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

Percentage Accurate: 99.3% → 99.8%

Time: 9.3s

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: 99.8% accurate, 0.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. frac-2neg N/A

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

   5. distribute-frac-neg N/A

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

   6. lower-neg.f64 N/A

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

   7. lower-/.f64 N/A

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

4. Applied rewrites 99.6%

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

   1. lift-neg.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. distribute-neg-frac2 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/r* N/A

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

   7. associate-/l/ N/A

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

   8. lift-fma.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. +-commutative N/A

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

   11. distribute-neg-in N/A

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

   12. metadata-eval N/A

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

   13. lift-*.f64 N/A

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

6. Applied rewrites 99.8%

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

Alternative 2: 98.4% accurate, 0.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in v around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-PI.f64 98.3

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

5. Applied rewrites 98.3%

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

6. Final simplification 98.3%

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

Alternative 3: 98.3% accurate, 0.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in v around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-PI.f64 98.3

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

5. Applied rewrites 98.3%

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

7. Applied rewrites 98.2%

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

8. Final simplification 98.2%

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

Alternative 4: 98.3% accurate, 0.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in v around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-PI.f64 98.3

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

5. Applied rewrites 98.3%

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

7. Applied rewrites 98.1%

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

8. Final simplification 98.1%

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

Alternative 5: 99.5% accurate, 0.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.3%

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

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

   5. distribute-frac-neg N/A

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

   6. lower-neg.f64 N/A

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

   7. lower-/.f64 N/A

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

4. Applied rewrites 99.6%

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

5. Final simplification 99.6%

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

Alternative 6: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. frac-2neg N/A

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

   5. distribute-frac-neg N/A

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

   6. lower-neg.f64 N/A

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

   7. lower-/.f64 N/A

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

4. Applied rewrites 99.6%

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

   1. lift-neg.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. distribute-neg-frac2 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/r* N/A

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

   7. associate-/l/ N/A

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

   8. lift-fma.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. +-commutative N/A

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

   11. distribute-neg-in N/A

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

   12. metadata-eval N/A

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

   13. lift-*.f64 N/A

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

6. Applied rewrites 99.8%

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

8. Applied rewrites 99.6%

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

Alternative 7: 99.4% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.3%

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

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

   5. distribute-frac-neg N/A

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

   6. lower-neg.f64 N/A

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

   7. lower-/.f64 N/A

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

4. Applied rewrites 99.6%

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

   1. lift-neg.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. distribute-neg-frac2 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/r* N/A

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

   7. associate-/l/ N/A

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

   8. lift-fma.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. +-commutative N/A

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

   11. distribute-neg-in N/A

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

   12. metadata-eval N/A

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

   13. lift-*.f64 N/A

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

6. Applied rewrites 99.8%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-/l/ N/A

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

   5. associate-/l/ N/A

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

   6. lower-/.f64 N/A

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

   7. lower-*.f64 N/A

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

8. Applied rewrites 99.4%

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

Alternative 8: 99.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\mathsf{fma}\left(-3.5, v \cdot v, 1\right)}{\sqrt{2} \cdot \mathsf{PI}\left(\right)}}{t \cdot \left(1 - v \cdot v\right)} \end{array} \]

FPCore

(FPCore (v t)
 :precision binary64
 (/ (/ (fma -3.5 (* v v) 1.0) (* (sqrt 2.0) (PI))) (* t (- 1.0 (* v v)))))

Derivation

1. Initial program 99.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. frac-2neg N/A

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

   5. distribute-frac-neg N/A

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

   6. lower-neg.f64 N/A

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

   7. lower-/.f64 N/A

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

4. Applied rewrites 99.6%

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

   1. lift-neg.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. distribute-neg-frac2 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/r* N/A

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

   7. associate-/l/ N/A

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

   8. lift-fma.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. +-commutative N/A

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

   11. distribute-neg-in N/A

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

   12. metadata-eval N/A

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

   13. lift-*.f64 N/A

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

6. Applied rewrites 99.8%

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

7. Taylor expanded in v around 0

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

   1. associate-*r/ N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{-7}{2} \cdot {v}^{2}}{\mathsf{PI}\left(\right) \cdot \sqrt{2}}} + \frac{1}{\mathsf{PI}\left(\right) \cdot \sqrt{2}}}{t \cdot \left(1 - v \cdot v\right)} \]

   2. div-add-rev N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{-7}{2} \cdot {v}^{2} + 1}{\mathsf{PI}\left(\right) \cdot \sqrt{2}}}}{t \cdot \left(1 - v \cdot v\right)} \]

   3. lower-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{-7}{2} \cdot {v}^{2} + 1}{\mathsf{PI}\left(\right) \cdot \sqrt{2}}}}{t \cdot \left(1 - v \cdot v\right)} \]

   4. lower-fma.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lower-PI.f64 99.4

       \[\leadsto \frac{\frac{\mathsf{fma}\left(-3.5, v \cdot v, 1\right)}{\sqrt{2} \cdot \color{blue}{\mathsf{PI}\left(\right)}}}{t \cdot \left(1 - v \cdot v\right)} \]

9. Applied rewrites 99.4%

   \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(-3.5, v \cdot v, 1\right)}{\sqrt{2} \cdot \mathsf{PI}\left(\right)}}}{t \cdot \left(1 - v \cdot v\right)} \]
10. Add Preprocessing

Alternative 9: 98.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\mathsf{fma}\left(-3.5, v \cdot v, 1\right)}{\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot t}}{-\mathsf{fma}\left(v, v, -1\right)} \end{array} \]

FPCore

(FPCore (v t)
 :precision binary64
 (/ (/ (fma -3.5 (* v v) 1.0) (* (* (sqrt 2.0) (PI)) t)) (- (fma v v -1.0))))

Derivation

1. Initial program 99.3%

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

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

   5. distribute-frac-neg N/A

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

   6. lower-neg.f64 N/A

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

   7. lower-/.f64 N/A

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in v around 0

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

   1. associate-*r/ N/A

      \[\leadsto -\frac{\color{blue}{\frac{\frac{-7}{2} \cdot {v}^{2}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)}} + \frac{1}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)}}{\mathsf{fma}\left(v, v, -1\right)} \]

   2. div-add-rev N/A

      \[\leadsto -\frac{\color{blue}{\frac{\frac{-7}{2} \cdot {v}^{2} + 1}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)}}}{\mathsf{fma}\left(v, v, -1\right)} \]

   3. lower-/.f64 N/A

      \[\leadsto -\frac{\color{blue}{\frac{\frac{-7}{2} \cdot {v}^{2} + 1}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)}}}{\mathsf{fma}\left(v, v, -1\right)} \]

   4. lower-fma.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-PI.f64 98.9

       \[\leadsto -\frac{\frac{\mathsf{fma}\left(-3.5, v \cdot v, 1\right)}{\left(\sqrt{2} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot t}}{\mathsf{fma}\left(v, v, -1\right)} \]

7. Applied rewrites 98.9%

   \[\leadsto -\frac{\color{blue}{\frac{\mathsf{fma}\left(-3.5, v \cdot v, 1\right)}{\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot t}}}{\mathsf{fma}\left(v, v, -1\right)} \]

8. Final simplification 98.9%

   \[\leadsto \frac{\frac{\mathsf{fma}\left(-3.5, v \cdot v, 1\right)}{\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot t}}{-\mathsf{fma}\left(v, v, -1\right)} \]
9. Add Preprocessing

Alternative 10: 98.9% accurate, 1.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in v around 0

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

   1. associate-*r/ N/A

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

   2. div-add-rev N/A

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

   3. lower-/.f64 N/A

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

   4. lower-fma.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-PI.f64 98.9

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

5. Applied rewrites 98.9%

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

Reproduce

herbie shell --seed 2024352 
(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)))))