Average Error: 5.7 → 0.3
Time: 28.7s
Precision: 64
\[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 3366613.9142424935:\\ \;\;\;\;\frac{0.083333333333333 + \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z}{x} + \left(\left(x - 0.5\right) \cdot \log \left(\sqrt{x}\right) + \left(0.91893853320467 + \left(\sqrt{\left(x - 0.5\right) \cdot \log \left(\sqrt{x}\right)} \cdot \sqrt{\left(x - 0.5\right) \cdot \log \left(\sqrt{x}\right)} - x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-z}{x}, 0.0027777777777778, \left(\frac{z}{x} \cdot z\right) \cdot \left(y + 0.0007936500793651\right)\right) + \left(\mathsf{fma}\left(x - 0.5, \log x, \sqrt[3]{x} \cdot \left(\left(-\sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)\right) + \left(\mathsf{fma}\left(x, -1, x\right) + 0.91893853320467\right)\right)\\ \end{array}\]
\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 + \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z}{x} + \left(\left(x - 0.5\right) \cdot \log \left(\sqrt{x}\right) + \left(0.91893853320467 + \left(\sqrt{\left(x - 0.5\right) \cdot \log \left(\sqrt{x}\right)} \cdot \sqrt{\left(x - 0.5\right) \cdot \log \left(\sqrt{x}\right)} - x\right)\right)\right)\\

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

\end{array}
double f(double x, double y, double z) {
        double r18595413 = x;
        double r18595414 = 0.5;
        double r18595415 = r18595413 - r18595414;
        double r18595416 = log(r18595413);
        double r18595417 = r18595415 * r18595416;
        double r18595418 = r18595417 - r18595413;
        double r18595419 = 0.91893853320467;
        double r18595420 = r18595418 + r18595419;
        double r18595421 = y;
        double r18595422 = 0.0007936500793651;
        double r18595423 = r18595421 + r18595422;
        double r18595424 = z;
        double r18595425 = r18595423 * r18595424;
        double r18595426 = 0.0027777777777778;
        double r18595427 = r18595425 - r18595426;
        double r18595428 = r18595427 * r18595424;
        double r18595429 = 0.083333333333333;
        double r18595430 = r18595428 + r18595429;
        double r18595431 = r18595430 / r18595413;
        double r18595432 = r18595420 + r18595431;
        return r18595432;
}
↓
double f(double x, double y, double z) {
        double r18595433 = x;
        double r18595434 = 3366613.9142424935;
        bool r18595435 = r18595433 <= r18595434;
        double r18595436 = 0.083333333333333;
        double r18595437 = y;
        double r18595438 = 0.0007936500793651;
        double r18595439 = r18595437 + r18595438;
        double r18595440 = z;
        double r18595441 = r18595439 * r18595440;
        double r18595442 = 0.0027777777777778;
        double r18595443 = r18595441 - r18595442;
        double r18595444 = r18595443 * r18595440;
        double r18595445 = r18595436 + r18595444;
        double r18595446 = r18595445 / r18595433;
        double r18595447 = 0.5;
        double r18595448 = r18595433 - r18595447;
        double r18595449 = sqrt(r18595433);
        double r18595450 = log(r18595449);
        double r18595451 = r18595448 * r18595450;
        double r18595452 = 0.91893853320467;
        double r18595453 = sqrt(r18595451);
        double r18595454 = r18595453 * r18595453;
        double r18595455 = r18595454 - r18595433;
        double r18595456 = r18595452 + r18595455;
        double r18595457 = r18595451 + r18595456;
        double r18595458 = r18595446 + r18595457;
        double r18595459 = -r18595440;
        double r18595460 = r18595459 / r18595433;
        double r18595461 = r18595440 / r18595433;
        double r18595462 = r18595461 * r18595440;
        double r18595463 = r18595462 * r18595439;
        double r18595464 = fma(r18595460, r18595442, r18595463);
        double r18595465 = log(r18595433);
        double r18595466 = cbrt(r18595433);
        double r18595467 = -r18595466;
        double r18595468 = r18595467 * r18595466;
        double r18595469 = r18595466 * r18595468;
        double r18595470 = fma(r18595448, r18595465, r18595469);
        double r18595471 = -1.0;
        double r18595472 = fma(r18595433, r18595471, r18595433);
        double r18595473 = r18595472 + r18595452;
        double r18595474 = r18595470 + r18595473;
        double r18595475 = r18595464 + r18595474;
        double r18595476 = r18595435 ? r18595458 : r18595475;
        return r18595476;
}

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 add-sqr-sqrt0.1

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

    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

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

  4. Final simplification0.3

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

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)))