Average Error: 0.0 → 0.0
Time: 2.9s
Precision: binary64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\frac{\sqrt{2} \cdot \sqrt{1 \cdot 1 - \left(3 \cdot 3\right) \cdot {v}^{4}}}{4 \cdot \sqrt{1 + 3 \cdot \left(v \cdot v\right)}} \cdot \left(1 - v \cdot v\right)\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\frac{\sqrt{2} \cdot \sqrt{1 \cdot 1 - \left(3 \cdot 3\right) \cdot {v}^{4}}}{4 \cdot \sqrt{1 + 3 \cdot \left(v \cdot v\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
 (*
  (/
   (* (sqrt 2.0) (sqrt (- (* 1.0 1.0) (* (* 3.0 3.0) (pow v 4.0)))))
   (* 4.0 (sqrt (+ 1.0 (* 3.0 (* v v))))))
  (- 1.0 (* v v))))
double code(double v) {
	return ((double) (((double) ((((double) sqrt(2.0)) / 4.0) * ((double) sqrt(((double) (1.0 - ((double) (3.0 * ((double) (v * v)))))))))) * ((double) (1.0 - ((double) (v * v))))));
}
↓
double code(double v) {
	return ((double) ((((double) (((double) sqrt(2.0)) * ((double) sqrt(((double) (((double) (1.0 * 1.0)) - ((double) (((double) (3.0 * 3.0)) * ((double) pow(v, 4.0)))))))))) / ((double) (4.0 * ((double) sqrt(((double) (1.0 + ((double) (3.0 * ((double) (v * v))))))))))) * ((double) (1.0 - ((double) (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 flip--_binary640.0

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

  4. Applied sqrt-div_binary640.0

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

  5. Applied frac-times_binary640.0

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

  6. Simplified0.0

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

  7. Simplified0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2020210 
(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))))