdot \frac{\mathsf{fma}\left(\frac{C}{A}, B, B\right)}{A}\right) \cdot 180}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(B, A - C\right)}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= A -3.2e+107)
   (/ (* (atan (* 0.5 (/ (fma (/ C A) B B) A))) 180.0) (PI))
   (/ (* (atan (/ (- (- C A) (hypot B (- A C))) B)) 180.0) (PI))))

Derivation

1. Split input into 2 regimes

2. if A < -3.20000000000000029e107

   1. Initial program 10.0%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   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 41.7%

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

   5. Taylor expanded in A around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-lft-out N/A

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

      3. associate-/l* N/A

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

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

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 80.7

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

   7. Applied rewrites 80.7%

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

   if -3.20000000000000029e107 < A

   1. Initial program 64.5%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   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 86.7%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -3.2 \cdot 10^{+107}:\\ \;\;\;\;\frac{\tan^{-1} \left(0.5 \cdot \frac{\mathsf{fma}\left(\frac{C}{A}, B, B\right)}{A}\right) \cdot 180}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(B, A - C\right)}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 72.8% accurate, 0.4× speedup

Math

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

FPCore

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

Derivation

1. Split input into 3 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)))))) < -0.5

   1. Initial program 60.2%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   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 75.4

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

   5. Applied rewrites 75.4%

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

   if -0.5 < (*.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)))))) < 0.0

   1. Initial program 15.2%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing
   3. 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 15.2%

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

   5. Taylor expanded in A around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 53.1

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

   7. Applied rewrites 53.1%

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

   if 0.0 < (*.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 59.2%

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

   3. Taylor expanded in B around -inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\left(1 + \frac{C}{B}\right) - \frac{A}{B}\right)}}{\mathsf{PI}\left(\right)} \]
   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 76.8

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

   5. Applied rewrites 76.8%

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

4. Final simplification 73.4%

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

Alternative 3: 77.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;A \leq -2.7 \cdot 10^{+106}:\\ \;\;\;\;\frac{\tan^{-1} \left(0.5 \cdot \frac{\mathsf{fma}\left(\frac{C}{A}, B, B\right)}{A}\right) \cdot 180}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;A \leq 1.25 \cdot 10^{-24}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{C - \mathsf{hypot}\left(B, C\right)}{B}\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{\mathsf{hypot}\left(B, A\right) + A}{-B}\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= A -2.7e+106)
   (/ (* (atan (* 0.5 (/ (fma (/ C A) B B) A))) 180.0) (PI))
   (if (<= A 1.25e-24)
     (* 180.0 (/ (atan (/ (- C (hypot B C)) B)) (PI)))
     (* 180.0 (/ (atan (/ (+ (hypot B A) A) (- B))) (PI))))))

Derivation

1. Split input into 3 regimes

2. if A < -2.70000000000000006e106

   1. Initial program 10.0%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   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 41.7%

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

   5. Taylor expanded in A around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-lft-out N/A

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

      3. associate-/l* N/A

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

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

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 80.7

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

   7. Applied rewrites 80.7%

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

   if -2.70000000000000006e106 < A < 1.24999999999999995e-24

   1. Initial program 59.8%

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

   3. Taylor expanded in 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 81.3

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

   5. Applied rewrites 81.3%

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

   if 1.24999999999999995e-24 < A

   1. Initial program 74.6%

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

   3. Taylor expanded in C around 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 91.1

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

   5. Applied rewrites 91.1%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -2.7 \cdot 10^{+106}:\\ \;\;\;\;\frac{\tan^{-1} \left(0.5 \cdot \frac{\mathsf{fma}\left(\frac{C}{A}, B, B\right)}{A}\right) \cdot 180}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;A \leq 1.25 \cdot 10^{-24}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{C - \mathsf{hypot}\left(B, C\right)}{B}\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{\mathsf{hypot}\left(B, A\right) + A}{-B}\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;A \leq -2.7 \cdot 10^{+106}:\\ \;\;\;\;\frac{\tan^{-1} \left(0.5 \cdot \frac{\mathsf{fma}\left(\frac{C}{A}, B, B\right)}{A}\right) \cdot 180}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;A \leq 1.6 \cdot 10^{+139}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{C - \mathsf{hypot}\left(B, C\right)}{B}\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{C - A}{B} + 1\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= A -2.7e+106)
   (/ (* (atan (* 0.5 (/ (fma (/ C A) B B) A))) 180.0) (PI))
   (if (<= A 1.6e+139)
     (* 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 < -2.70000000000000006e106

   1. Initial program 10.0%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   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 41.7%

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

   5. Taylor expanded in A around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-lft-out N/A

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

      3. associate-/l* N/A

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

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

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 80.7

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

   7. Applied rewrites 80.7%

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

   if -2.70000000000000006e106 < A < 1.6000000000000001e139

   1. Initial program 60.6%

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

   3. Taylor expanded in 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 80.1

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

   5. Applied rewrites 80.1%

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

   if 1.6000000000000001e139 < A

   1. Initial program 83.8%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   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 94.3

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

   5. Applied rewrites 94.3%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -2.7 \cdot 10^{+106}:\\ \;\;\;\;\frac{\tan^{-1} \left(0.5 \cdot \frac{\mathsf{fma}\left(\frac{C}{A}, B, B\right)}{A}\right) \cdot 180}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;A \leq 1.6 \cdot 10^{+139}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{C - \mathsf{hypot}\left(B, C\right)}{B}\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{C - A}{B} + 1\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 55.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;A \leq -1.8 \cdot 10^{+105}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{B}{A} \cdot 0.5\right) \cdot 180}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;A \leq 5.8 \cdot 10^{+122}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - B}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{A}{B} \cdot -2\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= A -1.8e+105)
   (/ (* (atan (* (/ B A) 0.5)) 180.0) (PI))
   (if (<= A 5.8e+122)
     (/ (* (atan (/ (- C B) B)) 180.0) (PI))
     (* 180.0 (/ (atan (* (/ A B) -2.0)) (PI))))))

Derivation

1. Split input into 3 regimes

2. if A < -1.7999999999999999e105

   1. Initial program 10.0%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   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 41.7%

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

   5. Taylor expanded in A around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 80.6

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

   7. Applied rewrites 80.6%

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

   if -1.7999999999999999e105 < A < 5.8000000000000002e122

   1. Initial program 61.3%

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

   3. Taylor expanded in A around inf

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

      1. lower-*.f64 13.0

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

   5. Applied rewrites 13.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   7. Applied rewrites 13.0%

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

   8. Taylor expanded in A around 0

      \[\leadsto \frac{\tan^{-1} \left(\frac{\color{blue}{C - \sqrt{{B}^{2} + {C}^{2}}}}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)} \]
   9. 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 80.5

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

   10. Applied rewrites 80.5%

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

   11. Taylor expanded in C around 0

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

   13. Applied rewrites 48.7%

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

   if 5.8000000000000002e122 < A

   1. Initial program 75.9%

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

   3. Taylor expanded in A around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 73.1

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

   5. Applied rewrites 73.1%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -1.8 \cdot 10^{+105}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{B}{A} \cdot 0.5\right) \cdot 180}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;A \leq 5.8 \cdot 10^{+122}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - B}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{A}{B} \cdot -2\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 55.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;A \leq -1.8 \cdot 10^{+105}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{B}{A} \cdot 0.5\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;A \leq 5.8 \cdot 10^{+122}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - B}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{A}{B} \cdot -2\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= A -1.8e+105)
   (* 180.0 (/ (atan (* (/ B A) 0.5)) (PI)))
   (if (<= A 5.8e+122)
     (/ (* (atan (/ (- C B) B)) 180.0) (PI))
     (* 180.0 (/ (atan (* (/ A B) -2.0)) (PI))))))

Derivation

1. Split input into 3 regimes

2. if A < -1.7999999999999999e105

   1. Initial program 10.0%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   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 80.5

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

   5. Applied rewrites 80.5%

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

   if -1.7999999999999999e105 < A < 5.8000000000000002e122

   1. Initial program 61.3%

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

   3. Taylor expanded in A around inf

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

      1. lower-*.f64 13.0

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

   5. Applied rewrites 13.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   7. Applied rewrites 13.0%

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

   8. Taylor expanded in A around 0

      \[\leadsto \frac{\tan^{-1} \left(\frac{\color{blue}{C - \sqrt{{B}^{2} + {C}^{2}}}}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)} \]
   9. 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 80.5

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

   10. Applied rewrites 80.5%

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

   11. Taylor expanded in C around 0

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

   13. Applied rewrites 48.7%

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

   if 5.8000000000000002e122 < A

   1. Initial program 75.9%

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

   3. Taylor expanded in A around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 73.1

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

   5. Applied rewrites 73.1%

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

Alternative 7: 60.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;A \leq -3.5 \cdot 10^{+105}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{B}{A} \cdot 0.5\right) \cdot 180}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{C - A}{B} + 1\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= A -3.5e+105)
   (/ (* (atan (* (/ B A) 0.5)) 180.0) (PI))
   (* 180.0 (/ (atan (+ (/ (- C A) B) 1.0)) (PI)))))

Derivation

1. Split input into 2 regimes

2. if A < -3.49999999999999991e105

   1. Initial program 10.0%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   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 41.7%

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

   5. Taylor expanded in A around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 80.6

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

   7. Applied rewrites 80.6%

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

   if -3.49999999999999991e105 < A

   1. Initial program 64.5%

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

   3. Taylor expanded in 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 64.0

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

   5. Applied rewrites 64.0%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -3.5 \cdot 10^{+105}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{B}{A} \cdot 0.5\right) \cdot 180}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{C - A}{B} + 1\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 50.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq -3.3 \cdot 10^{-24}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} 1}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - B}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= B -3.3e-24)
   (* 180.0 (/ (atan 1.0) (PI)))
   (/ (* (atan (/ (- C B) B)) 180.0) (PI))))

Derivation

1. Split input into 2 regimes

2. if B < -3.29999999999999984e-24

   1. Initial program 51.9%

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

   3. Taylor expanded in B around -inf

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

   5. Applied rewrites 60.0%

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

   if -3.29999999999999984e-24 < B

   1. Initial program 55.4%

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

   3. Taylor expanded in A around inf

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

      1. lower-*.f64 25.9

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

   5. Applied rewrites 25.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   7. Applied rewrites 25.9%

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

   8. Taylor expanded in A around 0

      \[\leadsto \frac{\tan^{-1} \left(\frac{\color{blue}{C - \sqrt{{B}^{2} + {C}^{2}}}}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)} \]
   9. 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 63.1

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

   10. Applied rewrites 63.1%

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

   11. Taylor expanded in C around 0

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

   13. Applied rewrites 49.0%

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

Alternative 9: 44.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq -3.4 \cdot 10^{-116}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} 1}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;B \leq 5.8 \cdot 10^{-143}:\\ \;\;\;\;\frac{\tan^{-1} 0 \cdot 180}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= B -3.4e-116)
   (* 180.0 (/ (atan 1.0) (PI)))
   (if (<= B 5.8e-143)
     (/ (* (atan 0.0) 180.0) (PI))
     (* 180.0 (/ (atan -1.0) (PI))))))

Derivation

1. Split input into 3 regimes

2. if B < -3.39999999999999992e-116

   1. Initial program 56.5%

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

   3. Taylor expanded in B around -inf

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

   5. Applied rewrites 53.1%

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

   if -3.39999999999999992e-116 < B < 5.8000000000000002e-143

   1. Initial program 45.3%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing
   3. 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 76.2%

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

   5. Taylor expanded in B around -inf

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

   7. Applied rewrites 6.3%

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

   8. Taylor expanded in C around inf

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

      1. div-add N/A

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

      2. associate-*r/ N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. distribute-lft1-in N/A

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

      6. metadata-eval N/A

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

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

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

      8. metadata-eval N/A

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

      9. mul0-lft 39.5

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

   10. Applied rewrites 39.5%

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

   if 5.8000000000000002e-143 < B

   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. 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 49.7%

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

4. Final simplification 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -3.4 \cdot 10^{-116}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} 1}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;B \leq 5.8 \cdot 10^{-143}:\\ \;\;\;\;\frac{\tan^{-1} 0 \cdot 180}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 39.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq -1.4 \cdot 10^{-306}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} 1}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= B -1.4e-306)
   (* 180.0 (/ (atan 1.0) (PI)))
   (* 180.0 (/ (atan -1.0) (PI)))))

Derivation

1. Split input into 2 regimes

2. if B < -1.4000000000000001e-306

   1. Initial program 53.6%

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

   3. Taylor expanded in B around -inf

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

   5. Applied rewrites 42.5%

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

   if -1.4000000000000001e-306 < B

   1. Initial program 55.1%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   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 38.5%

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

Alternative 11: 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 54.3%

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

3. Taylor expanded in B around inf

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

5. Applied rewrites 18.6%

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

Reproduce

herbie shell --seed 2024326 
(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))))