Average Error: 0.0 → 0.0
Time: 1.4min
Precision: binary64
Cost: 26688
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\frac{\sqrt{2}}{4} \cdot \left(\left(1 - v \cdot v\right) \cdot \sqrt{1 - \log \left(e^{\left(v \cdot v\right) \cdot 3}\right)}\right)\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\frac{\sqrt{2}}{4} \cdot \left(\left(1 - v \cdot v\right) \cdot \sqrt{1 - \log \left(e^{\left(v \cdot v\right) \cdot 3}\right)}\right)
(FPCore (v)
 :precision binary64
 (* (* (/ (sqrt 2.0) 4.0) (sqrt (- 1.0 (* 3.0 (* v v))))) (- 1.0 (* v v))))
↓
(FPCore (v)
 :precision binary64
 (*
  (/ (sqrt 2.0) 4.0)
  (* (- 1.0 (* v v)) (sqrt (- 1.0 (log (exp (* (* v v) 3.0))))))))
double code(double v) {
	return ((sqrt(2.0) / 4.0) * sqrt(1.0 - (3.0 * (v * v)))) * (1.0 - (v * v));
}
↓
double code(double v) {
	return (sqrt(2.0) / 4.0) * ((1.0 - (v * v)) * sqrt(1.0 - log(exp((v * v) * 3.0))));
}

Error

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Alternatives

Alternative 1
Error0.0
Cost13888
\[\frac{\sqrt{2}}{4} \cdot \left(\left(1 - v \cdot v\right) \cdot \sqrt{1 - \left(v \cdot v\right) \cdot 3}\right)\]
Alternative 2
Error0.0
Cost13888
\[\left(1 - v \cdot v\right) \cdot \left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - \left(v \cdot v\right) \cdot 3}\right)\]
Alternative 3
Error0.2
Cost13824
\[\frac{\sqrt{2}}{4} \cdot \left(1 + v \cdot \left({v}^{3} \cdot 0.375 - v \cdot 2.5\right)\right)\]
Alternative 4
Error0.2
Cost13696
\[\sqrt{2} \cdot \left(\left(0.25 + {v}^{4} \cdot 0.09375\right) - \left(v \cdot v\right) \cdot 0.625\right)\]
Alternative 5
Error0.2
Cost13696
\[\sqrt{2} \cdot \left({v}^{4} \cdot 0.09375 + \left(0.25 - \left(v \cdot v\right) \cdot 0.625\right)\right)\]
Alternative 6
Error0.3
Cost6976
\[\sqrt{2} \cdot \left(0.25 - \left(v \cdot v\right) \cdot 0.625\right)\]
Alternative 7
Error0.7
Cost6592
\[\sqrt{2} \cdot 0.25\]
Alternative 8
Error52.7
Cost64
\[1\]

Error

Derivation

  1. Initial program 0.0

    \[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
  2. Using strategy rm

  3. Applied associate-*l*_binary64_13830.0

    \[\leadsto \color{blue}{\frac{\sqrt{2}}{4} \cdot \left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \left(1 - v \cdot v\right)\right)}\]
  4. Using strategy rm

  5. Applied add-log-exp_binary64_14810.0

    \[\leadsto \frac{\sqrt{2}}{4} \cdot \left(\sqrt{1 - \color{blue}{\log \left(e^{3 \cdot \left(v \cdot v\right)}\right)}} \cdot \left(1 - v \cdot v\right)\right)\]
  6. Using strategy rm

  7. Applied *-un-lft-identity_binary64_14420.0

    \[\leadsto \frac{\sqrt{2}}{4} \cdot \color{blue}{\left(1 \cdot \left(\sqrt{1 - \log \left(e^{3 \cdot \left(v \cdot v\right)}\right)} \cdot \left(1 - v \cdot v\right)\right)\right)}\]

  8. Simplified0.0

    \[\leadsto \color{blue}{\frac{\sqrt{2}}{4} \cdot \left(\left(1 - v \cdot v\right) \cdot \sqrt{1 - \log \left(e^{\left(v \cdot v\right) \cdot 3}\right)}\right)}\]

  9. Final simplification0.0

    \[\leadsto \frac{\sqrt{2}}{4} \cdot \left(\left(1 - v \cdot v\right) \cdot \sqrt{1 - \log \left(e^{\left(v \cdot v\right) \cdot 3}\right)}\right)\]

Reproduce

herbie shell --seed 2021014 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  :precision binary64
  (* (* (/ (sqrt 2.0) 4.0) (sqrt (- 1.0 (* 3.0 (* v v))))) (- 1.0 (* v v))))