Falkner and Boettcher, Equation (22+)

Percentage Accurate: 98.5% → 100.0%

Time: 7.3s

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, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}\\ \frac{4}{\mathsf{fma}\left(3 \cdot t\_0, \mathsf{PI}\left(\right), \left(\left(-v\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 3\right) \cdot v\right)\right) \cdot t\_0\right)} \end{array} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (let* ((t_0 (sqrt (fma -6.0 (* v v) 2.0))))
   (/ 4.0 (fma (* 3.0 t_0) (PI) (* (* (- v) (* (* (PI) 3.0) v)) t_0)))))

Derivation

1. Initial program 98.5%

   \[\frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
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. lift--.f64 N/A

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

   5. sub-neg N/A

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

   6. distribute-lft-in N/A

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

   7. *-rgt-identity N/A

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

   8. distribute-rgt-in N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*r* N/A

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

   12. lower-fma.f64 N/A

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

4. Applied rewrites 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.5%

   \[\frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
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. lift--.f64 N/A

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

   5. sub-neg N/A

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

   6. distribute-lft-in N/A

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

   7. *-rgt-identity N/A

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

   8. distribute-rgt-in N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*r* N/A

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

   12. lower-fma.f64 N/A

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

4. Applied rewrites 100.0%

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

6. Applied rewrites 100.0%

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

   1. remove-double-neg N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

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

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

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

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

   7. lift-fma.f64 N/A

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

   8. distribute-neg-in N/A

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

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

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

   10. lift-neg.f64 N/A

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

   11. sqr-neg N/A

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

   12. lift-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. metadata-eval N/A

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

   15. lift-fma.f64 N/A

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

   16. lower-*.f64 N/A

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

8. Applied rewrites 100.0%

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

9. Final simplification 100.0%

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

Alternative 3: 100.0% accurate, 1.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.5%

   \[\frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
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. lift--.f64 N/A

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

   5. sub-neg N/A

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

   6. distribute-lft-in N/A

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

   7. *-rgt-identity N/A

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

   8. distribute-rgt-in N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*r* N/A

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

   12. lower-fma.f64 N/A

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

4. Applied rewrites 100.0%

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

6. Applied rewrites 100.0%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. lower-/.f64 N/A

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

   4. metadata-eval N/A

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

   5. lift-*.f64 N/A

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

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

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

8. Applied rewrites 100.0%

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

Alternative 4: 99.1% accurate, 1.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.5%

   \[\frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
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. lift--.f64 N/A

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

   5. sub-neg N/A

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

   6. distribute-lft-in N/A

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

   7. *-rgt-identity N/A

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

   8. distribute-rgt-in N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*r* N/A

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

   12. lower-fma.f64 N/A

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

4. Applied rewrites 100.0%

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

6. Applied rewrites 100.0%

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

7. Taylor expanded in v around 0

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

   1. lower-PI.f64 99.0

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

9. Applied rewrites 99.0%

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

10. Final simplification 99.0%

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

Alternative 5: 99.1% 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 98.5%

   \[\frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
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 99.0

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

7. Applied rewrites 99.0%

   \[\leadsto \frac{1.3333333333333333}{\color{blue}{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}} \]
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 98.5%

   \[\frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
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: 97.5% 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 98.5%

   \[\frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
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 97.4

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

5. Applied rewrites 97.4%

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

Reproduce

herbie shell --seed 2024250 
(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)))))))