Average Error: 32.8 → 0.0
Time: 1.9s
Precision: 64
\[\frac{x}{x} - \frac{1}{x} \cdot \sqrt{x \cdot x}\]
↓
\[2 \cdot \log \left(\sqrt[3]{e^{\mathsf{fma}\left(-\frac{1}{x}, \left|x\right|, 1\right)}}\right) + \log \left(\sqrt[3]{e^{\mathsf{fma}\left(-\frac{1}{x}, \left|x\right|, 1\right)}}\right)\]
\frac{x}{x} - \frac{1}{x} \cdot \sqrt{x \cdot x}
↓
2 \cdot \log \left(\sqrt[3]{e^{\mathsf{fma}\left(-\frac{1}{x}, \left|x\right|, 1\right)}}\right) + \log \left(\sqrt[3]{e^{\mathsf{fma}\left(-\frac{1}{x}, \left|x\right|, 1\right)}}\right)
double f(double x) {
        double r180457 = x;
        double r180458 = r180457 / r180457;
        double r180459 = 1.0;
        double r180460 = r180459 / r180457;
        double r180461 = r180457 * r180457;
        double r180462 = sqrt(r180461);
        double r180463 = r180460 * r180462;
        double r180464 = r180458 - r180463;
        return r180464;
}
↓
double f(double x) {
        double r180465 = 2.0;
        double r180466 = 1.0;
        double r180467 = x;
        double r180468 = r180466 / r180467;
        double r180469 = -r180468;
        double r180470 = fabs(r180467);
        double r180471 = 1.0;
        double r180472 = fma(r180469, r180470, r180471);
        double r180473 = exp(r180472);
        double r180474 = cbrt(r180473);
        double r180475 = log(r180474);
        double r180476 = r180465 * r180475;
        double r180477 = r180476 + r180475;
        return r180477;
}

Error

Bits error versus x

Target

Original32.8
Target0
Herbie0.0
\[\begin{array}{l} \mathbf{if}\;x \lt 0.0:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;0.0\\ \end{array}\]

Derivation

  1. Initial program 32.8

    \[\frac{x}{x} - \frac{1}{x} \cdot \sqrt{x \cdot x}\]

  2. Simplified30.7

    \[\leadsto \color{blue}{\mathsf{fma}\left(-\frac{1}{x}, \left|x\right|, 1\right)}\]
  3. Using strategy rm

  4. Applied add-log-exp4.5

    \[\leadsto \color{blue}{\log \left(e^{\mathsf{fma}\left(-\frac{1}{x}, \left|x\right|, 1\right)}\right)}\]
  5. Using strategy rm

  6. Applied add-cube-cbrt0.0

    \[\leadsto \log \color{blue}{\left(\left(\sqrt[3]{e^{\mathsf{fma}\left(-\frac{1}{x}, \left|x\right|, 1\right)}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(-\frac{1}{x}, \left|x\right|, 1\right)}}\right) \cdot \sqrt[3]{e^{\mathsf{fma}\left(-\frac{1}{x}, \left|x\right|, 1\right)}}\right)}\]

  7. Applied log-prod0.0

    \[\leadsto \color{blue}{\log \left(\sqrt[3]{e^{\mathsf{fma}\left(-\frac{1}{x}, \left|x\right|, 1\right)}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(-\frac{1}{x}, \left|x\right|, 1\right)}}\right) + \log \left(\sqrt[3]{e^{\mathsf{fma}\left(-\frac{1}{x}, \left|x\right|, 1\right)}}\right)}\]

  8. Simplified0.0

    \[\leadsto \color{blue}{2 \cdot \log \left(\sqrt[3]{e^{\mathsf{fma}\left(-\frac{1}{x}, \left|x\right|, 1\right)}}\right)} + \log \left(\sqrt[3]{e^{\mathsf{fma}\left(-\frac{1}{x}, \left|x\right|, 1\right)}}\right)\]

  9. Final simplification0.0

    \[\leadsto 2 \cdot \log \left(\sqrt[3]{e^{\mathsf{fma}\left(-\frac{1}{x}, \left|x\right|, 1\right)}}\right) + \log \left(\sqrt[3]{e^{\mathsf{fma}\left(-\frac{1}{x}, \left|x\right|, 1\right)}}\right)\]

Reproduce

herbie shell --seed 2020002 +o rules:numerics
(FPCore (x)
  :name "sqrt sqr"
  :precision binary64

  :herbie-target
  (if (< x 0.0) 2 0.0)

  (- (/ x x) (* (/ 1 x) (sqrt (* x x)))))