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

Percentage Accurate: 99.3% → 99.8%

Time: 6.2s

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, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.4%

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

   1. lift-/.f64 N/A

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

   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. lift--.f64 N/A

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

   5. flip-- N/A

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

   6. associate-/r/ N/A

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

   7. lower-*.f64 N/A

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

4. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\frac{\frac{\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{\mathsf{PI}\left(\right)}}{\sqrt{\mathsf{fma}\left(-3, v \cdot v, 1\right) \cdot 2} \cdot t}}{1 - {v}^{4}} \cdot \mathsf{fma}\left(v, v, 1\right)} \]

6. Applied rewrites 99.8%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.4%

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

   1. lift-/.f64 N/A

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

   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. lift--.f64 N/A

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

   5. flip-- N/A

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

   6. associate-/r/ N/A

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

   7. lower-*.f64 N/A

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

4. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\frac{\frac{\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{\mathsf{PI}\left(\right)}}{\sqrt{\mathsf{fma}\left(-3, v \cdot v, 1\right) \cdot 2} \cdot t}}{1 - {v}^{4}} \cdot \mathsf{fma}\left(v, v, 1\right)} \]

6. Applied rewrites 99.8%

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

8. Applied rewrites 99.6%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 99.6

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 99.6

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

   7. lift-*.f64 N/A

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

   8. lift-fma.f64 N/A

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

   9. distribute-lft-in N/A

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

   10. associate-*r* N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. lift-fma.f64 99.6

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

10. Applied rewrites 99.6%

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

Alternative 3: 99.3% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.4%

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

   1. lift-/.f64 N/A

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

   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. lift--.f64 N/A

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

   5. flip-- N/A

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

   6. associate-/r/ N/A

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

   7. lower-*.f64 N/A

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

4. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\frac{\frac{\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{\mathsf{PI}\left(\right)}}{\sqrt{\mathsf{fma}\left(-3, v \cdot v, 1\right) \cdot 2} \cdot t}}{1 - {v}^{4}} \cdot \mathsf{fma}\left(v, v, 1\right)} \]

6. Applied rewrites 99.8%

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

8. Applied rewrites 99.6%

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

10. Applied rewrites 99.4%

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

Alternative 4: 99.3% accurate, 1.6× speedup

Math

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

FPCore

(FPCore (v t)
 :precision binary64
 (/ (/ (fma (* v v) -2.5 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. 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. lift--.f64 N/A

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

   5. flip-- N/A

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

   6. associate-/r/ N/A

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

   7. lower-*.f64 N/A

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

4. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\frac{\frac{\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{\mathsf{PI}\left(\right)}}{\sqrt{\mathsf{fma}\left(-3, v \cdot v, 1\right) \cdot 2} \cdot t}}{1 - {v}^{4}} \cdot \mathsf{fma}\left(v, v, 1\right)} \]

6. Applied rewrites 99.8%

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

7. Taylor expanded in v around 0

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

9. Applied rewrites 99.3%

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

Alternative 5: 98.9% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (v t)
 :precision binary64
 (/ (fma (* v v) -2.5 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

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

5. Applied rewrites 98.9%

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

Alternative 6: 98.8% accurate, 2.0× speedup

Math

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

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. 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. lift--.f64 N/A

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

   5. flip-- N/A

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

   6. associate-/r/ N/A

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

   7. lower-*.f64 N/A

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

4. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\frac{\frac{\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{\mathsf{PI}\left(\right)}}{\sqrt{\mathsf{fma}\left(-3, v \cdot v, 1\right) \cdot 2} \cdot t}}{1 - {v}^{4}} \cdot \mathsf{fma}\left(v, v, 1\right)} \]

6. Applied rewrites 99.8%

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

7. Taylor expanded in v around 0

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

9. Applied rewrites 98.9%

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

Alternative 7: 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 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

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

5. Applied rewrites 98.6%

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

7. Applied rewrites 98.6%

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

Alternative 8: 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

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

5. Applied rewrites 98.6%

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

Alternative 9: 98.3% accurate, 2.4× speedup

Math

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

FPCore

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

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

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

5. Applied rewrites 98.6%

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

7. Applied rewrites 98.5%

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

Alternative 10: 98.3% accurate, 2.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in v around 0

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

5. Applied rewrites 98.6%

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

7. Applied rewrites 98.4%

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

Reproduce

herbie shell --seed 2025022 
(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)))))