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

Percentage Accurate: 99.3% → 99.6%

Time: 9.9s

Alternatives: 11

Speedup: 1.1×

Specification

Math

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

FPCore

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

Initial Program: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 99.6% 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 99.4%

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

   1. lift-/.f64 N/A

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

   2. *-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 2: 99.4% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.4%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

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

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

   6. lower-fma.f64 N/A

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

   7. metadata-eval 99.4

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*r* N/A

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

   12. lower-*.f64 N/A

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

4. Applied rewrites 99.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 99.5

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

6. Applied rewrites 99.5%

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

Alternative 3: 99.3% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.4%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

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

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

   6. lower-fma.f64 N/A

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

   7. metadata-eval 99.4

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*r* N/A

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

   12. lower-*.f64 N/A

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

4. Applied rewrites 99.4%

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

Alternative 4: 98.6% accurate, 1.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.4%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

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

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

   6. lower-fma.f64 N/A

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

   7. metadata-eval 99.4

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*r* N/A

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

   12. lower-*.f64 N/A

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in v around 0

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

   1. lower-PI.f64 98.2

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

7. Applied rewrites 98.2%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. associate-/r* N/A

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

   6. *-commutative N/A

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

   7. lower-/.f64 N/A

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

9. Applied rewrites 98.3%

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

10. Final simplification 98.3%

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

Alternative 5: 98.5% accurate, 1.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.4%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

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

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

   6. lower-fma.f64 N/A

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

   7. metadata-eval 99.4

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*r* N/A

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

   12. lower-*.f64 N/A

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in v around 0

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

   1. lower-PI.f64 98.2

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

7. Applied rewrites 98.2%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 98.2

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

   7. lift-*.f64 N/A

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

   8. lift-fma.f64 N/A

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

   9. associate-*r* N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-*.f64 98.2

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

9. Applied rewrites 98.2%

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

Alternative 6: 98.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{0.5}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}} \cdot \frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{t} \end{array} \]

FPCore

(FPCore (v t)
 :precision binary64
 (* (sqrt (/ 0.5 (* (PI) (PI)))) (/ (fma -5.0 (* v v) 1.0) t)))

Derivation

1. Initial program 99.4%

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

   1. lift-/.f64 N/A

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

   2. *-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. Taylor expanded in v around 0

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

   1. lower-/.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-PI.f64 97.6

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

7. Applied rewrites 97.6%

   \[\leadsto \frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{t} \cdot \color{blue}{\frac{\sqrt{0.5}}{\mathsf{PI}\left(\right)}} \]
8. Step-by-step derivation

9. Applied rewrites 98.2%

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

10. Final simplification 98.2%

    \[\leadsto \sqrt{\frac{0.5}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}} \cdot \frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{t} \]
11. Add Preprocessing

Alternative 7: 98.4% accurate, 1.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in v around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-PI.f64 98.2

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

7. Applied rewrites 98.2%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. cancel-sign-sub-inv N/A

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

   4. metadata-eval N/A

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

   5. *-commutative N/A

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

   6. +-commutative N/A

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

   7. lift-fma.f64 98.2

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

9. Applied rewrites 98.2%

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

Alternative 8: 98.4% accurate, 2.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in v around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-PI.f64 98.2

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

5. Applied rewrites 98.2%

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

Alternative 9: 98.3% accurate, 2.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in v around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-PI.f64 98.2

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

5. Applied rewrites 98.2%

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

7. Applied rewrites 98.1%

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

Alternative 10: 98.3% accurate, 2.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in v around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-PI.f64 98.2

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

5. Applied rewrites 98.2%

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

7. Applied rewrites 98.1%

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

Alternative 11: 97.9% accurate, 2.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.4%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. 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

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

   1. associate-/r* N/A

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

   2. lower-/.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-PI.f64 97.6

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

7. Applied rewrites 97.6%

   \[\leadsto \color{blue}{\frac{\frac{\sqrt{0.5}}{t}}{\mathsf{PI}\left(\right)}} \]
8. Step-by-step derivation

9. Applied rewrites 97.6%

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

10. Final simplification 97.6%

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

Reproduce

herbie shell --seed 2024250 
(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)))))