Falkner and Boettcher, Equation (22+)

Percentage Accurate: 98.5% → 100.0%

Time: 6.0s

Alternatives: 5

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

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

FPCore

(FPCore (v)
 :precision binary64
 (/ (/ 4.0 (* (fma -3.0 (* v v) 3.0) (PI))) (sqrt (fma (* v v) -6.0 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. 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. lower-/.f64 N/A

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

   5. lower-/.f64 100.0

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

   6. lift-sqrt.f64 N/A

      \[\leadsto \frac{\frac{4}{\mathsf{Rewrite=>}\left(lower-*.f64, \left(\mathsf{fma}\left(-3, v \cdot v, 3\right) \cdot \mathsf{PI}\left(\right)\right)\right)}}{\color{blue}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}} \]

   7. pow1/2 N/A

      \[\leadsto \frac{\frac{4}{\mathsf{Rewrite=>}\left(lower-*.f64, \left(\mathsf{fma}\left(-3, v \cdot v, 3\right) \cdot \mathsf{PI}\left(\right)\right)\right)}}{\color{blue}{{\left(2 - 6 \cdot \left(v \cdot v\right)\right)}^{\frac{1}{2}}}} \]

7. Applied rewrites 100.0%

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

Alternative 2: 99.5% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (v)
 :precision binary64
 (/
  (/ (fma (* v v) 1.3333333333333333 1.3333333333333333) (PI))
  (sqrt (fma (* v v) -6.0 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. 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. lower-/.f64 N/A

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

   5. lower-/.f64 100.0

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

   6. lift-sqrt.f64 N/A

      \[\leadsto \frac{\frac{4}{\mathsf{Rewrite=>}\left(lower-*.f64, \left(\mathsf{fma}\left(-3, v \cdot v, 3\right) \cdot \mathsf{PI}\left(\right)\right)\right)}}{\color{blue}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}} \]

   7. pow1/2 N/A

      \[\leadsto \frac{\frac{4}{\mathsf{Rewrite=>}\left(lower-*.f64, \left(\mathsf{fma}\left(-3, v \cdot v, 3\right) \cdot \mathsf{PI}\left(\right)\right)\right)}}{\color{blue}{{\left(2 - 6 \cdot \left(v \cdot v\right)\right)}^{\frac{1}{2}}}} \]

7. Applied rewrites 100.0%

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

8. Taylor expanded in v around 0

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

   1. associate-*r/ N/A

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

   2. associate-*r/ N/A

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

   3. metadata-eval N/A

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

   4. div-add-rev N/A

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

   5. lower-/.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-PI.f64 99.7

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

10. Applied rewrites 99.7%

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

Alternative 3: 99.0% accurate, 1.5× speedup

Math

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

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 4: 98.9% 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 5: 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 98.5%

   \[\frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
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 2024346 
(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)))))))