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

Percentage Accurate: 99.3% → 99.6%

Time: 8.3s

Alternatives: 7

Speedup: 1.2×

Specification

Math

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

FPCore

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

Initial Program: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (v t)
 :precision binary64
 (/
  (/ (fma -5.0 (* v v) 1.0) (PI))
  (* (* t (- 1.0 (* v v))) (sqrt (fma -6.0 (* v v) 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. 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. div-inv N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*l* N/A

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

   8. associate-/r* N/A

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

   9. frac-times N/A

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

4. Applied rewrites 99.6%

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

5. Final simplification 99.6%

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

Alternative 2: 99.5% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (v t)
 :precision binary64
 (*
  (/ (fma -5.0 (* v v) 1.0) t)
  (pow (* (* (PI) (- 1.0 (* v v))) (sqrt (fma -6.0 (* v v) 2.0))) -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. 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. *-rgt-identity N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

   9. times-frac N/A

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

   10. lower-*.f64 N/A

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

4. Applied rewrites 99.6%

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

5. Final simplification 99.6%

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

Alternative 3: 98.5% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (v t) :precision binary64 (/ (pow (PI) -1.0) (* (sqrt 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. 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. div-inv N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*l* N/A

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

   8. associate-/r* N/A

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

   9. frac-times N/A

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in v around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 98.1

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

7. Applied rewrites 98.1%

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

8. Taylor expanded in v around 0

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

   1. lower-/.f64 N/A

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

   2. lower-PI.f64 98.1

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

10. Applied rewrites 98.1%

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

11. Final simplification 98.1%

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

Alternative 4: 98.4% accurate, 0.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 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 97.9

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

5. Applied rewrites 97.9%

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

6. Final simplification 97.9%

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

Alternative 5: 98.3% accurate, 0.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 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 97.9

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

5. Applied rewrites 97.9%

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

7. Applied rewrites 97.8%

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

8. Final simplification 97.8%

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

Alternative 6: 99.3% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (v t)
 :precision binary64
 (/
  (fma 5.0 (* v v) -1.0)
  (* (* (PI) (fma (* t v) v (- t))) (sqrt (fma -6.0 (* v v) 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. 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. frac-2neg N/A

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

   3. lower-/.f64 N/A

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

   4. lift--.f64 N/A

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

   5. sub-neg N/A

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

   6. +-commutative N/A

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

   7. distribute-neg-in N/A

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

   8. remove-double-neg N/A

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

   9. metadata-eval N/A

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

   10. lift-*.f64 N/A

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

   11. lower-fma.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. lift-*.f64 N/A

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in v around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt-out N/A

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

   5. lower-*.f64 N/A

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

   6. lower-PI.f64 N/A

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

   7. unpow2 N/A

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

   8. associate-*r* N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. mul-1-neg N/A

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

   12. lower-neg.f64 99.4

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

7. Applied rewrites 99.4%

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

Alternative 7: 99.3% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (v t)
 :precision binary64
 (/
  (fma 5.0 (* v v) -1.0)
  (* (* (PI) (* (fma v v -1.0) t)) (sqrt (fma -6.0 (* v v) 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. 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. frac-2neg N/A

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

   3. lower-/.f64 N/A

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

   4. lift--.f64 N/A

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

   5. sub-neg N/A

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

   6. +-commutative N/A

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

   7. distribute-neg-in N/A

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

   8. remove-double-neg N/A

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

   9. metadata-eval N/A

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

   10. lift-*.f64 N/A

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

   11. lower-fma.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. lift-*.f64 N/A

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in v around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt-out N/A

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

   5. lower-*.f64 N/A

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

   6. lower-PI.f64 N/A

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

   7. unpow2 N/A

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

   8. associate-*r* N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. mul-1-neg N/A

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

   12. lower-neg.f64 99.4

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

7. Applied rewrites 99.4%

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

8. Taylor expanded in v around 0

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

10. Applied rewrites 99.4%

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

Reproduce

herbie shell --seed 2024309 
(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)))))