Falkner and Boettcher, Equation (22+)

Percentage Accurate: 98.5% → 100.0%

Time: 9.6s

Alternatives: 4

Speedup: 2.1×

Specification

Math

\[\begin{array}{l} \\ \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (/ 4.0 (* (* (* 3.0 (PI)) (- 1.0 (* v v))) (sqrt (- 2.0 (* 6.0 (* v v)))))))

Initial Program: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (/ 4.0 (* (* (* 3.0 (PI)) (- 1.0 (* v v))) (sqrt (- 2.0 (* 6.0 (* v v)))))))

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1.3333333333333333}{\mathsf{PI}\left(\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)}} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (/
  (/ 1.3333333333333333 (* (PI) (- 1.0 (* v v))))
  (sqrt (fma v (* v -6.0) 2.0))))

Derivation

1. Initial program 98.4%

   \[\frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \color{blue}{\left(v \cdot v\right)}}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - \color{blue}{6 \cdot \left(v \cdot v\right)}}} \]

   3. sub-neg N/A

      \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\color{blue}{2 + \left(\mathsf{neg}\left(6 \cdot \left(v \cdot v\right)\right)\right)}}} \]

   4. +-commutative N/A

      \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(6 \cdot \left(v \cdot v\right)\right)\right) + 2}}} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\left(\mathsf{neg}\left(\color{blue}{6 \cdot \left(v \cdot v\right)}\right)\right) + 2}} \]

   6. *-commutative N/A

      \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\left(\mathsf{neg}\left(\color{blue}{\left(v \cdot v\right) \cdot 6}\right)\right) + 2}} \]

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

      \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\color{blue}{\left(v \cdot v\right) \cdot \left(\mathsf{neg}\left(6\right)\right)} + 2}} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(v \cdot v, \mathsf{neg}\left(6\right), 2\right)}}} \]

   9. metadata-eval 98.4

      \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\mathsf{fma}\left(v \cdot v, \color{blue}{-6}, 2\right)}} \]

4. Applied rewrites 98.4%

   \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}} \]
5. Step-by-step derivation

   1. lift-PI.f64 N/A

      \[\leadsto \frac{4}{\left(\left(3 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\left(v \cdot v\right) \cdot -6 + 2}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{4}{\left(\color{blue}{\left(3 \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\left(v \cdot v\right) \cdot -6 + 2}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - \color{blue}{v \cdot v}\right)\right) \cdot \sqrt{\left(v \cdot v\right) \cdot -6 + 2}} \]

   4. lift--.f64 N/A

      \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left(1 - v \cdot v\right)}\right) \cdot \sqrt{\left(v \cdot v\right) \cdot -6 + 2}} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{4}{\color{blue}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right)} \cdot \sqrt{\left(v \cdot v\right) \cdot -6 + 2}} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\color{blue}{\left(v \cdot v\right)} \cdot -6 + 2}} \]

   7. lift-fma.f64 N/A

      \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}} \]

   8. lift-sqrt.f64 N/A

      \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \color{blue}{\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}} \]

   9. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{4}{\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}} \]

   10. lift-fma.f64 N/A

       \[\leadsto \frac{\frac{4}{\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{\color{blue}{\left(v \cdot v\right) \cdot -6 + 2}}} \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{\frac{4}{\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{\color{blue}{\left(v \cdot v\right)} \cdot -6 + 2}} \]

   12. associate-*r* N/A

       \[\leadsto \frac{\frac{4}{\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{\color{blue}{v \cdot \left(v \cdot -6\right)} + 2}} \]

   13. lift-*.f64 N/A

       \[\leadsto \frac{\frac{4}{\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{v \cdot \color{blue}{\left(v \cdot -6\right)} + 2}} \]

   14. lift-fma.f64 N/A

       \[\leadsto \frac{\frac{4}{\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{\color{blue}{\mathsf{fma}\left(v, v \cdot -6, 2\right)}}} \]

6. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{\frac{1.3333333333333333}{\mathsf{PI}\left(\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)}}} \]
7. Add Preprocessing

Alternative 2: 100.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{1.3333333333333333}{\mathsf{PI}\left(\right) \cdot \left(\left(1 - v \cdot v\right) \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}\right)} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (/
  1.3333333333333333
  (* (PI) (* (- 1.0 (* v v)) (sqrt (fma (* v v) -6.0 2.0))))))

Derivation

1. Initial program 98.4%

   \[\frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \color{blue}{\left(v \cdot v\right)}}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - \color{blue}{6 \cdot \left(v \cdot v\right)}}} \]

   3. sub-neg N/A

      \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\color{blue}{2 + \left(\mathsf{neg}\left(6 \cdot \left(v \cdot v\right)\right)\right)}}} \]

   4. +-commutative N/A

      \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(6 \cdot \left(v \cdot v\right)\right)\right) + 2}}} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\left(\mathsf{neg}\left(\color{blue}{6 \cdot \left(v \cdot v\right)}\right)\right) + 2}} \]

   6. *-commutative N/A

      \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\left(\mathsf{neg}\left(\color{blue}{\left(v \cdot v\right) \cdot 6}\right)\right) + 2}} \]

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

      \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\color{blue}{\left(v \cdot v\right) \cdot \left(\mathsf{neg}\left(6\right)\right)} + 2}} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(v \cdot v, \mathsf{neg}\left(6\right), 2\right)}}} \]

   9. metadata-eval 98.4

      \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\mathsf{fma}\left(v \cdot v, \color{blue}{-6}, 2\right)}} \]

4. Applied rewrites 98.4%

   \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}} \]
5. Step-by-step derivation

   1. lift-PI.f64 N/A

      \[\leadsto \frac{4}{\left(\left(3 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\left(v \cdot v\right) \cdot -6 + 2}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{4}{\left(\color{blue}{\left(3 \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\left(v \cdot v\right) \cdot -6 + 2}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - \color{blue}{v \cdot v}\right)\right) \cdot \sqrt{\left(v \cdot v\right) \cdot -6 + 2}} \]

   4. lift--.f64 N/A

      \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left(1 - v \cdot v\right)}\right) \cdot \sqrt{\left(v \cdot v\right) \cdot -6 + 2}} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{4}{\color{blue}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right)} \cdot \sqrt{\left(v \cdot v\right) \cdot -6 + 2}} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\color{blue}{\left(v \cdot v\right)} \cdot -6 + 2}} \]

   7. lift-fma.f64 N/A

      \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}} \]

   8. lift-sqrt.f64 N/A

      \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \color{blue}{\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}} \]

   9. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{4}{\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}} \]

   10. lift-fma.f64 N/A

       \[\leadsto \frac{\frac{4}{\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{\color{blue}{\left(v \cdot v\right) \cdot -6 + 2}}} \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{\frac{4}{\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{\color{blue}{\left(v \cdot v\right)} \cdot -6 + 2}} \]

   12. associate-*r* N/A

       \[\leadsto \frac{\frac{4}{\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{\color{blue}{v \cdot \left(v \cdot -6\right)} + 2}} \]

   13. lift-*.f64 N/A

       \[\leadsto \frac{\frac{4}{\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{v \cdot \color{blue}{\left(v \cdot -6\right)} + 2}} \]

   14. lift-fma.f64 N/A

       \[\leadsto \frac{\frac{4}{\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{\color{blue}{\mathsf{fma}\left(v, v \cdot -6, 2\right)}}} \]

6. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{\frac{1.3333333333333333}{\mathsf{PI}\left(\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)}}} \]
7. Step-by-step derivation

   1. lift-PI.f64 N/A

      \[\leadsto \frac{\frac{\frac{4}{3}}{\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(1 - v \cdot v\right)}}{\sqrt{v \cdot \left(v \cdot -6\right) + 2}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\frac{4}{3}}{\mathsf{PI}\left(\right) \cdot \left(1 - \color{blue}{v \cdot v}\right)}}{\sqrt{v \cdot \left(v \cdot -6\right) + 2}} \]

   3. lift--.f64 N/A

      \[\leadsto \frac{\frac{\frac{4}{3}}{\mathsf{PI}\left(\right) \cdot \color{blue}{\left(1 - v \cdot v\right)}}}{\sqrt{v \cdot \left(v \cdot -6\right) + 2}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\frac{4}{3}}{\color{blue}{\mathsf{PI}\left(\right) \cdot \left(1 - v \cdot v\right)}}}{\sqrt{v \cdot \left(v \cdot -6\right) + 2}} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\frac{4}{3}}{\color{blue}{\mathsf{PI}\left(\right) \cdot \left(1 - v \cdot v\right)}}}{\sqrt{v \cdot \left(v \cdot -6\right) + 2}} \]

   6. associate-/r* N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{\frac{4}{3}}{\mathsf{PI}\left(\right)}}{1 - v \cdot v}}}{\sqrt{v \cdot \left(v \cdot -6\right) + 2}} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\frac{\frac{4}{3}}{\mathsf{PI}\left(\right)}}{1 - v \cdot v}}{\sqrt{v \cdot \color{blue}{\left(v \cdot -6\right)} + 2}} \]

   8. lift-fma.f64 N/A

      \[\leadsto \frac{\frac{\frac{\frac{4}{3}}{\mathsf{PI}\left(\right)}}{1 - v \cdot v}}{\sqrt{\color{blue}{\mathsf{fma}\left(v, v \cdot -6, 2\right)}}} \]

   9. lift-sqrt.f64 N/A

      \[\leadsto \frac{\frac{\frac{\frac{4}{3}}{\mathsf{PI}\left(\right)}}{1 - v \cdot v}}{\color{blue}{\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)}}} \]

   10. associate-/l/ N/A

       \[\leadsto \color{blue}{\frac{\frac{\frac{4}{3}}{\mathsf{PI}\left(\right)}}{\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)} \cdot \left(1 - v \cdot v\right)}} \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{\frac{\frac{4}{3}}{\mathsf{PI}\left(\right)}}{\color{blue}{\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)} \cdot \left(1 - v \cdot v\right)}} \]

   12. associate-/r* N/A

       \[\leadsto \color{blue}{\frac{\frac{4}{3}}{\mathsf{PI}\left(\right) \cdot \left(\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)} \cdot \left(1 - v \cdot v\right)\right)}} \]

   13. lift-*.f64 N/A

       \[\leadsto \frac{\frac{4}{3}}{\color{blue}{\mathsf{PI}\left(\right) \cdot \left(\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)} \cdot \left(1 - v \cdot v\right)\right)}} \]

   14. lower-/.f64 100.0

       \[\leadsto \color{blue}{\frac{1.3333333333333333}{\mathsf{PI}\left(\right) \cdot \left(\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)} \cdot \left(1 - v \cdot v\right)\right)}} \]

8. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{1.3333333333333333}{\mathsf{fma}\left(\mathsf{PI}\left(\right), -v \cdot v, \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)}}} \]
9. Step-by-step derivation

   1. lift-PI.f64 N/A

      \[\leadsto \frac{\frac{4}{3}}{\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(\mathsf{neg}\left(v \cdot v\right)\right) + \mathsf{PI}\left(\right)\right) \cdot \sqrt{v \cdot \left(v \cdot -6\right) + 2}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\frac{4}{3}}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{neg}\left(\color{blue}{v \cdot v}\right)\right) + \mathsf{PI}\left(\right)\right) \cdot \sqrt{v \cdot \left(v \cdot -6\right) + 2}} \]

   3. lift-neg.f64 N/A

      \[\leadsto \frac{\frac{4}{3}}{\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{neg}\left(v \cdot v\right)\right)} + \mathsf{PI}\left(\right)\right) \cdot \sqrt{v \cdot \left(v \cdot -6\right) + 2}} \]

   4. lift-PI.f64 N/A

      \[\leadsto \frac{\frac{4}{3}}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{neg}\left(v \cdot v\right)\right) + \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{v \cdot \left(v \cdot -6\right) + 2}} \]

   5. lift-fma.f64 N/A

      \[\leadsto \frac{\frac{4}{3}}{\color{blue}{\mathsf{fma}\left(\mathsf{PI}\left(\right), \mathsf{neg}\left(v \cdot v\right), \mathsf{PI}\left(\right)\right)} \cdot \sqrt{v \cdot \left(v \cdot -6\right) + 2}} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\frac{4}{3}}{\mathsf{fma}\left(\mathsf{PI}\left(\right), \mathsf{neg}\left(v \cdot v\right), \mathsf{PI}\left(\right)\right) \cdot \sqrt{v \cdot \color{blue}{\left(v \cdot -6\right)} + 2}} \]

   7. lift-fma.f64 N/A

      \[\leadsto \frac{\frac{4}{3}}{\mathsf{fma}\left(\mathsf{PI}\left(\right), \mathsf{neg}\left(v \cdot v\right), \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(v, v \cdot -6, 2\right)}}} \]

   8. lift-sqrt.f64 N/A

      \[\leadsto \frac{\frac{4}{3}}{\mathsf{fma}\left(\mathsf{PI}\left(\right), \mathsf{neg}\left(v \cdot v\right), \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)}}} \]

   9. *-commutative N/A

      \[\leadsto \frac{\frac{4}{3}}{\color{blue}{\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)} \cdot \mathsf{fma}\left(\mathsf{PI}\left(\right), \mathsf{neg}\left(v \cdot v\right), \mathsf{PI}\left(\right)\right)}} \]

   10. lift-fma.f64 N/A

       \[\leadsto \frac{\frac{4}{3}}{\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{neg}\left(v \cdot v\right)\right) + \mathsf{PI}\left(\right)\right)}} \]

   11. *-commutative N/A

       \[\leadsto \frac{\frac{4}{3}}{\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)} \cdot \left(\color{blue}{\left(\mathsf{neg}\left(v \cdot v\right)\right) \cdot \mathsf{PI}\left(\right)} + \mathsf{PI}\left(\right)\right)} \]

   12. distribute-lft1-in N/A

       \[\leadsto \frac{\frac{4}{3}}{\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)} \cdot \color{blue}{\left(\left(\left(\mathsf{neg}\left(v \cdot v\right)\right) + 1\right) \cdot \mathsf{PI}\left(\right)\right)}} \]

   13. associate-*r* N/A

       \[\leadsto \frac{\frac{4}{3}}{\color{blue}{\left(\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)} \cdot \left(\left(\mathsf{neg}\left(v \cdot v\right)\right) + 1\right)\right) \cdot \mathsf{PI}\left(\right)}} \]

   14. lower-*.f64 N/A

       \[\leadsto \frac{\frac{4}{3}}{\color{blue}{\left(\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)} \cdot \left(\left(\mathsf{neg}\left(v \cdot v\right)\right) + 1\right)\right) \cdot \mathsf{PI}\left(\right)}} \]

10. Applied rewrites 100.0%

    \[\leadsto \frac{1.3333333333333333}{\color{blue}{\left(\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)} \cdot \left(1 - v \cdot v\right)\right) \cdot \mathsf{PI}\left(\right)}} \]

11. Final simplification 100.0%

    \[\leadsto \frac{1.3333333333333333}{\mathsf{PI}\left(\right) \cdot \left(\left(1 - v \cdot v\right) \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}\right)} \]
12. Add Preprocessing

Alternative 3: 99.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{1.3333333333333333}{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)}} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (/ 1.3333333333333333 (* (PI) (sqrt (fma v (* v -6.0) 2.0)))))

Derivation

1. Initial program 98.4%

   \[\frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
2. Add Preprocessing

4. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\frac{1.3333333333333333}{\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(v, v, 1\right)\right)}} \]

5. Taylor expanded in v around 0

   \[\leadsto \frac{\frac{4}{3}}{\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)} \cdot \color{blue}{\mathsf{PI}\left(\right)}} \]
6. Step-by-step derivation

   1. lower-PI.f64 99.4

      \[\leadsto \frac{1.3333333333333333}{\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)} \cdot \color{blue}{\mathsf{PI}\left(\right)}} \]

7. Applied rewrites 99.4%

   \[\leadsto \frac{1.3333333333333333}{\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)} \cdot \color{blue}{\mathsf{PI}\left(\right)}} \]

8. Final simplification 99.4%

   \[\leadsto \frac{1.3333333333333333}{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)}} \]
9. Add Preprocessing

Alternative 4: 98.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \frac{1.3333333333333333}{\mathsf{PI}\left(\right) \cdot \sqrt{2}} \end{array} \]

FPCore

(FPCore (v) :precision binary64 (/ 1.3333333333333333 (* (PI) (sqrt 2.0))))

Derivation

1. Initial program 98.4%

   \[\frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
2. Add Preprocessing

4. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\frac{1.3333333333333333}{\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(v, v, 1\right)\right)}} \]

5. Taylor expanded in v around 0

   \[\leadsto \frac{\frac{4}{3}}{\color{blue}{\mathsf{PI}\left(\right) \cdot \sqrt{2}}} \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{\frac{4}{3}}{\color{blue}{\mathsf{PI}\left(\right) \cdot \sqrt{2}}} \]

   2. lower-PI.f64 N/A

      \[\leadsto \frac{\frac{4}{3}}{\color{blue}{\mathsf{PI}\left(\right)} \cdot \sqrt{2}} \]

   3. lower-sqrt.f64 99.4

      \[\leadsto \frac{1.3333333333333333}{\mathsf{PI}\left(\right) \cdot \color{blue}{\sqrt{2}}} \]

7. Applied rewrites 99.4%

   \[\leadsto \frac{1.3333333333333333}{\color{blue}{\mathsf{PI}\left(\right) \cdot \sqrt{2}}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024216 
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4.0 (* (* (* 3.0 (PI)) (- 1.0 (* v v))) (sqrt (- 2.0 (* 6.0 (* v v)))))))