Falkner and Boettcher, Equation (22+)

Percentage Accurate: 98.5% → 100.0%

Time: 7.0s

Alternatives: 8

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

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

FPCore

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

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

4. Applied rewrites 98.5%

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

6. Applied rewrites 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (v)
 :precision binary64
 (/
  1.3333333333333333
  (* (fma (* (- v) v) (PI) (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. 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

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

4. Applied rewrites 98.5%

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

6. Applied rewrites 100.0%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. div-inv N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-/r* N/A

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

   8. metadata-eval N/A

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

   9. frac-times N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f64 N/A

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

   12. lower-*.f64 100.0

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

8. Applied rewrites 100.0%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-neg N/A

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

   4. +-commutative N/A

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

   5. distribute-rgt-in N/A

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

   6. *-lft-identity N/A

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

   7. lower-fma.f64 N/A

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

   8. lift-*.f64 N/A

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

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

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

   10. lower-*.f64 N/A

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

   11. lower-neg.f64 100.0

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

10. Applied rewrites 100.0%

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

11. Final simplification 100.0%

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

Alternative 3: 100.0% accurate, 1.1× speedup

Math

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

FPCore

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

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. 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

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

4. Applied rewrites 98.5%

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

6. Applied rewrites 100.0%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. div-inv N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-/r* N/A

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

   8. metadata-eval N/A

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

   9. frac-times N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f64 N/A

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

   12. lower-*.f64 100.0

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

8. Applied rewrites 100.0%

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

Alternative 4: 98.9% accurate, 1.2× speedup

Math

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

FPCore

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

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-PI.f64 97.2

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

5. Applied rewrites 97.2%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lift--.f64 N/A

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

   5. lift-*.f64 N/A

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

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

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

   7. metadata-eval N/A

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

   8. *-commutative N/A

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

   9. +-commutative N/A

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

   10. lift-fma.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 98.7

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

7. Applied rewrites 98.7%

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

Alternative 5: 98.9% accurate, 1.3× speedup

Math

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

FPCore

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

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

4. Applied rewrites 98.5%

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

6. Applied rewrites 100.0%

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

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 98.7

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

9. Applied rewrites 98.7%

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

10. Final simplification 98.7%

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

Alternative 6: 98.9% 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 98.6%

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

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 98.7

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

7. Applied rewrites 98.7%

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

Alternative 7: 98.8% 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 98.6%

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

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 98.6

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

7. Applied rewrites 98.6%

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

Alternative 8: 97.3% 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 97.1

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

5. Applied rewrites 97.1%

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

Reproduce

herbie shell --seed 2024270 
(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)))))))