ABCF->ab-angle angle

Percentage Accurate: 53.3% → 75.4%

Time: 8.1s

Alternatives: 15

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: 53.3% 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: 75.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;C \leq -1.02 \cdot 10^{+133}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{C + B}{B}\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;C \leq 5 \cdot 10^{+122}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{\mathsf{hypot}\left(A, B\right) + A}{-B}\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(\mathsf{fma}\left(-0.5, \frac{B}{C}, 0\right)\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= C -1.02e+133)
   (* 180.0 (/ (atan (/ (+ C B) B)) (PI)))
   (if (<= C 5e+122)
     (* 180.0 (/ (atan (/ (+ (hypot A B) A) (- B))) (PI)))
     (/ (* 180.0 (atan (fma -0.5 (/ B C) 0.0))) (PI)))))

Derivation

1. Split input into 3 regimes

2. if C < -1.02e133

   1. Initial program 90.4%

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

   3. Taylor expanded in B around -inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 94.7

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

   5. Applied rewrites 94.7%

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

   6. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 94.7

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

   8. Applied rewrites 94.7%

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

   9. Taylor expanded in A around 0

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

       1. +-commutative N/A

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

       2. lower-+.f64 94.8

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

   11. Applied rewrites 94.8%

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

   if -1.02e133 < C < 4.99999999999999989e122

   1. Initial program 48.2%

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

   3. Taylor expanded in 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

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 68.4

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

   5. Applied rewrites 68.4%

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

   if 4.99999999999999989e122 < C

   1. Initial program 10.9%

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

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 36.3

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

   5. Applied rewrites 36.3%

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

   6. Taylor expanded in C around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. distribute-rgt1-in N/A

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

      6. metadata-eval N/A

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

      7. mul0-lft N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 82.8

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

   8. Applied rewrites 82.8%

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan.f64 N/A

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

      5. *-commutative N/A

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

      6. div0 N/A

         \[\leadsto \frac{180 \cdot \tan^{-1} \left(\frac{-1}{2} \cdot \frac{B}{C} + 0\right)}{\mathsf{PI}\left(\right)} \]

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

   10. Applied rewrites 83.0%

       \[\leadsto \color{blue}{\frac{180 \cdot \tan^{-1} \left(\mathsf{fma}\left(-0.5, \frac{B}{C}, 0\right)\right)}{\mathsf{PI}\left(\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq -1.02 \cdot 10^{+133}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{C + B}{B}\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;C \leq 5 \cdot 10^{+122}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{\mathsf{hypot}\left(A, B\right) + A}{-B}\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(\mathsf{fma}\left(-0.5, \frac{B}{C}, 0\right)\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 62.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq -4.5 \cdot 10^{-87}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{\left(C + B\right) - A}{B}\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;B \leq 1.25 \cdot 10^{-158}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{0.5 \cdot \left(\left(\frac{B}{C} + \frac{B}{A}\right) \cdot C\right)}{A}\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;B \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{\mathsf{fma}\left(C, C, B \cdot B\right)}\right)\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= B -4.5e-87)
   (* 180.0 (/ (atan (/ (- (+ C B) A) B)) (PI)))
   (if (<= B 1.25e-158)
     (* 180.0 (/ (atan (/ (* 0.5 (* (+ (/ B C) (/ B A)) C)) A)) (PI)))
     (if (<= B 1.35e+154)
       (*
        180.0
        (/ (atan (* (/ 1.0 B) (- (- C A) (sqrt (fma C C (* B B)))))) (PI)))
       (* 180.0 (/ (atan -1.0) (PI)))))))

Derivation

1. Split input into 4 regimes

2. if B < -4.49999999999999958e-87

   1. Initial program 49.6%

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

   3. Taylor expanded in B around -inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 71.3

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

   5. Applied rewrites 71.3%

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

   6. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 71.3

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

   8. Applied rewrites 71.3%

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

   if -4.49999999999999958e-87 < B < 1.24999999999999993e-158

   1. Initial program 48.5%

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

   3. Taylor expanded in 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. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. distribute-lft-out N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 57.5

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

   5. Applied rewrites 57.5%

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

   6. Taylor expanded in A around inf

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

      1. fp-cancel-sub-sign-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-/.f64 N/A

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

      5. distribute-lft-out N/A

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

      6. associate-*r/ N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*r/ N/A

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

      9. +-commutative N/A

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

      10. associate-*r/ N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 57.5

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

   8. Applied rewrites 57.5%

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

   9. Taylor expanded in C around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. lower-+.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. lower-/.f64 60.3

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

   11. Applied rewrites 60.3%

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

   if 1.24999999999999993e-158 < B < 1.35000000000000003e154

   1. Initial program 59.4%

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

   3. Taylor expanded in A around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 59.2

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

   5. Applied rewrites 59.2%

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

   if 1.35000000000000003e154 < B

   1. Initial program 19.8%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   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 81.2%

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

Alternative 3: 62.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq -4.5 \cdot 10^{-87}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{\left(C + B\right) - A}{B}\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;B \leq 1.25 \cdot 10^{-158}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{0.5 \cdot \mathsf{fma}\left(\frac{C}{A}, B, B\right)}{A}\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;B \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{\mathsf{fma}\left(C, C, B \cdot B\right)}\right)\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= B -4.5e-87)
   (* 180.0 (/ (atan (/ (- (+ C B) A) B)) (PI)))
   (if (<= B 1.25e-158)
     (* 180.0 (/ (atan (/ (* 0.5 (fma (/ C A) B B)) A)) (PI)))
     (if (<= B 1.35e+154)
       (*
        180.0
        (/ (atan (* (/ 1.0 B) (- (- C A) (sqrt (fma C C (* B B)))))) (PI)))
       (* 180.0 (/ (atan -1.0) (PI)))))))

Derivation

1. Split input into 4 regimes

2. if B < -4.49999999999999958e-87

   1. Initial program 49.6%

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

   3. Taylor expanded in B around -inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 71.3

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

   5. Applied rewrites 71.3%

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

   6. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 71.3

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

   8. Applied rewrites 71.3%

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

   if -4.49999999999999958e-87 < B < 1.24999999999999993e-158

   1. Initial program 48.5%

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

   3. Taylor expanded in 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. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. distribute-lft-out N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 57.5

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

   5. Applied rewrites 57.5%

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

   6. Taylor expanded in A around inf

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

      1. fp-cancel-sub-sign-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-/.f64 N/A

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

      5. distribute-lft-out N/A

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

      6. associate-*r/ N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*r/ N/A

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

      9. +-commutative N/A

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

      10. associate-*r/ N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 57.5

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

   8. Applied rewrites 57.5%

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

   if 1.24999999999999993e-158 < B < 1.35000000000000003e154

   1. Initial program 59.4%

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

   3. Taylor expanded in A around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 59.2

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

   5. Applied rewrites 59.2%

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

   if 1.35000000000000003e154 < B

   1. Initial program 19.8%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   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 81.2%

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

Alternative 4: 59.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq -4.5 \cdot 10^{-87}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{\left(C + B\right) - A}{B}\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;B \leq 1.85 \cdot 10^{-50}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{0.5 \cdot \mathsf{fma}\left(\frac{C}{A}, B, B\right)}{A}\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;B \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(C - \sqrt{\mathsf{fma}\left(C, C, B \cdot B\right)}\right)\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= B -4.5e-87)
   (* 180.0 (/ (atan (/ (- (+ C B) A) B)) (PI)))
   (if (<= B 1.85e-50)
     (* 180.0 (/ (atan (/ (* 0.5 (fma (/ C A) B B)) A)) (PI)))
     (if (<= B 1.35e+154)
       (* 180.0 (/ (atan (* (/ 1.0 B) (- C (sqrt (fma C C (* B B)))))) (PI)))
       (* 180.0 (/ (atan -1.0) (PI)))))))

Derivation

1. Split input into 4 regimes

2. if B < -4.49999999999999958e-87

   1. Initial program 49.6%

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

   3. Taylor expanded in B around -inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 71.3

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

   5. Applied rewrites 71.3%

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

   6. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 71.3

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

   8. Applied rewrites 71.3%

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

   if -4.49999999999999958e-87 < B < 1.85e-50

   1. Initial program 49.2%

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

   3. Taylor expanded in 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. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. distribute-lft-out N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 54.5

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

   5. Applied rewrites 54.5%

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

   6. Taylor expanded in A around inf

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

      1. fp-cancel-sub-sign-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-/.f64 N/A

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

      5. distribute-lft-out N/A

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

      6. associate-*r/ N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*r/ N/A

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

      9. +-commutative N/A

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

      10. associate-*r/ N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 54.5

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

   8. Applied rewrites 54.5%

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

   if 1.85e-50 < B < 1.35000000000000003e154

   1. Initial program 62.6%

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

   3. Taylor expanded in A around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 62.4

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

   5. Applied rewrites 62.4%

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

   6. Taylor expanded in A around 0

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

   8. Applied rewrites 57.8%

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

   if 1.35000000000000003e154 < B

   1. Initial program 19.8%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   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 81.2%

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

Alternative 5: 61.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;A \leq -0.005:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{0.5 \cdot \mathsf{fma}\left(\frac{C}{A}, B, B\right)}{A}\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \left(-B\right)\right)\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= A -0.005)
   (* 180.0 (/ (atan (/ (* 0.5 (fma (/ C A) B B)) A)) (PI)))
   (* 180.0 (/ (atan (* (/ 1.0 B) (- (- C A) (- B)))) (PI)))))

Derivation

1. Split input into 2 regimes

2. if A < -0.0050000000000000001

   1. Initial program 20.2%

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

   3. Taylor expanded in 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. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. distribute-lft-out N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 66.1

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

   5. Applied rewrites 66.1%

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

   6. Taylor expanded in A around inf

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

      1. fp-cancel-sub-sign-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-/.f64 N/A

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

      5. distribute-lft-out N/A

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

      6. associate-*r/ N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*r/ N/A

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

      9. +-commutative N/A

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

      10. associate-*r/ N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 66.1

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

   8. Applied rewrites 66.1%

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

   if -0.0050000000000000001 < A

   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

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 58.3

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

   5. Applied rewrites 58.3%

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

Alternative 6: 57.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;C \leq -1.15 \cdot 10^{-26}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{C + B}{B}\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;C \leq -2.85 \cdot 10^{-213}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{B}{A} \cdot 0.5\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;C \leq 2 \cdot 10^{+64}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(1 - \frac{A}{B}\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{B}{C}\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= C -1.15e-26)
   (* 180.0 (/ (atan (/ (+ C B) B)) (PI)))
   (if (<= C -2.85e-213)
     (* 180.0 (/ (atan (* (/ B A) 0.5)) (PI)))
     (if (<= C 2e+64)
       (/ (* 180.0 (atan (- 1.0 (/ A B)))) (PI))
       (* 180.0 (/ (atan (* -0.5 (/ B C))) (PI)))))))

Derivation

1. Split input into 4 regimes

2. if C < -1.15000000000000004e-26

   1. Initial program 76.1%

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

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

      2. div-sub N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 77.3

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

   5. Applied rewrites 77.3%

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

   6. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 77.3

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

   8. Applied rewrites 77.3%

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

   9. Taylor expanded in A around 0

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

       1. +-commutative N/A

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

       2. lower-+.f64 77.6

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

   11. Applied rewrites 77.6%

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

   if -1.15000000000000004e-26 < C < -2.84999999999999997e-213

   1. Initial program 40.2%

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

   3. Taylor expanded in 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

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 46.9

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

   5. Applied rewrites 46.9%

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

   if -2.84999999999999997e-213 < C < 2.00000000000000004e64

   1. Initial program 50.0%

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

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

      2. div-sub N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 48.7

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

   5. Applied rewrites 48.7%

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

   6. Taylor expanded in C around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 48.1

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

   8. Applied rewrites 48.1%

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   10. Applied rewrites 48.1%

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

   if 2.00000000000000004e64 < C

   1. Initial program 13.4%

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

   3. Taylor expanded in C around 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

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 38.8

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

   5. Applied rewrites 38.8%

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

   6. Taylor expanded in C around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. distribute-rgt1-in N/A

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

      6. metadata-eval N/A

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

      7. mul0-lft N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 73.2

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

   8. Applied rewrites 73.2%

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

   9. Taylor expanded in B around 0

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 73.2

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

   11. Applied rewrites 73.2%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \color{blue}{\frac{B}{C}}\right)}{\mathsf{PI}\left(\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 7: 55.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;C \leq -6.6 \cdot 10^{-16}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{C + B}{B}\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;C \leq -3.5 \cdot 10^{-213}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;C \leq 2 \cdot 10^{+64}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(1 - \frac{A}{B}\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{B}{C}\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= C -6.6e-16)
   (* 180.0 (/ (atan (/ (+ C B) B)) (PI)))
   (if (<= C -3.5e-213)
     (* 180.0 (/ (atan -1.0) (PI)))
     (if (<= C 2e+64)
       (/ (* 180.0 (atan (- 1.0 (/ A B)))) (PI))
       (* 180.0 (/ (atan (* -0.5 (/ B C))) (PI)))))))

Derivation

1. Split input into 4 regimes

2. if C < -6.59999999999999976e-16

   1. Initial program 76.3%

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

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

      2. div-sub N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 79.2

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

   5. Applied rewrites 79.2%

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

   6. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 79.2

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

   8. Applied rewrites 79.2%

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

   9. Taylor expanded in A around 0

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

       1. +-commutative N/A

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

       2. lower-+.f64 79.5

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

   11. Applied rewrites 79.5%

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

   if -6.59999999999999976e-16 < C < -3.50000000000000017e-213

   1. Initial program 43.2%

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

   3. Taylor expanded in B around inf

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

   5. Applied rewrites 39.3%

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

   if -3.50000000000000017e-213 < C < 2.00000000000000004e64

   1. Initial program 50.0%

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

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

      2. div-sub N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 48.7

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

   5. Applied rewrites 48.7%

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

   6. Taylor expanded in C around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 48.1

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

   8. Applied rewrites 48.1%

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   10. Applied rewrites 48.1%

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

   if 2.00000000000000004e64 < C

   1. Initial program 13.4%

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

   3. Taylor expanded in C around 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

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 38.8

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

   5. Applied rewrites 38.8%

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

   6. Taylor expanded in C around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. distribute-rgt1-in N/A

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

      6. metadata-eval N/A

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

      7. mul0-lft N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 73.2

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

   8. Applied rewrites 73.2%

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

   9. Taylor expanded in B around 0

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 73.2

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

   11. Applied rewrites 73.2%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \color{blue}{\frac{B}{C}}\right)}{\mathsf{PI}\left(\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 8: 61.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;A \leq -4.1 \cdot 10^{-7}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{B}{A} \cdot 0.5\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \left(-B\right)\right)\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= A -4.1e-7)
   (* 180.0 (/ (atan (* (/ B A) 0.5)) (PI)))
   (* 180.0 (/ (atan (* (/ 1.0 B) (- (- C A) (- B)))) (PI)))))

Derivation

1. Split input into 2 regimes

2. if A < -4.0999999999999999e-7

   1. Initial program 20.2%

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

   3. Taylor expanded in 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

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 64.9

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

   5. Applied rewrites 64.9%

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

   if -4.0999999999999999e-7 < A

   1. Initial program 59.4%

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

   3. Taylor expanded in B around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 58.6

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

   5. Applied rewrites 58.6%

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

Alternative 9: 49.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq -4.8 \cdot 10^{-47}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(1 - \frac{A}{B}\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;B \leq 9.5 \cdot 10^{-126}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{C}{B}\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= B -4.8e-47)
   (* 180.0 (/ (atan (- 1.0 (/ A B))) (PI)))
   (if (<= B 9.5e-126)
     (* 180.0 (/ (atan (/ C B)) (PI)))
     (* 180.0 (/ (atan -1.0) (PI))))))

Derivation

1. Split input into 3 regimes

2. if B < -4.7999999999999999e-47

   1. Initial program 48.2%

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

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

      2. div-sub N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 72.0

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

   5. Applied rewrites 72.0%

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

   6. Taylor expanded in C around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 65.9

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

   8. Applied rewrites 65.9%

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

   if -4.7999999999999999e-47 < B < 9.5000000000000003e-126

   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

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

      1. associate--l+ N/A

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

      2. div-sub N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 50.1

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

   5. Applied rewrites 50.1%

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

   6. Taylor expanded in C around inf

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

      1. lower-/.f64 38.5

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

   8. Applied rewrites 38.5%

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

   if 9.5000000000000003e-126 < B

   1. Initial program 44.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   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 52.0%

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

Alternative 10: 47.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq -3.9 \cdot 10^{-42}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} 1}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;B \leq 9.5 \cdot 10^{-126}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{C}{B}\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= B -3.9e-42)
   (* 180.0 (/ (atan 1.0) (PI)))
   (if (<= B 9.5e-126)
     (* 180.0 (/ (atan (/ C B)) (PI)))
     (* 180.0 (/ (atan -1.0) (PI))))))

Derivation

1. Split input into 3 regimes

2. if B < -3.9000000000000002e-42

   1. Initial program 46.9%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   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 57.9%

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

   if -3.9000000000000002e-42 < B < 9.5000000000000003e-126

   1. Initial program 53.3%

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

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

      2. div-sub N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 51.4

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

   5. Applied rewrites 51.4%

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

   6. Taylor expanded in C around inf

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

      1. lower-/.f64 38.8

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

   8. Applied rewrites 38.8%

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

   if 9.5000000000000003e-126 < B

   1. Initial program 44.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   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 52.0%

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

Alternative 11: 61.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;A \leq -4.1 \cdot 10^{-7}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{B}{A} \cdot 0.5\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{\left(C + B\right) - A}{B}\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= A -4.1e-7)
   (* 180.0 (/ (atan (* (/ B A) 0.5)) (PI)))
   (* 180.0 (/ (atan (/ (- (+ C B) A) B)) (PI)))))

Derivation

1. Split input into 2 regimes

2. if A < -4.0999999999999999e-7

   1. Initial program 20.2%

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

   3. Taylor expanded in 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

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 64.9

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

   5. Applied rewrites 64.9%

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

   if -4.0999999999999999e-7 < A

   1. Initial program 59.4%

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

   3. Taylor expanded in B around -inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 58.6

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

   5. Applied rewrites 58.6%

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

   6. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 58.6

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

   8. Applied rewrites 58.6%

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

Alternative 12: 51.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 9.5 \cdot 10^{-126}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{C + B}{B}\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= B 9.5e-126)
   (* 180.0 (/ (atan (/ (+ C B) B)) (PI)))
   (* 180.0 (/ (atan -1.0) (PI)))))

Derivation

1. Split input into 2 regimes

2. if B < 9.5000000000000003e-126

   1. Initial program 50.1%

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

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

      2. div-sub N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 61.3

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

   5. Applied rewrites 61.3%

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

   6. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 61.3

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

   8. Applied rewrites 61.3%

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

   9. Taylor expanded in A around 0

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

       1. +-commutative N/A

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

       2. lower-+.f64 53.5

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

   11. Applied rewrites 53.5%

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

   if 9.5000000000000003e-126 < B

   1. Initial program 44.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   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 52.0%

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

Alternative 13: 45.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq -1.1 \cdot 10^{-159}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} 1}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;B \leq 1.25 \cdot 10^{-132}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} 0}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= B -1.1e-159)
   (* 180.0 (/ (atan 1.0) (PI)))
   (if (<= B 1.25e-132)
     (* 180.0 (/ (atan 0.0) (PI)))
     (* 180.0 (/ (atan -1.0) (PI))))))

Derivation

1. Split input into 3 regimes

2. if B < -1.1e-159

   1. Initial program 49.5%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   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 47.2%

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

   if -1.1e-159 < B < 1.25e-132

   1. Initial program 50.4%

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

   3. Taylor expanded in C around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. distribute-rgt1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 35.5

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

   5. Applied rewrites 35.5%

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

   6. Taylor expanded in A around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} 0}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 35.5%

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

   if 1.25e-132 < B

   1. Initial program 45.0%

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

   3. Taylor expanded in B around inf

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

   5. Applied rewrites 51.6%

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

Alternative 14: 40.2% accurate, 2.9× speedup

Math

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

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= B -1e-310)
   (* 180.0 (/ (atan 1.0) (PI)))
   (* 180.0 (/ (atan -1.0) (PI)))))

Derivation

1. Split input into 2 regimes

2. if B < -9.999999999999969e-311

   1. Initial program 50.0%

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

   3. Taylor expanded in B around -inf

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

   5. Applied rewrites 41.2%

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

   if -9.999999999999969e-311 < B

   1. Initial program 46.0%

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

   3. Taylor expanded in B around inf

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

   5. Applied rewrites 41.2%

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

Alternative 15: 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 48.0%

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

3. Taylor expanded in B around inf

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

5. Applied rewrites 21.2%

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

Reproduce

herbie shell --seed 2025044 
(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))))