ABCF->ab-angle angle

Percentage Accurate: 54.0% → 72.6%

Time: 10.6s

Alternatives: 13

Speedup: 2.5×

Specification

Math

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

FPCore

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

Initial Program: 54.0% 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: 72.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(C - A\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right) \cdot \frac{1}{B}\\ t_1 := \frac{C - A}{B}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-14}:\\ \;\;\;\;\frac{\tan^{-1} \left(t\_1 - 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-30}:\\ \;\;\;\;\frac{\tan^{-1} \left(\mathsf{fma}\left(\frac{B}{C}, -0.5, 0\right)\right)}{\sqrt[3]{{\mathsf{PI}\left(\right)}^{3}}} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(t\_1 + 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \end{array} \]

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 -5e-14)
     (* (/ (atan (- t_1 1.0)) (PI)) 180.0)
     (if (<= t_0 5e-30)
       (* (/ (atan (fma (/ B C) -0.5 0.0)) (cbrt (pow (PI) 3.0))) 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)))))) < -5.0000000000000002e-14

   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

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C}{B} - \left(1 + \frac{A}{B}\right)\right)}}{\mathsf{PI}\left(\right)} \]
   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 72.9

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

   5. Applied rewrites 72.9%

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

   if -5.0000000000000002e-14 < (*.f64 (/.f64 #s(literal 1 binary64) B) (-.f64 (-.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))) < 4.99999999999999972e-30

   1. Initial program 21.2%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-out-- N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower--.f64 2.5

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\mathsf{fma}\left(C - A, 2, \frac{\left(-0.5 \cdot B\right) \cdot B}{\color{blue}{A - C}}\right)}{B}\right)}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 2.5%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{\mathsf{fma}\left(C - A, 2, \frac{\left(-0.5 \cdot B\right) \cdot B}{A - C}\right)}{B}\right)}}{\mathsf{PI}\left(\right)} \]
   6. Step-by-step derivation

      1. lift-PI.f64 N/A

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

      2. add-cbrt-cube N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\mathsf{fma}\left(C - A, 2, \frac{\left(\frac{-1}{2} \cdot B\right) \cdot B}{A - C}\right)}{B}\right)}{\color{blue}{\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}}} \]

      3. lower-cbrt.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\mathsf{fma}\left(C - A, 2, \frac{\left(\frac{-1}{2} \cdot B\right) \cdot B}{A - C}\right)}{B}\right)}{\color{blue}{\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}}} \]

      4. rem-cube-cbrt N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\mathsf{fma}\left(C - A, 2, \frac{\left(\frac{-1}{2} \cdot B\right) \cdot B}{A - C}\right)}{B}\right)}{\sqrt[3]{\color{blue}{{\left(\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}\right)}^{3}}}} \]

      5. add-cbrt-cube N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\mathsf{fma}\left(C - A, 2, \frac{\left(\frac{-1}{2} \cdot B\right) \cdot B}{A - C}\right)}{B}\right)}{\sqrt[3]{{\color{blue}{\mathsf{PI}\left(\right)}}^{3}}} \]

      6. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\mathsf{fma}\left(C - A, 2, \frac{\left(\frac{-1}{2} \cdot B\right) \cdot B}{A - C}\right)}{B}\right)}{\sqrt[3]{{\color{blue}{\mathsf{PI}\left(\right)}}^{3}}} \]

      7. lower-pow.f64 2.5

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\mathsf{fma}\left(C - A, 2, \frac{\left(-0.5 \cdot B\right) \cdot B}{A - C}\right)}{B}\right)}{\sqrt[3]{\color{blue}{{\mathsf{PI}\left(\right)}^{3}}}} \]

   7. Applied rewrites 2.5%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\mathsf{fma}\left(C - A, 2, \frac{\left(-0.5 \cdot B\right) \cdot B}{A - C}\right)}{B}\right)}{\color{blue}{\sqrt[3]{{\mathsf{PI}\left(\right)}^{3}}}} \]

   8. Taylor expanded in C around inf

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

      1. +-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-1}{2} \cdot \frac{B}{C} + -1 \cdot \frac{A + -1 \cdot A}{B}\right)}}{\sqrt[3]{{\mathsf{PI}\left(\right)}^{3}}} \]

      2. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\frac{B}{C} \cdot \frac{-1}{2}} + -1 \cdot \frac{A + -1 \cdot A}{B}\right)}{\sqrt[3]{{\mathsf{PI}\left(\right)}^{3}}} \]

      3. distribute-rgt1-in N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{B}{C} \cdot \frac{-1}{2} + -1 \cdot \frac{\color{blue}{\left(-1 + 1\right) \cdot A}}{B}\right)}{\sqrt[3]{{\mathsf{PI}\left(\right)}^{3}}} \]

      4. metadata-eval N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{B}{C} \cdot \frac{-1}{2} + -1 \cdot \frac{\color{blue}{0} \cdot A}{B}\right)}{\sqrt[3]{{\mathsf{PI}\left(\right)}^{3}}} \]

      5. mul0-lft N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{B}{C} \cdot \frac{-1}{2} + -1 \cdot \frac{\color{blue}{0}}{B}\right)}{\sqrt[3]{{\mathsf{PI}\left(\right)}^{3}}} \]

      6. div0 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{B}{C} \cdot \frac{-1}{2} + -1 \cdot \color{blue}{0}\right)}{\sqrt[3]{{\mathsf{PI}\left(\right)}^{3}}} \]

      7. metadata-eval N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{B}{C} \cdot \frac{-1}{2} + \color{blue}{0}\right)}{\sqrt[3]{{\mathsf{PI}\left(\right)}^{3}}} \]

      8. lower-fma.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{fma}\left(\frac{B}{C}, \frac{-1}{2}, 0\right)\right)}}{\sqrt[3]{{\mathsf{PI}\left(\right)}^{3}}} \]

      9. lower-/.f64 64.8

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

   10. Applied rewrites 64.8%

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

   if 4.99999999999999972e-30 < (*.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 54.2%

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

   3. Taylor expanded in B around -inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 75.6

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

   5. Applied rewrites 75.6%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(C - A\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right) \cdot \frac{1}{B} \leq -5 \cdot 10^{-14}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - A}{B} - 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;\left(\left(C - A\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right) \cdot \frac{1}{B} \leq 5 \cdot 10^{-30}:\\ \;\;\;\;\frac{\tan^{-1} \left(\mathsf{fma}\left(\frac{B}{C}, -0.5, 0\right)\right)}{\sqrt[3]{{\mathsf{PI}\left(\right)}^{3}}} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - A}{B} + 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 72.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(C - A\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right) \cdot \frac{1}{B}\\ t_1 := \frac{C - A}{B}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-14}:\\ \;\;\;\;\frac{\tan^{-1} \left(t\_1 - 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-30}:\\ \;\;\;\;\frac{\tan^{-1} \left(\mathsf{fma}\left(\frac{B}{C}, -0.5, 0\right)\right) \cdot 180}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(t\_1 + 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (let* ((t_0
         (* (- (- C A) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0)))) (/ 1.0 B)))
        (t_1 (/ (- C A) B)))
   (if (<= t_0 -5e-14)
     (* (/ (atan (- t_1 1.0)) (PI)) 180.0)
     (if (<= t_0 5e-30)
       (/ (* (atan (fma (/ B C) -0.5 0.0)) 180.0) (PI))
       (* (/ (atan (+ t_1 1.0)) (PI)) 180.0)))))

Derivation

1. Split input into 3 regimes

2. if (*.f64 (/.f64 #s(literal 1 binary64) B) (-.f64 (-.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))) < -5.0000000000000002e-14

   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

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C}{B} - \left(1 + \frac{A}{B}\right)\right)}}{\mathsf{PI}\left(\right)} \]
   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 72.9

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

   5. Applied rewrites 72.9%

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

   if -5.0000000000000002e-14 < (*.f64 (/.f64 #s(literal 1 binary64) B) (-.f64 (-.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))) < 4.99999999999999972e-30

   1. Initial program 21.2%

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

   3. Taylor expanded in C around inf

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. mul0-lft N/A

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

      4. --rgt-identity N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 34.5

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

   5. Applied rewrites 34.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   7. Applied rewrites 34.4%

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

   8. Taylor expanded in C around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-rgt1-in N/A

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

      4. metadata-eval N/A

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

      5. mul0-lft N/A

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

      6. div0 N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 64.8

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

   10. Applied rewrites 64.8%

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

   if 4.99999999999999972e-30 < (*.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 54.2%

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

   3. Taylor expanded in B around -inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 75.6

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

   5. Applied rewrites 75.6%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(C - A\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right) \cdot \frac{1}{B} \leq -5 \cdot 10^{-14}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - A}{B} - 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;\left(\left(C - A\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right) \cdot \frac{1}{B} \leq 5 \cdot 10^{-30}:\\ \;\;\;\;\frac{\tan^{-1} \left(\mathsf{fma}\left(\frac{B}{C}, -0.5, 0\right)\right) \cdot 180}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - A}{B} + 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(C - A\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right) \cdot \frac{1}{B}\\ t_1 := \frac{C - A}{B}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-14}:\\ \;\;\;\;\frac{\tan^{-1} \left(t\_1 - 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-30}:\\ \;\;\;\;\frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \left(\mathsf{fma}\left(\frac{B}{C}, -0.5, 0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(t\_1 + 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (let* ((t_0
         (* (- (- C A) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0)))) (/ 1.0 B)))
        (t_1 (/ (- C A) B)))
   (if (<= t_0 -5e-14)
     (* (/ (atan (- t_1 1.0)) (PI)) 180.0)
     (if (<= t_0 5e-30)
       (* (/ 180.0 (PI)) (atan (fma (/ B C) -0.5 0.0)))
       (* (/ (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)))))) < -5.0000000000000002e-14

   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

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C}{B} - \left(1 + \frac{A}{B}\right)\right)}}{\mathsf{PI}\left(\right)} \]
   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 72.9

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

   5. Applied rewrites 72.9%

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

   if -5.0000000000000002e-14 < (*.f64 (/.f64 #s(literal 1 binary64) B) (-.f64 (-.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))) < 4.99999999999999972e-30

   1. Initial program 21.2%

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

   3. Taylor expanded in C around inf

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. mul0-lft N/A

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

      4. --rgt-identity N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 34.5

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

   5. Applied rewrites 34.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   7. Applied rewrites 34.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 34.4

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

   9. Applied rewrites 34.4%

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

   10. Taylor expanded in C around inf

       \[\leadsto \frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \color{blue}{\left(-1 \cdot \frac{A + -1 \cdot A}{B} + \frac{-1}{2} \cdot \frac{B}{C}\right)} \]
   11. 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 64.8

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

   12. Applied rewrites 64.8%

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

   if 4.99999999999999972e-30 < (*.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 54.2%

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

   3. Taylor expanded in B around -inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 75.6

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

   5. Applied rewrites 75.6%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(C - A\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right) \cdot \frac{1}{B} \leq -5 \cdot 10^{-14}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - A}{B} - 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;\left(\left(C - A\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right) \cdot \frac{1}{B} \leq 5 \cdot 10^{-30}:\\ \;\;\;\;\frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \left(\mathsf{fma}\left(\frac{B}{C}, -0.5, 0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - A}{B} + 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(C - A\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right) \cdot \frac{1}{B}\\ t_1 := \frac{C - A}{B}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-14}:\\ \;\;\;\;\frac{\tan^{-1} \left(t\_1 - 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-30}:\\ \;\;\;\;\frac{\tan^{-1} \left(\mathsf{fma}\left(\frac{B}{C}, -0.5, 0\right)\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(t\_1 + 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \end{array} \]

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 -5e-14)
     (* (/ (atan (- t_1 1.0)) (PI)) 180.0)
     (if (<= t_0 5e-30)
       (* (/ (atan (fma (/ B C) -0.5 0.0)) (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)))))) < -5.0000000000000002e-14

   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

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C}{B} - \left(1 + \frac{A}{B}\right)\right)}}{\mathsf{PI}\left(\right)} \]
   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 72.9

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

   5. Applied rewrites 72.9%

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

   if -5.0000000000000002e-14 < (*.f64 (/.f64 #s(literal 1 binary64) B) (-.f64 (-.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))) < 4.99999999999999972e-30

   1. Initial program 21.2%

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

   3. Taylor expanded in C around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-rgt1-in N/A

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

      4. metadata-eval N/A

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

      5. mul0-lft N/A

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

      6. div0 N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 64.7

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

   5. Applied rewrites 64.7%

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

   if 4.99999999999999972e-30 < (*.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 54.2%

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

   3. Taylor expanded in B around -inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 75.6

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

   5. Applied rewrites 75.6%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(C - A\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right) \cdot \frac{1}{B} \leq -5 \cdot 10^{-14}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - A}{B} - 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;\left(\left(C - A\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right) \cdot \frac{1}{B} \leq 5 \cdot 10^{-30}:\\ \;\;\;\;\frac{\tan^{-1} \left(\mathsf{fma}\left(\frac{B}{C}, -0.5, 0\right)\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - A}{B} + 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(C - A\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right) \cdot \frac{1}{B}\\ t_1 := \frac{C - A}{B}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-14}:\\ \;\;\;\;\frac{\tan^{-1} \left(t\_1 - 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-30}:\\ \;\;\;\;\frac{\tan^{-1} \left(0.5 \cdot \frac{B}{A}\right) \cdot 180}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(t\_1 + 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (let* ((t_0
         (* (- (- C A) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0)))) (/ 1.0 B)))
        (t_1 (/ (- C A) B)))
   (if (<= t_0 -5e-14)
     (* (/ (atan (- t_1 1.0)) (PI)) 180.0)
     (if (<= t_0 5e-30)
       (/ (* (atan (* 0.5 (/ B A))) 180.0) (PI))
       (* (/ (atan (+ t_1 1.0)) (PI)) 180.0)))))

Derivation

1. Split input into 3 regimes

2. if (*.f64 (/.f64 #s(literal 1 binary64) B) (-.f64 (-.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))) < -5.0000000000000002e-14

   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

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C}{B} - \left(1 + \frac{A}{B}\right)\right)}}{\mathsf{PI}\left(\right)} \]
   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 72.9

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

   5. Applied rewrites 72.9%

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

   if -5.0000000000000002e-14 < (*.f64 (/.f64 #s(literal 1 binary64) B) (-.f64 (-.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))) < 4.99999999999999972e-30

   1. Initial program 21.2%

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

   3. Taylor expanded in C around inf

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. mul0-lft N/A

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

      4. --rgt-identity N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 34.5

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

   5. Applied rewrites 34.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   7. Applied rewrites 34.4%

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

   8. Taylor expanded in A around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 37.3

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

   10. Applied rewrites 37.3%

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

   if 4.99999999999999972e-30 < (*.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 54.2%

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

   3. Taylor expanded in B around -inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 75.6

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

   5. Applied rewrites 75.6%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(C - A\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right) \cdot \frac{1}{B} \leq -5 \cdot 10^{-14}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - A}{B} - 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;\left(\left(C - A\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right) \cdot \frac{1}{B} \leq 5 \cdot 10^{-30}:\\ \;\;\;\;\frac{\tan^{-1} \left(0.5 \cdot \frac{B}{A}\right) \cdot 180}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - A}{B} + 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 46.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq -59000000000000:\\ \;\;\;\;\frac{\tan^{-1} 1}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;B \leq -1.45 \cdot 10^{-100}:\\ \;\;\;\;\tan^{-1} \left(0.5 \cdot \frac{B}{A}\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;B \leq 3.5 \cdot 10^{+40}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{A}{B} \cdot -2\right) \cdot 180}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= B -59000000000000.0)
   (* (/ (atan 1.0) (PI)) 180.0)
   (if (<= B -1.45e-100)
     (* (atan (* 0.5 (/ B A))) (/ 180.0 (PI)))
     (if (<= B 3.5e+40)
       (/ (* (atan (* (/ A B) -2.0)) 180.0) (PI))
       (* (/ (atan -1.0) (PI)) 180.0)))))

Derivation

1. Split input into 4 regimes

2. if B < -5.9e13

   1. Initial program 46.5%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   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 73.4%

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

   if -5.9e13 < B < -1.44999999999999988e-100

   1. Initial program 41.6%

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

   3. Taylor expanded in C around inf

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. mul0-lft N/A

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

      4. --rgt-identity N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 12.2

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

   5. Applied rewrites 12.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   7. Applied rewrites 12.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 12.2

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

   9. Applied rewrites 12.2%

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

   10. Taylor expanded in A around -inf

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 63.9

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

   12. Applied rewrites 63.9%

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

   if -1.44999999999999988e-100 < B < 3.4999999999999999e40

   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 inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 36.9

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

   5. Applied rewrites 36.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   7. Applied rewrites 36.9%

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

   if 3.4999999999999999e40 < B

   1. Initial program 42.3%

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

   3. Taylor expanded in B around inf

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

   5. Applied rewrites 63.7%

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

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -59000000000000:\\ \;\;\;\;\frac{\tan^{-1} 1}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;B \leq -1.45 \cdot 10^{-100}:\\ \;\;\;\;\tan^{-1} \left(0.5 \cdot \frac{B}{A}\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;B \leq 3.5 \cdot 10^{+40}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{A}{B} \cdot -2\right) \cdot 180}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 46.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq -59000000000000:\\ \;\;\;\;\frac{\tan^{-1} 1}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;B \leq -1.45 \cdot 10^{-100}:\\ \;\;\;\;\tan^{-1} \left(0.5 \cdot \frac{B}{A}\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;B \leq 3.5 \cdot 10^{+40}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{A}{B} \cdot -2\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= B -59000000000000.0)
   (* (/ (atan 1.0) (PI)) 180.0)
   (if (<= B -1.45e-100)
     (* (atan (* 0.5 (/ B A))) (/ 180.0 (PI)))
     (if (<= B 3.5e+40)
       (* (/ (atan (* (/ A B) -2.0)) (PI)) 180.0)
       (* (/ (atan -1.0) (PI)) 180.0)))))

Derivation

1. Split input into 4 regimes

2. if B < -5.9e13

   1. Initial program 46.5%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   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 73.4%

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

   if -5.9e13 < B < -1.44999999999999988e-100

   1. Initial program 41.6%

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

   3. Taylor expanded in C around inf

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. mul0-lft N/A

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

      4. --rgt-identity N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 12.2

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

   5. Applied rewrites 12.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   7. Applied rewrites 12.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 12.2

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

   9. Applied rewrites 12.2%

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

   10. Taylor expanded in A around -inf

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 63.9

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

   12. Applied rewrites 63.9%

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

   if -1.44999999999999988e-100 < B < 3.4999999999999999e40

   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 inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 36.9

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

   5. Applied rewrites 36.9%

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

   if 3.4999999999999999e40 < B

   1. Initial program 42.3%

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

   3. Taylor expanded in B around inf

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

   5. Applied rewrites 63.7%

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

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -59000000000000:\\ \;\;\;\;\frac{\tan^{-1} 1}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;B \leq -1.45 \cdot 10^{-100}:\\ \;\;\;\;\tan^{-1} \left(0.5 \cdot \frac{B}{A}\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}\\ \mathbf{elif}\;B \leq 3.5 \cdot 10^{+40}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{A}{B} \cdot -2\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 46.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq -59000000000000:\\ \;\;\;\;\frac{\tan^{-1} 1}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;B \leq -1.45 \cdot 10^{-100}:\\ \;\;\;\;\frac{\tan^{-1} \left(0.5 \cdot \frac{B}{A}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;B \leq 3.5 \cdot 10^{+40}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{A}{B} \cdot -2\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= B -59000000000000.0)
   (* (/ (atan 1.0) (PI)) 180.0)
   (if (<= B -1.45e-100)
     (* (/ (atan (* 0.5 (/ B A))) (PI)) 180.0)
     (if (<= B 3.5e+40)
       (* (/ (atan (* (/ A B) -2.0)) (PI)) 180.0)
       (* (/ (atan -1.0) (PI)) 180.0)))))

Derivation

1. Split input into 4 regimes

2. if B < -5.9e13

   1. Initial program 46.5%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   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 73.4%

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

   if -5.9e13 < B < -1.44999999999999988e-100

   1. Initial program 41.6%

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

   3. Taylor expanded in A around -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 63.8

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

   5. Applied rewrites 63.8%

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

   if -1.44999999999999988e-100 < B < 3.4999999999999999e40

   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 inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 36.9

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

   5. Applied rewrites 36.9%

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

   if 3.4999999999999999e40 < B

   1. Initial program 42.3%

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

   3. Taylor expanded in B around inf

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

   5. Applied rewrites 63.7%

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

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -59000000000000:\\ \;\;\;\;\frac{\tan^{-1} 1}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;B \leq -1.45 \cdot 10^{-100}:\\ \;\;\;\;\frac{\tan^{-1} \left(0.5 \cdot \frac{B}{A}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;B \leq 3.5 \cdot 10^{+40}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{A}{B} \cdot -2\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 46.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq -2.2 \cdot 10^{-53}:\\ \;\;\;\;\frac{\tan^{-1} 1}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;B \leq 3.5 \cdot 10^{+40}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{A}{B} \cdot -2\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= B -2.2e-53)
   (* (/ (atan 1.0) (PI)) 180.0)
   (if (<= B 3.5e+40)
     (* (/ (atan (* (/ A B) -2.0)) (PI)) 180.0)
     (* (/ (atan -1.0) (PI)) 180.0))))

Derivation

1. Split input into 3 regimes

2. if B < -2.20000000000000018e-53

   1. Initial program 48.2%

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

   3. Taylor expanded in B around -inf

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

   5. Applied rewrites 65.6%

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

   if -2.20000000000000018e-53 < B < 3.4999999999999999e40

   1. Initial program 56.3%

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

   3. Taylor expanded in A around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 35.3

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

   5. Applied rewrites 35.3%

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

   if 3.4999999999999999e40 < B

   1. Initial program 42.3%

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

   3. Taylor expanded in B around inf

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

   5. Applied rewrites 63.7%

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

4. Final simplification 51.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -2.2 \cdot 10^{-53}:\\ \;\;\;\;\frac{\tan^{-1} 1}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;B \leq 3.5 \cdot 10^{+40}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{A}{B} \cdot -2\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 61.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;A \leq -3.7 \cdot 10^{+14}:\\ \;\;\;\;\tan^{-1} \left(0.5 \cdot \frac{B}{A}\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - A}{B} + 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= A -3.7e+14)
   (* (atan (* 0.5 (/ B A))) (/ 180.0 (PI)))
   (* (/ (atan (+ (/ (- C A) B) 1.0)) (PI)) 180.0)))

Derivation

1. Split input into 2 regimes

2. if A < -3.7e14

   1. Initial program 21.0%

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

   3. Taylor expanded in C around inf

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. mul0-lft N/A

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

      4. --rgt-identity N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 20.4

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

   5. Applied rewrites 20.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   7. Applied rewrites 20.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 20.4

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

   9. Applied rewrites 20.4%

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

   10. Taylor expanded in A around -inf

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 76.0

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

   12. Applied rewrites 76.0%

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

   if -3.7e14 < A

   1. Initial program 60.3%

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

   3. Taylor expanded in B around -inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\left(1 + \frac{C}{B}\right) - \frac{A}{B}\right)}}{\mathsf{PI}\left(\right)} \]
   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 58.6

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

   5. Applied rewrites 58.6%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -3.7 \cdot 10^{+14}:\\ \;\;\;\;\tan^{-1} \left(0.5 \cdot \frac{B}{A}\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - A}{B} + 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 45.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq -2.9 \cdot 10^{-64}:\\ \;\;\;\;\frac{\tan^{-1} 1}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;B \leq 1.4 \cdot 10^{-78}:\\ \;\;\;\;\frac{\tan^{-1} 0}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= B -2.9e-64)
   (* (/ (atan 1.0) (PI)) 180.0)
   (if (<= B 1.4e-78)
     (* (/ (atan 0.0) (PI)) 180.0)
     (* (/ (atan -1.0) (PI)) 180.0))))

Derivation

1. Split input into 3 regimes

2. if B < -2.8999999999999999e-64

   1. Initial program 48.8%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   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 65.0%

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

   if -2.8999999999999999e-64 < B < 1.40000000000000012e-78

   1. Initial program 52.7%

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

   3. 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 27.0

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

   5. Applied rewrites 27.0%

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

   if 1.40000000000000012e-78 < B

   1. Initial program 49.2%

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

   3. Taylor expanded in B around inf

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

   5. Applied rewrites 52.9%

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

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -2.9 \cdot 10^{-64}:\\ \;\;\;\;\frac{\tan^{-1} 1}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;B \leq 1.4 \cdot 10^{-78}:\\ \;\;\;\;\frac{\tan^{-1} 0}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 29.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 1.4 \cdot 10^{-78}:\\ \;\;\;\;\frac{\tan^{-1} 0}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= B 1.4e-78)
   (* (/ (atan 0.0) (PI)) 180.0)
   (* (/ (atan -1.0) (PI)) 180.0)))

Derivation

1. Split input into 2 regimes

2. if B < 1.40000000000000012e-78

   1. Initial program 50.9%

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

   3. Taylor expanded in 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 16.3

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

   5. Applied rewrites 16.3%

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

   if 1.40000000000000012e-78 < B

   1. Initial program 49.2%

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

   3. Taylor expanded in B around inf

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

   5. Applied rewrites 52.9%

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

4. Final simplification 28.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.4 \cdot 10^{-78}:\\ \;\;\;\;\frac{\tan^{-1} 0}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 21.1% accurate, 3.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 50.3%

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

3. Taylor expanded in B around inf

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

5. Applied rewrites 21.4%

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

6. Final simplification 21.4%

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

Reproduce

herbie shell --seed 2024249 
(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))))