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

Percentage Accurate: 99.3% → 99.5%

Time: 3.8s

Alternatives: 5

Speedup: 1.0×

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, 1.0× speedup

Math

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

FPCore

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

   \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{\color{blue}{2 + -6 \cdot {v}^{2}}}\right) \cdot \left(1 - v \cdot v\right)} \]
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. fp-cancel-sub-sign-inv N/A

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

   6. metadata-eval N/A

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

   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

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

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. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lower-*.f64 99.5

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

   8. lift-*.f64 N/A

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

   9. lift--.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

9. Applied rewrites 99.5%

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

    1. lift-/.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-fma.f64 N/A

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

    4. lift-PI.f64 N/A

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

    5. lift-*.f64 N/A

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

    6. associate-/r* N/A

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

    7. lower-/.f64 N/A

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

11. Applied rewrites 99.6%

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

Alternative 2: 99.3% accurate, 1.1× speedup

Math

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

FPCore

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

   \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{\color{blue}{2 + -6 \cdot {v}^{2}}}\right) \cdot \left(1 - v \cdot v\right)} \]
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. fp-cancel-sub-sign-inv N/A

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

   6. metadata-eval N/A

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

   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

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

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. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lower-*.f64 99.5

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

   8. lift-*.f64 N/A

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

   9. lift--.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

9. Applied rewrites 99.5%

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

Alternative 3: 98.4% 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 98.7

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

5. Applied rewrites 98.7%

   \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot t}} \]
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. fp-cancel-sub-sign-inv N/A

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

   6. metadata-eval N/A

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

   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 98.7

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

7. Applied rewrites 98.7%

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

Alternative 4: 98.4% 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 98.7

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

5. Applied rewrites 98.7%

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

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 98.7%

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

Alternative 5: 98.3% 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 98.7

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

5. Applied rewrites 98.7%

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

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 98.7%

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

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

   3. lift-PI.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. *-commutative N/A

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

   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. associate-*l* N/A

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

   10. lower-*.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lift-sqrt.f64 98.6

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

10. Applied rewrites 98.6%

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

Reproduce

herbie shell --seed 2025053 
(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)))))