Average Error: 5.8 → 5.8
Time: 28.3s
Precision: 64
\[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.9189385332046700050057097541866824030876\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\]
↓
\[\left(\log \left(\sqrt[3]{x} \cdot {x}^{\frac{1}{3}}\right) \cdot \left(x - 0.5\right) + \mathsf{fma}\left(x - 0.5, \log \left(\sqrt[3]{x}\right), 0.9189385332046700050057097541866824030876 - x\right)\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\]
\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.9189385332046700050057097541866824030876\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}
↓
\left(\log \left(\sqrt[3]{x} \cdot {x}^{\frac{1}{3}}\right) \cdot \left(x - 0.5\right) + \mathsf{fma}\left(x - 0.5, \log \left(\sqrt[3]{x}\right), 0.9189385332046700050057097541866824030876 - x\right)\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}
double f(double x, double y, double z) {
        double r319476 = x;
        double r319477 = 0.5;
        double r319478 = r319476 - r319477;
        double r319479 = log(r319476);
        double r319480 = r319478 * r319479;
        double r319481 = r319480 - r319476;
        double r319482 = 0.91893853320467;
        double r319483 = r319481 + r319482;
        double r319484 = y;
        double r319485 = 0.0007936500793651;
        double r319486 = r319484 + r319485;
        double r319487 = z;
        double r319488 = r319486 * r319487;
        double r319489 = 0.0027777777777778;
        double r319490 = r319488 - r319489;
        double r319491 = r319490 * r319487;
        double r319492 = 0.083333333333333;
        double r319493 = r319491 + r319492;
        double r319494 = r319493 / r319476;
        double r319495 = r319483 + r319494;
        return r319495;
}
↓
double f(double x, double y, double z) {
        double r319496 = x;
        double r319497 = cbrt(r319496);
        double r319498 = 0.3333333333333333;
        double r319499 = pow(r319496, r319498);
        double r319500 = r319497 * r319499;
        double r319501 = log(r319500);
        double r319502 = 0.5;
        double r319503 = r319496 - r319502;
        double r319504 = r319501 * r319503;
        double r319505 = log(r319497);
        double r319506 = 0.91893853320467;
        double r319507 = r319506 - r319496;
        double r319508 = fma(r319503, r319505, r319507);
        double r319509 = r319504 + r319508;
        double r319510 = y;
        double r319511 = 0.0007936500793651;
        double r319512 = r319510 + r319511;
        double r319513 = z;
        double r319514 = r319512 * r319513;
        double r319515 = 0.0027777777777778;
        double r319516 = r319514 - r319515;
        double r319517 = r319516 * r319513;
        double r319518 = 0.083333333333333;
        double r319519 = r319517 + r319518;
        double r319520 = r319519 / r319496;
        double r319521 = r319509 + r319520;
        return r319521;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original5.8
Target1.4
Herbie5.8
\[\left(\left(\left(x - 0.5\right) \cdot \log x + \left(0.9189385332046700050057097541866824030876 - x\right)\right) + \frac{0.08333333333333299564049667651488562114537}{x}\right) + \frac{z}{x} \cdot \left(z \cdot \left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) - 0.002777777777777800001512975569539776188321\right)\]

Derivation

  1. Initial program 5.8

    \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.9189385332046700050057097541866824030876\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\]
  2. Using strategy rm

  3. Applied add-cube-cbrt5.8

    \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log \color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)} - x\right) + 0.9189385332046700050057097541866824030876\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\]

  4. Applied log-prod5.8

    \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \color{blue}{\left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \log \left(\sqrt[3]{x}\right)\right)} - x\right) + 0.9189385332046700050057097541866824030876\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\]

  5. Applied distribute-rgt-in5.8

    \[\leadsto \left(\left(\color{blue}{\left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right) + \log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right)\right)} - x\right) + 0.9189385332046700050057097541866824030876\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\]

  6. Applied associate--l+5.8

    \[\leadsto \left(\color{blue}{\left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right) + \left(\log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right) - x\right)\right)} + 0.9189385332046700050057097541866824030876\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\]

  7. Applied associate-+l+5.8

    \[\leadsto \color{blue}{\left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right) + \left(\left(\log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right) - x\right) + 0.9189385332046700050057097541866824030876\right)\right)} + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\]

  8. Simplified5.8

    \[\leadsto \left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right) + \color{blue}{\mathsf{fma}\left(x - 0.5, \log \left(\sqrt[3]{x}\right), 0.9189385332046700050057097541866824030876 - x\right)}\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\]
  9. Using strategy rm

  10. Applied pow1/35.8

    \[\leadsto \left(\log \left(\sqrt[3]{x} \cdot \color{blue}{{x}^{\frac{1}{3}}}\right) \cdot \left(x - 0.5\right) + \mathsf{fma}\left(x - 0.5, \log \left(\sqrt[3]{x}\right), 0.9189385332046700050057097541866824030876 - x\right)\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\]

  11. Final simplification5.8

    \[\leadsto \left(\log \left(\sqrt[3]{x} \cdot {x}^{\frac{1}{3}}\right) \cdot \left(x - 0.5\right) + \mathsf{fma}\left(x - 0.5, \log \left(\sqrt[3]{x}\right), 0.9189385332046700050057097541866824030876 - x\right)\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\]

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(FPCore (x y z)
  :name "Numeric.SpecFunctions:$slogFactorial from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (+ (+ (+ (* (- x 0.5) (log x)) (- 0.91893853320467001 x)) (/ 0.0833333333333329956 x)) (* (/ z x) (- (* z (+ y 7.93650079365100015e-4)) 0.0027777777777778)))

  (+ (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467001) (/ (+ (* (- (* (+ y 7.93650079365100015e-4) z) 0.0027777777777778) z) 0.0833333333333329956) x)))