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

Percentage Accurate: 99.3% → 99.8%

Time: 6.1s

Alternatives: 10

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: 99.8% accurate, 0.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 98.7

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 98.7

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

   11. lift--.f64 N/A

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

   12. lift-*.f64 N/A

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

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

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

   14. +-commutative N/A

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

   15. lower-fma.f64 N/A

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

   16. metadata-eval 98.7

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

4. Applied rewrites 98.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

6. Applied rewrites 99.8%

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

Alternative 2: 99.6% accurate, 0.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 98.7

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 98.7

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

   11. lift--.f64 N/A

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

   12. lift-*.f64 N/A

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

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

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

   14. +-commutative N/A

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

   15. lower-fma.f64 N/A

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

   16. metadata-eval 98.7

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

4. Applied rewrites 98.7%

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

6. Applied rewrites 99.6%

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

Alternative 3: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 98.7

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 98.7

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

   11. lift--.f64 N/A

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

   12. lift-*.f64 N/A

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

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

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

   14. +-commutative N/A

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

   15. lower-fma.f64 N/A

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

   16. metadata-eval 98.7

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

4. Applied rewrites 98.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

6. Applied rewrites 99.2%

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

7. Taylor expanded in v around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 99.2

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

9. Applied rewrites 99.2%

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

Alternative 4: 99.3% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.8%

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

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 98.8

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

5. Applied rewrites 98.8%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

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

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

   4. metadata-eval N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 98.8

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

7. Applied rewrites 98.8%

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

Alternative 5: 98.8% accurate, 1.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 98.7

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 98.7

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

   11. lift--.f64 N/A

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

   12. lift-*.f64 N/A

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

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

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

   14. +-commutative N/A

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

   15. lower-fma.f64 N/A

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

   16. metadata-eval 98.7

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

4. Applied rewrites 98.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

6. Applied rewrites 99.8%

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

7. Taylor expanded in v around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-PI.f64 98.5

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

9. Applied rewrites 98.5%

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

Alternative 6: 99.1% accurate, 1.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 98.7

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 98.7

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

   11. lift--.f64 N/A

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

   12. lift-*.f64 N/A

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

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

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

   14. +-commutative N/A

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

   15. lower-fma.f64 N/A

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

   16. metadata-eval 98.7

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

4. Applied rewrites 98.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

6. Applied rewrites 99.8%

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

7. Taylor expanded in v around 0

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

   1. associate-*r/ N/A

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

   2. div-add-rev N/A

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

   3. lower-/.f64 N/A

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

   4. lower-fma.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-PI.f64 98.2

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

9. Applied rewrites 98.2%

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

11. Applied rewrites 98.5%

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

Alternative 7: 98.6% accurate, 1.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in v around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-PI.f64 97.6

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

5. Applied rewrites 97.6%

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

7. Applied rewrites 97.8%

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

8. Taylor expanded in t around 0

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

10. Applied rewrites 98.2%

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

Alternative 8: 98.5% accurate, 2.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in v around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-PI.f64 97.6

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

5. Applied rewrites 97.6%

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

7. Applied rewrites 97.5%

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

9. Applied rewrites 98.2%

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

Alternative 9: 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 98.8%

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

3. Taylor expanded in v around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-PI.f64 97.6

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

5. Applied rewrites 97.6%

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

Alternative 10: 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 98.8%

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

3. Taylor expanded in v around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-PI.f64 97.6

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

5. Applied rewrites 97.6%

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

7. Applied rewrites 97.5%

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

Reproduce

herbie shell --seed 2025012 
(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)))))