Falkner and Boettcher, Equation (22+)

Percentage Accurate: 98.5% → 100.0%

Time: 7.4s

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

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

FPCore

(FPCore (v)
 :precision binary64
 (let* ((t_0 (sqrt (fma -6.0 (* v v) 2.0))))
   (/ 4.0 (fma (* t_0 3.0) (PI) (* (* (* (* v v) (PI)) -3.0) t_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. 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. lift-*.f64 N/A

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

   6. fp-cancel-sub-sign-inv N/A

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

   7. distribute-lft-in N/A

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

   8. distribute-rgt-in N/A

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

   9. *-rgt-identity N/A

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

   10. *-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. associate-*r* N/A

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

   13. lower-fma.f64 N/A

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

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. Taylor expanded in v around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-PI.f64 100.0

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

7. Applied rewrites 100.0%

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

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

3. Taylor expanded in v around 0

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

   1. associate-*r* N/A

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

   2. distribute-rgt-out N/A

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

   3. lower-*.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 98.5

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

5. Applied rewrites 98.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lift-sqrt.f64 N/A

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

   5. pow1/2 N/A

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

   6. lift--.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. fp-cancel-sub-sign-inv N/A

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

   9. metadata-eval N/A

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

   10. *-commutative N/A

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

   11. +-commutative N/A

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

   12. lift-fma.f64 N/A

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

   13. pow1/2 N/A

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

   14. lift-sqrt.f64 N/A

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

7. Applied rewrites 100.0%

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

Alternative 3: 98.5% accurate, 1.1× speedup

Math

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

FPCore

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

3. Taylor expanded in v around 0

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

   1. associate-*r* N/A

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

   2. distribute-rgt-out N/A

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

   3. lower-*.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 98.5

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

5. Applied rewrites 98.5%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-sqrt.f64 N/A

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

   4. pow1/2 N/A

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

   5. lift--.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. fp-cancel-sub-sign-inv N/A

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

   8. metadata-eval N/A

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

   9. *-commutative N/A

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

   10. +-commutative N/A

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

   11. lift-fma.f64 N/A

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

   12. pow1/2 N/A

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

   13. lift-sqrt.f64 N/A

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

   14. lower-*.f64 98.5

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

7. Applied rewrites 98.5%

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

Alternative 4: 99.1% accurate, 1.2× speedup

Math

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

FPCore

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

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. Taylor expanded in v around 0

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

   1. lower-*.f64 N/A

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

   2. lower-PI.f64 96.7

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

5. Applied rewrites 96.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift--.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. fp-cancel-sub-sign-inv N/A

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

   7. metadata-eval N/A

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

   8. +-commutative N/A

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

   9. lift-fma.f64 N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

7. Applied rewrites 98.2%

   \[\leadsto \color{blue}{\frac{\frac{4}{\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}}{\mathsf{PI}\left(\right) \cdot 3}} \]
8. 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 98.4%

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

4. Applied rewrites 98.2%

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

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 98.2

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

7. Applied rewrites 98.2%

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

Alternative 6: 99.1% 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 98.2%

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

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 98.1

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

7. Applied rewrites 98.1%

   \[\leadsto \frac{1.3333333333333333}{\color{blue}{\sqrt{2} \cdot \mathsf{PI}\left(\right)}} \]
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 98.4%

   \[\frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
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 96.6

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

5. Applied rewrites 96.6%

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

Reproduce

herbie shell --seed 2024361 
(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)))))))