Average Error: 0.0 → 0.0
Time: 2.4s
Precision: binary64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right) \]
↓
\[\log \left(e^{\frac{\sqrt{2}}{4} \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right)}}\right) \cdot \left(1 - v \cdot v\right) \]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\log \left(e^{\frac{\sqrt{2}}{4} \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right)}}\right) \cdot \left(1 - v \cdot v\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
 (*
  (log (exp (* (/ (sqrt 2.0) 4.0) (sqrt (fma (* v v) -3.0 1.0)))))
  (- 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 log(exp((sqrt(2.0) / 4.0) * sqrt(fma((v * v), -3.0, 1.0)))) * (1.0 - (v * v));
}

Error

Bits error versus v

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. Applied add-log-exp_binary640.0

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

  3. Applied add-log-exp_binary640.0

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2022088 
(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))))