Average Error: 0.0 → 0.0
Time: 25.0s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\left(1 - v \cdot v\right) \cdot \left(\left(\sqrt{\sqrt{1 - \left(v \cdot v\right) \cdot 3}} \cdot \sqrt{\sqrt{1 - \left(v \cdot v\right) \cdot 3}}\right) \cdot \frac{\sqrt{2}}{4}\right)\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\left(1 - v \cdot v\right) \cdot \left(\left(\sqrt{\sqrt{1 - \left(v \cdot v\right) \cdot 3}} \cdot \sqrt{\sqrt{1 - \left(v \cdot v\right) \cdot 3}}\right) \cdot \frac{\sqrt{2}}{4}\right)
double f(double v) {
        double r6675392 = 2.0;
        double r6675393 = sqrt(r6675392);
        double r6675394 = 4.0;
        double r6675395 = r6675393 / r6675394;
        double r6675396 = 1.0;
        double r6675397 = 3.0;
        double r6675398 = v;
        double r6675399 = r6675398 * r6675398;
        double r6675400 = r6675397 * r6675399;
        double r6675401 = r6675396 - r6675400;
        double r6675402 = sqrt(r6675401);
        double r6675403 = r6675395 * r6675402;
        double r6675404 = r6675396 - r6675399;
        double r6675405 = r6675403 * r6675404;
        return r6675405;
}
↓
double f(double v) {
        double r6675406 = 1.0;
        double r6675407 = v;
        double r6675408 = r6675407 * r6675407;
        double r6675409 = r6675406 - r6675408;
        double r6675410 = 3.0;
        double r6675411 = r6675408 * r6675410;
        double r6675412 = r6675406 - r6675411;
        double r6675413 = sqrt(r6675412);
        double r6675414 = sqrt(r6675413);
        double r6675415 = r6675414 * r6675414;
        double r6675416 = 2.0;
        double r6675417 = sqrt(r6675416);
        double r6675418 = 4.0;
        double r6675419 = r6675417 / r6675418;
        double r6675420 = r6675415 * r6675419;
        double r6675421 = r6675409 * r6675420;
        return r6675421;
}

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-sqrt0.0

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

  4. Applied sqrt-prod0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))