ABCF->ab-angle angle

Percentage Accurate: 54.5% → 80.7%

Time: 9.8s

Alternatives: 17

Speedup: 1.5×

Specification

Math

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

FPCore

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

Initial Program: 54.5% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 80.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;C \leq 2.6 \cdot 10^{+54}:\\ \;\;\;\;\frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(A - C, B\right)}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\mathsf{fma}\left(\frac{B}{C}, -0.5, 0\right)\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= C 2.6e+54)
   (* (/ 180.0 (PI)) (atan (/ (- (- C A) (hypot (- A C) B)) B)))
   (* 180.0 (/ (atan (fma (/ B C) -0.5 0.0)) (PI)))))

Derivation

1. Split input into 2 regimes

2. if C < 2.60000000000000007e54

   1. Initial program 63.7%

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

      1. lift-*.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift-/.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. /-rgt-identity N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 63.7

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

   4. Applied rewrites 87.4%

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

   if 2.60000000000000007e54 < C

   1. Initial program 25.3%

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

   3. Taylor expanded in C around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r/ N/A

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

      4. distribute-rgt1-in N/A

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

      5. metadata-eval N/A

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

      6. mul0-lft N/A

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

      7. mul-1-neg N/A

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

      8. metadata-eval N/A

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

      9. div0 N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 79.8

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

   5. Applied rewrites 79.8%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{fma}\left(\frac{B}{C}, -0.5, 0\right)\right)}}{\mathsf{PI}\left(\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 62.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1} \left({B}^{-1} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-9}:\\ \;\;\;\;\frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \left(\frac{C - B}{B}\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-68}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{B}{A} \cdot 0.5\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{C}{B} + 1\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (let* ((t_0
         (atan
          (*
           (pow B -1.0)
           (- (- C A) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))))
   (if (<= t_0 -1e-9)
     (* (/ 180.0 (PI)) (atan (/ (- C B) B)))
     (if (<= t_0 2e-68)
       (* 180.0 (/ (atan (* (/ B A) 0.5)) (PI)))
       (* 180.0 (/ (atan (+ (/ C B) 1.0)) (PI)))))))

Derivation

1. Split input into 3 regimes

2. if (atan.f64 (*.f64 (/.f64 #s(literal 1 binary64) B) (-.f64 (-.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64))))))) < -1.00000000000000006e-9

   1. Initial program 58.9%

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

      1. lift-*.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift-/.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. /-rgt-identity N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 58.9

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

   4. Applied rewrites 89.2%

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

   5. Taylor expanded in A around 0

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

      1. lower--.f64 N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. lower-hypot.f64 75.0

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

   7. Applied rewrites 75.0%

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

   8. Taylor expanded in C around 0

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

   10. Applied rewrites 67.1%

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

   if -1.00000000000000006e-9 < (atan.f64 (*.f64 (/.f64 #s(literal 1 binary64) B) (-.f64 (-.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64))))))) < 2.00000000000000013e-68

   1. Initial program 26.5%

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

   3. Taylor expanded in A around -inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{B}{A}\right)}}{\mathsf{PI}\left(\right)} \]
   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 58.0

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

   5. Applied rewrites 58.0%

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

   if 2.00000000000000013e-68 < (atan.f64 (*.f64 (/.f64 #s(literal 1 binary64) B) (-.f64 (-.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))

   1. Initial program 67.2%

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

   3. Taylor expanded in B around -inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 87.3

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

   5. Applied rewrites 87.3%

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

   6. Taylor expanded in A around 0

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

   8. Applied rewrites 77.4%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan^{-1} \left({B}^{-1} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right) \leq -1 \cdot 10^{-9}:\\ \;\;\;\;\frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \left(\frac{C - B}{B}\right)\\ \mathbf{elif}\;\tan^{-1} \left({B}^{-1} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right) \leq 2 \cdot 10^{-68}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{B}{A} \cdot 0.5\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{C}{B} + 1\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {B}^{-1} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\\ t_1 := \frac{C - A}{B}\\ t_2 := \frac{180}{\mathsf{PI}\left(\right)}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-9}:\\ \;\;\;\;\tan^{-1} \left(t\_1 - 1\right) \cdot t\_2\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-68}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{\mathsf{fma}\left(\frac{C}{A}, B, B\right)}{A} \cdot 0.5\right) \cdot 180}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(t\_1 + 1\right) \cdot t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (let* ((t_0
         (* (pow B -1.0) (- (- C A) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))
        (t_1 (/ (- C A) B))
        (t_2 (/ 180.0 (PI))))
   (if (<= t_0 -1e-9)
     (* (atan (- t_1 1.0)) t_2)
     (if (<= t_0 2e-68)
       (/ (* (atan (* (/ (fma (/ C A) B B) A) 0.5)) 180.0) (PI))
       (* (atan (+ t_1 1.0)) t_2)))))

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)))))) < -1.00000000000000006e-9

   1. Initial program 58.9%

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

   3. Taylor expanded in B around -inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\left(1 + \frac{C}{B}\right) - \frac{A}{B}\right)}}{\mathsf{PI}\left(\right)} \]
   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 36.1

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

   5. Applied rewrites 36.1%

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

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. associate-/r/ N/A

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

      7. clear-num N/A

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

      8. lift-/.f64 N/A

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

      9. lower-*.f64 36.1

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

   7. Applied rewrites 36.1%

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

   8. Taylor expanded in B around inf

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 76.3

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

   10. Applied rewrites 76.3%

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

   if -1.00000000000000006e-9 < (*.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)))))) < 2.00000000000000013e-68

   1. Initial program 26.5%

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

   3. Taylor expanded in A around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-lft-out N/A

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

      3. associate-/l* N/A

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

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

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 58.0

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

   5. Applied rewrites 58.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   7. Applied rewrites 58.1%

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

   if 2.00000000000000013e-68 < (*.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 67.2%

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

   3. Taylor expanded in B around -inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 87.3

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

   5. Applied rewrites 87.3%

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

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. associate-/r/ N/A

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

      7. clear-num N/A

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

      8. lift-/.f64 N/A

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

      9. lower-*.f64 87.3

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

   7. Applied rewrites 87.3%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{-1} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \leq -1 \cdot 10^{-9}:\\ \;\;\;\;\tan^{-1} \left(\frac{C - A}{B} - 1\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;{B}^{-1} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \leq 2 \cdot 10^{-68}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{\mathsf{fma}\left(\frac{C}{A}, B, B\right)}{A} \cdot 0.5\right) \cdot 180}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{C - A}{B} + 1\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {B}^{-1} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\\ t_1 := \frac{C - A}{B}\\ t_2 := \frac{180}{\mathsf{PI}\left(\right)}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-9}:\\ \;\;\;\;\tan^{-1} \left(t\_1 - 1\right) \cdot t\_2\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-68}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{\mathsf{fma}\left(\frac{C}{A}, B, B\right)}{A}\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(t\_1 + 1\right) \cdot t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (let* ((t_0
         (* (pow B -1.0) (- (- C A) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))
        (t_1 (/ (- C A) B))
        (t_2 (/ 180.0 (PI))))
   (if (<= t_0 -1e-9)
     (* (atan (- t_1 1.0)) t_2)
     (if (<= t_0 2e-68)
       (* 180.0 (/ (atan (* 0.5 (/ (fma (/ C A) B B) A))) (PI)))
       (* (atan (+ t_1 1.0)) t_2)))))

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)))))) < -1.00000000000000006e-9

   1. Initial program 58.9%

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

   3. Taylor expanded in B around -inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\left(1 + \frac{C}{B}\right) - \frac{A}{B}\right)}}{\mathsf{PI}\left(\right)} \]
   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 36.1

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

   5. Applied rewrites 36.1%

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

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. associate-/r/ N/A

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

      7. clear-num N/A

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

      8. lift-/.f64 N/A

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

      9. lower-*.f64 36.1

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

   7. Applied rewrites 36.1%

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

   8. Taylor expanded in B around inf

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 76.3

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

   10. Applied rewrites 76.3%

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

   if -1.00000000000000006e-9 < (*.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)))))) < 2.00000000000000013e-68

   1. Initial program 26.5%

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

   3. Taylor expanded in A around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-lft-out N/A

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

      3. associate-/l* N/A

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

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

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 58.0

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

   5. Applied rewrites 58.0%

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

   if 2.00000000000000013e-68 < (*.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 67.2%

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

   3. Taylor expanded in B around -inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 87.3

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

   5. Applied rewrites 87.3%

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

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. associate-/r/ N/A

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

      7. clear-num N/A

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

      8. lift-/.f64 N/A

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

      9. lower-*.f64 87.3

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

   7. Applied rewrites 87.3%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{-1} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \leq -1 \cdot 10^{-9}:\\ \;\;\;\;\tan^{-1} \left(\frac{C - A}{B} - 1\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;{B}^{-1} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \leq 2 \cdot 10^{-68}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{\mathsf{fma}\left(\frac{C}{A}, B, B\right)}{A}\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{C - A}{B} + 1\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {B}^{-1} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\\ t_1 := \frac{C - A}{B}\\ t_2 := \frac{180}{\mathsf{PI}\left(\right)}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-9}:\\ \;\;\;\;\tan^{-1} \left(t\_1 - 1\right) \cdot t\_2\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-68}:\\ \;\;\;\;\tan^{-1} \left(\frac{B}{A} \cdot 0.5\right) \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(t\_1 + 1\right) \cdot t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (let* ((t_0
         (* (pow B -1.0) (- (- C A) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))
        (t_1 (/ (- C A) B))
        (t_2 (/ 180.0 (PI))))
   (if (<= t_0 -1e-9)
     (* (atan (- t_1 1.0)) t_2)
     (if (<= t_0 2e-68)
       (* (atan (* (/ B A) 0.5)) t_2)
       (* (atan (+ t_1 1.0)) t_2)))))

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)))))) < -1.00000000000000006e-9

   1. Initial program 58.9%

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

   3. Taylor expanded in B around -inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\left(1 + \frac{C}{B}\right) - \frac{A}{B}\right)}}{\mathsf{PI}\left(\right)} \]
   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 36.1

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

   5. Applied rewrites 36.1%

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

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. associate-/r/ N/A

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

      7. clear-num N/A

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

      8. lift-/.f64 N/A

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

      9. lower-*.f64 36.1

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

   7. Applied rewrites 36.1%

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

   8. Taylor expanded in B around inf

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 76.3

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

   10. Applied rewrites 76.3%

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

   if -1.00000000000000006e-9 < (*.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)))))) < 2.00000000000000013e-68

   1. Initial program 26.5%

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

   3. Taylor expanded in B around -inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 2.4

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

   5. Applied rewrites 2.4%

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

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. associate-/r/ N/A

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

      7. clear-num N/A

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

      8. lift-/.f64 N/A

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

      9. lower-*.f64 2.4

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

   7. Applied rewrites 2.4%

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

   8. Taylor expanded in A around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 58.0

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

   10. Applied rewrites 58.0%

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

   if 2.00000000000000013e-68 < (*.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 67.2%

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

   3. Taylor expanded in B around -inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 87.3

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

   5. Applied rewrites 87.3%

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

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. associate-/r/ N/A

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

      7. clear-num N/A

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

      8. lift-/.f64 N/A

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

      9. lower-*.f64 87.3

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

   7. Applied rewrites 87.3%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{-1} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \leq -1 \cdot 10^{-9}:\\ \;\;\;\;\tan^{-1} \left(\frac{C - A}{B} - 1\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;{B}^{-1} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \leq 2 \cdot 10^{-68}:\\ \;\;\;\;\tan^{-1} \left(\frac{B}{A} \cdot 0.5\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{C - A}{B} + 1\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{180}{\mathsf{PI}\left(\right)}\\ t_1 := {B}^{-1} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\\ t_2 := \frac{C - A}{B}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-9}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(t\_2 - 1\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-68}:\\ \;\;\;\;\tan^{-1} \left(\frac{B}{A} \cdot 0.5\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(t\_2 + 1\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (let* ((t_0 (/ 180.0 (PI)))
        (t_1
         (* (pow B -1.0) (- (- C A) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))
        (t_2 (/ (- C A) B)))
   (if (<= t_1 -1e-9)
     (* 180.0 (/ (atan (- t_2 1.0)) (PI)))
     (if (<= t_1 2e-68)
       (* (atan (* (/ B A) 0.5)) t_0)
       (* (atan (+ t_2 1.0)) t_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)))))) < -1.00000000000000006e-9

   1. Initial program 58.9%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   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 76.3

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

   5. Applied rewrites 76.3%

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

   if -1.00000000000000006e-9 < (*.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)))))) < 2.00000000000000013e-68

   1. Initial program 26.5%

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

   3. Taylor expanded in B around -inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 2.4

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

   5. Applied rewrites 2.4%

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

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. associate-/r/ N/A

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

      7. clear-num N/A

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

      8. lift-/.f64 N/A

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

      9. lower-*.f64 2.4

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

   7. Applied rewrites 2.4%

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

   8. Taylor expanded in A around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 58.0

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

   10. Applied rewrites 58.0%

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

   if 2.00000000000000013e-68 < (*.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 67.2%

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

   3. Taylor expanded in B around -inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 87.3

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

   5. Applied rewrites 87.3%

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

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. associate-/r/ N/A

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

      7. clear-num N/A

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

      8. lift-/.f64 N/A

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

      9. lower-*.f64 87.3

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

   7. Applied rewrites 87.3%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{-1} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \leq -1 \cdot 10^{-9}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{C - A}{B} - 1\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;{B}^{-1} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \leq 2 \cdot 10^{-68}:\\ \;\;\;\;\tan^{-1} \left(\frac{B}{A} \cdot 0.5\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{C - A}{B} + 1\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {B}^{-1} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\\ t_1 := \frac{C - A}{B}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-9}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(t\_1 - 1\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-68}:\\ \;\;\;\;\tan^{-1} \left(\frac{B}{A} \cdot 0.5\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(t\_1 + 1\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (let* ((t_0
         (* (pow B -1.0) (- (- C A) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))
        (t_1 (/ (- C A) B)))
   (if (<= t_0 -1e-9)
     (* 180.0 (/ (atan (- t_1 1.0)) (PI)))
     (if (<= t_0 2e-68)
       (* (atan (* (/ B A) 0.5)) (/ 180.0 (PI)))
       (* 180.0 (/ (atan (+ t_1 1.0)) (PI)))))))

Derivation

1. Split input into 3 regimes

2. if (*.f64 (/.f64 #s(literal 1 binary64) B) (-.f64 (-.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))) < -1.00000000000000006e-9

   1. Initial program 58.9%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   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 76.3

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

   5. Applied rewrites 76.3%

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

   if -1.00000000000000006e-9 < (*.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)))))) < 2.00000000000000013e-68

   1. Initial program 26.5%

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

   3. Taylor expanded in B around -inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 2.4

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

   5. Applied rewrites 2.4%

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

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. associate-/r/ N/A

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

      7. clear-num N/A

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

      8. lift-/.f64 N/A

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

      9. lower-*.f64 2.4

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

   7. Applied rewrites 2.4%

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

   8. Taylor expanded in A around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 58.0

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

   10. Applied rewrites 58.0%

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

   if 2.00000000000000013e-68 < (*.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 67.2%

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

   3. Taylor expanded in B around -inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 87.3

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

   5. Applied rewrites 87.3%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{-1} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \leq -1 \cdot 10^{-9}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{C - A}{B} - 1\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;{B}^{-1} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \leq 2 \cdot 10^{-68}:\\ \;\;\;\;\tan^{-1} \left(\frac{B}{A} \cdot 0.5\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{C - A}{B} + 1\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {B}^{-1} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\\ t_1 := \frac{180}{\mathsf{PI}\left(\right)}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-9}:\\ \;\;\;\;t\_1 \cdot \tan^{-1} \left(\frac{C - B}{B}\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-68}:\\ \;\;\;\;\tan^{-1} \left(\frac{B}{A} \cdot 0.5\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{C - A}{B} + 1\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (let* ((t_0
         (* (pow B -1.0) (- (- C A) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))
        (t_1 (/ 180.0 (PI))))
   (if (<= t_0 -1e-9)
     (* t_1 (atan (/ (- C B) B)))
     (if (<= t_0 2e-68)
       (* (atan (* (/ B A) 0.5)) t_1)
       (* 180.0 (/ (atan (+ (/ (- C A) B) 1.0)) (PI)))))))

Derivation

1. Split input into 3 regimes

2. if (*.f64 (/.f64 #s(literal 1 binary64) B) (-.f64 (-.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))) < -1.00000000000000006e-9

   1. Initial program 58.9%

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

      1. lift-*.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift-/.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. /-rgt-identity N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 58.9

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

   4. Applied rewrites 89.2%

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

   5. Taylor expanded in A around 0

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

      1. lower--.f64 N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. lower-hypot.f64 75.0

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

   7. Applied rewrites 75.0%

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

   8. Taylor expanded in C around 0

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

   10. Applied rewrites 67.1%

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

   if -1.00000000000000006e-9 < (*.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)))))) < 2.00000000000000013e-68

   1. Initial program 26.5%

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

   3. Taylor expanded in B around -inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 2.4

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

   5. Applied rewrites 2.4%

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

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. associate-/r/ N/A

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

      7. clear-num N/A

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

      8. lift-/.f64 N/A

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

      9. lower-*.f64 2.4

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

   7. Applied rewrites 2.4%

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

   8. Taylor expanded in A around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 58.0

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

   10. Applied rewrites 58.0%

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

   if 2.00000000000000013e-68 < (*.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 67.2%

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

   3. Taylor expanded in B around -inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 87.3

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

   5. Applied rewrites 87.3%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{-1} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \leq -1 \cdot 10^{-9}:\\ \;\;\;\;\frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \left(\frac{C - B}{B}\right)\\ \mathbf{elif}\;{B}^{-1} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \leq 2 \cdot 10^{-68}:\\ \;\;\;\;\tan^{-1} \left(\frac{B}{A} \cdot 0.5\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{C - A}{B} + 1\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {B}^{-1} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\\ t_1 := \frac{180}{\mathsf{PI}\left(\right)}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-9}:\\ \;\;\;\;t\_1 \cdot \tan^{-1} \left(\frac{C - B}{B}\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-68}:\\ \;\;\;\;\tan^{-1} \left(\frac{B}{A} \cdot 0.5\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{C}{B} + 1\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (let* ((t_0
         (* (pow B -1.0) (- (- C A) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))
        (t_1 (/ 180.0 (PI))))
   (if (<= t_0 -1e-9)
     (* t_1 (atan (/ (- C B) B)))
     (if (<= t_0 2e-68)
       (* (atan (* (/ B A) 0.5)) t_1)
       (* 180.0 (/ (atan (+ (/ C B) 1.0)) (PI)))))))

Derivation

1. Split input into 3 regimes

2. if (*.f64 (/.f64 #s(literal 1 binary64) B) (-.f64 (-.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))) < -1.00000000000000006e-9

   1. Initial program 58.9%

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

      1. lift-*.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift-/.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. /-rgt-identity N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 58.9

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

   4. Applied rewrites 89.2%

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

   5. Taylor expanded in A around 0

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

      1. lower--.f64 N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. lower-hypot.f64 75.0

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

   7. Applied rewrites 75.0%

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

   8. Taylor expanded in C around 0

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

   10. Applied rewrites 67.1%

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

   if -1.00000000000000006e-9 < (*.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)))))) < 2.00000000000000013e-68

   1. Initial program 26.5%

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

   3. Taylor expanded in B around -inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 2.4

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

   5. Applied rewrites 2.4%

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

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. associate-/r/ N/A

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

      7. clear-num N/A

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

      8. lift-/.f64 N/A

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

      9. lower-*.f64 2.4

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

   7. Applied rewrites 2.4%

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

   8. Taylor expanded in A around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 58.0

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

   10. Applied rewrites 58.0%

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

   if 2.00000000000000013e-68 < (*.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 67.2%

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

   3. Taylor expanded in B around -inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 87.3

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

   5. Applied rewrites 87.3%

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

   6. Taylor expanded in A around 0

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

   8. Applied rewrites 77.4%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{-1} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \leq -1 \cdot 10^{-9}:\\ \;\;\;\;\frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \left(\frac{C - B}{B}\right)\\ \mathbf{elif}\;{B}^{-1} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \leq 2 \cdot 10^{-68}:\\ \;\;\;\;\tan^{-1} \left(\frac{B}{A} \cdot 0.5\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{C}{B} + 1\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {B}^{-1} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-9}:\\ \;\;\;\;\frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \left(\frac{C - B}{B}\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-68}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} 0}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{C}{B} + 1\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (let* ((t_0
         (*
          (pow B -1.0)
          (- (- C A) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0)))))))
   (if (<= t_0 -1e-9)
     (* (/ 180.0 (PI)) (atan (/ (- C B) B)))
     (if (<= t_0 2e-68)
       (* 180.0 (/ (atan 0.0) (PI)))
       (* 180.0 (/ (atan (+ (/ C B) 1.0)) (PI)))))))

Derivation

1. Split input into 3 regimes

2. if (*.f64 (/.f64 #s(literal 1 binary64) B) (-.f64 (-.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))) < -1.00000000000000006e-9

   1. Initial program 58.9%

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

      1. lift-*.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift-/.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. /-rgt-identity N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 58.9

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

   4. Applied rewrites 89.2%

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

   5. Taylor expanded in A around 0

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

      1. lower--.f64 N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. lower-hypot.f64 75.0

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

   7. Applied rewrites 75.0%

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

   8. Taylor expanded in C around 0

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

   10. Applied rewrites 67.1%

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

   if -1.00000000000000006e-9 < (*.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)))))) < 2.00000000000000013e-68

   1. Initial program 26.5%

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

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. mul-1-neg N/A

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

      6. metadata-eval N/A

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

      7. div0 26.5

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

   5. Applied rewrites 26.5%

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

   if 2.00000000000000013e-68 < (*.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 67.2%

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

   3. Taylor expanded in B around -inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 87.3

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

   5. Applied rewrites 87.3%

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

   6. Taylor expanded in A around 0

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

   8. Applied rewrites 77.4%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{-1} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \leq -1 \cdot 10^{-9}:\\ \;\;\;\;\frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \left(\frac{C - B}{B}\right)\\ \mathbf{elif}\;{B}^{-1} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \leq 2 \cdot 10^{-68}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} 0}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{C}{B} + 1\right)}{\mathsf{PI}\left(\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 74.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{180}{\mathsf{PI}\left(\right)}\\ \mathbf{if}\;A \leq -5.5 \cdot 10^{+81}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{\mathsf{fma}\left(\frac{C}{A}, B, B\right)}{A} \cdot 0.5\right) \cdot 180}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;A \leq 6 \cdot 10^{+163}:\\ \;\;\;\;t\_0 \cdot \tan^{-1} \left(\frac{C - \mathsf{hypot}\left(B, C\right)}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{C - A}{B} - 1\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (let* ((t_0 (/ 180.0 (PI))))
   (if (<= A -5.5e+81)
     (/ (* (atan (* (/ (fma (/ C A) B B) A) 0.5)) 180.0) (PI))
     (if (<= A 6e+163)
       (* t_0 (atan (/ (- C (hypot B C)) B)))
       (* (atan (- (/ (- C A) B) 1.0)) t_0)))))

Derivation

1. Split input into 3 regimes

2. if A < -5.5000000000000003e81

   1. Initial program 21.3%

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

   3. Taylor expanded in A around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-lft-out N/A

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

      3. associate-/l* N/A

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

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

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 70.9

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

   5. Applied rewrites 70.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   7. Applied rewrites 71.1%

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

   if -5.5000000000000003e81 < A < 6.00000000000000027e163

   1. Initial program 61.0%

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

      1. lift-*.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift-/.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. /-rgt-identity N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 61.0

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

   4. Applied rewrites 83.9%

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

   5. Taylor expanded in A around 0

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

      1. lower--.f64 N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. lower-hypot.f64 82.7

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

   7. Applied rewrites 82.7%

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

   if 6.00000000000000027e163 < A

   1. Initial program 84.6%

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

   3. Taylor expanded in B around -inf

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

      1. associate--l+ N/A

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

      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 82.2

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

   5. Applied rewrites 82.2%

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

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. associate-/r/ N/A

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

      7. clear-num N/A

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

      8. lift-/.f64 N/A

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

      9. lower-*.f64 82.2

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

   7. Applied rewrites 82.2%

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

   8. Taylor expanded in B around inf

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 96.2

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

   10. Applied rewrites 96.2%

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

Alternative 12: 74.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;A \leq -5.5 \cdot 10^{+81}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{\mathsf{fma}\left(\frac{C}{A}, B, B\right)}{A} \cdot 0.5\right) \cdot 180}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;A \leq 6 \cdot 10^{+163}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{C - \mathsf{hypot}\left(C, B\right)}{B}\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{C - A}{B} - 1\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= A -5.5e+81)
   (/ (* (atan (* (/ (fma (/ C A) B B) A) 0.5)) 180.0) (PI))
   (if (<= A 6e+163)
     (* 180.0 (/ (atan (/ (- C (hypot C B)) B)) (PI)))
     (* (atan (- (/ (- C A) B) 1.0)) (/ 180.0 (PI))))))

Derivation

1. Split input into 3 regimes

2. if A < -5.5000000000000003e81

   1. Initial program 21.3%

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

   3. Taylor expanded in A around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-lft-out N/A

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

      3. associate-/l* N/A

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

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

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 70.9

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

   5. Applied rewrites 70.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   7. Applied rewrites 71.1%

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

   if -5.5000000000000003e81 < A < 6.00000000000000027e163

   1. Initial program 61.0%

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

   3. Taylor expanded in A around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C - \sqrt{{B}^{2} + {C}^{2}}}{B}\right)}}{\mathsf{PI}\left(\right)} \]
   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 82.7

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

   5. Applied rewrites 82.7%

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

   if 6.00000000000000027e163 < A

   1. Initial program 84.6%

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

   3. Taylor expanded in B around -inf

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

      1. associate--l+ N/A

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

      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 82.2

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

   5. Applied rewrites 82.2%

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

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. associate-/r/ N/A

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

      7. clear-num N/A

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

      8. lift-/.f64 N/A

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

      9. lower-*.f64 82.2

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

   7. Applied rewrites 82.2%

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

   8. Taylor expanded in B around inf

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 96.2

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

   10. Applied rewrites 96.2%

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

Alternative 13: 45.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq -2.7 \cdot 10^{-83}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} 1}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;B \leq -2 \cdot 10^{-222}:\\ \;\;\;\;\tan^{-1} \left(\frac{-A}{B}\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;B \leq 3.25 \cdot 10^{-136}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} 0}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= B -2.7e-83)
   (* 180.0 (/ (atan 1.0) (PI)))
   (if (<= B -2e-222)
     (* (atan (/ (- A) B)) (/ 180.0 (PI)))
     (if (<= B 3.25e-136)
       (* 180.0 (/ (atan 0.0) (PI)))
       (* 180.0 (/ (atan -1.0) (PI)))))))

Derivation

1. Split input into 4 regimes

2. if B < -2.69999999999999991e-83

   1. Initial program 58.9%

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

   3. Taylor expanded in B around -inf

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

   5. Applied rewrites 66.1%

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

   if -2.69999999999999991e-83 < B < -2.0000000000000001e-222

   1. Initial program 79.2%

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

   3. Taylor expanded in B around -inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 79.2

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

   5. Applied rewrites 79.2%

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

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. associate-/r/ N/A

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

      7. clear-num N/A

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

      8. lift-/.f64 N/A

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

      9. lower-*.f64 79.2

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

   7. Applied rewrites 79.2%

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

   8. Taylor expanded in A around inf

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

   10. Applied rewrites 58.8%

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

   if -2.0000000000000001e-222 < B < 3.25000000000000005e-136

   1. Initial program 56.2%

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

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. mul-1-neg N/A

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

      6. metadata-eval N/A

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

      7. div0 39.5

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

   5. Applied rewrites 39.5%

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

   if 3.25000000000000005e-136 < B

   1. Initial program 53.8%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   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 53.9%

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

Alternative 14: 51.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 3.2 \cdot 10^{-58}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{C}{B} + 1\right)}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= B 3.2e-58)
   (* 180.0 (/ (atan (+ (/ C B) 1.0)) (PI)))
   (* 180.0 (/ (atan -1.0) (PI)))))

Derivation

1. Split input into 2 regimes

2. if B < 3.2000000000000001e-58

   1. Initial program 59.7%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   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 63.9

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

   5. Applied rewrites 63.9%

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

   6. Taylor expanded in A around 0

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

   8. Applied rewrites 56.8%

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

   if 3.2000000000000001e-58 < B

   1. Initial program 51.7%

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

   3. Taylor expanded in B around inf

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

   5. Applied rewrites 61.7%

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

Alternative 15: 44.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq -1.05 \cdot 10^{-107}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} 1}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;B \leq 3.25 \cdot 10^{-136}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} 0}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= B -1.05e-107)
   (* 180.0 (/ (atan 1.0) (PI)))
   (if (<= B 3.25e-136)
     (* 180.0 (/ (atan 0.0) (PI)))
     (* 180.0 (/ (atan -1.0) (PI))))))

Derivation

1. Split input into 3 regimes

2. if B < -1.05e-107

   1. Initial program 60.3%

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

   3. Taylor expanded in B around -inf

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

   5. Applied rewrites 64.5%

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

   if -1.05e-107 < B < 3.25000000000000005e-136

   1. Initial program 58.7%

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

   3. Taylor expanded in C around 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. associate-*r/ N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. mul-1-neg N/A

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

      6. metadata-eval N/A

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

      7. div0 38.1

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

   5. Applied rewrites 38.1%

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

   if 3.25000000000000005e-136 < B

   1. Initial program 53.8%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   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 53.9%

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

Alternative 16: 28.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 3.25 \cdot 10^{-136}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} 0}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= B 3.25e-136)
   (* 180.0 (/ (atan 0.0) (PI)))
   (* 180.0 (/ (atan -1.0) (PI)))))

Derivation

1. Split input into 2 regimes

2. if B < 3.25000000000000005e-136

   1. Initial program 59.6%

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

   3. Taylor expanded in C around inf

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

      1. associate-*r/ N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. mul-1-neg N/A

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

      6. metadata-eval N/A

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

      7. div0 20.2

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

   5. Applied rewrites 20.2%

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

   if 3.25000000000000005e-136 < B

   1. Initial program 53.8%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   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 53.9%

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

Alternative 17: 20.8% accurate, 3.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 57.4%

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

3. Taylor expanded in B around inf

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

5. Applied rewrites 23.2%

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

Reproduce

herbie shell --seed 2024313 
(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))))