Average Error: 0.0 → 0.0
Time: 4.1s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\frac{\sqrt{2} \cdot \sqrt{1 \cdot 1 - \left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right)}}{4 \cdot \sqrt{1 + 3 \cdot \left(v \cdot v\right)}} \cdot \left(1 - v \cdot v\right)\]
\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 \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)
double f(double v) {
        double r326546 = 2.0;
        double r326547 = sqrt(r326546);
        double r326548 = 4.0;
        double r326549 = r326547 / r326548;
        double r326550 = 1.0;
        double r326551 = 3.0;
        double r326552 = v;
        double r326553 = r326552 * r326552;
        double r326554 = r326551 * r326553;
        double r326555 = r326550 - r326554;
        double r326556 = sqrt(r326555);
        double r326557 = r326549 * r326556;
        double r326558 = r326550 - r326553;
        double r326559 = r326557 * r326558;
        return r326559;
}
↓
double f(double v) {
        double r326560 = 2.0;
        double r326561 = sqrt(r326560);
        double r326562 = 1.0;
        double r326563 = r326562 * r326562;
        double r326564 = 3.0;
        double r326565 = v;
        double r326566 = r326565 * r326565;
        double r326567 = r326564 * r326566;
        double r326568 = r326567 * r326567;
        double r326569 = r326563 - r326568;
        double r326570 = sqrt(r326569);
        double r326571 = r326561 * r326570;
        double r326572 = 4.0;
        double r326573 = r326562 + r326567;
        double r326574 = sqrt(r326573);
        double r326575 = r326572 * r326574;
        double r326576 = r326571 / r326575;
        double r326577 = r326562 - r326566;
        double r326578 = r326576 * r326577;
        return r326578;
}

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--0.0

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

  4. Applied sqrt-div0.0

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

  5. Applied frac-times0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019354 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  :precision binary64
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))