Average Error: 0.0 → 0.0
Time: 3.1s
Precision: binary64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[e^{\log \left(\left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \frac{\sqrt{2}}{4}\right) \cdot \left(1 - v \cdot v\right)\right)}\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
e^{\log \left(\left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \frac{\sqrt{2}}{4}\right) \cdot \left(1 - v \cdot v\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
 (exp
  (log
   (* (* (sqrt (- 1.0 (* 3.0 (* v v)))) (/ (sqrt 2.0) 4.0)) (- 1.0 (* v v))))))
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 exp(log((sqrt(1.0 - (3.0 * (v * v))) * (sqrt(2.0) / 4.0)) * (1.0 - (v * v))));
}

Error

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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 add-exp-log_binary64_17970.0

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

  4. Applied add-exp-log_binary64_17970.0

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

  5. Applied add-exp-log_binary64_17970.0

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

  6. Applied add-exp-log_binary64_17970.0

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

  7. Applied div-exp_binary64_18100.0

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

  8. Applied prod-exp_binary64_18080.0

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

  9. Applied prod-exp_binary64_18080.0

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

  10. Simplified0.0

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

  11. Final simplification0.0

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

Reproduce

herbie shell --seed 2020288 
(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))))