Average Error: 5.7 → 0.3
Time: 28.4s
Precision: 64
\[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 3366613.9142424935:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)}{x} + \mathsf{fma}\left(\log \left(\sqrt{x}\right), x - 0.5, 0.91893853320467 + \left(\left(\sqrt[3]{\log \left(\sqrt{x}\right)} \cdot \left(x - 0.5\right)\right) \cdot \left(\sqrt[3]{\log \left(\sqrt{x}\right)} \cdot \sqrt[3]{\log \left(\sqrt{x}\right)}\right) - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{z}{x} \cdot z\right) \cdot \left(y + 0.0007936500793651\right) - \frac{z}{x} \cdot 0.0027777777777778\right) + \left(\left(0.91893853320467 + \mathsf{fma}\left(x, -1, x\right)\right) + \mathsf{fma}\left(x - 0.5, \log x, \left(\left(-\sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)\right)\\ \end{array}\]
\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le 3366613.9142424935:\\
\;\;\;\;\frac{0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)}{x} + \mathsf{fma}\left(\log \left(\sqrt{x}\right), x - 0.5, 0.91893853320467 + \left(\left(\sqrt[3]{\log \left(\sqrt{x}\right)} \cdot \left(x - 0.5\right)\right) \cdot \left(\sqrt[3]{\log \left(\sqrt{x}\right)} \cdot \sqrt[3]{\log \left(\sqrt{x}\right)}\right) - x\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\left(\frac{z}{x} \cdot z\right) \cdot \left(y + 0.0007936500793651\right) - \frac{z}{x} \cdot 0.0027777777777778\right) + \left(\left(0.91893853320467 + \mathsf{fma}\left(x, -1, x\right)\right) + \mathsf{fma}\left(x - 0.5, \log x, \left(\left(-\sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)\right)\\

\end{array}
double f(double x, double y, double z) {
        double r24222555 = x;
        double r24222556 = 0.5;
        double r24222557 = r24222555 - r24222556;
        double r24222558 = log(r24222555);
        double r24222559 = r24222557 * r24222558;
        double r24222560 = r24222559 - r24222555;
        double r24222561 = 0.91893853320467;
        double r24222562 = r24222560 + r24222561;
        double r24222563 = y;
        double r24222564 = 0.0007936500793651;
        double r24222565 = r24222563 + r24222564;
        double r24222566 = z;
        double r24222567 = r24222565 * r24222566;
        double r24222568 = 0.0027777777777778;
        double r24222569 = r24222567 - r24222568;
        double r24222570 = r24222569 * r24222566;
        double r24222571 = 0.083333333333333;
        double r24222572 = r24222570 + r24222571;
        double r24222573 = r24222572 / r24222555;
        double r24222574 = r24222562 + r24222573;
        return r24222574;
}
↓
double f(double x, double y, double z) {
        double r24222575 = x;
        double r24222576 = 3366613.9142424935;
        bool r24222577 = r24222575 <= r24222576;
        double r24222578 = 0.083333333333333;
        double r24222579 = z;
        double r24222580 = y;
        double r24222581 = 0.0007936500793651;
        double r24222582 = r24222580 + r24222581;
        double r24222583 = r24222579 * r24222582;
        double r24222584 = 0.0027777777777778;
        double r24222585 = r24222583 - r24222584;
        double r24222586 = r24222579 * r24222585;
        double r24222587 = r24222578 + r24222586;
        double r24222588 = r24222587 / r24222575;
        double r24222589 = sqrt(r24222575);
        double r24222590 = log(r24222589);
        double r24222591 = 0.5;
        double r24222592 = r24222575 - r24222591;
        double r24222593 = 0.91893853320467;
        double r24222594 = cbrt(r24222590);
        double r24222595 = r24222594 * r24222592;
        double r24222596 = r24222594 * r24222594;
        double r24222597 = r24222595 * r24222596;
        double r24222598 = r24222597 - r24222575;
        double r24222599 = r24222593 + r24222598;
        double r24222600 = fma(r24222590, r24222592, r24222599);
        double r24222601 = r24222588 + r24222600;
        double r24222602 = r24222579 / r24222575;
        double r24222603 = r24222602 * r24222579;
        double r24222604 = r24222603 * r24222582;
        double r24222605 = r24222602 * r24222584;
        double r24222606 = r24222604 - r24222605;
        double r24222607 = -1.0;
        double r24222608 = fma(r24222575, r24222607, r24222575);
        double r24222609 = r24222593 + r24222608;
        double r24222610 = log(r24222575);
        double r24222611 = cbrt(r24222575);
        double r24222612 = -r24222611;
        double r24222613 = r24222612 * r24222611;
        double r24222614 = r24222613 * r24222611;
        double r24222615 = fma(r24222592, r24222610, r24222614);
        double r24222616 = r24222609 + r24222615;
        double r24222617 = r24222606 + r24222616;
        double r24222618 = r24222577 ? r24222601 : r24222617;
        return r24222618;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original5.7
Target1.2
Herbie0.3
\[\left(\left(\left(x - 0.5\right) \cdot \log x + \left(0.91893853320467 - x\right)\right) + \frac{0.083333333333333}{x}\right) + \frac{z}{x} \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)\]

Derivation

  1. Split input into 2 regimes

  2. if x < 3366613.9142424935

    1. Initial program 0.1

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt0.1

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\]

    4. Applied log-prod0.1

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \color{blue}{\left(\log \left(\sqrt{x}\right) + \log \left(\sqrt{x}\right)\right)} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\]

    5. Applied distribute-rgt-in0.1

      \[\leadsto \left(\left(\color{blue}{\left(\log \left(\sqrt{x}\right) \cdot \left(x - 0.5\right) + \log \left(\sqrt{x}\right) \cdot \left(x - 0.5\right)\right)} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\]

    6. Applied associate--l+0.1

      \[\leadsto \left(\color{blue}{\left(\log \left(\sqrt{x}\right) \cdot \left(x - 0.5\right) + \left(\log \left(\sqrt{x}\right) \cdot \left(x - 0.5\right) - x\right)\right)} + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\]

    7. Applied associate-+l+0.1

      \[\leadsto \color{blue}{\left(\log \left(\sqrt{x}\right) \cdot \left(x - 0.5\right) + \left(\left(\log \left(\sqrt{x}\right) \cdot \left(x - 0.5\right) - x\right) + 0.91893853320467\right)\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\]
    8. Using strategy rm

    9. Applied fma-def0.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log \left(\sqrt{x}\right), x - 0.5, \left(\log \left(\sqrt{x}\right) \cdot \left(x - 0.5\right) - x\right) + 0.91893853320467\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\]
    10. Using strategy rm

    11. Applied add-cube-cbrt0.1

      \[\leadsto \mathsf{fma}\left(\log \left(\sqrt{x}\right), x - 0.5, \left(\color{blue}{\left(\left(\sqrt[3]{\log \left(\sqrt{x}\right)} \cdot \sqrt[3]{\log \left(\sqrt{x}\right)}\right) \cdot \sqrt[3]{\log \left(\sqrt{x}\right)}\right)} \cdot \left(x - 0.5\right) - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\]

    12. Applied associate-*l*0.1

      \[\leadsto \mathsf{fma}\left(\log \left(\sqrt{x}\right), x - 0.5, \left(\color{blue}{\left(\sqrt[3]{\log \left(\sqrt{x}\right)} \cdot \sqrt[3]{\log \left(\sqrt{x}\right)}\right) \cdot \left(\sqrt[3]{\log \left(\sqrt{x}\right)} \cdot \left(x - 0.5\right)\right)} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\]

    if 3366613.9142424935 < x

    1. Initial program 9.7

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\]
    2. Using strategy rm

    3. Applied add-cube-cbrt9.7

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\]

    4. Applied prod-diff9.6

      \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(x - 0.5, \log x, -\sqrt[3]{x} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{x}, \sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right)\right)} + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\]

    5. Applied associate-+l+9.6

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(x - 0.5, \log x, -\sqrt[3]{x} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right) + \left(\mathsf{fma}\left(-\sqrt[3]{x}, \sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right) + 0.91893853320467\right)\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\]

    6. Simplified9.6

      \[\leadsto \left(\mathsf{fma}\left(x - 0.5, \log x, -\sqrt[3]{x} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right) + \color{blue}{\left(0.91893853320467 + \mathsf{fma}\left(x, -1, x\right)\right)}\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\]

    7. Taylor expanded around inf 9.7

      \[\leadsto \left(\mathsf{fma}\left(x - 0.5, \log x, -\sqrt[3]{x} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right) + \left(0.91893853320467 + \mathsf{fma}\left(x, -1, x\right)\right)\right) + \color{blue}{\left(\left(0.0007936500793651 \cdot \frac{{z}^{2}}{x} + \frac{{z}^{2} \cdot y}{x}\right) - 0.0027777777777778 \cdot \frac{z}{x}\right)}\]

    8. Simplified0.3

      \[\leadsto \left(\mathsf{fma}\left(x - 0.5, \log x, -\sqrt[3]{x} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right) + \left(0.91893853320467 + \mathsf{fma}\left(x, -1, x\right)\right)\right) + \color{blue}{\left(\left(\frac{z}{x} \cdot z\right) \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778 \cdot \frac{z}{x}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 3366613.9142424935:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)}{x} + \mathsf{fma}\left(\log \left(\sqrt{x}\right), x - 0.5, 0.91893853320467 + \left(\left(\sqrt[3]{\log \left(\sqrt{x}\right)} \cdot \left(x - 0.5\right)\right) \cdot \left(\sqrt[3]{\log \left(\sqrt{x}\right)} \cdot \sqrt[3]{\log \left(\sqrt{x}\right)}\right) - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{z}{x} \cdot z\right) \cdot \left(y + 0.0007936500793651\right) - \frac{z}{x} \cdot 0.0027777777777778\right) + \left(\left(0.91893853320467 + \mathsf{fma}\left(x, -1, x\right)\right) + \mathsf{fma}\left(x - 0.5, \log x, \left(\left(-\sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (x y z)
  :name "Numeric.SpecFunctions:$slogFactorial from math-functions-0.1.5.2, B"

  :herbie-target
  (+ (+ (+ (* (- x 0.5) (log x)) (- 0.91893853320467 x)) (/ 0.083333333333333 x)) (* (/ z x) (- (* z (+ y 0.0007936500793651)) 0.0027777777777778)))

  (+ (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467) (/ (+ (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z) 0.083333333333333) x)))