Average Error: 0.0 → 0.0
Time: 2.1s
Precision: binary64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\sqrt{0.125 + \left(\left(\left(v \cdot v\right) \cdot -3\right) \cdot 0.25\right) \cdot 0.5} \cdot \left(1 - v \cdot v\right)\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\sqrt{0.125 + \left(\left(\left(v \cdot v\right) \cdot -3\right) \cdot 0.25\right) \cdot 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
 (* (sqrt (+ 0.125 (* (* (* (* v v) -3.0) 0.25) 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 sqrt(0.125 + ((((v * v) * -3.0) * 0.25) * 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_14641.0

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

  4. Applied associate-*l*_binary64_13831.0

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

  5. Simplified1.0

    \[\leadsto \left(\sqrt{\frac{\sqrt{2}}{4}} \cdot \color{blue}{\left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \sqrt{0.25 \cdot \sqrt{2}}\right)}\right) \cdot \left(1 - v \cdot v\right)\]
  6. Using strategy rm

  7. Applied sqrt-unprod_binary64_14621.0

    \[\leadsto \left(\sqrt{\frac{\sqrt{2}}{4}} \cdot \color{blue}{\sqrt{\left(1 - 3 \cdot \left(v \cdot v\right)\right) \cdot \left(0.25 \cdot \sqrt{2}\right)}}\right) \cdot \left(1 - v \cdot v\right)\]

  8. Applied sqrt-unprod_binary64_14620.0

    \[\leadsto \color{blue}{\sqrt{\frac{\sqrt{2}}{4} \cdot \left(\left(1 - 3 \cdot \left(v \cdot v\right)\right) \cdot \left(0.25 \cdot \sqrt{2}\right)\right)}} \cdot \left(1 - v \cdot v\right)\]

  9. Simplified0.0

    \[\leadsto \sqrt{\color{blue}{0.125 + \left(\left(\left(v \cdot v\right) \cdot -3\right) \cdot 0.25\right) \cdot 0.5}} \cdot \left(1 - v \cdot v\right)\]

  10. Final simplification0.0

    \[\leadsto \sqrt{0.125 + \left(\left(\left(v \cdot v\right) \cdot -3\right) \cdot 0.25\right) \cdot 0.5} \cdot \left(1 - v \cdot v\right)\]

Reproduce

herbie shell --seed 2021015 
(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))))