Average Error: 0.0 → 0.0
Time: 2.0s
Precision: binary64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[{\left(0.125 + 0.125 \cdot \left(\left(v \cdot v\right) \cdot -3\right)\right)}^{0.5} \cdot \left(1 - v \cdot v\right)\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
{\left(0.125 + 0.125 \cdot \left(\left(v \cdot v\right) \cdot -3\right)\right)}^{0.5} \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
 (* (pow (+ 0.125 (* 0.125 (* (* v v) -3.0))) 0.5) (- 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 pow((0.125 + (0.125 * ((v * v) * -3.0))), 0.5) * (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-sqr-sqrt_binary64_14771.0

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

  4. Applied associate-*l*_binary64_13961.0

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

  5. Simplified1.0

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

  7. Applied pow1/2_binary64_15351.0

    \[\leadsto \left(\sqrt{\frac{\sqrt{2}}{4}} \cdot \left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \color{blue}{{\left(\frac{\sqrt{2}}{4}\right)}^{0.5}}\right)\right) \cdot \left(1 - v \cdot v\right)\]

  8. Applied pow1/2_binary64_15351.0

    \[\leadsto \left(\sqrt{\frac{\sqrt{2}}{4}} \cdot \left(\color{blue}{{\left(1 - 3 \cdot \left(v \cdot v\right)\right)}^{0.5}} \cdot {\left(\frac{\sqrt{2}}{4}\right)}^{0.5}\right)\right) \cdot \left(1 - v \cdot v\right)\]

  9. Applied pow-prod-down_binary64_15261.0

    \[\leadsto \left(\sqrt{\frac{\sqrt{2}}{4}} \cdot \color{blue}{{\left(\left(1 - 3 \cdot \left(v \cdot v\right)\right) \cdot \frac{\sqrt{2}}{4}\right)}^{0.5}}\right) \cdot \left(1 - v \cdot v\right)\]

  10. Applied pow1/2_binary64_15351.0

    \[\leadsto \left(\color{blue}{{\left(\frac{\sqrt{2}}{4}\right)}^{0.5}} \cdot {\left(\left(1 - 3 \cdot \left(v \cdot v\right)\right) \cdot \frac{\sqrt{2}}{4}\right)}^{0.5}\right) \cdot \left(1 - v \cdot v\right)\]

  11. Applied pow-prod-down_binary64_15260.0

    \[\leadsto \color{blue}{{\left(\frac{\sqrt{2}}{4} \cdot \left(\left(1 - 3 \cdot \left(v \cdot v\right)\right) \cdot \frac{\sqrt{2}}{4}\right)\right)}^{0.5}} \cdot \left(1 - v \cdot v\right)\]

  12. Simplified0.0

    \[\leadsto {\color{blue}{\left(0.125 + \left(\left(v \cdot v\right) \cdot -3\right) \cdot 0.125\right)}}^{0.5} \cdot \left(1 - v \cdot v\right)\]

  13. Final simplification0.0

    \[\leadsto {\left(0.125 + 0.125 \cdot \left(\left(v \cdot v\right) \cdot -3\right)\right)}^{0.5} \cdot \left(1 - v \cdot v\right)\]

Reproduce

herbie shell --seed 2020280 
(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))))