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

Percentage Accurate: 99.3% → 99.8%

Time: 7.4s

Alternatives: 9

Speedup: 1.2×

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 98.5%

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

   1. lift-/.f64 N/A

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

   2. 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. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

4. Applied rewrites 99.7%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/r* N/A

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

   5. associate-/l/ N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lift-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-*.f64 99.9

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

6. Applied rewrites 99.9%

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

Alternative 2: 98.5% accurate, 0.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.5%

   \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
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 97.7

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

5. Applied rewrites 97.7%

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

7. Applied rewrites 97.6%

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

9. Applied rewrites 98.7%

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

10. Final simplification 98.7%

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

Alternative 3: 98.8% accurate, 0.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.5%

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

   1. lift-/.f64 N/A

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

   2. 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. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

4. Applied rewrites 99.7%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/r* N/A

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

   5. associate-/l/ N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lift-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-*.f64 99.9

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

6. Applied rewrites 99.9%

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

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 98.9

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

9. Applied rewrites 98.9%

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

10. Final simplification 98.9%

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

Alternative 4: 98.5% accurate, 0.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.5%

   \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
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 97.7

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

5. Applied rewrites 97.7%

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

7. Applied rewrites 98.6%

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

8. Final simplification 98.6%

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

Alternative 5: 98.4% accurate, 0.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.5%

   \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
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 97.7

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

5. Applied rewrites 97.7%

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

6. Final simplification 97.7%

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

Alternative 6: 98.3% accurate, 0.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.5%

   \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
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 97.7

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

5. Applied rewrites 97.7%

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

7. Applied rewrites 97.7%

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

8. Final simplification 97.7%

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

Alternative 7: 98.3% accurate, 0.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.5%

   \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
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 97.7

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

5. Applied rewrites 97.7%

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

7. Applied rewrites 97.6%

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

8. Final simplification 97.6%

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

Alternative 8: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.5%

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

   1. lift-/.f64 N/A

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

   2. 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. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

4. Applied rewrites 99.7%

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

6. Applied rewrites 99.6%

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

Alternative 9: 99.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\mathsf{fma}\left(-3.5, v \cdot v, 1\right)}{\sqrt{2} \cdot \mathsf{PI}\left(\right)}}{t \cdot \left(1 - v \cdot v\right)} \end{array} \]

FPCore

(FPCore (v t)
 :precision binary64
 (/ (/ (fma -3.5 (* v v) 1.0) (* (sqrt 2.0) (PI))) (* t (- 1.0 (* v v)))))

Derivation

1. Initial program 98.5%

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

   1. lift-/.f64 N/A

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

   2. 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. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

4. Applied rewrites 99.7%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/r* N/A

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

   5. associate-/l/ N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lift-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-*.f64 99.9

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

6. Applied rewrites 99.9%

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

7. Taylor expanded in v around 0

   \[\leadsto \frac{\color{blue}{\frac{-7}{2} \cdot \frac{{v}^{2}}{\mathsf{PI}\left(\right) \cdot \sqrt{2}} + \frac{1}{\mathsf{PI}\left(\right) \cdot \sqrt{2}}}}{t \cdot \left(1 - v \cdot v\right)} \]
8. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{-7}{2} \cdot {v}^{2}}{\mathsf{PI}\left(\right) \cdot \sqrt{2}}} + \frac{1}{\mathsf{PI}\left(\right) \cdot \sqrt{2}}}{t \cdot \left(1 - v \cdot v\right)} \]

   2. div-add-rev N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{-7}{2} \cdot {v}^{2} + 1}{\mathsf{PI}\left(\right) \cdot \sqrt{2}}}}{t \cdot \left(1 - v \cdot v\right)} \]

   3. lower-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{-7}{2} \cdot {v}^{2} + 1}{\mathsf{PI}\left(\right) \cdot \sqrt{2}}}}{t \cdot \left(1 - v \cdot v\right)} \]

   4. lower-fma.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lower-PI.f64 99.5

       \[\leadsto \frac{\frac{\mathsf{fma}\left(-3.5, v \cdot v, 1\right)}{\sqrt{2} \cdot \color{blue}{\mathsf{PI}\left(\right)}}}{t \cdot \left(1 - v \cdot v\right)} \]

9. Applied rewrites 99.5%

   \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(-3.5, v \cdot v, 1\right)}{\sqrt{2} \cdot \mathsf{PI}\left(\right)}}}{t \cdot \left(1 - v \cdot v\right)} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024357 
(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)))))