ABCF->ab-angle angle

Percentage Accurate: 53.3% → 81.0%

Time: 7.4s

Alternatives: 17

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: 81.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;C \leq 3.6 \cdot 10^{+16}:\\ \;\;\;\;\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(B, A - C\right)}{B}\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{B}{C} \cdot -0.5\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= C 3.6e+16)
   (* (atan (/ (- (- C A) (hypot B (- A C))) B)) (/ 180.0 (PI)))
   (* 180.0 (/ (atan (* (/ B C) -0.5)) (PI)))))

Derivation

1. Split input into 2 regimes

2. if C < 3.6e16

   1. Initial program 61.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 85.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-hypot.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-hypot.f64 N/A

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

      10. lower-/.f64 85.6

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

   6. Applied rewrites 85.6%

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

   if 3.6e16 < C

   1. Initial program 17.3%

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

   3. Taylor expanded in C around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. distribute-frac-neg N/A

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

      7. distribute-rgt1-in N/A

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

      8. metadata-eval N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. distribute-rgt1-in N/A

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

      13. lower-/.f64 N/A

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

      14. distribute-rgt1-in N/A

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

      15. metadata-eval N/A

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

      16. mul0-lft 70.7

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

   5. Applied rewrites 70.7%

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

   6. Taylor expanded in B around 0

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

   8. Applied rewrites 70.7%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{B}{C} \cdot \color{blue}{-0.5}\right)}{\mathsf{PI}\left(\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 72.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}\\ t_1 := \frac{C - A}{B}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-13}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(t\_1 - 1\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(\mathsf{fma}\left(-0.5, \frac{B}{C}, 0\right)\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(1 + t\_1\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (let* ((t_0
         (*
          180.0
          (/
           (atan
            (* (/ 1.0 B) (- (- C A) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))
           (PI))))
        (t_1 (/ (- C A) B)))
   (if (<= t_0 -2e-13)
     (* 180.0 (/ (atan (- t_1 1.0)) (PI)))
     (if (<= t_0 0.0)
       (/ (* 180.0 (atan (fma -0.5 (/ B C) 0.0))) (PI))
       (* (atan (+ 1.0 t_1)) (/ 180.0 (PI)))))))

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal 180 binary64) (/.f64 (atan.f64 (*.f64 (/.f64 #s(literal 1 binary64) B) (-.f64 (-.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64))))))) (PI.f64))) < -2.0000000000000001e-13

   1. Initial program 53.2%

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

   3. Taylor expanded in B around inf

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 71.2

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

   5. Applied rewrites 71.2%

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

   if -2.0000000000000001e-13 < (*.f64 #s(literal 180 binary64) (/.f64 (atan.f64 (*.f64 (/.f64 #s(literal 1 binary64) B) (-.f64 (-.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64))))))) (PI.f64))) < 0.0

   1. Initial program 22.0%

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

   3. Taylor expanded in C around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. distribute-frac-neg N/A

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

      7. distribute-rgt1-in N/A

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

      8. metadata-eval N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. distribute-rgt1-in N/A

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

      13. lower-/.f64 N/A

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

      14. distribute-rgt1-in N/A

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

      15. metadata-eval N/A

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

      16. mul0-lft 62.8

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

   5. Applied rewrites 62.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   7. Applied rewrites 62.9%

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

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

   1. Initial program 57.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 83.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-hypot.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-hypot.f64 N/A

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

      10. lower-/.f64 83.0

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

   6. Applied rewrites 83.0%

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

   7. Taylor expanded in B around -inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 74.9

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

   9. Applied rewrites 74.9%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \leq -2 \cdot 10^{-13}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{C - A}{B} - 1\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \leq 0:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(\mathsf{fma}\left(-0.5, \frac{B}{C}, 0\right)\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(1 + \frac{C - A}{B}\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}\\ t_1 := \frac{C - A}{B}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-13}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(t\_1 - 1\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{B}{C} \cdot -0.5\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(1 + t\_1\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (let* ((t_0
         (*
          180.0
          (/
           (atan
            (* (/ 1.0 B) (- (- C A) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))
           (PI))))
        (t_1 (/ (- C A) B)))
   (if (<= t_0 -2e-13)
     (* 180.0 (/ (atan (- t_1 1.0)) (PI)))
     (if (<= t_0 0.0)
       (* 180.0 (/ (atan (* (/ B C) -0.5)) (PI)))
       (* (atan (+ 1.0 t_1)) (/ 180.0 (PI)))))))

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal 180 binary64) (/.f64 (atan.f64 (*.f64 (/.f64 #s(literal 1 binary64) B) (-.f64 (-.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64))))))) (PI.f64))) < -2.0000000000000001e-13

   1. Initial program 53.2%

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

   3. Taylor expanded in B around inf

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 71.2

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

   5. Applied rewrites 71.2%

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

   if -2.0000000000000001e-13 < (*.f64 #s(literal 180 binary64) (/.f64 (atan.f64 (*.f64 (/.f64 #s(literal 1 binary64) B) (-.f64 (-.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64))))))) (PI.f64))) < 0.0

   1. Initial program 22.0%

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

   3. Taylor expanded in C around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. distribute-frac-neg N/A

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

      7. distribute-rgt1-in N/A

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

      8. metadata-eval N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. distribute-rgt1-in N/A

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

      13. lower-/.f64 N/A

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

      14. distribute-rgt1-in N/A

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

      15. metadata-eval N/A

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

      16. mul0-lft 62.8

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

   5. Applied rewrites 62.8%

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

   6. Taylor expanded in B around 0

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

   8. Applied rewrites 62.8%

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

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

   1. Initial program 57.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 83.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-hypot.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-hypot.f64 N/A

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

      10. lower-/.f64 83.0

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

   6. Applied rewrites 83.0%

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

   7. Taylor expanded in B around -inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 74.9

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

   9. Applied rewrites 74.9%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \leq -2 \cdot 10^{-13}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{C - A}{B} - 1\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \leq 0:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{B}{C} \cdot -0.5\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(1 + \frac{C - A}{B}\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}\\ t_1 := \frac{C - A}{B}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-13}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(t\_1 - 1\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{B}{C} \cdot -0.5\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(1 + t\_1\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (let* ((t_0
         (*
          180.0
          (/
           (atan
            (* (/ 1.0 B) (- (- C A) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))
           (PI))))
        (t_1 (/ (- C A) B)))
   (if (<= t_0 -2e-13)
     (* 180.0 (/ (atan (- t_1 1.0)) (PI)))
     (if (<= t_0 0.0)
       (* 180.0 (/ (atan (* (/ B C) -0.5)) (PI)))
       (* 180.0 (/ (atan (+ 1.0 t_1)) (PI)))))))

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal 180 binary64) (/.f64 (atan.f64 (*.f64 (/.f64 #s(literal 1 binary64) B) (-.f64 (-.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64))))))) (PI.f64))) < -2.0000000000000001e-13

   1. Initial program 53.2%

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

   3. Taylor expanded in B around inf

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 71.2

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

   5. Applied rewrites 71.2%

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

   if -2.0000000000000001e-13 < (*.f64 #s(literal 180 binary64) (/.f64 (atan.f64 (*.f64 (/.f64 #s(literal 1 binary64) B) (-.f64 (-.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64))))))) (PI.f64))) < 0.0

   1. Initial program 22.0%

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

   3. Taylor expanded in C around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. distribute-frac-neg N/A

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

      7. distribute-rgt1-in N/A

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

      8. metadata-eval N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. distribute-rgt1-in N/A

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

      13. lower-/.f64 N/A

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

      14. distribute-rgt1-in N/A

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

      15. metadata-eval N/A

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

      16. mul0-lft 62.8

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

   5. Applied rewrites 62.8%

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

   6. Taylor expanded in B around 0

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

   8. Applied rewrites 62.8%

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

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

   1. Initial program 57.8%

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

   3. Taylor expanded in B around -inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 74.9

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

   5. Applied rewrites 74.9%

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

Alternative 5: 67.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-15}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{-A}{B} - 1\right) \cdot 180}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{B}{C} \cdot -0.5\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(1 + \frac{C - A}{B}\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (let* ((t_0
         (* (/ 1.0 B) (- (- C A) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0)))))))
   (if (<= t_0 -4e-15)
     (/ (* (atan (- (/ (- A) B) 1.0)) 180.0) (PI))
     (if (<= t_0 0.0)
       (* 180.0 (/ (atan (* (/ B C) -0.5)) (PI)))
       (* 180.0 (/ (atan (+ 1.0 (/ (- C A) B))) (PI)))))))

Derivation

1. Split input into 3 regimes

2. if (*.f64 (/.f64 #s(literal 1 binary64) B) (-.f64 (-.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))) < -4.0000000000000003e-15

   1. Initial program 53.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 85.2%

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

   5. Taylor expanded in B around inf

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 71.2

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

   7. Applied rewrites 71.2%

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

   8. Taylor expanded in A around inf

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

   10. Applied rewrites 59.3%

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

   if -4.0000000000000003e-15 < (*.f64 (/.f64 #s(literal 1 binary64) B) (-.f64 (-.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))) < 0.0

   1. Initial program 22.0%

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

   3. Taylor expanded in C around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. distribute-frac-neg N/A

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

      7. distribute-rgt1-in N/A

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

      8. metadata-eval N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. distribute-rgt1-in N/A

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

      13. lower-/.f64 N/A

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

      14. distribute-rgt1-in N/A

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

      15. metadata-eval N/A

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

      16. mul0-lft 62.8

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

   5. Applied rewrites 62.8%

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

   6. Taylor expanded in B around 0

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

   8. Applied rewrites 62.8%

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

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

   1. Initial program 57.8%

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

   3. Taylor expanded in B around -inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 74.9

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

   5. Applied rewrites 74.9%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \leq -4 \cdot 10^{-15}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{-A}{B} - 1\right) \cdot 180}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \leq 0:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{B}{C} \cdot -0.5\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(1 + \frac{C - A}{B}\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;C \leq -2.25 \cdot 10^{-7}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - \mathsf{hypot}\left(B, C\right)}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;C \leq 3.6 \cdot 10^{+16}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{\mathsf{hypot}\left(A, B\right) + A}{-B}\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{B}{C} \cdot -0.5\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= C -2.25e-7)
   (/ (* (atan (/ (- C (hypot B C)) B)) 180.0) (PI))
   (if (<= C 3.6e+16)
     (* 180.0 (/ (atan (/ (+ (hypot A B) A) (- B))) (PI)))
     (* 180.0 (/ (atan (* (/ B C) -0.5)) (PI))))))

Derivation

1. Split input into 3 regimes

2. if C < -2.2499999999999999e-7

   1. Initial program 71.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 95.2%

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

   5. Taylor expanded in A around 0

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

      1. lower--.f64 N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. lower-hypot.f64 92.2

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

   7. Applied rewrites 92.2%

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

   if -2.2499999999999999e-7 < C < 3.6e16

   1. Initial program 56.6%

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

   3. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 79.7

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

   5. Applied rewrites 79.7%

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

   if 3.6e16 < C

   1. Initial program 17.3%

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

   3. Taylor expanded in C around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. distribute-frac-neg N/A

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

      7. distribute-rgt1-in N/A

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

      8. metadata-eval N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. distribute-rgt1-in N/A

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

      13. lower-/.f64 N/A

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

      14. distribute-rgt1-in N/A

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

      15. metadata-eval N/A

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

      16. mul0-lft 70.7

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

   5. Applied rewrites 70.7%

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

   6. Taylor expanded in B around 0

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

   8. Applied rewrites 70.7%

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

Alternative 7: 73.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;A \leq -1.25 \cdot 10^{+187}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{B}{A} \cdot 0.5\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;A \leq 3.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - \mathsf{hypot}\left(B, C\right)}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(1 + \frac{C - A}{B}\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= A -1.25e+187)
   (* 180.0 (/ (atan (* (/ B A) 0.5)) (PI)))
   (if (<= A 3.5e-8)
     (/ (* (atan (/ (- C (hypot B C)) B)) 180.0) (PI))
     (* 180.0 (/ (atan (+ 1.0 (/ (- C A) B))) (PI))))))

Derivation

1. Split input into 3 regimes

2. if A < -1.25e187

   1. Initial program 4.9%

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

   3. Taylor expanded in A around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 96.5

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

   5. Applied rewrites 96.5%

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

   if -1.25e187 < A < 3.50000000000000024e-8

   1. Initial program 47.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 72.9%

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

   5. Taylor expanded in A around 0

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

      1. lower--.f64 N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. lower-hypot.f64 69.2

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

   7. Applied rewrites 69.2%

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

   if 3.50000000000000024e-8 < A

   1. Initial program 77.2%

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

   3. Taylor expanded in B around -inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 80.5

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

   5. Applied rewrites 80.5%

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

Alternative 8: 73.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;A \leq -1.25 \cdot 10^{+187}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{B}{A} \cdot 0.5\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;A \leq 3.5 \cdot 10^{-8}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{C - \mathsf{hypot}\left(C, B\right)}{B}\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(1 + \frac{C - A}{B}\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= A -1.25e+187)
   (* 180.0 (/ (atan (* (/ B A) 0.5)) (PI)))
   (if (<= A 3.5e-8)
     (* 180.0 (/ (atan (/ (- C (hypot C B)) B)) (PI)))
     (* 180.0 (/ (atan (+ 1.0 (/ (- C A) B))) (PI))))))

Derivation

1. Split input into 3 regimes

2. if A < -1.25e187

   1. Initial program 4.9%

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

   3. Taylor expanded in A around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 96.5

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

   5. Applied rewrites 96.5%

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

   if -1.25e187 < A < 3.50000000000000024e-8

   1. Initial program 47.6%

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

   3. Taylor expanded in A around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 69.2

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

   5. Applied rewrites 69.2%

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

   if 3.50000000000000024e-8 < A

   1. Initial program 77.2%

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

   3. Taylor expanded in B around -inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 80.5

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

   5. Applied rewrites 80.5%

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

Alternative 9: 55.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{180}{\mathsf{PI}\left(\right)}\\ \mathbf{if}\;C \leq -9.5 \cdot 10^{+171}:\\ \;\;\;\;\tan^{-1} \left(1 + \frac{C}{B}\right) \cdot t\_0\\ \mathbf{elif}\;C \leq -2.09 \cdot 10^{-74}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{-A}{B} - 1\right) \cdot 180}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;C \leq 10^{-69}:\\ \;\;\;\;\tan^{-1} \left(1 - \frac{A}{B}\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{B}{C} \cdot -0.5\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (let* ((t_0 (/ 180.0 (PI))))
   (if (<= C -9.5e+171)
     (* (atan (+ 1.0 (/ C B))) t_0)
     (if (<= C -2.09e-74)
       (/ (* (atan (- (/ (- A) B) 1.0)) 180.0) (PI))
       (if (<= C 1e-69)
         (* (atan (- 1.0 (/ A B))) t_0)
         (* 180.0 (/ (atan (* (/ B C) -0.5)) (PI))))))))

Derivation

1. Split input into 4 regimes

2. if C < -9.49999999999999924e171

   1. Initial program 83.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 95.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-hypot.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-hypot.f64 N/A

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

      10. lower-/.f64 95.4

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

   6. Applied rewrites 95.4%

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

   7. Taylor expanded in B around -inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 84.6

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

   9. Applied rewrites 84.6%

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

   10. Taylor expanded in A around 0

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

   12. Applied rewrites 84.7%

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

   if -9.49999999999999924e171 < C < -2.09000000000000004e-74

   1. Initial program 61.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 96.2%

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

   5. Taylor expanded in B around inf

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 82.3

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

   7. Applied rewrites 82.3%

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

   8. Taylor expanded in A around inf

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

   10. Applied rewrites 66.0%

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

   if -2.09000000000000004e-74 < C < 9.9999999999999996e-70

   1. Initial program 57.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 80.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-hypot.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-hypot.f64 N/A

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

      10. lower-/.f64 80.2

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

   6. Applied rewrites 80.2%

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

   7. Taylor expanded in B around -inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 56.8

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

   9. Applied rewrites 56.8%

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

   10. Taylor expanded in C around 0

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

   12. Applied rewrites 56.8%

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

   if 9.9999999999999996e-70 < C

   1. Initial program 23.1%

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

   3. Taylor expanded in C around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. distribute-frac-neg N/A

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

      7. distribute-rgt1-in N/A

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

      8. metadata-eval N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. distribute-rgt1-in N/A

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

      13. lower-/.f64 N/A

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

      14. distribute-rgt1-in N/A

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

      15. metadata-eval N/A

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

      16. mul0-lft 67.2

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

   5. Applied rewrites 67.2%

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

   6. Taylor expanded in B around 0

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

   8. Applied rewrites 67.2%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq -9.5 \cdot 10^{+171}:\\ \;\;\;\;\tan^{-1} \left(1 + \frac{C}{B}\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;C \leq -2.09 \cdot 10^{-74}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{-A}{B} - 1\right) \cdot 180}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;C \leq 10^{-69}:\\ \;\;\;\;\tan^{-1} \left(1 - \frac{A}{B}\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{B}{C} \cdot -0.5\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{180}{\mathsf{PI}\left(\right)}\\ \mathbf{if}\;C \leq -1.12 \cdot 10^{-20}:\\ \;\;\;\;\tan^{-1} \left(1 + \frac{C}{B}\right) \cdot t\_0\\ \mathbf{elif}\;C \leq -2.09 \cdot 10^{-74}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;C \leq 10^{-69}:\\ \;\;\;\;\tan^{-1} \left(1 - \frac{A}{B}\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{B}{C} \cdot -0.5\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (let* ((t_0 (/ 180.0 (PI))))
   (if (<= C -1.12e-20)
     (* (atan (+ 1.0 (/ C B))) t_0)
     (if (<= C -2.09e-74)
       (* 180.0 (/ (atan -1.0) (PI)))
       (if (<= C 1e-69)
         (* (atan (- 1.0 (/ A B))) t_0)
         (* 180.0 (/ (atan (* (/ B C) -0.5)) (PI))))))))

Derivation

1. Split input into 4 regimes

2. if C < -1.12000000000000002e-20

   1. Initial program 71.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 95.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-hypot.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-hypot.f64 N/A

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

      10. lower-/.f64 95.3

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

   6. Applied rewrites 95.3%

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

   7. Taylor expanded in B around -inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 74.0

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

   9. Applied rewrites 74.0%

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

   10. Taylor expanded in A around 0

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

   12. Applied rewrites 70.2%

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

   if -1.12000000000000002e-20 < C < -2.09000000000000004e-74

   1. Initial program 57.8%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   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 74.1%

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

   if -2.09000000000000004e-74 < C < 9.9999999999999996e-70

   1. Initial program 57.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 80.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-hypot.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-hypot.f64 N/A

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

      10. lower-/.f64 80.2

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

   6. Applied rewrites 80.2%

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

   7. Taylor expanded in B around -inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 56.8

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

   9. Applied rewrites 56.8%

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

   10. Taylor expanded in C around 0

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

   12. Applied rewrites 56.8%

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

   if 9.9999999999999996e-70 < C

   1. Initial program 23.1%

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

   3. Taylor expanded in C around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. distribute-frac-neg N/A

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

      7. distribute-rgt1-in N/A

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

      8. metadata-eval N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. distribute-rgt1-in N/A

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

      13. lower-/.f64 N/A

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

      14. distribute-rgt1-in N/A

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

      15. metadata-eval N/A

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

      16. mul0-lft 67.2

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

   5. Applied rewrites 67.2%

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

   6. Taylor expanded in B around 0

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

   8. Applied rewrites 67.2%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq -1.12 \cdot 10^{-20}:\\ \;\;\;\;\tan^{-1} \left(1 + \frac{C}{B}\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;C \leq -2.09 \cdot 10^{-74}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;C \leq 10^{-69}:\\ \;\;\;\;\tan^{-1} \left(1 - \frac{A}{B}\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{B}{C} \cdot -0.5\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 49.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{180}{\mathsf{PI}\left(\right)}\\ \mathbf{if}\;C \leq -1.12 \cdot 10^{-20}:\\ \;\;\;\;\tan^{-1} \left(1 + \frac{C}{B}\right) \cdot t\_0\\ \mathbf{elif}\;C \leq -2.09 \cdot 10^{-74}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;C \leq 8 \cdot 10^{+117}:\\ \;\;\;\;\tan^{-1} \left(1 - \frac{A}{B}\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{0}{B}\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (let* ((t_0 (/ 180.0 (PI))))
   (if (<= C -1.12e-20)
     (* (atan (+ 1.0 (/ C B))) t_0)
     (if (<= C -2.09e-74)
       (* 180.0 (/ (atan -1.0) (PI)))
       (if (<= C 8e+117)
         (* (atan (- 1.0 (/ A B))) t_0)
         (* 180.0 (/ (atan (/ 0.0 B)) (PI))))))))

Derivation

1. Split input into 4 regimes

2. if C < -1.12000000000000002e-20

   1. Initial program 71.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 95.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-hypot.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-hypot.f64 N/A

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

      10. lower-/.f64 95.3

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

   6. Applied rewrites 95.3%

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

   7. Taylor expanded in B around -inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 74.0

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

   9. Applied rewrites 74.0%

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

   10. Taylor expanded in A around 0

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

   12. Applied rewrites 70.2%

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

   if -1.12000000000000002e-20 < C < -2.09000000000000004e-74

   1. Initial program 57.8%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   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 74.1%

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

   if -2.09000000000000004e-74 < C < 8.0000000000000004e117

   1. Initial program 51.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 72.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-hypot.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-hypot.f64 N/A

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

      10. lower-/.f64 72.4

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

   6. Applied rewrites 72.4%

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

   7. Taylor expanded in B around -inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 50.7

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

   9. Applied rewrites 50.7%

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

   10. Taylor expanded in C around 0

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

   12. Applied rewrites 50.9%

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

   if 8.0000000000000004e117 < C

   1. Initial program 11.2%

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

   3. Taylor expanded in C around inf

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

      1. mul-1-neg N/A

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

      2. distribute-frac-neg N/A

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

      3. distribute-rgt1-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. distribute-rgt1-in N/A

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

      9. lower-/.f64 N/A

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

      10. distribute-rgt1-in N/A

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

      11. metadata-eval N/A

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

      12. mul0-lft 35.6

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

   5. Applied rewrites 35.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{0}{B}\right)}}{\mathsf{PI}\left(\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq -1.12 \cdot 10^{-20}:\\ \;\;\;\;\tan^{-1} \left(1 + \frac{C}{B}\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;C \leq -2.09 \cdot 10^{-74}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;C \leq 8 \cdot 10^{+117}:\\ \;\;\;\;\tan^{-1} \left(1 - \frac{A}{B}\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{0}{B}\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 55.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq -2 \cdot 10^{+17}:\\ \;\;\;\;\tan^{-1} \left(1 + \frac{C}{B}\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;B \leq 5.6 \cdot 10^{+98}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - A}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= B -2e+17)
   (* (atan (+ 1.0 (/ C B))) (/ 180.0 (PI)))
   (if (<= B 5.6e+98)
     (/ (* (atan (/ (- C A) B)) 180.0) (PI))
     (* 180.0 (/ (atan -1.0) (PI))))))

Derivation

1. Split input into 3 regimes

2. if B < -2e17

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 77.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-hypot.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-hypot.f64 N/A

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

      10. lower-/.f64 77.5

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

   6. Applied rewrites 77.5%

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

   7. Taylor expanded in B around -inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 73.1

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

   9. Applied rewrites 73.1%

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

   10. Taylor expanded in A around 0

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

   12. Applied rewrites 68.5%

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

   if -2e17 < B < 5.6000000000000001e98

   1. Initial program 58.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 72.3%

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

   5. Taylor expanded in B around inf

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 51.3

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

   7. Applied rewrites 51.3%

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

   8. Taylor expanded in B around 0

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

   10. Applied rewrites 47.6%

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

   if 5.6000000000000001e98 < B

   1. Initial program 37.1%

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

   3. Taylor expanded in B around inf

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

   5. Applied rewrites 74.0%

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

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -2 \cdot 10^{+17}:\\ \;\;\;\;\tan^{-1} \left(1 + \frac{C}{B}\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;B \leq 5.6 \cdot 10^{+98}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - A}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 45.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq -7 \cdot 10^{+15}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} 1}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;B \leq 1.4 \cdot 10^{+94}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{-A}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= B -7e+15)
   (* 180.0 (/ (atan 1.0) (PI)))
   (if (<= B 1.4e+94)
     (/ (* (atan (/ (- A) B)) 180.0) (PI))
     (* 180.0 (/ (atan -1.0) (PI))))))

Derivation

1. Split input into 3 regimes

2. if B < -7e15

   1. Initial program 42.5%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   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 62.0%

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

   if -7e15 < B < 1.39999999999999999e94

   1. Initial program 58.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 72.7%

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

   5. Taylor expanded in B around inf

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 51.7

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

   7. Applied rewrites 51.7%

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

   8. Taylor expanded in A around inf

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

   10. Applied rewrites 34.7%

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

   if 1.39999999999999999e94 < B

   1. Initial program 37.1%

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

   3. Taylor expanded in B around inf

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

   5. Applied rewrites 74.0%

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

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -7 \cdot 10^{+15}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} 1}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;B \leq 1.4 \cdot 10^{+94}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{-A}{B}\right) \cdot 180}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 45.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq -1.65 \cdot 10^{-161}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} 1}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;B \leq 2.35 \cdot 10^{-79}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{0}{B}\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= B -1.65e-161)
   (* 180.0 (/ (atan 1.0) (PI)))
   (if (<= B 2.35e-79)
     (* 180.0 (/ (atan (/ 0.0 B)) (PI)))
     (* 180.0 (/ (atan -1.0) (PI))))))

Derivation

1. Split input into 3 regimes

2. if B < -1.6499999999999999e-161

   1. Initial program 49.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   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 49.0%

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

   if -1.6499999999999999e-161 < B < 2.3500000000000001e-79

   1. Initial program 55.4%

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

   3. Taylor expanded in C around inf

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

      1. mul-1-neg N/A

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

      2. distribute-frac-neg N/A

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

      3. distribute-rgt1-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. distribute-rgt1-in N/A

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

      9. lower-/.f64 N/A

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

      10. distribute-rgt1-in N/A

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

      11. metadata-eval N/A

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

      12. mul0-lft 30.7

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

   5. Applied rewrites 30.7%

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

   if 2.3500000000000001e-79 < B

   1. Initial program 47.4%

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

   3. Taylor expanded in B around inf

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

   5. Applied rewrites 54.9%

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

Alternative 15: 51.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 1.85 \cdot 10^{-81}:\\ \;\;\;\;\tan^{-1} \left(1 - \frac{A}{B}\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= B 1.85e-81)
   (* (atan (- 1.0 (/ A B))) (/ 180.0 (PI)))
   (* 180.0 (/ (atan -1.0) (PI)))))

Derivation

1. Split input into 2 regimes

2. if B < 1.84999999999999993e-81

   1. Initial program 52.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 75.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-hypot.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-hypot.f64 N/A

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

      10. lower-/.f64 75.1

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

   6. Applied rewrites 75.1%

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

   7. Taylor expanded in B around -inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 59.4

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

   9. Applied rewrites 59.4%

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

   10. Taylor expanded in C around 0

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

   12. Applied rewrites 48.8%

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

   if 1.84999999999999993e-81 < B

   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 54.3%

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

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.85 \cdot 10^{-81}:\\ \;\;\;\;\tan^{-1} \left(1 - \frac{A}{B}\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 40.7% accurate, 2.9× speedup

Math

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

FPCore

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

Derivation

1. Split input into 2 regimes

2. if B < -4.999999999999985e-310

   1. Initial program 52.2%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   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 38.7%

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

   if -4.999999999999985e-310 < B

   1. Initial program 49.7%

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

   3. Taylor expanded in B around inf

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

   5. Applied rewrites 37.0%

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

Alternative 17: 22.2% 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 50.9%

   \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
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 20.4%

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

Reproduce

herbie shell --seed 2025017 
(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))))