Falkner and Boettcher, Equation (20:1,3)

Percentage Accurate: 99.3% → 98.7%

Time: 3.9s

Alternatives: 5

Speedup: N/A×

Specification

Math

\[\begin{array}{l} \\ \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \end{array} \]

FPCore

(FPCore (v t)
 :precision binary64
 (/
  (- 1.0 (* 5.0 (* v v)))
  (* (* (* (PI) t) (sqrt (* 2.0 (- 1.0 (* 3.0 (* v v)))))) (- 1.0 (* v v)))))

Initial Program: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \end{array} \]

FPCore

(FPCore (v t)
 :precision binary64
 (/
  (- 1.0 (* 5.0 (* v v)))
  (* (* (* (PI) t) (sqrt (* 2.0 (- 1.0 (* 3.0 (* v v)))))) (- 1.0 (* v v)))))

Alternative 1: 98.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right)}^{-1}}{t} \end{array} \]

FPCore

(FPCore (v t) :precision binary64 (/ (pow (* (sqrt 2.0) (PI)) -1.0) t))

Derivation

1. Initial program 99.3%

   \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
2. Add Preprocessing

3. Taylor expanded in v around 0

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

   1. inv-pow N/A

      \[\leadsto {\left(t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)\right)}^{\color{blue}{-1}} \]

   2. lower-pow.f64 N/A

      \[\leadsto {\left(t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)\right)}^{\color{blue}{-1}} \]

   3. *-commutative N/A

      \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right) \cdot t\right)}^{-1} \]

   4. lower-*.f64 N/A

      \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right) \cdot t\right)}^{-1} \]

   5. *-commutative N/A

      \[\leadsto {\left(\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot t\right)}^{-1} \]

   6. lower-*.f64 N/A

      \[\leadsto {\left(\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot t\right)}^{-1} \]

   7. lower-sqrt.f64 N/A

      \[\leadsto {\left(\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot t\right)}^{-1} \]

   8. lift-PI.f64 99.4

      \[\leadsto {\left(\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot t\right)}^{-1} \]

5. Applied rewrites 99.4%

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

   1. lift-pow.f64 N/A

      \[\leadsto {\left(\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot t\right)}^{\color{blue}{-1}} \]

   2. lift-*.f64 N/A

      \[\leadsto {\left(\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot t\right)}^{-1} \]

   3. unpow-prod-down N/A

      \[\leadsto {\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right)}^{-1} \cdot \color{blue}{{t}^{-1}} \]

   4. lower-*.f64 N/A

      \[\leadsto {\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right)}^{-1} \cdot \color{blue}{{t}^{-1}} \]

   5. lift-PI.f64 N/A

      \[\leadsto {\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right)}^{-1} \cdot {t}^{-1} \]

   6. lift-*.f64 N/A

      \[\leadsto {\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right)}^{-1} \cdot {t}^{-1} \]

   7. lift-sqrt.f64 N/A

      \[\leadsto {\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right)}^{-1} \cdot {t}^{-1} \]

   8. *-commutative N/A

      \[\leadsto {\left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)}^{-1} \cdot {t}^{-1} \]

   9. lower-pow.f64 N/A

      \[\leadsto {\left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)}^{-1} \cdot {\color{blue}{t}}^{-1} \]

   10. *-commutative N/A

       \[\leadsto {\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right)}^{-1} \cdot {t}^{-1} \]

   11. lift-sqrt.f64 N/A

       \[\leadsto {\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right)}^{-1} \cdot {t}^{-1} \]

   12. lift-*.f64 N/A

       \[\leadsto {\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right)}^{-1} \cdot {t}^{-1} \]

   13. lift-PI.f64 N/A

       \[\leadsto {\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right)}^{-1} \cdot {t}^{-1} \]

   14. lower-pow.f64 99.5

       \[\leadsto {\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right)}^{-1} \cdot {t}^{\color{blue}{-1}} \]

7. Applied rewrites 99.5%

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

   1. lift-pow.f64 N/A

      \[\leadsto {\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right)}^{-1} \cdot {t}^{\color{blue}{-1}} \]

   2. inv-pow N/A

      \[\leadsto {\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right)}^{-1} \cdot \frac{1}{\color{blue}{t}} \]

   3. lower-/.f64 99.5

      \[\leadsto {\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right)}^{-1} \cdot \frac{1}{\color{blue}{t}} \]

9. Applied rewrites 99.5%

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

    1. lift-*.f64 N/A

       \[\leadsto {\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right)}^{-1} \cdot \color{blue}{\frac{1}{t}} \]

    2. lift-/.f64 N/A

       \[\leadsto {\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right)}^{-1} \cdot \frac{1}{\color{blue}{t}} \]

    3. associate-*r/ N/A

       \[\leadsto \frac{{\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right)}^{-1} \cdot 1}{\color{blue}{t}} \]

    4. lower-/.f64 N/A

       \[\leadsto \frac{{\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right)}^{-1} \cdot 1}{\color{blue}{t}} \]

    5. lower-*.f64 99.8

       \[\leadsto \frac{{\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right)}^{-1} \cdot 1}{t} \]

11. Applied rewrites 99.8%

    \[\leadsto \frac{{\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right)}^{-1} \cdot 1}{\color{blue}{t}} \]

12. Final simplification 99.8%

    \[\leadsto \frac{{\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right)}^{-1}}{t} \]
13. Add Preprocessing

Alternative 2: 98.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\sqrt{2} \cdot \mathsf{PI}\left(\right)} \cdot \frac{1}{t} \end{array} \]

FPCore

(FPCore (v t) :precision binary64 (* (/ 1.0 (* (sqrt 2.0) (PI))) (/ 1.0 t)))

Derivation

1. Initial program 99.3%

   \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
2. Add Preprocessing

3. Taylor expanded in v around 0

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

   1. inv-pow N/A

      \[\leadsto {\left(t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)\right)}^{\color{blue}{-1}} \]

   2. lower-pow.f64 N/A

      \[\leadsto {\left(t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)\right)}^{\color{blue}{-1}} \]

   3. *-commutative N/A

      \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right) \cdot t\right)}^{-1} \]

   4. lower-*.f64 N/A

      \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right) \cdot t\right)}^{-1} \]

   5. *-commutative N/A

      \[\leadsto {\left(\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot t\right)}^{-1} \]

   6. lower-*.f64 N/A

      \[\leadsto {\left(\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot t\right)}^{-1} \]

   7. lower-sqrt.f64 N/A

      \[\leadsto {\left(\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot t\right)}^{-1} \]

   8. lift-PI.f64 99.4

      \[\leadsto {\left(\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot t\right)}^{-1} \]

5. Applied rewrites 99.4%

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

   1. lift-pow.f64 N/A

      \[\leadsto {\left(\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot t\right)}^{\color{blue}{-1}} \]

   2. lift-*.f64 N/A

      \[\leadsto {\left(\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot t\right)}^{-1} \]

   3. unpow-prod-down N/A

      \[\leadsto {\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right)}^{-1} \cdot \color{blue}{{t}^{-1}} \]

   4. lower-*.f64 N/A

      \[\leadsto {\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right)}^{-1} \cdot \color{blue}{{t}^{-1}} \]

   5. lift-PI.f64 N/A

      \[\leadsto {\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right)}^{-1} \cdot {t}^{-1} \]

   6. lift-*.f64 N/A

      \[\leadsto {\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right)}^{-1} \cdot {t}^{-1} \]

   7. lift-sqrt.f64 N/A

      \[\leadsto {\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right)}^{-1} \cdot {t}^{-1} \]

   8. *-commutative N/A

      \[\leadsto {\left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)}^{-1} \cdot {t}^{-1} \]

   9. lower-pow.f64 N/A

      \[\leadsto {\left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)}^{-1} \cdot {\color{blue}{t}}^{-1} \]

   10. *-commutative N/A

       \[\leadsto {\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right)}^{-1} \cdot {t}^{-1} \]

   11. lift-sqrt.f64 N/A

       \[\leadsto {\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right)}^{-1} \cdot {t}^{-1} \]

   12. lift-*.f64 N/A

       \[\leadsto {\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right)}^{-1} \cdot {t}^{-1} \]

   13. lift-PI.f64 N/A

       \[\leadsto {\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right)}^{-1} \cdot {t}^{-1} \]

   14. lower-pow.f64 99.5

       \[\leadsto {\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right)}^{-1} \cdot {t}^{\color{blue}{-1}} \]

7. Applied rewrites 99.5%

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

   1. lift-pow.f64 N/A

      \[\leadsto {\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right)}^{-1} \cdot {t}^{\color{blue}{-1}} \]

   2. inv-pow N/A

      \[\leadsto {\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right)}^{-1} \cdot \frac{1}{\color{blue}{t}} \]

   3. lower-/.f64 99.5

      \[\leadsto {\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right)}^{-1} \cdot \frac{1}{\color{blue}{t}} \]

9. Applied rewrites 99.5%

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

    1. lift-pow.f64 N/A

       \[\leadsto {\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right)}^{-1} \cdot \frac{\color{blue}{1}}{t} \]

    2. unpow-1 N/A

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

    3. lift-PI.f64 N/A

       \[\leadsto \frac{1}{\sqrt{2} \cdot \mathsf{PI}\left(\right)} \cdot \frac{1}{t} \]

    4. lift-*.f64 N/A

       \[\leadsto \frac{1}{\sqrt{2} \cdot \mathsf{PI}\left(\right)} \cdot \frac{1}{t} \]

    5. lift-sqrt.f64 N/A

       \[\leadsto \frac{1}{\sqrt{2} \cdot \mathsf{PI}\left(\right)} \cdot \frac{1}{t} \]

    6. *-commutative N/A

       \[\leadsto \frac{1}{\mathsf{PI}\left(\right) \cdot \sqrt{2}} \cdot \frac{1}{t} \]

    7. lower-/.f64 N/A

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

    8. *-commutative N/A

       \[\leadsto \frac{1}{\sqrt{2} \cdot \mathsf{PI}\left(\right)} \cdot \frac{1}{t} \]

    9. lift-sqrt.f64 N/A

       \[\leadsto \frac{1}{\sqrt{2} \cdot \mathsf{PI}\left(\right)} \cdot \frac{1}{t} \]

    10. lift-*.f64 N/A

        \[\leadsto \frac{1}{\sqrt{2} \cdot \mathsf{PI}\left(\right)} \cdot \frac{1}{t} \]

    11. lift-PI.f64 99.5

        \[\leadsto \frac{1}{\sqrt{2} \cdot \mathsf{PI}\left(\right)} \cdot \frac{1}{t} \]

11. Applied rewrites 99.5%

    \[\leadsto \frac{1}{\sqrt{2} \cdot \mathsf{PI}\left(\right)} \cdot \frac{\color{blue}{1}}{t} \]
12. Add Preprocessing

Alternative 3: 98.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{t}}{\sqrt{2} \cdot \mathsf{PI}\left(\right)} \end{array} \]

FPCore

(FPCore (v t) :precision binary64 (/ (/ 1.0 t) (* (sqrt 2.0) (PI))))

Derivation

1. Initial program 99.3%

   \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
2. Add Preprocessing

3. Taylor expanded in v around 0

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

   1. inv-pow N/A

      \[\leadsto {\left(t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)\right)}^{\color{blue}{-1}} \]

   2. lower-pow.f64 N/A

      \[\leadsto {\left(t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)\right)}^{\color{blue}{-1}} \]

   3. *-commutative N/A

      \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right) \cdot t\right)}^{-1} \]

   4. lower-*.f64 N/A

      \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right) \cdot t\right)}^{-1} \]

   5. *-commutative N/A

      \[\leadsto {\left(\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot t\right)}^{-1} \]

   6. lower-*.f64 N/A

      \[\leadsto {\left(\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot t\right)}^{-1} \]

   7. lower-sqrt.f64 N/A

      \[\leadsto {\left(\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot t\right)}^{-1} \]

   8. lift-PI.f64 99.4

      \[\leadsto {\left(\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot t\right)}^{-1} \]

5. Applied rewrites 99.4%

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

   1. lift-pow.f64 N/A

      \[\leadsto {\left(\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot t\right)}^{\color{blue}{-1}} \]

   2. unpow-1 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-PI.f64 N/A

      \[\leadsto \frac{1}{\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot t} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{1}{\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot t} \]

   6. lift-sqrt.f64 N/A

      \[\leadsto \frac{1}{\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot t} \]

   7. *-commutative N/A

      \[\leadsto \frac{1}{\left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right) \cdot t} \]

   8. *-commutative N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. *-commutative N/A

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

   13. lift-sqrt.f64 N/A

       \[\leadsto \frac{\frac{1}{t}}{\sqrt{2} \cdot \mathsf{PI}\left(\right)} \]

   14. lift-*.f64 N/A

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

   15. lift-PI.f64 99.5

       \[\leadsto \frac{\frac{1}{t}}{\sqrt{2} \cdot \mathsf{PI}\left(\right)} \]

7. Applied rewrites 99.5%

   \[\leadsto \frac{\frac{1}{t}}{\color{blue}{\sqrt{2} \cdot \mathsf{PI}\left(\right)}} \]
8. Add Preprocessing

Alternative 4: 98.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot t} \end{array} \]

FPCore

(FPCore (v t) :precision binary64 (/ 1.0 (* (* (sqrt 2.0) (PI)) t)))

Derivation

1. Initial program 99.3%

   \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
2. Add Preprocessing

3. Taylor expanded in v around 0

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

   1. *-commutative N/A

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right) \cdot \color{blue}{t}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right) \cdot \color{blue}{t}} \]

   3. *-commutative N/A

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot t} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot t} \]

   5. lower-sqrt.f64 N/A

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot t} \]

   6. lift-PI.f64 99.4

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot t} \]

5. Applied rewrites 99.4%

   \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot t}} \]

6. Taylor expanded in v around 0

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

8. Applied rewrites 99.4%

   \[\leadsto \frac{\color{blue}{1}}{\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot t} \]
9. Add Preprocessing

Alternative 5: 98.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(t \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{2}} \end{array} \]

FPCore

(FPCore (v t) :precision binary64 (/ 1.0 (* (* t (PI)) (sqrt 2.0))))

Derivation

1. Initial program 99.3%

   \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
2. Add Preprocessing

3. Taylor expanded in v around 0

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

   1. *-commutative N/A

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right) \cdot \color{blue}{t}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right) \cdot \color{blue}{t}} \]

   3. *-commutative N/A

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot t} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot t} \]

   5. lower-sqrt.f64 N/A

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot t} \]

   6. lift-PI.f64 99.4

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot t} \]

5. Applied rewrites 99.4%

   \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot t}} \]

6. Taylor expanded in v around 0

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

8. Applied rewrites 99.4%

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

   1. lift-*.f64 N/A

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

   2. lift-PI.f64 N/A

      \[\leadsto \frac{1}{\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot t} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{1}{\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot t} \]

   4. lift-sqrt.f64 N/A

      \[\leadsto \frac{1}{\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot t} \]

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. lift-PI.f64 N/A

      \[\leadsto \frac{1}{\left(t \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{2}} \]

   10. lift-sqrt.f64 N/A

       \[\leadsto \frac{1}{\left(t \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{2}} \]

   11. lift-*.f64 99.3

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

10. Applied rewrites 99.3%

    \[\leadsto \frac{1}{\left(t \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\sqrt{2}}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025037 
(FPCore (v t)
  :name "Falkner and Boettcher, Equation (20:1,3)"
  :precision binary64
  (/ (- 1.0 (* 5.0 (* v v))) (* (* (* (PI) t) (sqrt (* 2.0 (- 1.0 (* 3.0 (* v v)))))) (- 1.0 (* v v)))))