ABCF->ab-angle angle

Percentage Accurate: 53.7% → 80.8%

Time: 11.1s

Alternatives: 16

Speedup: 1.5×

Specification

Math

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

FPCore

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

Initial Program: 53.7% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 80.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{180}{\mathsf{PI}\left(\right)}\\ \mathbf{if}\;C \leq 9.6 \cdot 10^{+70}:\\ \;\;\;\;\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(A - C, B\right)}{B}\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\mathsf{fma}\left(\frac{B}{C}, -0.5, 0\right)\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (let* ((t_0 (/ 180.0 (PI))))
   (if (<= C 9.6e+70)
     (* (atan (/ (- (- C A) (hypot (- A C) B)) B)) t_0)
     (* (atan (fma (/ B C) -0.5 0.0)) t_0))))

Derivation

1. Split input into 2 regimes

2. if C < 9.59999999999999947e70

   1. Initial program 66.7%

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

      1. lift-*.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift-/.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. /-rgt-identity N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 66.7

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

   4. Applied rewrites 89.0%

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

   if 9.59999999999999947e70 < C

   1. Initial program 12.3%

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

      1. lift-*.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift-/.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. /-rgt-identity N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 12.3

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

   4. Applied rewrites 47.4%

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

   5. Taylor expanded in C around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-rgt1-in N/A

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

      4. metadata-eval N/A

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

      5. mul0-lft N/A

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

      6. div0 N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 73.4

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

   7. Applied rewrites 73.4%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 9.6 \cdot 10^{+70}:\\ \;\;\;\;\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(A - C, B\right)}{B}\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\mathsf{fma}\left(\frac{B}{C}, -0.5, 0\right)\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 73.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1} \left(\left(\left(C - A\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right) \cdot \frac{1}{B}\right)\\ t_1 := \frac{C - A}{B}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;\frac{\tan^{-1} \left(t\_1 - 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;t\_0 \leq 0.02:\\ \;\;\;\;\frac{180}{\frac{\mathsf{PI}\left(\right)}{\tan^{-1} \left(\frac{B}{A} \cdot 0.5\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{PI}\left(\right)}{\tan^{-1} \left(t\_1 + 1\right) \cdot 180}}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (let* ((t_0
         (atan
          (* (- (- C A) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0)))) (/ 1.0 B))))
        (t_1 (/ (- C A) B)))
   (if (<= t_0 -0.5)
     (* (/ (atan (- t_1 1.0)) (PI)) 180.0)
     (if (<= t_0 0.02)
       (/ 180.0 (/ (PI) (atan (* (/ B A) 0.5))))
       (/ 1.0 (/ (PI) (* (atan (+ t_1 1.0)) 180.0)))))))

Derivation

1. Split input into 3 regimes

2. if (atan.f64 (*.f64 (/.f64 #s(literal 1 binary64) B) (-.f64 (-.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64))))))) < -0.5

   1. Initial program 58.3%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   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 74.8

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

   5. Applied rewrites 74.8%

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

   if -0.5 < (atan.f64 (*.f64 (/.f64 #s(literal 1 binary64) B) (-.f64 (-.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64))))))) < 0.0200000000000000004

   1. Initial program 29.4%

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

   3. Taylor expanded in A around -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 55.2

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

   5. Applied rewrites 55.2%

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

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 55.2

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

   7. Applied rewrites 55.2%

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

   if 0.0200000000000000004 < (atan.f64 (*.f64 (/.f64 #s(literal 1 binary64) B) (-.f64 (-.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))

   1. Initial program 60.1%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. inv-pow N/A

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

      17. lower-pow.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. un-div-inv N/A

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

      20. lower-/.f64 80.0

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

   4. Applied rewrites 80.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 80.0

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

   6. Applied rewrites 88.1%

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

   7. Taylor expanded in B around -inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 78.2

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

   9. Applied rewrites 78.2%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan^{-1} \left(\left(\left(C - A\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right) \cdot \frac{1}{B}\right) \leq -0.5:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - A}{B} - 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;\tan^{-1} \left(\left(\left(C - A\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right) \cdot \frac{1}{B}\right) \leq 0.02:\\ \;\;\;\;\frac{180}{\frac{\mathsf{PI}\left(\right)}{\tan^{-1} \left(\frac{B}{A} \cdot 0.5\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{PI}\left(\right)}{\tan^{-1} \left(\frac{C - A}{B} + 1\right) \cdot 180}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1} \left(\left(\left(C - A\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right) \cdot \frac{1}{B}\right)\\ t_1 := \frac{C - A}{B}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;\frac{\tan^{-1} \left(t\_1 - 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;t\_0 \leq 0.02:\\ \;\;\;\;\frac{180}{\frac{\mathsf{PI}\left(\right)}{\tan^{-1} \left(\frac{B}{A} \cdot 0.5\right)}}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(t\_1 + 1\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (let* ((t_0
         (atan
          (* (- (- C A) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0)))) (/ 1.0 B))))
        (t_1 (/ (- C A) B)))
   (if (<= t_0 -0.5)
     (* (/ (atan (- t_1 1.0)) (PI)) 180.0)
     (if (<= t_0 0.02)
       (/ 180.0 (/ (PI) (atan (* (/ B A) 0.5))))
       (* (atan (+ t_1 1.0)) (/ 180.0 (PI)))))))

Derivation

1. Split input into 3 regimes

2. if (atan.f64 (*.f64 (/.f64 #s(literal 1 binary64) B) (-.f64 (-.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64))))))) < -0.5

   1. Initial program 58.3%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   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 74.8

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

   5. Applied rewrites 74.8%

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

   if -0.5 < (atan.f64 (*.f64 (/.f64 #s(literal 1 binary64) B) (-.f64 (-.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64))))))) < 0.0200000000000000004

   1. Initial program 29.4%

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

   3. Taylor expanded in A around -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 55.2

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

   5. Applied rewrites 55.2%

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

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 55.2

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

   7. Applied rewrites 55.2%

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

   if 0.0200000000000000004 < (atan.f64 (*.f64 (/.f64 #s(literal 1 binary64) B) (-.f64 (-.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))

   1. Initial program 60.1%

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

      1. lift-*.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift-/.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. /-rgt-identity N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 60.1

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

   4. Applied rewrites 88.1%

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

   5. Taylor expanded in B around -inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 78.2

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

   7. Applied rewrites 78.2%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan^{-1} \left(\left(\left(C - A\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right) \cdot \frac{1}{B}\right) \leq -0.5:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - A}{B} - 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;\tan^{-1} \left(\left(\left(C - A\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right) \cdot \frac{1}{B}\right) \leq 0.02:\\ \;\;\;\;\frac{180}{\frac{\mathsf{PI}\left(\right)}{\tan^{-1} \left(\frac{B}{A} \cdot 0.5\right)}}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{C - A}{B} + 1\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(C - A\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right) \cdot \frac{1}{B}\\ t_1 := \frac{C - A}{B}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;\frac{\tan^{-1} \left(t\_1 - 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;t\_0 \leq 0.02:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{B}{A} \cdot 0.5\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(t\_1 + 1\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (let* ((t_0
         (* (- (- C A) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0)))) (/ 1.0 B)))
        (t_1 (/ (- C A) B)))
   (if (<= t_0 -0.5)
     (* (/ (atan (- t_1 1.0)) (PI)) 180.0)
     (if (<= t_0 0.02)
       (* (/ (atan (* (/ B A) 0.5)) (PI)) 180.0)
       (* (atan (+ t_1 1.0)) (/ 180.0 (PI)))))))

Derivation

1. Split input into 3 regimes

2. if (*.f64 (/.f64 #s(literal 1 binary64) B) (-.f64 (-.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))) < -0.5

   1. Initial program 58.3%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   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 74.8

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

   5. Applied rewrites 74.8%

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

   if -0.5 < (*.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)))))) < 0.0200000000000000004

   1. Initial program 29.4%

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

   3. Taylor expanded in A around -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 55.2

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

   5. Applied rewrites 55.2%

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

   if 0.0200000000000000004 < (*.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 60.1%

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

      1. lift-*.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift-/.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. /-rgt-identity N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 60.1

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

   4. Applied rewrites 88.1%

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

   5. Taylor expanded in B around -inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 78.2

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

   7. Applied rewrites 78.2%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(C - A\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right) \cdot \frac{1}{B} \leq -0.5:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - A}{B} - 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;\left(\left(C - A\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right) \cdot \frac{1}{B} \leq 0.02:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{B}{A} \cdot 0.5\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{C - A}{B} + 1\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(C - A\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right) \cdot \frac{1}{B}\\ t_1 := \frac{C - A}{B}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;\frac{\tan^{-1} \left(t\_1 - 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;t\_0 \leq 0.02:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{B}{A} \cdot 0.5\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(t\_1 + 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (let* ((t_0
         (* (- (- C A) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0)))) (/ 1.0 B)))
        (t_1 (/ (- C A) B)))
   (if (<= t_0 -0.5)
     (* (/ (atan (- t_1 1.0)) (PI)) 180.0)
     (if (<= t_0 0.02)
       (* (/ (atan (* (/ B A) 0.5)) (PI)) 180.0)
       (* (/ (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)))))) < -0.5

   1. Initial program 58.3%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   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 74.8

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

   5. Applied rewrites 74.8%

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

   if -0.5 < (*.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)))))) < 0.0200000000000000004

   1. Initial program 29.4%

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

   3. Taylor expanded in A around -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 55.2

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

   5. Applied rewrites 55.2%

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

   if 0.0200000000000000004 < (*.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 60.1%

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

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

   5. Applied rewrites 78.2%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(C - A\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right) \cdot \frac{1}{B} \leq -0.5:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - A}{B} - 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;\left(\left(C - A\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right) \cdot \frac{1}{B} \leq 0.02:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{B}{A} \cdot 0.5\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - A}{B} + 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(C - A\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right) \cdot \frac{1}{B}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;\tan^{-1} \left(\frac{-A}{B} - 1\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;t\_0 \leq 0.02:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{B}{A} \cdot 0.5\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - A}{B} + 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (let* ((t_0
         (* (- (- C A) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0)))) (/ 1.0 B))))
   (if (<= t_0 -0.5)
     (* (atan (- (/ (- A) B) 1.0)) (/ 180.0 (PI)))
     (if (<= t_0 0.02)
       (* (/ (atan (* (/ B A) 0.5)) (PI)) 180.0)
       (* (/ (atan (+ (/ (- C A) B) 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)))))) < -0.5

   1. Initial program 58.3%

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

      1. lift-*.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift-/.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. /-rgt-identity N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 58.3

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

   4. Applied rewrites 86.3%

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

   5. Taylor expanded in B around inf

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 74.8

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

   7. Applied rewrites 74.8%

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

   8. Taylor expanded in C around 0

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

   10. Applied rewrites 65.4%

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

   if -0.5 < (*.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)))))) < 0.0200000000000000004

   1. Initial program 29.4%

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

   3. Taylor expanded in A around -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 55.2

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

   5. Applied rewrites 55.2%

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

   if 0.0200000000000000004 < (*.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 60.1%

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

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

   5. Applied rewrites 78.2%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(C - A\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right) \cdot \frac{1}{B} \leq -0.5:\\ \;\;\;\;\tan^{-1} \left(\frac{-A}{B} - 1\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;\left(\left(C - A\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right) \cdot \frac{1}{B} \leq 0.02:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{B}{A} \cdot 0.5\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - A}{B} + 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(C - A\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right) \cdot \frac{1}{B}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;\tan^{-1} \left(\frac{-A}{B} - 1\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;t\_0 \leq 0.02:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{B}{A} \cdot 0.5\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C}{B} + 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (let* ((t_0
         (* (- (- C A) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0)))) (/ 1.0 B))))
   (if (<= t_0 -0.5)
     (* (atan (- (/ (- A) B) 1.0)) (/ 180.0 (PI)))
     (if (<= t_0 0.02)
       (* (/ (atan (* (/ B A) 0.5)) (PI)) 180.0)
       (* (/ (atan (+ (/ C B) 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)))))) < -0.5

   1. Initial program 58.3%

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

      1. lift-*.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift-/.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. /-rgt-identity N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 58.3

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

   4. Applied rewrites 86.3%

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

   5. Taylor expanded in B around inf

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 74.8

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

   7. Applied rewrites 74.8%

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

   8. Taylor expanded in C around 0

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

   10. Applied rewrites 65.4%

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

   if -0.5 < (*.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)))))) < 0.0200000000000000004

   1. Initial program 29.4%

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

   3. Taylor expanded in A around -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 55.2

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

   5. Applied rewrites 55.2%

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

   if 0.0200000000000000004 < (*.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 60.1%

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

   3. Taylor expanded in A around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 67.7

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

   5. Applied rewrites 67.7%

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

   6. Taylor expanded in B around -inf

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

   8. Applied rewrites 62.7%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(C - A\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right) \cdot \frac{1}{B} \leq -0.5:\\ \;\;\;\;\tan^{-1} \left(\frac{-A}{B} - 1\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;\left(\left(C - A\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right) \cdot \frac{1}{B} \leq 0.02:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{B}{A} \cdot 0.5\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C}{B} + 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(C - A\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right) \cdot \frac{1}{B}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - B}{B}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;t\_0 \leq 0.02:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{B}{A} \cdot 0.5\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C}{B} + 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (let* ((t_0
         (* (- (- C A) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0)))) (/ 1.0 B))))
   (if (<= t_0 -0.5)
     (* (/ (atan (/ (- C B) B)) (PI)) 180.0)
     (if (<= t_0 0.02)
       (* (/ (atan (* (/ B A) 0.5)) (PI)) 180.0)
       (* (/ (atan (+ (/ C B) 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)))))) < -0.5

   1. Initial program 58.3%

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

   3. Taylor expanded in A around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 69.1

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

   5. Applied rewrites 69.1%

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

   6. Taylor expanded in C around 0

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

   8. Applied rewrites 63.2%

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

   if -0.5 < (*.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)))))) < 0.0200000000000000004

   1. Initial program 29.4%

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

   3. Taylor expanded in A around -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 55.2

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

   5. Applied rewrites 55.2%

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

   if 0.0200000000000000004 < (*.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 60.1%

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

   3. Taylor expanded in A around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 67.7

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

   5. Applied rewrites 67.7%

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

   6. Taylor expanded in B around -inf

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

   8. Applied rewrites 62.7%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(C - A\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right) \cdot \frac{1}{B} \leq -0.5:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - B}{B}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;\left(\left(C - A\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right) \cdot \frac{1}{B} \leq 0.02:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{B}{A} \cdot 0.5\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C}{B} + 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.5% accurate, 0.6× speedup

Math

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

FPCore

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

   1. Initial program 58.3%

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

   3. Taylor expanded in A around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 69.1

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

   5. Applied rewrites 69.1%

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

   6. Taylor expanded in C around 0

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

   8. Applied rewrites 63.2%

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

   if -0.5 < (*.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)))))) < 1.00000000000000008e-5

   1. Initial program 27.4%

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

   3. Taylor expanded in A around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 21.1

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

   5. Applied rewrites 21.1%

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

   6. Taylor expanded in C around inf

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

   8. Applied rewrites 45.0%

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

   if 1.00000000000000008e-5 < (*.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 60.3%

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

   3. Taylor expanded in A around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 67.3

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

   5. Applied rewrites 67.3%

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

   6. Taylor expanded in B around -inf

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

   8. Applied rewrites 62.3%

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

4. Final simplification 60.8%

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

Alternative 10: 58.1% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (A B C)
 :precision binary64
 (let* ((t_0
         (* (- (- C A) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0)))) (/ 1.0 B))))
   (if (<= t_0 -0.5)
     (* (/ (atan (/ (- C B) B)) (PI)) 180.0)
     (if (<= t_0 1e-5)
       (* (/ (atan 0.0) (PI)) 180.0)
       (* (/ (atan (+ (/ C B) 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)))))) < -0.5

   1. Initial program 58.3%

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

   3. Taylor expanded in A around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 69.1

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

   5. Applied rewrites 69.1%

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

   6. Taylor expanded in C around 0

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

   8. Applied rewrites 63.2%

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

   if -0.5 < (*.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)))))) < 1.00000000000000008e-5

   1. Initial program 27.4%

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

   3. Taylor expanded in C around inf

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. mul0-lft N/A

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

      4. div0 N/A

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

      5. metadata-eval 25.7

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

   5. Applied rewrites 25.7%

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

   if 1.00000000000000008e-5 < (*.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 60.3%

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

   3. Taylor expanded in A around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 67.3

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

   5. Applied rewrites 67.3%

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

   6. Taylor expanded in B around -inf

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

   8. Applied rewrites 62.3%

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

4. Final simplification 58.7%

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

Alternative 11: 77.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;C \leq -9.2 \cdot 10^{-88}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - \mathsf{hypot}\left(C, B\right)}{B}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;C \leq 8.8 \cdot 10^{+70}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{\mathsf{hypot}\left(B, A\right) + A}{-B}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\mathsf{fma}\left(\frac{B}{C}, -0.5, 0\right)\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= C -9.2e-88)
   (* (/ (atan (/ (- C (hypot C B)) B)) (PI)) 180.0)
   (if (<= C 8.8e+70)
     (* (/ (atan (/ (+ (hypot B A) A) (- B))) (PI)) 180.0)
     (* (atan (fma (/ B C) -0.5 0.0)) (/ 180.0 (PI))))))

Derivation

1. Split input into 3 regimes

2. if C < -9.19999999999999945e-88

   1. Initial program 72.7%

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

   3. Taylor expanded in A around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 87.4

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

   5. Applied rewrites 87.4%

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

   if -9.19999999999999945e-88 < C < 8.80000000000000003e70

   1. Initial program 63.0%

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

   3. Taylor expanded in C around 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 84.8

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

   5. Applied rewrites 84.8%

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

   if 8.80000000000000003e70 < C

   1. Initial program 12.3%

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

      1. lift-*.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift-/.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. /-rgt-identity N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 12.3

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

   4. Applied rewrites 47.4%

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

   5. Taylor expanded in C around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-rgt1-in N/A

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

      4. metadata-eval N/A

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

      5. mul0-lft N/A

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

      6. div0 N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 73.4

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

   7. Applied rewrites 73.4%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq -9.2 \cdot 10^{-88}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - \mathsf{hypot}\left(C, B\right)}{B}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;C \leq 8.8 \cdot 10^{+70}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{\mathsf{hypot}\left(B, A\right) + A}{-B}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\mathsf{fma}\left(\frac{B}{C}, -0.5, 0\right)\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 73.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;A \leq -8.5 \cdot 10^{+97}:\\ \;\;\;\;\tan^{-1} \left(\frac{\mathsf{fma}\left(\frac{C}{A}, B, B\right)}{A} \cdot 0.5\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;A \leq 10^{-105}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - \mathsf{hypot}\left(C, B\right)}{B}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;A \leq 9.5 \cdot 10^{-47}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{\mathsf{fma}\left(A, \frac{B}{C}, B\right)}{C} \cdot -0.5\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{PI}\left(\right)}{\tan^{-1} \left(\frac{C - A}{B} + 1\right) \cdot 180}}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= A -8.5e+97)
   (* (atan (* (/ (fma (/ C A) B B) A) 0.5)) (/ 180.0 (PI)))
   (if (<= A 1e-105)
     (* (/ (atan (/ (- C (hypot C B)) B)) (PI)) 180.0)
     (if (<= A 9.5e-47)
       (* (/ (atan (* (/ (fma A (/ B C) B) C) -0.5)) (PI)) 180.0)
       (/ 1.0 (/ (PI) (* (atan (+ (/ (- C A) B) 1.0)) 180.0)))))))

Derivation

1. Split input into 4 regimes

2. if A < -8.4999999999999993e97

   1. Initial program 16.9%

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

      1. lift-*.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift-/.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. /-rgt-identity N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 16.9

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

   4. Applied rewrites 62.8%

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

   5. Taylor expanded in A around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-lft-out N/A

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

      3. associate-/l* N/A

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

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

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 80.3

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

   7. Applied rewrites 80.3%

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

   if -8.4999999999999993e97 < A < 9.99999999999999965e-106

   1. Initial program 59.3%

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

   3. Taylor expanded in A around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 78.1

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

   5. Applied rewrites 78.1%

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

   if 9.99999999999999965e-106 < A < 9.4999999999999991e-47

   1. Initial program 59.3%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. inv-pow N/A

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

      17. lower-pow.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. un-div-inv N/A

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

      20. lower-/.f64 58.6

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

   4. Applied rewrites 58.6%

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

   5. Taylor expanded in C around inf

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

      1. distribute-lft-out N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 75.8

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

   7. Applied rewrites 75.8%

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

   if 9.4999999999999991e-47 < A

   1. Initial program 67.5%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. inv-pow N/A

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

      17. lower-pow.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. un-div-inv N/A

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

      20. lower-/.f64 89.0

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

   4. Applied rewrites 89.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 88.9

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

   6. Applied rewrites 91.4%

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

   7. Taylor expanded in B around -inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 74.2

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

   9. Applied rewrites 74.2%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -8.5 \cdot 10^{+97}:\\ \;\;\;\;\tan^{-1} \left(\frac{\mathsf{fma}\left(\frac{C}{A}, B, B\right)}{A} \cdot 0.5\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;A \leq 10^{-105}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - \mathsf{hypot}\left(C, B\right)}{B}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;A \leq 9.5 \cdot 10^{-47}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{\mathsf{fma}\left(A, \frac{B}{C}, B\right)}{C} \cdot -0.5\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{PI}\left(\right)}{\tan^{-1} \left(\frac{C - A}{B} + 1\right) \cdot 180}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 51.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 2.7 \cdot 10^{-159}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C}{B} + 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= B 2.7e-159)
   (* (/ (atan (+ (/ C B) 1.0)) (PI)) 180.0)
   (* (/ (atan -1.0) (PI)) 180.0)))

Derivation

1. Split input into 2 regimes

2. if B < 2.7e-159

   1. Initial program 56.5%

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

   3. Taylor expanded in A around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 61.6

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

   5. Applied rewrites 61.6%

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

   6. Taylor expanded in B around -inf

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

   8. Applied rewrites 52.2%

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

   if 2.7e-159 < B

   1. Initial program 54.6%

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

   3. Taylor expanded in B around inf

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

   5. Applied rewrites 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 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.7 \cdot 10^{-159}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C}{B} + 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 44.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq -8.8 \cdot 10^{-147}:\\ \;\;\;\;\frac{\tan^{-1} 1}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;B \leq 1.1 \cdot 10^{-202}:\\ \;\;\;\;\frac{\tan^{-1} 0}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= B -8.8e-147)
   (* (/ (atan 1.0) (PI)) 180.0)
   (if (<= B 1.1e-202)
     (* (/ (atan 0.0) (PI)) 180.0)
     (* (/ (atan -1.0) (PI)) 180.0))))

Derivation

1. Split input into 3 regimes

2. if B < -8.8000000000000004e-147

   1. Initial program 55.0%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   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 52.1%

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

   if -8.8000000000000004e-147 < B < 1.10000000000000004e-202

   1. Initial program 56.1%

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

   3. Taylor expanded in C around inf

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. mul0-lft N/A

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

      4. div0 N/A

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

      5. metadata-eval 36.6

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

   5. Applied rewrites 36.6%

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

   if 1.10000000000000004e-202 < B

   1. Initial program 56.5%

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

   3. Taylor expanded in B around inf

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

   5. Applied rewrites 51.3%

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

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -8.8 \cdot 10^{-147}:\\ \;\;\;\;\frac{\tan^{-1} 1}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;B \leq 1.1 \cdot 10^{-202}:\\ \;\;\;\;\frac{\tan^{-1} 0}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 28.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 1.1 \cdot 10^{-202}:\\ \;\;\;\;\frac{\tan^{-1} 0}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= B 1.1e-202)
   (* (/ (atan 0.0) (PI)) 180.0)
   (* (/ (atan -1.0) (PI)) 180.0)))

Derivation

1. Split input into 2 regimes

2. if B < 1.10000000000000004e-202

   1. Initial program 55.5%

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

   3. Taylor expanded in C around inf

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. mul0-lft N/A

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

      4. div0 N/A

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

      5. metadata-eval 16.9

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

   5. Applied rewrites 16.9%

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

   if 1.10000000000000004e-202 < B

   1. Initial program 56.5%

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

   3. Taylor expanded in B around inf

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

   5. Applied rewrites 51.3%

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

4. Final simplification 29.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.1 \cdot 10^{-202}:\\ \;\;\;\;\frac{\tan^{-1} 0}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 20.8% 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 55.8%

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

3. Taylor expanded in B around inf

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

5. Applied rewrites 21.1%

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

6. Final simplification 21.1%

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

Reproduce

herbie shell --seed 2024264 
(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))))