ABCF->ab-angle angle

Percentage Accurate: 53.2% → 81.5%

Time: 10.3s

Alternatives: 12

Speedup: 2.5×

Specification

Math

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

FPCore

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

Initial Program: 53.2% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 81.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;A \leq -5 \cdot 10^{+128}:\\ \;\;\;\;\frac{\tan^{-1} \left(0.5 \cdot \frac{B}{A}\right)}{0.005555555555555556 \cdot \mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(B, A - C\right)}{B}\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= A -5e+128)
   (/ (atan (* 0.5 (/ B A))) (* 0.005555555555555556 (PI)))
   (* (atan (/ (- (- C A) (hypot B (- A C))) B)) (/ 180.0 (PI)))))

Derivation

1. Split input into 2 regimes

2. if A < -5e128

   1. Initial program 11.9%

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

   3. Taylor expanded in A around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 86.5

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

   5. Applied rewrites 86.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   7. Applied rewrites 86.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

   9. Applied rewrites 86.6%

      \[\leadsto \color{blue}{\frac{\tan^{-1} \left(\frac{B}{A} \cdot 0.5\right)}{\mathsf{PI}\left(\right) \cdot 0.005555555555555556}} \]

   if -5e128 < A

   1. Initial program 58.8%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 58.8

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 58.8

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

   4. Applied rewrites 86.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 86.8

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

      8. lift-hypot.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-hypot.f64 86.8

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

   6. Applied rewrites 86.8%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -5 \cdot 10^{+128}:\\ \;\;\;\;\frac{\tan^{-1} \left(0.5 \cdot \frac{B}{A}\right)}{0.005555555555555556 \cdot \mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(B, A - C\right)}{B}\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 71.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(C - A\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right) \cdot {B}^{-1}\\ t_1 := \frac{-1}{\sqrt{\mathsf{PI}\left(\right)}}\\ t_2 := \frac{C - A}{B}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-32}:\\ \;\;\;\;\frac{\tan^{-1} \left(t\_2 - 1\right) \cdot 180}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(\left(\tan^{-1} \left(\frac{B}{C} \cdot -0.5\right) \cdot t\_1\right) \cdot t\_1\right) \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(t\_2 + 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (let* ((t_0
         (* (- (- C A) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0)))) (pow B -1.0)))
        (t_1 (/ -1.0 (sqrt (PI))))
        (t_2 (/ (- C A) B)))
   (if (<= t_0 -2e-32)
     (/ (* (atan (- t_2 1.0)) 180.0) (PI))
     (if (<= t_0 0.0)
       (* (* (* (atan (* (/ B C) -0.5)) t_1) t_1) 180.0)
       (* (/ (atan (+ t_2 1.0)) (PI)) 180.0)))))

Derivation

1. Split input into 3 regimes

2. if (*.f64 (/.f64 #s(literal 1 binary64) B) (-.f64 (-.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))) < -2.00000000000000011e-32

   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 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 29.6

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

   5. Applied rewrites 29.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   7. Applied rewrites 29.5%

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

   8. Taylor expanded in B around inf

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 73.1

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

   10. Applied rewrites 73.1%

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

   if -2.00000000000000011e-32 < (*.f64 (/.f64 #s(literal 1 binary64) B) (-.f64 (-.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))) < 0.0

   1. Initial program 11.8%

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

   3. Taylor expanded in C around inf

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

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

   5. Applied rewrites 52.0%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. neg-mul-1 N/A

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

      4. lift-PI.f64 N/A

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

      5. add-sqr-sqrt N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

   7. Applied rewrites 52.1%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg2 N/A

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

      4. distribute-frac-neg N/A

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

      5. neg-mul-1 N/A

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

      6. associate-*l/ N/A

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

      7. lift-/.f64 N/A

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

      8. lower-*.f64 52.2

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

   9. Applied rewrites 52.2%

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

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

   1. Initial program 60.9%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   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 81.7

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

   5. Applied rewrites 81.7%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(C - A\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right) \cdot {B}^{-1} \leq -2 \cdot 10^{-32}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - A}{B} - 1\right) \cdot 180}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;\left(\left(C - A\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right) \cdot {B}^{-1} \leq 0:\\ \;\;\;\;\left(\left(\tan^{-1} \left(\frac{B}{C} \cdot -0.5\right) \cdot \frac{-1}{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot \frac{-1}{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - A}{B} + 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(C - A\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right) \cdot {B}^{-1}\\ t_1 := \frac{C - A}{B}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-32}:\\ \;\;\;\;\frac{\tan^{-1} \left(t\_1 - 1\right) \cdot 180}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\tan^{-1} \left(\frac{B}{C} \cdot -0.5\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(t\_1 + 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (let* ((t_0
         (* (- (- C A) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0)))) (pow B -1.0)))
        (t_1 (/ (- C A) B)))
   (if (<= t_0 -2e-32)
     (/ (* (atan (- t_1 1.0)) 180.0) (PI))
     (if (<= t_0 0.0)
       (* (atan (* (/ B C) -0.5)) (/ 180.0 (PI)))
       (* (/ (atan (+ t_1 1.0)) (PI)) 180.0)))))

Derivation

1. Split input into 3 regimes

2. if (*.f64 (/.f64 #s(literal 1 binary64) B) (-.f64 (-.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))) < -2.00000000000000011e-32

   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 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 29.6

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

   5. Applied rewrites 29.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   7. Applied rewrites 29.5%

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

   8. Taylor expanded in B around inf

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 73.1

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

   10. Applied rewrites 73.1%

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

   if -2.00000000000000011e-32 < (*.f64 (/.f64 #s(literal 1 binary64) B) (-.f64 (-.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))) < 0.0

   1. Initial program 11.8%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 11.8

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 11.8

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

   4. Applied rewrites 11.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 11.8

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

      8. lift-hypot.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-hypot.f64 11.8

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

   6. Applied rewrites 11.8%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 4.4

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

      9. lift--.f64 N/A

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

      10. lift-hypot.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-hypot.f64 N/A

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

      13. lift--.f64 4.4

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

   8. Applied rewrites 4.4%

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

   9. Taylor expanded in C around inf

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

       1. distribute-lft1-in N/A

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

       2. metadata-eval N/A

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

       3. mul0-lft N/A

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

       4. --rgt-identity N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. lower-/.f64 52.2

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

   11. Applied rewrites 52.2%

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

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

   1. Initial program 60.9%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   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 81.7

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

   5. Applied rewrites 81.7%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(C - A\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right) \cdot {B}^{-1} \leq -2 \cdot 10^{-32}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - A}{B} - 1\right) \cdot 180}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;\left(\left(C - A\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right) \cdot {B}^{-1} \leq 0:\\ \;\;\;\;\tan^{-1} \left(\frac{B}{C} \cdot -0.5\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - A}{B} + 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(C - A\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right) \cdot {B}^{-1}\\ t_1 := \frac{C - A}{B}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-32}:\\ \;\;\;\;\frac{\tan^{-1} \left(t\_1 - 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\tan^{-1} \left(\frac{B}{C} \cdot -0.5\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(t\_1 + 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (let* ((t_0
         (* (- (- C A) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0)))) (pow B -1.0)))
        (t_1 (/ (- C A) B)))
   (if (<= t_0 -2e-32)
     (* (/ (atan (- t_1 1.0)) (PI)) 180.0)
     (if (<= t_0 0.0)
       (* (atan (* (/ B C) -0.5)) (/ 180.0 (PI)))
       (* (/ (atan (+ t_1 1.0)) (PI)) 180.0)))))

Derivation

1. Split input into 3 regimes

2. if (*.f64 (/.f64 #s(literal 1 binary64) B) (-.f64 (-.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))) < -2.00000000000000011e-32

   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

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C}{B} - \left(1 + \frac{A}{B}\right)\right)}}{\mathsf{PI}\left(\right)} \]
   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 73.1

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

   5. Applied rewrites 73.1%

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

   if -2.00000000000000011e-32 < (*.f64 (/.f64 #s(literal 1 binary64) B) (-.f64 (-.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))) < 0.0

   1. Initial program 11.8%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 11.8

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 11.8

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

   4. Applied rewrites 11.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 11.8

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

      8. lift-hypot.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-hypot.f64 11.8

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

   6. Applied rewrites 11.8%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 4.4

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

      9. lift--.f64 N/A

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

      10. lift-hypot.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-hypot.f64 N/A

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

      13. lift--.f64 4.4

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

   8. Applied rewrites 4.4%

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

   9. Taylor expanded in C around inf

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

       1. distribute-lft1-in N/A

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

       2. metadata-eval N/A

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

       3. mul0-lft N/A

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

       4. --rgt-identity N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. lower-/.f64 52.2

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

   11. Applied rewrites 52.2%

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

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

   1. Initial program 60.9%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   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 81.7

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

   5. Applied rewrites 81.7%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(C - A\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right) \cdot {B}^{-1} \leq -2 \cdot 10^{-32}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - A}{B} - 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;\left(\left(C - A\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right) \cdot {B}^{-1} \leq 0:\\ \;\;\;\;\tan^{-1} \left(\frac{B}{C} \cdot -0.5\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - A}{B} + 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{\sqrt{\mathsf{PI}\left(\right)}}\\ \mathbf{if}\;C \leq -4.2 \cdot 10^{+47}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - A}{B} - 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;C \leq 1.35 \cdot 10^{+135}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{\mathsf{hypot}\left(B, A\right) + A}{-B}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\tan^{-1} \left(\frac{B}{C} \cdot -0.5\right) \cdot t\_0\right) \cdot 180\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (let* ((t_0 (/ -1.0 (sqrt (PI)))))
   (if (<= C -4.2e+47)
     (* (/ (atan (- (/ (- C A) B) 1.0)) (PI)) 180.0)
     (if (<= C 1.35e+135)
       (* (/ (atan (/ (+ (hypot B A) A) (- B))) (PI)) 180.0)
       (* (* (* (atan (* (/ B C) -0.5)) t_0) 180.0) t_0)))))

Derivation

1. Split input into 3 regimes

2. if C < -4.2e47

   1. Initial program 80.3%

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

   3. Taylor expanded in B around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C}{B} - \left(1 + \frac{A}{B}\right)\right)}}{\mathsf{PI}\left(\right)} \]
   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 87.5

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

   5. Applied rewrites 87.5%

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

   if -4.2e47 < C < 1.34999999999999992e135

   1. Initial program 50.5%

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

   3. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 75.0

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

   5. Applied rewrites 75.0%

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

   if 1.34999999999999992e135 < C

   1. Initial program 8.8%

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

   3. Taylor expanded in C around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-1 \cdot \frac{A + -1 \cdot A}{B} + \frac{-1}{2} \cdot \frac{B}{C}\right)}}{\mathsf{PI}\left(\right)} \]
   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 79.5

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

   5. Applied rewrites 79.5%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. neg-mul-1 N/A

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

      4. lift-PI.f64 N/A

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

      5. add-sqr-sqrt N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

   7. Applied rewrites 79.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 79.6

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

   9. Applied rewrites 79.6%

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

       1. lift-/.f64 N/A

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

       2. clear-num N/A

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

       3. associate-/r/ N/A

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

       4. metadata-eval N/A

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

       5. lift-neg.f64 N/A

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

       6. frac-2neg N/A

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

       7. lift-/.f64 N/A

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

       8. lower-*.f64 79.8

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

   11. Applied rewrites 79.8%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq -4.2 \cdot 10^{+47}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - A}{B} - 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;C \leq 1.35 \cdot 10^{+135}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{\mathsf{hypot}\left(B, A\right) + A}{-B}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\tan^{-1} \left(\frac{B}{C} \cdot -0.5\right) \cdot \frac{-1}{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot 180\right) \cdot \frac{-1}{\sqrt{\mathsf{PI}\left(\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 47.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\tan^{-1} \left(0.5 \cdot \frac{B}{A}\right)}{0.005555555555555556 \cdot \mathsf{PI}\left(\right)}\\ \mathbf{if}\;B \leq -5.2 \cdot 10^{+54}:\\ \;\;\;\;\frac{\tan^{-1} 1}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;B \leq -7 \cdot 10^{-101}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;B \leq 7 \cdot 10^{-294}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{A}{B} \cdot -2\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;B \leq 1.45:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (let* ((t_0 (/ (atan (* 0.5 (/ B A))) (* 0.005555555555555556 (PI)))))
   (if (<= B -5.2e+54)
     (* (/ (atan 1.0) (PI)) 180.0)
     (if (<= B -7e-101)
       t_0
       (if (<= B 7e-294)
         (* (/ (atan (* (/ A B) -2.0)) (PI)) 180.0)
         (if (<= B 1.45) t_0 (* (/ (atan -1.0) (PI)) 180.0)))))))

Derivation

1. Split input into 4 regimes

2. if B < -5.20000000000000013e54

   1. Initial program 43.0%

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

   3. Taylor expanded in B around -inf

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

   5. Applied rewrites 75.1%

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

   if -5.20000000000000013e54 < B < -6.99999999999999989e-101 or 7.00000000000000064e-294 < B < 1.44999999999999996

   1. Initial program 46.7%

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

   3. Taylor expanded in 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 43.7

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

   5. Applied rewrites 43.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   7. Applied rewrites 43.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

   9. Applied rewrites 43.8%

      \[\leadsto \color{blue}{\frac{\tan^{-1} \left(\frac{B}{A} \cdot 0.5\right)}{\mathsf{PI}\left(\right) \cdot 0.005555555555555556}} \]

   if -6.99999999999999989e-101 < B < 7.00000000000000064e-294

   1. Initial program 79.5%

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

   3. Taylor expanded in A around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 62.9

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

   5. Applied rewrites 62.9%

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

   if 1.44999999999999996 < B

   1. Initial program 45.9%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   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 65.3%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -5.2 \cdot 10^{+54}:\\ \;\;\;\;\frac{\tan^{-1} 1}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;B \leq -7 \cdot 10^{-101}:\\ \;\;\;\;\frac{\tan^{-1} \left(0.5 \cdot \frac{B}{A}\right)}{0.005555555555555556 \cdot \mathsf{PI}\left(\right)}\\ \mathbf{elif}\;B \leq 7 \cdot 10^{-294}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{A}{B} \cdot -2\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;B \leq 1.45:\\ \;\;\;\;\frac{\tan^{-1} \left(0.5 \cdot \frac{B}{A}\right)}{0.005555555555555556 \cdot \mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 47.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\tan^{-1} \left(0.5 \cdot \frac{B}{A}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{if}\;B \leq -5.2 \cdot 10^{+54}:\\ \;\;\;\;\frac{\tan^{-1} 1}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;B \leq -7 \cdot 10^{-101}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;B \leq 7 \cdot 10^{-294}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{A}{B} \cdot -2\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;B \leq 1.45:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (let* ((t_0 (* (/ (atan (* 0.5 (/ B A))) (PI)) 180.0)))
   (if (<= B -5.2e+54)
     (* (/ (atan 1.0) (PI)) 180.0)
     (if (<= B -7e-101)
       t_0
       (if (<= B 7e-294)
         (* (/ (atan (* (/ A B) -2.0)) (PI)) 180.0)
         (if (<= B 1.45) t_0 (* (/ (atan -1.0) (PI)) 180.0)))))))

Derivation

1. Split input into 4 regimes

2. if B < -5.20000000000000013e54

   1. Initial program 43.0%

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

   3. Taylor expanded in B around -inf

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

   5. Applied rewrites 75.1%

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

   if -5.20000000000000013e54 < B < -6.99999999999999989e-101 or 7.00000000000000064e-294 < B < 1.44999999999999996

   1. Initial program 46.7%

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

   3. Taylor expanded in 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 43.7

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

   5. Applied rewrites 43.7%

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

   if -6.99999999999999989e-101 < B < 7.00000000000000064e-294

   1. Initial program 79.5%

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

   3. Taylor expanded in A around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 62.9

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

   5. Applied rewrites 62.9%

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

   if 1.44999999999999996 < B

   1. Initial program 45.9%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   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 65.3%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -5.2 \cdot 10^{+54}:\\ \;\;\;\;\frac{\tan^{-1} 1}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;B \leq -7 \cdot 10^{-101}:\\ \;\;\;\;\frac{\tan^{-1} \left(0.5 \cdot \frac{B}{A}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;B \leq 7 \cdot 10^{-294}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{A}{B} \cdot -2\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;B \leq 1.45:\\ \;\;\;\;\frac{\tan^{-1} \left(0.5 \cdot \frac{B}{A}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 46.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq -8.8 \cdot 10^{-100}:\\ \;\;\;\;\frac{\tan^{-1} 1}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;B \leq 4.6 \cdot 10^{-296}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{A}{B} \cdot -2\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;B \leq 3 \cdot 10^{-107}:\\ \;\;\;\;\frac{\tan^{-1} 0}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= B -8.8e-100)
   (* (/ (atan 1.0) (PI)) 180.0)
   (if (<= B 4.6e-296)
     (* (/ (atan (* (/ A B) -2.0)) (PI)) 180.0)
     (if (<= B 3e-107)
       (* (/ (atan 0.0) (PI)) 180.0)
       (* (/ (atan -1.0) (PI)) 180.0)))))

Derivation

1. Split input into 4 regimes

2. if B < -8.79999999999999957e-100

   1. Initial program 41.1%

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

   3. Taylor expanded in B around -inf

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

   5. Applied rewrites 57.8%

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

   if -8.79999999999999957e-100 < B < 4.60000000000000008e-296

   1. Initial program 79.5%

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

   3. Taylor expanded in A around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 62.9

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

   5. Applied rewrites 62.9%

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

   if 4.60000000000000008e-296 < B < 2.9999999999999997e-107

   1. Initial program 42.3%

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

   3. Taylor expanded in C around inf

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

      1. 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 51.9

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

   5. Applied rewrites 51.9%

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

   if 2.9999999999999997e-107 < B

   1. Initial program 50.3%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   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 52.7%

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

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -8.8 \cdot 10^{-100}:\\ \;\;\;\;\frac{\tan^{-1} 1}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;B \leq 4.6 \cdot 10^{-296}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{A}{B} \cdot -2\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;B \leq 3 \cdot 10^{-107}:\\ \;\;\;\;\frac{\tan^{-1} 0}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;A \leq -5800000000000:\\ \;\;\;\;\frac{\tan^{-1} \left(0.5 \cdot \frac{B}{A}\right)}{0.005555555555555556 \cdot \mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - A}{B} + 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= A -5800000000000.0)
   (/ (atan (* 0.5 (/ B A))) (* 0.005555555555555556 (PI)))
   (* (/ (atan (+ (/ (- C A) B) 1.0)) (PI)) 180.0)))

Derivation

1. Split input into 2 regimes

2. if A < -5.8e12

   1. Initial program 20.7%

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

   3. Taylor expanded in 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 72.6

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

   5. Applied rewrites 72.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   7. Applied rewrites 72.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

   9. Applied rewrites 72.7%

      \[\leadsto \color{blue}{\frac{\tan^{-1} \left(\frac{B}{A} \cdot 0.5\right)}{\mathsf{PI}\left(\right) \cdot 0.005555555555555556}} \]

   if -5.8e12 < A

   1. Initial program 61.1%

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

   3. Taylor expanded in B around -inf

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

      1. associate--l+ N/A

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

      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 62.1

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

   5. Applied rewrites 62.1%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -5800000000000:\\ \;\;\;\;\frac{\tan^{-1} \left(0.5 \cdot \frac{B}{A}\right)}{0.005555555555555556 \cdot \mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - A}{B} + 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 43.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq -1.9 \cdot 10^{-295}:\\ \;\;\;\;\frac{\tan^{-1} 1}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;B \leq 3 \cdot 10^{-107}:\\ \;\;\;\;\frac{\tan^{-1} 0}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= B -1.9e-295)
   (* (/ (atan 1.0) (PI)) 180.0)
   (if (<= B 3e-107)
     (* (/ (atan 0.0) (PI)) 180.0)
     (* (/ (atan -1.0) (PI)) 180.0))))

Derivation

1. Split input into 3 regimes

2. if B < -1.90000000000000009e-295

   1. Initial program 54.1%

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

   3. Taylor expanded in B around -inf

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

   5. Applied rewrites 44.6%

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

   if -1.90000000000000009e-295 < B < 2.9999999999999997e-107

   1. Initial program 45.1%

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

   3. Taylor expanded in C around inf

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

      1. 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 49.6

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

   5. Applied rewrites 49.6%

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

   if 2.9999999999999997e-107 < B

   1. Initial program 50.3%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   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 52.7%

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

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -1.9 \cdot 10^{-295}:\\ \;\;\;\;\frac{\tan^{-1} 1}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;B \leq 3 \cdot 10^{-107}:\\ \;\;\;\;\frac{\tan^{-1} 0}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 29.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 3 \cdot 10^{-107}:\\ \;\;\;\;\frac{\tan^{-1} 0}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= B 3e-107)
   (* (/ (atan 0.0) (PI)) 180.0)
   (* (/ (atan -1.0) (PI)) 180.0)))

Derivation

1. Split input into 2 regimes

2. if B < 2.9999999999999997e-107

   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 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 19.0

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

   5. Applied rewrites 19.0%

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

   if 2.9999999999999997e-107 < B

   1. Initial program 50.3%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   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 52.7%

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

4. Final simplification 32.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3 \cdot 10^{-107}:\\ \;\;\;\;\frac{\tan^{-1} 0}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 20.9% accurate, 3.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 51.1%

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

3. Taylor expanded in B around inf

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

5. Applied rewrites 23.1%

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

6. Final simplification 23.1%

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

Reproduce

herbie shell --seed 2024271 
(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))))