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

Percentage Accurate: 99.3% → 99.6%

Time: 6.0s

Alternatives: 10

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

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

FPCore

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

Derivation

1. Initial program 99.2%

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

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.2%

   \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
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 99.5%

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

5. Final simplification 99.5%

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

Alternative 3: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.2%

   \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
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. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

4. Applied rewrites 99.4%

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

5. Final simplification 99.4%

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

Alternative 4: 99.3% accurate, 1.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.2%

   \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
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. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 99.2

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

   7. lift--.f64 N/A

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

   8. sub-neg N/A

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

   9. +-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. distribute-lft-neg-in N/A

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

   12. lower-fma.f64 N/A

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

   13. metadata-eval 99.2

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

4. Applied rewrites 99.2%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. frac-2neg N/A

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

   4. associate-/l/ N/A

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

   5. lower-/.f64 N/A

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

   6. lift-fma.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. distribute-neg-in N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. distribute-lft-neg-in N/A

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

   13. metadata-eval N/A

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

   14. lift-*.f64 N/A

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

   15. associate-*r* N/A

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

   16. lift-*.f64 N/A

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

   17. metadata-eval N/A

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

   18. lower-fma.f64 N/A

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

6. Applied rewrites 99.2%

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

7. Taylor expanded in v around 0

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

   1. sub-neg N/A

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

   2. unpow2 N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f64 99.2

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

9. Applied rewrites 99.2%

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

10. Final simplification 99.2%

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

Alternative 5: 99.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\mathsf{fma}\left(-4, v \cdot v, -4\right), v \cdot v, 1\right)}{\left(t \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}} \end{array} \]

FPCore

(FPCore (v t)
 :precision binary64
 (/
  (fma (fma -4.0 (* v v) -4.0) (* v v) 1.0)
  (* (* t (PI)) (sqrt (fma -6.0 (* v v) 2.0)))))

Derivation

1. Initial program 99.2%

   \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
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. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 99.2

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

   7. lift--.f64 N/A

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

   8. sub-neg N/A

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

   9. +-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. distribute-lft-neg-in N/A

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

   12. lower-fma.f64 N/A

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

   13. metadata-eval 99.2

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

4. Applied rewrites 99.2%

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

5. Taylor expanded in v around 0

   \[\leadsto \frac{\color{blue}{1 + {v}^{2} \cdot \left(-4 \cdot {v}^{2} - 4\right)}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \left(t \cdot \mathsf{PI}\left(\right)\right)} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{{v}^{2} \cdot \left(-4 \cdot {v}^{2} - 4\right) + 1}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \left(t \cdot \mathsf{PI}\left(\right)\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(-4 \cdot {v}^{2} - 4\right) \cdot {v}^{2}} + 1}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \left(t \cdot \mathsf{PI}\left(\right)\right)} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-4 \cdot {v}^{2} - 4, {v}^{2}, 1\right)}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \left(t \cdot \mathsf{PI}\left(\right)\right)} \]

   4. sub-neg N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{-4 \cdot {v}^{2} + \left(\mathsf{neg}\left(4\right)\right)}, {v}^{2}, 1\right)}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \left(t \cdot \mathsf{PI}\left(\right)\right)} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot {v}^{2} + \color{blue}{-4}, {v}^{2}, 1\right)}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \left(t \cdot \mathsf{PI}\left(\right)\right)} \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-4, {v}^{2}, -4\right)}, {v}^{2}, 1\right)}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \left(t \cdot \mathsf{PI}\left(\right)\right)} \]

   7. unpow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-4, \color{blue}{v \cdot v}, -4\right), {v}^{2}, 1\right)}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \left(t \cdot \mathsf{PI}\left(\right)\right)} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-4, \color{blue}{v \cdot v}, -4\right), {v}^{2}, 1\right)}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \left(t \cdot \mathsf{PI}\left(\right)\right)} \]

   9. unpow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-4, v \cdot v, -4\right), \color{blue}{v \cdot v}, 1\right)}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \left(t \cdot \mathsf{PI}\left(\right)\right)} \]

   10. lower-*.f64 97.7

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-4, v \cdot v, -4\right), \color{blue}{v \cdot v}, 1\right)}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \left(t \cdot \mathsf{PI}\left(\right)\right)} \]

7. Applied rewrites 97.7%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-4, v \cdot v, -4\right), v \cdot v, 1\right)}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \left(t \cdot \mathsf{PI}\left(\right)\right)} \]

8. Final simplification 97.7%

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-4, v \cdot v, -4\right), v \cdot v, 1\right)}{\left(t \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}} \]
9. Add Preprocessing

Alternative 6: 98.5% accurate, 1.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.2%

   \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
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 97.4

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

7. Applied rewrites 97.4%

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

8. Final simplification 97.4%

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

Alternative 7: 98.5% accurate, 2.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.2%

   \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
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 97.4

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

7. Applied rewrites 97.4%

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

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 97.4

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

10. Applied rewrites 97.4%

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

Alternative 8: 98.4% accurate, 2.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.2%

   \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
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 97.0

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

5. Applied rewrites 97.0%

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

Alternative 9: 98.3% accurate, 2.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.2%

   \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
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 97.0

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

5. Applied rewrites 97.0%

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

7. Applied rewrites 96.9%

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

8. Final simplification 96.9%

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

Alternative 10: 97.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt{0.5}}{t \cdot \mathsf{PI}\left(\right)} \end{array} \]

FPCore

(FPCore (v t) :precision binary64 (/ (sqrt 0.5) (* t (PI))))

Derivation

1. Initial program 99.2%

   \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
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. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in v around 0

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

   1. lower-sqrt.f64 97.2

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

7. Applied rewrites 97.2%

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

9. Applied rewrites 97.2%

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

10. Taylor expanded in v around 0

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

    1. lower-sqrt.f64 96.7

       \[\leadsto \frac{\color{blue}{\sqrt{0.5}}}{\mathsf{PI}\left(\right) \cdot t} \]

12. Applied rewrites 96.7%

    \[\leadsto \frac{\color{blue}{\sqrt{0.5}}}{\mathsf{PI}\left(\right) \cdot t} \]

13. Final simplification 96.7%

    \[\leadsto \frac{\sqrt{0.5}}{t \cdot \mathsf{PI}\left(\right)} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024304 
(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)))))