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

Percentage Accurate: 99.3% → 99.3%

Time: 3.5s

Alternatives: 5

Speedup: 0.4×

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: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{2} \cdot \mathsf{PI}\left(\right)\\ \frac{\frac{2.5 \cdot \left(v \cdot v\right)}{t\_1} - {t\_1}^{-1}}{-t} \end{array} \end{array} \]

FPCore

(FPCore (v t)
 :precision binary64
 (let* ((t_1 (* (sqrt 2.0) (PI))))
   (/ (- (/ (* 2.5 (* v v)) t_1) (pow t_1 -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 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

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

   2. lower-fma.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. inv-pow N/A

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

   13. lower-pow.f64 N/A

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

5. Applied rewrites 99.2%

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

6. Taylor expanded in t around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-/.f64 N/A

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

8. Applied rewrites 99.6%

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

9. Final simplification 99.6%

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

Alternative 2: 98.8% accurate, 1.9× 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 99.3%

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

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

   2. lower-fma.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. inv-pow N/A

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

   13. lower-pow.f64 N/A

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

5. Applied rewrites 99.2%

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

6. Taylor expanded in t around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-/.f64 N/A

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

8. Applied rewrites 99.6%

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

9. Taylor expanded in v around 0

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

    1. lower-/.f64 N/A

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

    2. *-commutative N/A

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

    3. lift-sqrt.f64 N/A

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

    4. lift-*.f64 N/A

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

    5. lift-PI.f64 99.5

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

11. Applied rewrites 99.5%

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

12. Final simplification 99.5%

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

Alternative 3: 98.5% accurate, 1.9× 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 99.3%

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

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

   2. lower-fma.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. inv-pow N/A

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

   13. lower-pow.f64 N/A

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

5. Applied rewrites 99.2%

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

6. Taylor expanded in t around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-/.f64 N/A

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

8. Applied rewrites 99.6%

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

9. Taylor expanded in v around 0

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

    1. lower-/.f64 N/A

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

    2. *-commutative N/A

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

    3. *-commutative N/A

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

    4. lift-sqrt.f64 N/A

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

    5. lift-*.f64 N/A

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

    6. lift-PI.f64 N/A

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

    7. lift-*.f64 99.1

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

11. Applied rewrites 99.1%

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

    1. lift-/.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-PI.f64 N/A

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

    4. lift-*.f64 N/A

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

    5. lift-sqrt.f64 N/A

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

    6. *-commutative N/A

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

    7. *-commutative N/A

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

    8. associate-/r* N/A

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

    9. lower-/.f64 N/A

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

    10. lower-/.f64 N/A

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

    11. *-commutative N/A

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

    12. lift-sqrt.f64 N/A

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

    13. lift-*.f64 N/A

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

    14. lift-PI.f64 99.2

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

13. Applied rewrites 99.2%

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

Alternative 4: 98.4% accurate, 2.2× 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

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

   1. *-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. inv-pow N/A

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

   13. lower-pow.f64 N/A

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

5. Applied rewrites 99.2%

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

6. Taylor expanded in t around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-/.f64 N/A

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

8. Applied rewrites 99.6%

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

9. Taylor expanded in v around 0

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

    1. lower-/.f64 N/A

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

    2. *-commutative N/A

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

    3. *-commutative N/A

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

    4. lift-sqrt.f64 N/A

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

    5. lift-*.f64 N/A

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

    6. lift-PI.f64 N/A

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

    7. lift-*.f64 99.1

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

11. Applied rewrites 99.1%

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

Alternative 5: 98.3% accurate, 2.2× 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 99.3%

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

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

   2. lower-fma.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. inv-pow N/A

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

   13. lower-pow.f64 N/A

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

5. Applied rewrites 99.2%

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

6. Taylor expanded in t around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-/.f64 N/A

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

8. Applied rewrites 99.6%

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

9. Taylor expanded in v around 0

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

    1. lower-/.f64 N/A

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

    2. *-commutative N/A

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

    3. *-commutative N/A

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

    4. lift-sqrt.f64 N/A

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

    5. lift-*.f64 N/A

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

    6. lift-PI.f64 N/A

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

    7. lift-*.f64 99.1

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

11. Applied rewrites 99.1%

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

    1. lift-*.f64 N/A

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

    2. lift-PI.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. lift-sqrt.f64 N/A

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

    5. *-commutative N/A

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

    6. *-commutative N/A

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

    7. associate-*r* N/A

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

    8. lower-*.f64 N/A

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

    9. lift-*.f64 N/A

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

    10. lift-PI.f64 N/A

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

    11. lift-sqrt.f64 99.0

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

13. Applied rewrites 99.0%

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

Reproduce

herbie shell --seed 2025054 
(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)))))