ABCF->ab-angle angle

Percentage Accurate: 54.3% → 89.3%

Time: 12.1s

Alternatives: 19

Speedup: 2.4×

Specification

Math

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

FPCore

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

Initial Program: 54.3% 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: 89.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(C - A\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right) \cdot \frac{1}{B}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-49}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{PI}\left(\right)}{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(A - C, B\right)}{B}\right)}} \cdot 180\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{\tan^{-1} \left(-0.5 \cdot \frac{B}{C - A}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(B, A - C\right)}{B}\right)}{\mathsf{PI}\left(\right)}}} \cdot 180\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (let* ((t_0
         (* (- (- C A) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0)))) (/ 1.0 B))))
   (if (<= t_0 -5e-49)
     (* (/ 1.0 (/ (PI) (atan (/ (- (- C A) (hypot (- A C) B)) B)))) 180.0)
     (if (<= t_0 0.0)
       (* (/ (atan (* -0.5 (/ B (- C A)))) (PI)) 180.0)
       (*
        (/ 1.0 (/ 1.0 (/ (atan (/ (- (- C A) (hypot B (- A C))) B)) (PI))))
        180.0)))))

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)))))) < -4.9999999999999999e-49

   1. Initial program 52.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 53.0

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 53.0

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

   4. Applied rewrites 89.8%

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

   if -4.9999999999999999e-49 < (*.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 19.1%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 19.1

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 19.1

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

   4. Applied rewrites 19.1%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. clear-num N/A

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

      5. frac-sub N/A

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

      6. lower-/.f64 N/A

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

      7. metadata-eval N/A

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

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

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

      9. neg-mul-1 N/A

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

      10. remove-double-neg N/A

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

      11. lower--.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-hypot.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-hypot.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 2.6

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

   6. Applied rewrites 2.6%

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

   7. Taylor expanded in A around -inf

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

      1. lower-/.f64 N/A

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

   9. Applied rewrites 3.3%

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

   10. Taylor expanded in B around 0

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

   12. Applied rewrites 99.5%

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

   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 58.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 58.0

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 58.0

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

   4. Applied rewrites 88.6%

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

      1. /-rgt-identity N/A

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

      2. clear-num N/A

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

      3. lift-/.f64 N/A

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

      4. lower-/.f64 88.6

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. lower-/.f64 88.7

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

      9. lift-hypot.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-hypot.f64 88.7

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

   6. Applied rewrites 88.7%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(C - A\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right) \cdot \frac{1}{B} \leq -5 \cdot 10^{-49}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{PI}\left(\right)}{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(A - C, B\right)}{B}\right)}} \cdot 180\\ \mathbf{elif}\;\left(\left(C - A\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right) \cdot \frac{1}{B} \leq 0:\\ \;\;\;\;\frac{\tan^{-1} \left(-0.5 \cdot \frac{B}{C - A}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(B, A - C\right)}{B}\right)}{\mathsf{PI}\left(\right)}}} \cdot 180\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 67.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(C - A\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right) \cdot \frac{1}{B}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+286}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C}{B} - 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;t\_0 \leq -5 \cdot 10^{-49}:\\ \;\;\;\;\frac{\tan^{-1} \left(-1 - \frac{A}{B}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;t\_0 \leq 0.05:\\ \;\;\;\;\tan^{-1} \left(\frac{B}{C} \cdot -0.5\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - A}{B} + 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (let* ((t_0
         (* (- (- C A) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0)))) (/ 1.0 B))))
   (if (<= t_0 -1e+286)
     (* (/ (atan (- (/ C B) 1.0)) (PI)) 180.0)
     (if (<= t_0 -5e-49)
       (* (/ (atan (- -1.0 (/ A B))) (PI)) 180.0)
       (if (<= t_0 0.05)
         (* (atan (* (/ B C) -0.5)) (/ 180.0 (PI)))
         (* (/ (atan (+ (/ (- C A) B) 1.0)) (PI)) 180.0))))))

Derivation

1. Split input into 4 regimes

2. if (*.f64 (/.f64 #s(literal 1 binary64) B) (-.f64 (-.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))) < -1.00000000000000003e286

   1. Initial program 38.7%

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

   3. Taylor expanded in B around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C}{B} - \left(1 + \frac{A}{B}\right)\right)}}{\mathsf{PI}\left(\right)} \]
   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 67.4

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

   5. Applied rewrites 67.4%

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

   6. Taylor expanded in A around 0

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

   8. Applied rewrites 59.7%

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

   if -1.00000000000000003e286 < (*.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)))))) < -4.9999999999999999e-49

   1. Initial program 90.9%

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

   3. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 88.4

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

   5. Applied rewrites 88.4%

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

   6. Taylor expanded in B around inf

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

   8. Applied rewrites 85.9%

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

   if -4.9999999999999999e-49 < (*.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)))))) < 0.050000000000000003

   1. Initial program 21.7%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 21.7

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 21.7

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

   4. Applied rewrites 21.7%

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

   5. Taylor expanded in C around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. distribute-rgt1-in N/A

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

      7. metadata-eval N/A

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

      8. mul0-lft N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 52.4

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

   7. Applied rewrites 52.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 52.3

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

   9. Applied rewrites 52.3%

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

   if 0.050000000000000003 < (*.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 57.7%

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

   3. Taylor expanded in B around -inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 74.7

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

   5. Applied rewrites 74.7%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(C - A\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right) \cdot \frac{1}{B} \leq -1 \cdot 10^{+286}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C}{B} - 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;\left(\left(C - A\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right) \cdot \frac{1}{B} \leq -5 \cdot 10^{-49}:\\ \;\;\;\;\frac{\tan^{-1} \left(-1 - \frac{A}{B}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;\left(\left(C - A\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right) \cdot \frac{1}{B} \leq 0.05:\\ \;\;\;\;\tan^{-1} \left(\frac{B}{C} \cdot -0.5\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - A}{B} + 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\frac{\mathsf{PI}\left(\right)}{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(A - C, B\right)}{B}\right)}} \cdot 180\\ t_1 := \left(\left(C - A\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right) \cdot \frac{1}{B}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-49}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\tan^{-1} \left(-0.5 \cdot \frac{B}{C - A}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (let* ((t_0
         (* (/ 1.0 (/ (PI) (atan (/ (- (- C A) (hypot (- A C) B)) B)))) 180.0))
        (t_1
         (* (- (- C A) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0)))) (/ 1.0 B))))
   (if (<= t_1 -5e-49)
     t_0
     (if (<= t_1 0.0) (* (/ (atan (* -0.5 (/ B (- C A)))) (PI)) 180.0) t_0))))

Derivation

1. Split input into 2 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)))))) < -4.9999999999999999e-49 or 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 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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 55.4

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 55.4

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

   4. Applied rewrites 89.3%

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

   if -4.9999999999999999e-49 < (*.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 19.1%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 19.1

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 19.1

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

   4. Applied rewrites 19.1%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. clear-num N/A

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

      5. frac-sub N/A

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

      6. lower-/.f64 N/A

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

      7. metadata-eval N/A

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

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

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

      9. neg-mul-1 N/A

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

      10. remove-double-neg N/A

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

      11. lower--.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-hypot.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-hypot.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 2.6

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

   6. Applied rewrites 2.6%

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

   7. Taylor expanded in A around -inf

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

      1. lower-/.f64 N/A

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

   9. Applied rewrites 3.3%

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

   10. Taylor expanded in B around 0

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

   12. Applied rewrites 99.5%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(C - A\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right) \cdot \frac{1}{B} \leq -5 \cdot 10^{-49}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{PI}\left(\right)}{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(A - C, B\right)}{B}\right)}} \cdot 180\\ \mathbf{elif}\;\left(\left(C - A\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right) \cdot \frac{1}{B} \leq 0:\\ \;\;\;\;\frac{\tan^{-1} \left(-0.5 \cdot \frac{B}{C - A}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{PI}\left(\right)}{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(A - C, B\right)}{B}\right)}} \cdot 180\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(C - A\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right) \cdot \frac{1}{B}\\ t_1 := \frac{C - A}{B}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-49}:\\ \;\;\;\;\frac{\tan^{-1} \left(t\_1 - 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;t\_0 \leq 0.05:\\ \;\;\;\;\frac{\tan^{-1} \left(-0.5 \cdot \frac{B}{C - A}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(t\_1 + 1\right) \cdot 180}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (let* ((t_0
         (* (- (- C A) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0)))) (/ 1.0 B)))
        (t_1 (/ (- C A) B)))
   (if (<= t_0 -5e-49)
     (* (/ (atan (- t_1 1.0)) (PI)) 180.0)
     (if (<= t_0 0.05)
       (* (/ (atan (* -0.5 (/ B (- C A)))) (PI)) 180.0)
       (/ (* (atan (+ t_1 1.0)) 180.0) (PI))))))

Derivation

1. Split input into 3 regimes

2. if (*.f64 (/.f64 #s(literal 1 binary64) B) (-.f64 (-.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))) < -4.9999999999999999e-49

   1. Initial program 52.9%

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

   3. Taylor expanded in B around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C}{B} - \left(1 + \frac{A}{B}\right)\right)}}{\mathsf{PI}\left(\right)} \]
   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 74.6

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

   5. Applied rewrites 74.6%

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

   if -4.9999999999999999e-49 < (*.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)))))) < 0.050000000000000003

   1. Initial program 21.7%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 21.7

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 21.7

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

   4. Applied rewrites 21.7%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. clear-num N/A

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

      5. frac-sub N/A

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

      6. lower-/.f64 N/A

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

      7. metadata-eval N/A

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

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

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

      9. neg-mul-1 N/A

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

      10. remove-double-neg N/A

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

      11. lower--.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-hypot.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-hypot.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 5.7

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

   6. Applied rewrites 5.7%

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

   7. Taylor expanded in A around -inf

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

      1. lower-/.f64 N/A

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

   9. Applied rewrites 6.4%

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

   10. Taylor expanded in B around 0

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

   12. Applied rewrites 97.1%

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

   if 0.050000000000000003 < (*.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 57.7%

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

   3. Taylor expanded in B around -inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 74.7

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

   5. Applied rewrites 74.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 74.7

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

   7. Applied rewrites 74.7%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(C - A\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right) \cdot \frac{1}{B} \leq -5 \cdot 10^{-49}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - A}{B} - 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;\left(\left(C - A\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right) \cdot \frac{1}{B} \leq 0.05:\\ \;\;\;\;\frac{\tan^{-1} \left(-0.5 \cdot \frac{B}{C - A}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - A}{B} + 1\right) \cdot 180}{\mathsf{PI}\left(\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(C - A\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right) \cdot \frac{1}{B}\\ t_1 := \frac{C - A}{B}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-49}:\\ \;\;\;\;\frac{\tan^{-1} \left(t\_1 - 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;t\_0 \leq 0.05:\\ \;\;\;\;\tan^{-1} \left(\frac{B}{C} \cdot -0.5\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(t\_1 + 1\right) \cdot 180}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (let* ((t_0
         (* (- (- C A) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0)))) (/ 1.0 B)))
        (t_1 (/ (- C A) B)))
   (if (<= t_0 -5e-49)
     (* (/ (atan (- t_1 1.0)) (PI)) 180.0)
     (if (<= t_0 0.05)
       (* (atan (* (/ B C) -0.5)) (/ 180.0 (PI)))
       (/ (* (atan (+ t_1 1.0)) 180.0) (PI))))))

Derivation

1. Split input into 3 regimes

2. if (*.f64 (/.f64 #s(literal 1 binary64) B) (-.f64 (-.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))) < -4.9999999999999999e-49

   1. Initial program 52.9%

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

   3. Taylor expanded in B around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C}{B} - \left(1 + \frac{A}{B}\right)\right)}}{\mathsf{PI}\left(\right)} \]
   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 74.6

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

   5. Applied rewrites 74.6%

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

   if -4.9999999999999999e-49 < (*.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)))))) < 0.050000000000000003

   1. Initial program 21.7%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 21.7

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 21.7

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

   4. Applied rewrites 21.7%

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

   5. Taylor expanded in C around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. distribute-rgt1-in N/A

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

      7. metadata-eval N/A

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

      8. mul0-lft N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 52.4

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

   7. Applied rewrites 52.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 52.3

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

   9. Applied rewrites 52.3%

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

   if 0.050000000000000003 < (*.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 57.7%

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

   3. Taylor expanded in B around -inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 74.7

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

   5. Applied rewrites 74.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 74.7

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

   7. Applied rewrites 74.7%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(C - A\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right) \cdot \frac{1}{B} \leq -5 \cdot 10^{-49}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - A}{B} - 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;\left(\left(C - A\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right) \cdot \frac{1}{B} \leq 0.05:\\ \;\;\;\;\tan^{-1} \left(\frac{B}{C} \cdot -0.5\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - A}{B} + 1\right) \cdot 180}{\mathsf{PI}\left(\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(C - A\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right) \cdot \frac{1}{B}\\ t_1 := \frac{C - A}{B}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-49}:\\ \;\;\;\;\frac{\tan^{-1} \left(t\_1 - 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;t\_0 \leq 0.05:\\ \;\;\;\;\tan^{-1} \left(\frac{B}{C} \cdot -0.5\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(t\_1 + 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (let* ((t_0
         (* (- (- C A) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0)))) (/ 1.0 B)))
        (t_1 (/ (- C A) B)))
   (if (<= t_0 -5e-49)
     (* (/ (atan (- t_1 1.0)) (PI)) 180.0)
     (if (<= t_0 0.05)
       (* (atan (* (/ B C) -0.5)) (/ 180.0 (PI)))
       (* (/ (atan (+ t_1 1.0)) (PI)) 180.0)))))

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)))))) < -4.9999999999999999e-49

   1. Initial program 52.9%

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

   3. Taylor expanded in B around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C}{B} - \left(1 + \frac{A}{B}\right)\right)}}{\mathsf{PI}\left(\right)} \]
   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 74.6

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

   5. Applied rewrites 74.6%

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

   if -4.9999999999999999e-49 < (*.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)))))) < 0.050000000000000003

   1. Initial program 21.7%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 21.7

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 21.7

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

   4. Applied rewrites 21.7%

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

   5. Taylor expanded in C around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. distribute-rgt1-in N/A

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

      7. metadata-eval N/A

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

      8. mul0-lft N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 52.4

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

   7. Applied rewrites 52.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 52.3

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

   9. Applied rewrites 52.3%

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

   if 0.050000000000000003 < (*.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 57.7%

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

   3. Taylor expanded in B around -inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 74.7

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

   5. Applied rewrites 74.7%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(C - A\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right) \cdot \frac{1}{B} \leq -5 \cdot 10^{-49}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - A}{B} - 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;\left(\left(C - A\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right) \cdot \frac{1}{B} \leq 0.05:\\ \;\;\;\;\tan^{-1} \left(\frac{B}{C} \cdot -0.5\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - A}{B} + 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;A \leq -9.2 \cdot 10^{+110}:\\ \;\;\;\;\frac{\tan^{-1} \left(-0.5 \cdot \frac{B}{C - A}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;A \leq 5 \cdot 10^{-29}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - \mathsf{hypot}\left(C, B\right)}{B}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{180}{\frac{\mathsf{PI}\left(\right)}{\tan^{-1} \left(\frac{\mathsf{hypot}\left(B, A\right) + A}{-B}\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= A -9.2e+110)
   (* (/ (atan (* -0.5 (/ B (- C A)))) (PI)) 180.0)
   (if (<= A 5e-29)
     (* (/ (atan (/ (- C (hypot C B)) B)) (PI)) 180.0)
     (/ 180.0 (/ (PI) (atan (/ (+ (hypot B A) A) (- B))))))))

Derivation

1. Split input into 3 regimes

2. if A < -9.2000000000000001e110

   1. Initial program 15.5%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 15.5

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 15.5

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

   4. Applied rewrites 61.1%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. clear-num N/A

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

      5. frac-sub N/A

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

      6. lower-/.f64 N/A

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

      7. metadata-eval N/A

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

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

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

      9. neg-mul-1 N/A

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

      10. remove-double-neg N/A

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

      11. lower--.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-hypot.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-hypot.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 10.5

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

   6. Applied rewrites 10.5%

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

   7. Taylor expanded in A around -inf

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

      1. lower-/.f64 N/A

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

   9. Applied rewrites 5.5%

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

   10. Taylor expanded in B around 0

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

   12. Applied rewrites 85.6%

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

   if -9.2000000000000001e110 < A < 4.99999999999999986e-29

   1. Initial program 52.4%

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

   3. Taylor expanded in A around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. +-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 82.2

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

   5. Applied rewrites 82.2%

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

   if 4.99999999999999986e-29 < A

   1. Initial program 71.8%

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

   3. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 82.2

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

   5. Applied rewrites 82.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 82.2

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

   7. Applied rewrites 82.2%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -9.2 \cdot 10^{+110}:\\ \;\;\;\;\frac{\tan^{-1} \left(-0.5 \cdot \frac{B}{C - A}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;A \leq 5 \cdot 10^{-29}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - \mathsf{hypot}\left(C, B\right)}{B}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{180}{\frac{\mathsf{PI}\left(\right)}{\tan^{-1} \left(\frac{\mathsf{hypot}\left(B, A\right) + A}{-B}\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;A \leq -9.2 \cdot 10^{+110}:\\ \;\;\;\;\frac{\tan^{-1} \left(-0.5 \cdot \frac{B}{C - A}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;A \leq 5 \cdot 10^{-29}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - \mathsf{hypot}\left(C, B\right)}{B}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{\mathsf{hypot}\left(B, A\right) + A}{-B}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= A -9.2e+110)
   (* (/ (atan (* -0.5 (/ B (- C A)))) (PI)) 180.0)
   (if (<= A 5e-29)
     (* (/ (atan (/ (- C (hypot C B)) B)) (PI)) 180.0)
     (* (/ (atan (/ (+ (hypot B A) A) (- B))) (PI)) 180.0))))

Derivation

1. Split input into 3 regimes

2. if A < -9.2000000000000001e110

   1. Initial program 15.5%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 15.5

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 15.5

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

   4. Applied rewrites 61.1%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. clear-num N/A

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

      5. frac-sub N/A

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

      6. lower-/.f64 N/A

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

      7. metadata-eval N/A

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

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

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

      9. neg-mul-1 N/A

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

      10. remove-double-neg N/A

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

      11. lower--.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-hypot.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-hypot.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 10.5

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

   6. Applied rewrites 10.5%

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

   7. Taylor expanded in A around -inf

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

      1. lower-/.f64 N/A

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

   9. Applied rewrites 5.5%

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

   10. Taylor expanded in B around 0

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

   12. Applied rewrites 85.6%

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

   if -9.2000000000000001e110 < A < 4.99999999999999986e-29

   1. Initial program 52.4%

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

   3. Taylor expanded in A around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. +-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 82.2

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

   5. Applied rewrites 82.2%

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

   if 4.99999999999999986e-29 < A

   1. Initial program 71.8%

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

   3. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 82.2

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

   5. Applied rewrites 82.2%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -9.2 \cdot 10^{+110}:\\ \;\;\;\;\frac{\tan^{-1} \left(-0.5 \cdot \frac{B}{C - A}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;A \leq 5 \cdot 10^{-29}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - \mathsf{hypot}\left(C, B\right)}{B}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{\mathsf{hypot}\left(B, A\right) + A}{-B}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;A \leq -9.2 \cdot 10^{+110}:\\ \;\;\;\;\frac{\tan^{-1} \left(-0.5 \cdot \frac{B}{C - A}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;A \leq 1.45 \cdot 10^{+30}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - \mathsf{hypot}\left(C, B\right)}{B}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - A}{B} - 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= A -9.2e+110)
   (* (/ (atan (* -0.5 (/ B (- C A)))) (PI)) 180.0)
   (if (<= A 1.45e+30)
     (* (/ (atan (/ (- C (hypot C B)) B)) (PI)) 180.0)
     (* (/ (atan (- (/ (- C A) B) 1.0)) (PI)) 180.0))))

Derivation

1. Split input into 3 regimes

2. if A < -9.2000000000000001e110

   1. Initial program 15.5%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 15.5

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 15.5

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

   4. Applied rewrites 61.1%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. clear-num N/A

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

      5. frac-sub N/A

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

      6. lower-/.f64 N/A

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

      7. metadata-eval N/A

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

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

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

      9. neg-mul-1 N/A

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

      10. remove-double-neg N/A

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

      11. lower--.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-hypot.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-hypot.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 10.5

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

   6. Applied rewrites 10.5%

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

   7. Taylor expanded in A around -inf

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

      1. lower-/.f64 N/A

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

   9. Applied rewrites 5.5%

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

   10. Taylor expanded in B around 0

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

   12. Applied rewrites 85.6%

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

   if -9.2000000000000001e110 < A < 1.4499999999999999e30

   1. Initial program 52.1%

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

   3. Taylor expanded in 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 80.6

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

   5. Applied rewrites 80.6%

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

   if 1.4499999999999999e30 < A

   1. Initial program 78.5%

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

   3. Taylor expanded in B around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C}{B} - \left(1 + \frac{A}{B}\right)\right)}}{\mathsf{PI}\left(\right)} \]
   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 81.9

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

   5. Applied rewrites 81.9%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -9.2 \cdot 10^{+110}:\\ \;\;\;\;\frac{\tan^{-1} \left(-0.5 \cdot \frac{B}{C - A}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;A \leq 1.45 \cdot 10^{+30}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - \mathsf{hypot}\left(C, B\right)}{B}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - A}{B} - 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 46.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq -1.9 \cdot 10^{+75}:\\ \;\;\;\;\frac{\tan^{-1} 1}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;B \leq -8 \cdot 10^{-195}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C}{B}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;B \leq 2.5 \cdot 10^{-158}:\\ \;\;\;\;\frac{\tan^{-1} 0}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;B \leq 9.5 \cdot 10^{-31}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{-A}{B}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= B -1.9e+75)
   (* (/ (atan 1.0) (PI)) 180.0)
   (if (<= B -8e-195)
     (* (/ (atan (/ C B)) (PI)) 180.0)
     (if (<= B 2.5e-158)
       (* (/ (atan 0.0) (PI)) 180.0)
       (if (<= B 9.5e-31)
         (* (/ (atan (/ (- A) B)) (PI)) 180.0)
         (* (/ (atan -1.0) (PI)) 180.0))))))

Derivation

1. Split input into 5 regimes

2. if B < -1.9000000000000001e75

   1. Initial program 46.8%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   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 67.9%

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

   if -1.9000000000000001e75 < B < -8.0000000000000007e-195

   1. Initial program 69.4%

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

   3. Taylor expanded in B around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C}{B} - \left(1 + \frac{A}{B}\right)\right)}}{\mathsf{PI}\left(\right)} \]
   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 59.8

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

   5. Applied rewrites 59.8%

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

   6. Taylor expanded in C around inf

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

   8. Applied rewrites 42.0%

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

   if -8.0000000000000007e-195 < B < 2.49999999999999986e-158

   1. Initial program 45.3%

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

   3. 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. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. mul0-lft N/A

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

      4. div0 N/A

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

      5. metadata-eval 52.0

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

   5. Applied rewrites 52.0%

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

   if 2.49999999999999986e-158 < B < 9.5000000000000008e-31

   1. Initial program 52.8%

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

   3. Taylor expanded in B around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C}{B} - \left(1 + \frac{A}{B}\right)\right)}}{\mathsf{PI}\left(\right)} \]
   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 49.9

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

   5. Applied rewrites 49.9%

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

   6. Taylor expanded in A around inf

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

   8. Applied rewrites 36.5%

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

   if 9.5000000000000008e-31 < B

   1. Initial program 46.1%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   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 65.8%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{-1}}{\mathsf{PI}\left(\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -1.9 \cdot 10^{+75}:\\ \;\;\;\;\frac{\tan^{-1} 1}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;B \leq -8 \cdot 10^{-195}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C}{B}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;B \leq 2.5 \cdot 10^{-158}:\\ \;\;\;\;\frac{\tan^{-1} 0}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;B \leq 9.5 \cdot 10^{-31}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{-A}{B}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 60.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;A \leq -3.2 \cdot 10^{+18}:\\ \;\;\;\;\frac{\tan^{-1} \left(0.5 \cdot \frac{B}{A}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;A \leq 6.5 \cdot 10^{-84}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C}{B} - 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;A \leq 5 \cdot 10^{+155}:\\ \;\;\;\;\frac{\tan^{-1} \left(1 - \frac{A}{B}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{A + B}{-B}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= A -3.2e+18)
   (* (/ (atan (* 0.5 (/ B A))) (PI)) 180.0)
   (if (<= A 6.5e-84)
     (* (/ (atan (- (/ C B) 1.0)) (PI)) 180.0)
     (if (<= A 5e+155)
       (* (/ (atan (- 1.0 (/ A B))) (PI)) 180.0)
       (* (/ (atan (/ (+ A B) (- B))) (PI)) 180.0)))))

Derivation

1. Split input into 4 regimes

2. if A < -3.2e18

   1. Initial program 20.0%

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

   3. Taylor expanded in A around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 71.6

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

   5. Applied rewrites 71.6%

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

   if -3.2e18 < A < 6.50000000000000022e-84

   1. Initial program 54.1%

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

   3. Taylor expanded in B around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C}{B} - \left(1 + \frac{A}{B}\right)\right)}}{\mathsf{PI}\left(\right)} \]
   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 58.2

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

   5. Applied rewrites 58.2%

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

   6. Taylor expanded in A around 0

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

   8. Applied rewrites 57.6%

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

   if 6.50000000000000022e-84 < A < 4.9999999999999999e155

   1. Initial program 66.6%

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

   3. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 76.0

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

   5. Applied rewrites 76.0%

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

   6. Taylor expanded in B around -inf

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

   8. Applied rewrites 71.1%

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

   if 4.9999999999999999e155 < A

   1. Initial program 75.5%

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

   3. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 88.2

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

   5. Applied rewrites 88.2%

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

   6. Taylor expanded in A around 0

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

   8. Applied rewrites 88.2%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -3.2 \cdot 10^{+18}:\\ \;\;\;\;\frac{\tan^{-1} \left(0.5 \cdot \frac{B}{A}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;A \leq 6.5 \cdot 10^{-84}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C}{B} - 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;A \leq 5 \cdot 10^{+155}:\\ \;\;\;\;\frac{\tan^{-1} \left(1 - \frac{A}{B}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{A + B}{-B}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 60.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;A \leq -3.2 \cdot 10^{+18}:\\ \;\;\;\;\frac{\tan^{-1} \left(0.5 \cdot \frac{B}{A}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;A \leq 6.5 \cdot 10^{-84}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C}{B} - 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;A \leq 5 \cdot 10^{+155}:\\ \;\;\;\;\frac{\tan^{-1} \left(1 - \frac{A}{B}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(-1 - \frac{A}{B}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= A -3.2e+18)
   (* (/ (atan (* 0.5 (/ B A))) (PI)) 180.0)
   (if (<= A 6.5e-84)
     (* (/ (atan (- (/ C B) 1.0)) (PI)) 180.0)
     (if (<= A 5e+155)
       (* (/ (atan (- 1.0 (/ A B))) (PI)) 180.0)
       (* (/ (atan (- -1.0 (/ A B))) (PI)) 180.0)))))

Derivation

1. Split input into 4 regimes

2. if A < -3.2e18

   1. Initial program 20.0%

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

   3. Taylor expanded in A around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 71.6

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

   5. Applied rewrites 71.6%

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

   if -3.2e18 < A < 6.50000000000000022e-84

   1. Initial program 54.1%

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

   3. Taylor expanded in B around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C}{B} - \left(1 + \frac{A}{B}\right)\right)}}{\mathsf{PI}\left(\right)} \]
   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 58.2

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

   5. Applied rewrites 58.2%

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

   6. Taylor expanded in A around 0

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

   8. Applied rewrites 57.6%

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

   if 6.50000000000000022e-84 < A < 4.9999999999999999e155

   1. Initial program 66.6%

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

   3. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 76.0

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

   5. Applied rewrites 76.0%

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

   6. Taylor expanded in B around -inf

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

   8. Applied rewrites 71.1%

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

   if 4.9999999999999999e155 < A

   1. Initial program 75.5%

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

   3. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 88.2

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

   5. Applied rewrites 88.2%

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

   6. Taylor expanded in B around inf

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

   8. Applied rewrites 88.2%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -3.2 \cdot 10^{+18}:\\ \;\;\;\;\frac{\tan^{-1} \left(0.5 \cdot \frac{B}{A}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;A \leq 6.5 \cdot 10^{-84}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C}{B} - 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;A \leq 5 \cdot 10^{+155}:\\ \;\;\;\;\frac{\tan^{-1} \left(1 - \frac{A}{B}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(-1 - \frac{A}{B}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 51.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;A \leq -6.5 \cdot 10^{+83}:\\ \;\;\;\;\frac{\tan^{-1} 0}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;A \leq 6.5 \cdot 10^{-84}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C}{B} - 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;A \leq 5 \cdot 10^{+155}:\\ \;\;\;\;\frac{\tan^{-1} \left(1 - \frac{A}{B}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(-1 - \frac{A}{B}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= A -6.5e+83)
   (* (/ (atan 0.0) (PI)) 180.0)
   (if (<= A 6.5e-84)
     (* (/ (atan (- (/ C B) 1.0)) (PI)) 180.0)
     (if (<= A 5e+155)
       (* (/ (atan (- 1.0 (/ A B))) (PI)) 180.0)
       (* (/ (atan (- -1.0 (/ A B))) (PI)) 180.0)))))

Derivation

1. Split input into 4 regimes

2. if A < -6.5000000000000003e83

   1. Initial program 18.8%

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

   3. Taylor expanded in 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. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. mul0-lft N/A

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

      4. div0 N/A

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

      5. metadata-eval 43.1

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

   5. Applied rewrites 43.1%

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

   if -6.5000000000000003e83 < A < 6.50000000000000022e-84

   1. Initial program 51.4%

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

   3. Taylor expanded in B around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C}{B} - \left(1 + \frac{A}{B}\right)\right)}}{\mathsf{PI}\left(\right)} \]
   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 56.0

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

   5. Applied rewrites 56.0%

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

   6. Taylor expanded in A around 0

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

   8. Applied rewrites 55.5%

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

   if 6.50000000000000022e-84 < A < 4.9999999999999999e155

   1. Initial program 66.6%

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

   3. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 76.0

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

   5. Applied rewrites 76.0%

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

   6. Taylor expanded in B around -inf

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

   8. Applied rewrites 71.1%

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

   if 4.9999999999999999e155 < A

   1. Initial program 75.5%

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

   3. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 88.2

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

   5. Applied rewrites 88.2%

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

   6. Taylor expanded in B around inf

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

   8. Applied rewrites 88.2%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -6.5 \cdot 10^{+83}:\\ \;\;\;\;\frac{\tan^{-1} 0}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;A \leq 6.5 \cdot 10^{-84}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C}{B} - 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;A \leq 5 \cdot 10^{+155}:\\ \;\;\;\;\frac{\tan^{-1} \left(1 - \frac{A}{B}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(-1 - \frac{A}{B}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 50.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq -1.9 \cdot 10^{+75}:\\ \;\;\;\;\frac{\tan^{-1} 1}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;B \leq -8 \cdot 10^{-195}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C}{B}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;B \leq 8.8 \cdot 10^{-207}:\\ \;\;\;\;\frac{\tan^{-1} 0}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C}{B} - 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= B -1.9e+75)
   (* (/ (atan 1.0) (PI)) 180.0)
   (if (<= B -8e-195)
     (* (/ (atan (/ C B)) (PI)) 180.0)
     (if (<= B 8.8e-207)
       (* (/ (atan 0.0) (PI)) 180.0)
       (* (/ (atan (- (/ C B) 1.0)) (PI)) 180.0)))))

Derivation

1. Split input into 4 regimes

2. if B < -1.9000000000000001e75

   1. Initial program 46.8%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   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 67.9%

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

   if -1.9000000000000001e75 < B < -8.0000000000000007e-195

   1. Initial program 69.4%

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

   3. Taylor expanded in B around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C}{B} - \left(1 + \frac{A}{B}\right)\right)}}{\mathsf{PI}\left(\right)} \]
   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 59.8

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

   5. Applied rewrites 59.8%

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

   6. Taylor expanded in C around inf

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

   8. Applied rewrites 42.0%

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

   if -8.0000000000000007e-195 < B < 8.7999999999999995e-207

   1. Initial program 47.2%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   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. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. mul0-lft N/A

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

      4. div0 N/A

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

      5. metadata-eval 61.9

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

   5. Applied rewrites 61.9%

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

   if 8.7999999999999995e-207 < 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

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C}{B} - \left(1 + \frac{A}{B}\right)\right)}}{\mathsf{PI}\left(\right)} \]
   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 68.4

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

   5. Applied rewrites 68.4%

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

   6. Taylor expanded in A around 0

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

   8. Applied rewrites 58.3%

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

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -1.9 \cdot 10^{+75}:\\ \;\;\;\;\frac{\tan^{-1} 1}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;B \leq -8 \cdot 10^{-195}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C}{B}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;B \leq 8.8 \cdot 10^{-207}:\\ \;\;\;\;\frac{\tan^{-1} 0}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C}{B} - 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 51.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;C \leq -9.2 \cdot 10^{-101}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C}{B} - 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;C \leq 3.6 \cdot 10^{+208}:\\ \;\;\;\;\frac{\tan^{-1} \left(-1 - \frac{A}{B}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} 0}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= C -9.2e-101)
   (* (/ (atan (- (/ C B) 1.0)) (PI)) 180.0)
   (if (<= C 3.6e+208)
     (* (/ (atan (- -1.0 (/ A B))) (PI)) 180.0)
     (* (/ (atan 0.0) (PI)) 180.0))))

Derivation

1. Split input into 3 regimes

2. if C < -9.1999999999999998e-101

   1. Initial program 73.2%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   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 78.2

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

   5. Applied rewrites 78.2%

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

   6. Taylor expanded in A around 0

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

   8. Applied rewrites 77.1%

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

   if -9.1999999999999998e-101 < C < 3.60000000000000003e208

   1. Initial program 47.3%

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

   3. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 72.9

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

   5. Applied rewrites 72.9%

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

   6. Taylor expanded in B around inf

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

   8. Applied rewrites 45.9%

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

   if 3.60000000000000003e208 < C

   1. Initial program 5.5%

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

   3. Taylor expanded in C around 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. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. mul0-lft N/A

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

      4. div0 N/A

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

      5. metadata-eval 55.9

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

   5. Applied rewrites 55.9%

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

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq -9.2 \cdot 10^{-101}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C}{B} - 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;C \leq 3.6 \cdot 10^{+208}:\\ \;\;\;\;\frac{\tan^{-1} \left(-1 - \frac{A}{B}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} 0}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 44.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq -1.9 \cdot 10^{+75}:\\ \;\;\;\;\frac{\tan^{-1} 1}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;B \leq -8 \cdot 10^{-195}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C}{B}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;B \leq 2 \cdot 10^{-51}:\\ \;\;\;\;\frac{\tan^{-1} 0}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= B -1.9e+75)
   (* (/ (atan 1.0) (PI)) 180.0)
   (if (<= B -8e-195)
     (* (/ (atan (/ C B)) (PI)) 180.0)
     (if (<= B 2e-51)
       (* (/ (atan 0.0) (PI)) 180.0)
       (* (/ (atan -1.0) (PI)) 180.0)))))

Derivation

1. Split input into 4 regimes

2. if B < -1.9000000000000001e75

   1. Initial program 46.8%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   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 67.9%

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

   if -1.9000000000000001e75 < B < -8.0000000000000007e-195

   1. Initial program 69.4%

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

   3. Taylor expanded in B around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C}{B} - \left(1 + \frac{A}{B}\right)\right)}}{\mathsf{PI}\left(\right)} \]
   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 59.8

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

   5. Applied rewrites 59.8%

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

   6. Taylor expanded in C around inf

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

   8. Applied rewrites 42.0%

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

   if -8.0000000000000007e-195 < B < 2e-51

   1. Initial program 47.0%

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

   3. Taylor expanded in C around inf

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. mul0-lft N/A

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

      4. div0 N/A

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

      5. metadata-eval 40.7

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

   5. Applied rewrites 40.7%

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

   if 2e-51 < B

   1. Initial program 47.8%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   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 63.0%

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

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -1.9 \cdot 10^{+75}:\\ \;\;\;\;\frac{\tan^{-1} 1}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;B \leq -8 \cdot 10^{-195}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C}{B}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;B \leq 2 \cdot 10^{-51}:\\ \;\;\;\;\frac{\tan^{-1} 0}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 44.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq -4.8 \cdot 10^{-78}:\\ \;\;\;\;\frac{\tan^{-1} 1}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;B \leq 2 \cdot 10^{-51}:\\ \;\;\;\;\frac{\tan^{-1} 0}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= B -4.8e-78)
   (* (/ (atan 1.0) (PI)) 180.0)
   (if (<= B 2e-51)
     (* (/ (atan 0.0) (PI)) 180.0)
     (* (/ (atan -1.0) (PI)) 180.0))))

Derivation

1. Split input into 3 regimes

2. if B < -4.79999999999999999e-78

   1. Initial program 56.8%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   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 54.0%

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

   if -4.79999999999999999e-78 < B < 2e-51

   1. Initial program 50.1%

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

   3. Taylor expanded in 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. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. mul0-lft N/A

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

      4. div0 N/A

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

      5. metadata-eval 35.6

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

   5. Applied rewrites 35.6%

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

   if 2e-51 < B

   1. Initial program 47.8%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   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 63.0%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -4.8 \cdot 10^{-78}:\\ \;\;\;\;\frac{\tan^{-1} 1}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;B \leq 2 \cdot 10^{-51}:\\ \;\;\;\;\frac{\tan^{-1} 0}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 28.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 2 \cdot 10^{-51}:\\ \;\;\;\;\frac{\tan^{-1} 0}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= B 2e-51)
   (* (/ (atan 0.0) (PI)) 180.0)
   (* (/ (atan -1.0) (PI)) 180.0)))

Derivation

1. Split input into 2 regimes

2. if B < 2e-51

   1. Initial program 53.1%

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

   3. Taylor expanded in 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. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. mul0-lft N/A

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

      4. div0 N/A

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

      5. metadata-eval 21.2

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

   5. Applied rewrites 21.2%

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

   if 2e-51 < B

   1. Initial program 47.8%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   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 63.0%

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

4. Final simplification 33.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2 \cdot 10^{-51}:\\ \;\;\;\;\frac{\tan^{-1} 0}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 21.1% accurate, 3.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 51.6%

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

3. Taylor expanded in B around inf

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

5. Applied rewrites 22.1%

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

6. Final simplification 22.1%

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

Reproduce

herbie shell --seed 2024277 
(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))))