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

Percentage Accurate: 99.3% → 99.3%

Time: 9.1s

Alternatives: 9

Speedup: 1.4×

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: 99.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\left(\left(\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)} \cdot t\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \end{array} \]

FPCore

(FPCore (v t)
 :precision binary64
 (/
  (fma (* v v) -5.0 1.0)
  (* (* (* (sqrt (fma (* v v) -6.0 2.0)) t) (PI)) (- 1.0 (* v v)))))

Derivation

1. Initial program 99.4%

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

   1. lift-*.f64 N/A

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right)} \cdot \left(1 - v \cdot v\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot t\right)} \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]

   3. associate-*l* N/A

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(t \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right)\right)} \cdot \left(1 - v \cdot v\right)} \]

   4. *-commutative N/A

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(\left(t \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(1 - v \cdot v\right)} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(\left(t \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(1 - v \cdot v\right)} \]

   6. *-commutative N/A

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\color{blue}{\left(\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)} \cdot t\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \]

   7. lower-*.f64 99.5

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\color{blue}{\left(\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)} \cdot t\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\sqrt{\color{blue}{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}} \cdot t\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \]

   9. *-commutative N/A

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\sqrt{\color{blue}{\left(1 - 3 \cdot \left(v \cdot v\right)\right) \cdot 2}} \cdot t\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \]

   10. lower-*.f64 99.5

       \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\sqrt{\color{blue}{\left(1 - 3 \cdot \left(v \cdot v\right)\right) \cdot 2}} \cdot t\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \]

   11. lift--.f64 N/A

       \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\sqrt{\color{blue}{\left(1 - 3 \cdot \left(v \cdot v\right)\right)} \cdot 2} \cdot t\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \]

   12. lift-*.f64 N/A

       \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\sqrt{\left(1 - \color{blue}{3 \cdot \left(v \cdot v\right)}\right) \cdot 2} \cdot t\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \]

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

       \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\sqrt{\color{blue}{\left(1 + \left(\mathsf{neg}\left(3\right)\right) \cdot \left(v \cdot v\right)\right)} \cdot 2} \cdot t\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \]

   14. +-commutative N/A

       \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\sqrt{\color{blue}{\left(\left(\mathsf{neg}\left(3\right)\right) \cdot \left(v \cdot v\right) + 1\right)} \cdot 2} \cdot t\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \]

   15. lower-fma.f64 N/A

       \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(3\right), v \cdot v, 1\right)} \cdot 2} \cdot t\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \]

   16. metadata-eval 99.5

       \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\sqrt{\mathsf{fma}\left(\color{blue}{-3}, v \cdot v, 1\right) \cdot 2} \cdot t\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \]

4. Applied rewrites 99.5%

   \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(\left(\sqrt{\mathsf{fma}\left(-3, v \cdot v, 1\right) \cdot 2} \cdot t\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(1 - v \cdot v\right)} \]

5. Taylor expanded in v around 0

   \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\sqrt{\color{blue}{2 + -6 \cdot {v}^{2}}} \cdot t\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\sqrt{\color{blue}{-6 \cdot {v}^{2} + 2}} \cdot t\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\sqrt{\color{blue}{{v}^{2} \cdot -6} + 2} \cdot t\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\sqrt{\color{blue}{\mathsf{fma}\left({v}^{2}, -6, 2\right)}} \cdot t\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \]

   4. unpow2 N/A

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\sqrt{\mathsf{fma}\left(\color{blue}{v \cdot v}, -6, 2\right)} \cdot t\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \]

   5. lower-*.f64 99.5

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\sqrt{\mathsf{fma}\left(\color{blue}{v \cdot v}, -6, 2\right)} \cdot t\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \]

7. Applied rewrites 99.5%

   \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\sqrt{\color{blue}{\mathsf{fma}\left(v \cdot v, -6, 2\right)}} \cdot t\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \]
8. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \frac{\color{blue}{1 - 5 \cdot \left(v \cdot v\right)}}{\left(\left(\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)} \cdot t\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{1 - \color{blue}{5 \cdot \left(v \cdot v\right)}}{\left(\left(\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)} \cdot t\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \]

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

      \[\leadsto \frac{\color{blue}{1 + \left(\mathsf{neg}\left(5\right)\right) \cdot \left(v \cdot v\right)}}{\left(\left(\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)} \cdot t\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \]

   4. metadata-eval N/A

      \[\leadsto \frac{1 + \color{blue}{-5} \cdot \left(v \cdot v\right)}{\left(\left(\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)} \cdot t\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \]

   5. +-commutative N/A

      \[\leadsto \frac{\color{blue}{-5 \cdot \left(v \cdot v\right) + 1}}{\left(\left(\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)} \cdot t\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \]

   6. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(v \cdot v\right) \cdot -5} + 1}{\left(\left(\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)} \cdot t\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \]

   7. lower-fma.f64 99.5

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(v \cdot v, -5, 1\right)}}{\left(\left(\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)} \cdot t\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \]

9. Applied rewrites 99.5%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(v \cdot v, -5, 1\right)}}{\left(\left(\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)} \cdot t\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \]
10. Add Preprocessing

Alternative 2: 98.5% accurate, 0.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.4%

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

   1. lift-*.f64 N/A

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right)} \cdot \left(1 - v \cdot v\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot t\right)} \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]

   3. associate-*l* N/A

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(t \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right)\right)} \cdot \left(1 - v \cdot v\right)} \]

   4. *-commutative N/A

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(\left(t \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(1 - v \cdot v\right)} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(\left(t \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(1 - v \cdot v\right)} \]

   6. *-commutative N/A

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\color{blue}{\left(\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)} \cdot t\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \]

   7. lower-*.f64 99.5

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\color{blue}{\left(\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)} \cdot t\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\sqrt{\color{blue}{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}} \cdot t\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \]

   9. *-commutative N/A

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\sqrt{\color{blue}{\left(1 - 3 \cdot \left(v \cdot v\right)\right) \cdot 2}} \cdot t\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \]

   10. lower-*.f64 99.5

       \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\sqrt{\color{blue}{\left(1 - 3 \cdot \left(v \cdot v\right)\right) \cdot 2}} \cdot t\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \]

   11. lift--.f64 N/A

       \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\sqrt{\color{blue}{\left(1 - 3 \cdot \left(v \cdot v\right)\right)} \cdot 2} \cdot t\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \]

   12. lift-*.f64 N/A

       \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\sqrt{\left(1 - \color{blue}{3 \cdot \left(v \cdot v\right)}\right) \cdot 2} \cdot t\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \]

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

       \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\sqrt{\color{blue}{\left(1 + \left(\mathsf{neg}\left(3\right)\right) \cdot \left(v \cdot v\right)\right)} \cdot 2} \cdot t\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \]

   14. +-commutative N/A

       \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\sqrt{\color{blue}{\left(\left(\mathsf{neg}\left(3\right)\right) \cdot \left(v \cdot v\right) + 1\right)} \cdot 2} \cdot t\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \]

   15. lower-fma.f64 N/A

       \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(3\right), v \cdot v, 1\right)} \cdot 2} \cdot t\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \]

   16. metadata-eval 99.5

       \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\sqrt{\mathsf{fma}\left(\color{blue}{-3}, v \cdot v, 1\right) \cdot 2} \cdot t\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \]

4. Applied rewrites 99.5%

   \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(\left(\sqrt{\mathsf{fma}\left(-3, v \cdot v, 1\right) \cdot 2} \cdot t\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(1 - v \cdot v\right)} \]

5. Taylor expanded in v around 0

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

   1. associate-*r/ N/A

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

   2. div-add-rev N/A

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

   3. lower-/.f64 N/A

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

   4. lower-fma.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-PI.f64 98.9

       \[\leadsto \frac{\mathsf{fma}\left(-2.5, v \cdot v, 1\right)}{\left(\sqrt{2} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot t} \]

7. Applied rewrites 98.9%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-2.5, v \cdot v, 1\right)}{\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot t}} \]
8. Step-by-step derivation

9. Applied rewrites 98.9%

   \[\leadsto \frac{\frac{\mathsf{fma}\left(-2.5, v \cdot v, 1\right)}{\mathsf{PI}\left(\right)}}{\color{blue}{\sqrt{2} \cdot t}} \]

10. Taylor expanded in v around 0

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

12. Applied rewrites 98.4%

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

13. Final simplification 98.4%

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

Alternative 3: 98.4% accurate, 0.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.4%

   \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
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. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-PI.f64 98.4

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

5. Applied rewrites 98.4%

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

6. Final simplification 98.4%

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

Alternative 4: 98.3% accurate, 0.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.4%

   \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
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. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-PI.f64 98.4

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

5. Applied rewrites 98.4%

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

7. Applied rewrites 98.3%

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

8. Final simplification 98.3%

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

Alternative 5: 98.3% accurate, 0.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.4%

   \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
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. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-PI.f64 98.4

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

5. Applied rewrites 98.4%

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

7. Applied rewrites 98.3%

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

8. Final simplification 98.3%

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

Alternative 6: 99.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{\mathsf{fma}\left(-2.5, v \cdot v, 1\right)}{\mathsf{PI}\left(\right)}}{\sqrt{2}}}{t} \end{array} \]

FPCore

(FPCore (v t)
 :precision binary64
 (/ (/ (/ (fma -2.5 (* v v) 1.0) (PI)) (sqrt 2.0)) t))

Derivation

1. Initial program 99.4%

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

   1. lift-*.f64 N/A

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right)} \cdot \left(1 - v \cdot v\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot t\right)} \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]

   3. associate-*l* N/A

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(t \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right)\right)} \cdot \left(1 - v \cdot v\right)} \]

   4. *-commutative N/A

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(\left(t \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(1 - v \cdot v\right)} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(\left(t \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(1 - v \cdot v\right)} \]

   6. *-commutative N/A

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\color{blue}{\left(\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)} \cdot t\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \]

   7. lower-*.f64 99.5

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\color{blue}{\left(\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)} \cdot t\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\sqrt{\color{blue}{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}} \cdot t\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \]

   9. *-commutative N/A

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\sqrt{\color{blue}{\left(1 - 3 \cdot \left(v \cdot v\right)\right) \cdot 2}} \cdot t\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \]

   10. lower-*.f64 99.5

       \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\sqrt{\color{blue}{\left(1 - 3 \cdot \left(v \cdot v\right)\right) \cdot 2}} \cdot t\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \]

   11. lift--.f64 N/A

       \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\sqrt{\color{blue}{\left(1 - 3 \cdot \left(v \cdot v\right)\right)} \cdot 2} \cdot t\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \]

   12. lift-*.f64 N/A

       \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\sqrt{\left(1 - \color{blue}{3 \cdot \left(v \cdot v\right)}\right) \cdot 2} \cdot t\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \]

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

       \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\sqrt{\color{blue}{\left(1 + \left(\mathsf{neg}\left(3\right)\right) \cdot \left(v \cdot v\right)\right)} \cdot 2} \cdot t\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \]

   14. +-commutative N/A

       \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\sqrt{\color{blue}{\left(\left(\mathsf{neg}\left(3\right)\right) \cdot \left(v \cdot v\right) + 1\right)} \cdot 2} \cdot t\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \]

   15. lower-fma.f64 N/A

       \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(3\right), v \cdot v, 1\right)} \cdot 2} \cdot t\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \]

   16. metadata-eval 99.5

       \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\sqrt{\mathsf{fma}\left(\color{blue}{-3}, v \cdot v, 1\right) \cdot 2} \cdot t\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \]

4. Applied rewrites 99.5%

   \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(\left(\sqrt{\mathsf{fma}\left(-3, v \cdot v, 1\right) \cdot 2} \cdot t\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(1 - v \cdot v\right)} \]

5. Taylor expanded in v around 0

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

   1. associate-*r/ N/A

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

   2. div-add-rev N/A

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

   3. lower-/.f64 N/A

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

   4. lower-fma.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-PI.f64 98.9

       \[\leadsto \frac{\mathsf{fma}\left(-2.5, v \cdot v, 1\right)}{\left(\sqrt{2} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot t} \]

7. Applied rewrites 98.9%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-2.5, v \cdot v, 1\right)}{\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot t}} \]
8. Step-by-step derivation

9. Applied rewrites 99.2%

   \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(-2.5, v \cdot v, 1\right)}{\mathsf{PI}\left(\right)}}{\sqrt{2}}}{\color{blue}{t}} \]
10. Add Preprocessing

Alternative 7: 99.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{\mathsf{fma}\left(-2.5, v \cdot v, 1\right)}{\mathsf{PI}\left(\right)}}{t}}{\sqrt{2}} \end{array} \]

FPCore

(FPCore (v t)
 :precision binary64
 (/ (/ (/ (fma -2.5 (* v v) 1.0) (PI)) t) (sqrt 2.0)))

Derivation

1. Initial program 99.4%

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

   1. lift-*.f64 N/A

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right)} \cdot \left(1 - v \cdot v\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot t\right)} \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]

   3. associate-*l* N/A

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(t \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right)\right)} \cdot \left(1 - v \cdot v\right)} \]

   4. *-commutative N/A

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(\left(t \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(1 - v \cdot v\right)} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(\left(t \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(1 - v \cdot v\right)} \]

   6. *-commutative N/A

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\color{blue}{\left(\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)} \cdot t\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \]

   7. lower-*.f64 99.5

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\color{blue}{\left(\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)} \cdot t\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\sqrt{\color{blue}{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}} \cdot t\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \]

   9. *-commutative N/A

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\sqrt{\color{blue}{\left(1 - 3 \cdot \left(v \cdot v\right)\right) \cdot 2}} \cdot t\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \]

   10. lower-*.f64 99.5

       \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\sqrt{\color{blue}{\left(1 - 3 \cdot \left(v \cdot v\right)\right) \cdot 2}} \cdot t\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \]

   11. lift--.f64 N/A

       \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\sqrt{\color{blue}{\left(1 - 3 \cdot \left(v \cdot v\right)\right)} \cdot 2} \cdot t\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \]

   12. lift-*.f64 N/A

       \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\sqrt{\left(1 - \color{blue}{3 \cdot \left(v \cdot v\right)}\right) \cdot 2} \cdot t\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \]

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

       \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\sqrt{\color{blue}{\left(1 + \left(\mathsf{neg}\left(3\right)\right) \cdot \left(v \cdot v\right)\right)} \cdot 2} \cdot t\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \]

   14. +-commutative N/A

       \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\sqrt{\color{blue}{\left(\left(\mathsf{neg}\left(3\right)\right) \cdot \left(v \cdot v\right) + 1\right)} \cdot 2} \cdot t\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \]

   15. lower-fma.f64 N/A

       \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(3\right), v \cdot v, 1\right)} \cdot 2} \cdot t\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \]

   16. metadata-eval 99.5

       \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\sqrt{\mathsf{fma}\left(\color{blue}{-3}, v \cdot v, 1\right) \cdot 2} \cdot t\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \]

4. Applied rewrites 99.5%

   \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(\left(\sqrt{\mathsf{fma}\left(-3, v \cdot v, 1\right) \cdot 2} \cdot t\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(1 - v \cdot v\right)} \]

5. Taylor expanded in v around 0

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

   1. associate-*r/ N/A

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

   2. div-add-rev N/A

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

   3. lower-/.f64 N/A

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

   4. lower-fma.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-PI.f64 98.9

       \[\leadsto \frac{\mathsf{fma}\left(-2.5, v \cdot v, 1\right)}{\left(\sqrt{2} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot t} \]

7. Applied rewrites 98.9%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-2.5, v \cdot v, 1\right)}{\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot t}} \]
8. Step-by-step derivation

9. Applied rewrites 98.9%

   \[\leadsto \frac{\frac{\mathsf{fma}\left(-2.5, v \cdot v, 1\right)}{\mathsf{PI}\left(\right)}}{\color{blue}{\sqrt{2} \cdot t}} \]
10. Step-by-step derivation

11. Applied rewrites 99.0%

    \[\leadsto \color{blue}{\frac{\frac{\frac{\mathsf{fma}\left(-2.5, v \cdot v, 1\right)}{\mathsf{PI}\left(\right)}}{t}}{\sqrt{2}}} \]
12. Add Preprocessing

Alternative 8: 99.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\mathsf{fma}\left(-2.5, v \cdot v, 1\right)}{\mathsf{PI}\left(\right)}}{\sqrt{2} \cdot t} \end{array} \]

FPCore

(FPCore (v t)
 :precision binary64
 (/ (/ (fma -2.5 (* v v) 1.0) (PI)) (* (sqrt 2.0) t)))

Derivation

1. Initial program 99.4%

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

   1. lift-*.f64 N/A

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right)} \cdot \left(1 - v \cdot v\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot t\right)} \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]

   3. associate-*l* N/A

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(t \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right)\right)} \cdot \left(1 - v \cdot v\right)} \]

   4. *-commutative N/A

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(\left(t \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(1 - v \cdot v\right)} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(\left(t \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(1 - v \cdot v\right)} \]

   6. *-commutative N/A

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\color{blue}{\left(\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)} \cdot t\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \]

   7. lower-*.f64 99.5

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\color{blue}{\left(\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)} \cdot t\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\sqrt{\color{blue}{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}} \cdot t\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \]

   9. *-commutative N/A

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\sqrt{\color{blue}{\left(1 - 3 \cdot \left(v \cdot v\right)\right) \cdot 2}} \cdot t\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \]

   10. lower-*.f64 99.5

       \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\sqrt{\color{blue}{\left(1 - 3 \cdot \left(v \cdot v\right)\right) \cdot 2}} \cdot t\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \]

   11. lift--.f64 N/A

       \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\sqrt{\color{blue}{\left(1 - 3 \cdot \left(v \cdot v\right)\right)} \cdot 2} \cdot t\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \]

   12. lift-*.f64 N/A

       \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\sqrt{\left(1 - \color{blue}{3 \cdot \left(v \cdot v\right)}\right) \cdot 2} \cdot t\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \]

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

       \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\sqrt{\color{blue}{\left(1 + \left(\mathsf{neg}\left(3\right)\right) \cdot \left(v \cdot v\right)\right)} \cdot 2} \cdot t\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \]

   14. +-commutative N/A

       \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\sqrt{\color{blue}{\left(\left(\mathsf{neg}\left(3\right)\right) \cdot \left(v \cdot v\right) + 1\right)} \cdot 2} \cdot t\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \]

   15. lower-fma.f64 N/A

       \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(3\right), v \cdot v, 1\right)} \cdot 2} \cdot t\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \]

   16. metadata-eval 99.5

       \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\sqrt{\mathsf{fma}\left(\color{blue}{-3}, v \cdot v, 1\right) \cdot 2} \cdot t\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)} \]

4. Applied rewrites 99.5%

   \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(\left(\sqrt{\mathsf{fma}\left(-3, v \cdot v, 1\right) \cdot 2} \cdot t\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(1 - v \cdot v\right)} \]

5. Taylor expanded in v around 0

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

   1. associate-*r/ N/A

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

   2. div-add-rev N/A

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

   3. lower-/.f64 N/A

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

   4. lower-fma.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-PI.f64 98.9

       \[\leadsto \frac{\mathsf{fma}\left(-2.5, v \cdot v, 1\right)}{\left(\sqrt{2} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot t} \]

7. Applied rewrites 98.9%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-2.5, v \cdot v, 1\right)}{\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot t}} \]
8. Step-by-step derivation

9. Applied rewrites 98.9%

   \[\leadsto \frac{\frac{\mathsf{fma}\left(-2.5, v \cdot v, 1\right)}{\mathsf{PI}\left(\right)}}{\color{blue}{\sqrt{2} \cdot t}} \]
10. Add Preprocessing

Alternative 9: 98.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(-2.5, v \cdot v, 1\right)}{\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot t} \end{array} \]

FPCore

(FPCore (v t)
 :precision binary64
 (/ (fma -2.5 (* v v) 1.0) (* (* (sqrt 2.0) (PI)) t)))

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in v around 0

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

   1. associate-*r/ N/A

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

   2. div-add-rev N/A

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

   3. lower-/.f64 N/A

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

   4. lower-fma.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-PI.f64 98.9

       \[\leadsto \frac{\mathsf{fma}\left(-2.5, v \cdot v, 1\right)}{\left(\sqrt{2} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot t} \]

5. Applied rewrites 98.9%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-2.5, v \cdot v, 1\right)}{\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot t}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024346 
(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)))))