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

Percentage Accurate: 99.3% → 99.8%

Time: 6.7s

Alternatives: 9

Speedup: 1.1×

Specification

Math

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

FPCore

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

Initial Program: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

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 99.3%

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

3. Taylor expanded in t around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

5. Applied rewrites 99.5%

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

7. Applied rewrites 99.5%

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

9. Applied rewrites 99.9%

   \[\leadsto \frac{\mathsf{fma}\left(v \cdot v, -5, 1\right) \cdot \frac{{\left(\mathsf{fma}\left(-3, v \cdot v, 1\right)\right)}^{-0.5}}{\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)}}{\color{blue}{t}} \]

11. Applied rewrites 99.9%

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

12. Final simplification 99.9%

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

Alternative 2: 99.4% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in t around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-sqrt.f64 N/A

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

   13. lower-PI.f64 99.6

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

5. Applied rewrites 99.6%

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

7. Applied rewrites 99.4%

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

9. Applied rewrites 99.6%

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

10. Final simplification 99.6%

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

Alternative 3: 99.4% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in t around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

5. Applied rewrites 99.5%

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

7. Applied rewrites 99.5%

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

9. Applied rewrites 99.6%

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

11. Applied rewrites 99.6%

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

12. Final simplification 99.6%

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

Alternative 4: 98.8% accurate, 1.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.3%

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

   1. *-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. lower-PI.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. lower-sqrt.f64 N/A

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

   18. lower-PI.f64 99.1

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

5. Applied rewrites 99.1%

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

7. Applied rewrites 99.1%

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

8. Final simplification 99.1%

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

Alternative 5: 98.7% accurate, 1.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in t around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

5. Applied rewrites 99.5%

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

7. Applied rewrites 99.5%

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

9. Applied rewrites 99.9%

   \[\leadsto \frac{\mathsf{fma}\left(v \cdot v, -5, 1\right) \cdot \frac{{\left(\mathsf{fma}\left(-3, v \cdot v, 1\right)\right)}^{-0.5}}{\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)}}{\color{blue}{t}} \]

10. Taylor expanded in v around 0

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

12. Applied rewrites 98.5%

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

13. Final simplification 98.5%

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

Alternative 6: 98.2% 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 99.3%

   \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
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 98.2

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

5. Applied rewrites 98.2%

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

Alternative 7: 98.1% 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 99.3%

   \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
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 98.2

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

5. Applied rewrites 98.2%

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

7. Applied rewrites 98.1%

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

8. Final simplification 98.1%

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

Alternative 8: 98.1% accurate, 2.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.3%

   \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
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 98.2

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

5. Applied rewrites 98.2%

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

7. Applied rewrites 98.0%

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

8. Final simplification 98.0%

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

Alternative 9: 3.8% accurate, 76.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (v t) :precision binary64 0.0)

C

double code(double v, double t) {
	return 0.0;
}

Fortran

real(8) function code(v, t)
    real(8), intent (in) :: v
    real(8), intent (in) :: t
    code = 0.0d0
end function

Java

public static double code(double v, double t) {
	return 0.0;
}

Python

def code(v, t):
	return 0.0

Julia

function code(v, t)
	return 0.0
end

MATLAB

function tmp = code(v, t)
	tmp = 0.0;
end

Wolfram

code[v_, t_] := 0.0

Derivation

1. Initial program 99.3%

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

4. Applied rewrites 3.8%

   \[\leadsto \color{blue}{\frac{v \cdot v}{\mathsf{fma}\left(v, v, 1\right)} \cdot 0} \]

5. Taylor expanded in v around 0

   \[\leadsto \color{blue}{0} \]
6. Step-by-step derivation

7. Applied rewrites 3.8%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024312 
(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)))))