Average Error: 32.4 → 0.0
Time: 2.6s
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 r133576 = x;
        double r133577 = r133576 / r133576;
        double r133578 = 1.0;
        double r133579 = r133578 / r133576;
        double r133580 = r133576 * r133576;
        double r133581 = sqrt(r133580);
        double r133582 = r133579 * r133581;
        double r133583 = r133577 - r133582;
        return r133583;
}
↓
double f(double x) {
        double r133584 = 2.0;
        double r133585 = 1.0;
        double r133586 = x;
        double r133587 = r133585 / r133586;
        double r133588 = -r133587;
        double r133589 = fabs(r133586);
        double r133590 = 1.0;
        double r133591 = fma(r133588, r133589, r133590);
        double r133592 = exp(r133591);
        double r133593 = cbrt(r133592);
        double r133594 = log(r133593);
        double r133595 = r133584 * r133594;
        double r133596 = r133595 + r133594;
        return r133596;
}

Error

Bits error versus x

Target

Original32.4
Target0
Herbie0.0
\[\begin{array}{l} \mathbf{if}\;x \lt 0.0:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;0.0\\ \end{array}\]

Derivation

  1. Initial program 32.4

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

  2. Simplified30.3

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

  4. Applied add-log-exp4.9

    \[\leadsto \color{blue}{\log \left(e^{\mathsf{fma}\left(-\frac{1}{x}, \left|x\right|, 1\right)}\right)}\]
  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 2020081 +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)))))