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

Percentage Accurate: 99.3% → 99.5%

Time: 3.4s

Alternatives: 9

Speedup: 1.1×

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

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

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 99.4%

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

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

   3. lift-PI.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift--.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift--.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. associate-*l* N/A

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

   13. lower-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. lift-PI.f64 N/A

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

4. Applied rewrites 99.4%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-fma.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift--.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

6. Applied rewrites 99.4%

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

7. Taylor expanded in t around 0

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

9. Applied rewrites 99.5%

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

Alternative 2: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

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

   3. lift-PI.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift--.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift--.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. associate-*l* N/A

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

   13. lower-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. lift-PI.f64 N/A

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

4. Applied rewrites 99.4%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-fma.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift--.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

6. Applied rewrites 99.4%

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

7. Taylor expanded in v around 0

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

9. Applied rewrites 98.7%

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

10. Taylor expanded in v around 0

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

    1. +-commutative N/A

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

    2. lower-fma.f64 N/A

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

    3. pow2 N/A

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

    4. lift-*.f64 99.4

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

12. Applied rewrites 99.4%

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

Alternative 3: 99.3% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (v t)
 :precision binary64
 (/
  (fma -5.0 (* v v) 1.0)
  (* (* (PI) t) (* (sqrt (fma (* v v) -6.0 2.0)) (- 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. 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 99.4

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

5. Applied rewrites 99.4%

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

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

   8. lower-fma.f64 N/A

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

   9. pow2 N/A

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

   10. lift-*.f64 99.4

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

   11. lift-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. associate-*l* N/A

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

   14. lower-*.f64 N/A

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

   15. lower-*.f64 99.4

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

7. Applied rewrites 99.4%

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

Alternative 4: 98.6% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (v t)
 :precision binary64
 (/ (/ (/ (fma (* v v) -5.0 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

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

   3. lift-PI.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift--.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift--.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. associate-*l* N/A

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

   13. lower-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. lift-PI.f64 N/A

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in v around 0

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

   1. lift-sqrt.f64 98.7

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

7. Applied rewrites 98.7%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/r* N/A

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

9. Applied rewrites 98.8%

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

Alternative 5: 98.6% accurate, 1.6× speedup

Math

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

FPCore

(FPCore (v t)
 :precision binary64
 (/ (/ (fma (* v v) -5.0 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

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

   3. lift-PI.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift--.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift--.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. associate-*l* N/A

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

   13. lower-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. lift-PI.f64 N/A

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in v around 0

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

   1. lift-sqrt.f64 98.7

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

7. Applied rewrites 98.7%

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

   1. lift-*.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lower-*.f64 98.7

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

9. Applied rewrites 98.7%

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

    1. lift-/.f64 N/A

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

    2. lift--.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. lift-*.f64 N/A

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

    5. lift-PI.f64 N/A

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

    6. lift-*.f64 N/A

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

    7. associate-/r* N/A

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

    8. lower-/.f64 N/A

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

11. Applied rewrites 98.8%

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

Alternative 6: 98.5% 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 99.4%

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

3. Taylor expanded in v around 0

   \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)}} \]
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 98.8

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

5. Applied rewrites 98.8%

   \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot t}} \]
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 98.8

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

7. Applied rewrites 98.8%

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

Alternative 7: 98.5% 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 99.4%

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

3. Taylor expanded in v around 0

   \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)}} \]
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 98.8

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

5. Applied rewrites 98.8%

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

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 98.8%

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

Alternative 8: 98.4% accurate, 2.4× speedup

Math

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

FPCore

(FPCore (v t) :precision binary64 (/ 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. 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 98.8

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

5. Applied rewrites 98.8%

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

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 98.8%

   \[\leadsto \frac{\color{blue}{1}}{\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot t} \]
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-sqrt.f64 98.6

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

10. Applied rewrites 98.6%

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

Alternative 9: 98.4% accurate, 2.4× speedup

Math

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

FPCore

(FPCore (v t) :precision binary64 (/ 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. 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 98.8

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

5. Applied rewrites 98.8%

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

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 98.8%

   \[\leadsto \frac{\color{blue}{1}}{\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot t} \]
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-sqrt.f64 98.6

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

10. Applied rewrites 98.6%

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

    1. lift-*.f64 N/A

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

    2. lift-PI.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. lift-sqrt.f64 N/A

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

    5. associate-*l* N/A

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

    6. lower-*.f64 N/A

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

    7. lift-PI.f64 N/A

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

    8. *-commutative N/A

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

    9. lower-*.f64 N/A

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

    10. lift-sqrt.f64 98.6

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

12. Applied rewrites 98.6%

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

Reproduce

herbie shell --seed 2025073 
(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)))))