Falkner and Boettcher, Equation (22+)

Percentage Accurate: 98.5% → 100.0%

Time: 7.0s

Alternatives: 7

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

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

FPCore

(FPCore (v)
 :precision binary64
 (/
  (/ 1.3333333333333333 (* (fma (- v) v 1.0) (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
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. lift--.f64 N/A

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

   6. flip-- N/A

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

   7. associate-*l/ N/A

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

   8. lower-/.f64 N/A

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

4. Applied rewrites 100.0%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-/l* N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*l* N/A

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

   9. lift-PI.f64 N/A

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

6. Applied rewrites 100.0%

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

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (v)
 :precision binary64
 (/ 4.0 (* (PI) (* 3.0 (* (fma (- v) v 1.0) (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

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lift--.f64 N/A

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

   6. sub-neg N/A

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

   7. +-commutative N/A

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

   8. distribute-rgt-in N/A

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

   9. *-commutative N/A

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

   10. associate-*l* N/A

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

   11. *-lft-identity N/A

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

   12. *-commutative N/A

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

4. Applied rewrites 100.0%

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

   1. lift-fma.f64 N/A

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

   2. distribute-lft1-in N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. lift-PI.f64 N/A

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

   7. associate-*r* N/A

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

6. Applied rewrites 100.0%

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

Alternative 3: 99.5% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (v)
 :precision binary64
 (/
  (* (fma v v 1.0) (/ 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
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. lift--.f64 N/A

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

   6. flip-- N/A

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

   7. associate-*l/ N/A

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

   8. lower-/.f64 N/A

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

4. Applied rewrites 100.0%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-/l* N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*l* N/A

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

   9. lift-PI.f64 N/A

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

6. Applied rewrites 100.0%

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

7. Taylor expanded in v around 0

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

   1. *-commutative N/A

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

   2. associate-*l/ N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. associate-*r/ N/A

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

   6. distribute-lft1-in N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-fma.f64 N/A

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

   10. associate-*r/ N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f64 N/A

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

   13. lower-PI.f64 99.3

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

9. Applied rewrites 99.3%

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

Alternative 4: 99.0% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (v)
 :precision binary64
 (/ 4.0 (* (PI) (* 3.0 (* 1.0 (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

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lift--.f64 N/A

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

   6. sub-neg N/A

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

   7. +-commutative N/A

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

   8. distribute-rgt-in N/A

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

   9. *-commutative N/A

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

   10. associate-*l* N/A

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

   11. *-lft-identity N/A

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

   12. *-commutative N/A

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

4. Applied rewrites 100.0%

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

   1. lift-fma.f64 N/A

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

   2. distribute-lft1-in N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. lift-PI.f64 N/A

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

   7. associate-*r* N/A

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

6. Applied rewrites 100.0%

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

7. Taylor expanded in v around 0

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

9. Applied rewrites 98.9%

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

Alternative 5: 99.0% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (v)
 :precision binary64
 (/ 1.3333333333333333 (* (PI) (sqrt (fma -6.0 (* v v) 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 98.9%

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

5. Taylor expanded in v around 0

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

   1. lower-PI.f64 98.9

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

7. Applied rewrites 98.9%

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

Alternative 6: 99.0% accurate, 2.1× speedup

Math

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

FPCore

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

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 98.9%

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

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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-PI.f64 98.9

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

7. Applied rewrites 98.9%

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

Alternative 7: 97.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt{0.5}}{\mathsf{PI}\left(\right)} \cdot 1.3333333333333333 \end{array} \]

FPCore

(FPCore (v) :precision binary64 (* (/ (sqrt 0.5) (PI)) 1.3333333333333333))

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. Taylor expanded in v around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-PI.f64 97.3

      \[\leadsto \frac{\sqrt{0.5}}{\color{blue}{\mathsf{PI}\left(\right)}} \cdot 1.3333333333333333 \]

5. Applied rewrites 97.3%

   \[\leadsto \color{blue}{\frac{\sqrt{0.5}}{\mathsf{PI}\left(\right)} \cdot 1.3333333333333333} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024304 
(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)))))))