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

Percentage Accurate: 99.3% → 99.5%

Time: 4.6s

Alternatives: 11

Speedup: 0.9×

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: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{2} \cdot \mathsf{PI}\left(\right)\\ \frac{\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{t}}{\mathsf{fma}\left(\left(-v\right) \cdot v, t\_1, t\_1\right)} \cdot \sqrt{{\left(\mathsf{fma}\left(v \cdot v, -3, 1\right)\right)}^{-1}} \end{array} \end{array} \]

FPCore

(FPCore (v t)
 :precision binary64
 (let* ((t_1 (* (sqrt 2.0) (PI))))
   (*
    (/ (/ (fma (* v v) -5.0 1.0) t) (fma (* (- v) v) t_1 t_1))
    (sqrt (pow (fma (* v v) -3.0 1.0) -1.0)))))

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in t around 0

   \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(\left(t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)\right) \cdot \sqrt{1 - 3 \cdot {v}^{2}}\right)} \cdot \left(1 - v \cdot v\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. pow2 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. pow2 N/A

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

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

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

   7. metadata-eval N/A

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

   8. +-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. pow2 N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lower-sqrt.f64 N/A

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

   17. lift-PI.f64 99.2

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

5. Applied rewrites 99.2%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. pow2 N/A

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

   5. metadata-eval N/A

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

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

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. pow2 N/A

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

   11. lift-*.f64 99.2

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

7. Applied rewrites 99.2%

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

8. Taylor expanded in t around 0

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

10. Applied rewrites 99.5%

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

11. Taylor expanded in v around 0

    \[\leadsto \frac{\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{t}}{-1 \cdot \left({v}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)\right) + \mathsf{PI}\left(\right) \cdot \sqrt{2}} \cdot \sqrt{{\left(\mathsf{fma}\left(v \cdot v, -3, 1\right)\right)}^{\color{blue}{-1}}} \]
12. Step-by-step derivation

    1. associate-*r* N/A

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

    2. mul-1-neg N/A

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

    3. lower-fma.f64 N/A

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

    4. lower-neg.f64 N/A

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

    5. pow2 N/A

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

    6. lift-*.f64 N/A

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

    7. *-commutative N/A

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

    8. lift-sqrt.f64 N/A

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

    9. lift-*.f64 N/A

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

    10. lift-PI.f64 N/A

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

    11. *-commutative N/A

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

    12. lift-sqrt.f64 N/A

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

    13. lift-*.f64 N/A

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

    14. lift-PI.f64 99.5

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

13. Applied rewrites 99.5%

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

14. Final simplification 99.5%

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

Alternative 2: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in t around 0

   \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(\left(t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)\right) \cdot \sqrt{1 - 3 \cdot {v}^{2}}\right)} \cdot \left(1 - v \cdot v\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. pow2 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. pow2 N/A

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

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

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

   7. metadata-eval N/A

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

   8. +-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. pow2 N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lower-sqrt.f64 N/A

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

   17. lift-PI.f64 99.2

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

5. Applied rewrites 99.2%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. pow2 N/A

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

   5. metadata-eval N/A

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

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

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. pow2 N/A

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

   11. lift-*.f64 99.2

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

7. Applied rewrites 99.2%

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

8. Taylor expanded in t around 0

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

10. Applied rewrites 99.5%

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

Alternative 3: 99.4% accurate, 0.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in t around 0

   \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(\left(t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)\right) \cdot \sqrt{1 - 3 \cdot {v}^{2}}\right)} \cdot \left(1 - v \cdot v\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. pow2 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. pow2 N/A

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

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

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

   7. metadata-eval N/A

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

   8. +-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. pow2 N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lower-sqrt.f64 N/A

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

   17. lift-PI.f64 99.2

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

5. Applied rewrites 99.2%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. pow2 N/A

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

   5. metadata-eval N/A

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

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

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. pow2 N/A

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

   11. lift-*.f64 99.2

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

7. Applied rewrites 99.2%

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

Alternative 4: 99.3% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.1%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. pow2 N/A

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

   5. metadata-eval N/A

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

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

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

   7. +-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. pow2 N/A

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

   10. lift-*.f64 99.2

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

   11. lift-*.f64 N/A

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

4. Applied rewrites 99.2%

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

Alternative 5: 99.3% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in v around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 99.1

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

5. Applied rewrites 99.1%

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

Alternative 6: 99.3% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in v around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 99.1

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

5. Applied rewrites 99.1%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. pow2 N/A

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

   5. metadata-eval N/A

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

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

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

   7. *-commutative N/A

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

   8. pow2 N/A

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

   9. lift-*.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-fma.f64 99.2

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

   12. lift-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. associate-*l* N/A

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

   15. lower-*.f64 N/A

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

7. Applied rewrites 99.1%

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

Alternative 7: 98.4% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in t around 0

   \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(\left(t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)\right) \cdot \sqrt{1 - 3 \cdot {v}^{2}}\right)} \cdot \left(1 - v \cdot v\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. pow2 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. pow2 N/A

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

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

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

   7. metadata-eval N/A

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

   8. +-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. pow2 N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lower-sqrt.f64 N/A

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

   17. lift-PI.f64 99.2

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

5. Applied rewrites 99.2%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. pow2 N/A

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

   5. metadata-eval N/A

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

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

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. pow2 N/A

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

   11. lift-*.f64 99.2

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

7. Applied rewrites 99.2%

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

8. Taylor expanded in t around 0

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

10. Applied rewrites 99.5%

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

11. Taylor expanded in v around 0

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

13. Applied rewrites 97.6%

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

Alternative 8: 98.3% accurate, 1.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in t around 0

   \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(\left(t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)\right) \cdot \sqrt{1 - 3 \cdot {v}^{2}}\right)} \cdot \left(1 - v \cdot v\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. pow2 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. pow2 N/A

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

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

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

   7. metadata-eval N/A

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

   8. +-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. pow2 N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lower-sqrt.f64 N/A

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

   17. lift-PI.f64 99.2

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

5. Applied rewrites 99.2%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. pow2 N/A

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

   5. metadata-eval N/A

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

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

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. pow2 N/A

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

   11. lift-*.f64 99.2

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

7. Applied rewrites 99.2%

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

8. Taylor expanded in v around 0

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-*.f64 97.3

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

10. Applied rewrites 97.3%

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

Alternative 9: 98.2% accurate, 1.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.1%

   \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
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 97.3

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

5. Applied rewrites 97.3%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. pow2 N/A

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

   5. metadata-eval N/A

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

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

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. pow2 N/A

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

   11. lift-*.f64 97.3

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

7. Applied rewrites 97.3%

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

Alternative 10: 98.2% 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 99.1%

   \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
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 97.3

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

5. Applied rewrites 97.3%

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

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 97.3%

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

Alternative 11: 98.1% 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 99.1%

   \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
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 97.3

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

5. Applied rewrites 97.3%

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

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 97.3%

   \[\leadsto \frac{\color{blue}{1}}{\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot t} \]
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. *-commutative N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lift-PI.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lift-PI.f64 N/A

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

   17. lift-sqrt.f64 97.2

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

10. Applied rewrites 97.2%

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

Reproduce

herbie shell --seed 2025040 
(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)))))