Average Error: 32.5 → 0
Time: 3.3s
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 r467 = x;
        double r468 = r467 / r467;
        double r469 = 1.0;
        double r470 = r469 / r467;
        double r471 = r467 * r467;
        double r472 = sqrt(r471);
        double r473 = r470 * r472;
        double r474 = r468 - r473;
        return r474;
}
↓
double f(double x) {
        double r475 = 2.0;
        double r476 = 1.0;
        double r477 = x;
        double r478 = r476 / r477;
        double r479 = -r478;
        double r480 = fabs(r477);
        double r481 = 1.0;
        double r482 = fma(r479, r480, r481);
        double r483 = exp(r482);
        double r484 = cbrt(r483);
        double r485 = log(r484);
        double r486 = r475 * r485;
        double r487 = r486 + r485;
        return r487;
}

Error

Bits error versus x

Target

Original32.5
Target0
Herbie0
\[\begin{array}{l} \mathbf{if}\;x \lt 0.0:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;0.0\\ \end{array}\]

Derivation

  1. Initial program 32.5

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

  2. Simplified30.4

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

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

  7. Applied log-prod0

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

  8. Simplified0

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

  9. Final simplification0

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

Reproduce

herbie shell --seed 2020025 +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)))))