Numeric.SpecFunctions:$slogFactorial from math-functions-0.1.5.2, B

Average Error: 5.9 → 0.3

Time: 20.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le 29348759.7692681215703487396240234375:\\ \;\;\;\;\left(\frac{\left(\left(-z\right) \cdot 0.002777777777777800001512975569539776188321 + \left(7.936500793651000149400709382518925849581 \cdot 10^{-4} + y\right) \cdot \left(z \cdot z\right)\right) + 0.08333333333333299564049667651488562114537}{x} + 0.9189385332046700050057097541866824030876\right) + \left(\left(x - 0.5\right) \cdot \log x - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.9189385332046700050057097541866824030876 + \left(\left(7.936500793651000149400709382518925849581 \cdot 10^{-4} + y\right) \cdot \frac{z}{\frac{x}{z}} - \frac{0.002777777777777800001512975569539776188321 \cdot z}{x}\right)\right) + \left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \left(\log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right) - x\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r371446 = x;
        double r371447 = 0.5;
        double r371448 = r371446 - r371447;
        double r371449 = log(r371446);
        double r371450 = r371448 * r371449;
        double r371451 = r371450 - r371446;
        double r371452 = 0.91893853320467;
        double r371453 = r371451 + r371452;
        double r371454 = y;
        double r371455 = 0.0007936500793651;
        double r371456 = r371454 + r371455;
        double r371457 = z;
        double r371458 = r371456 * r371457;
        double r371459 = 0.0027777777777778;
        double r371460 = r371458 - r371459;
        double r371461 = r371460 * r371457;
        double r371462 = 0.083333333333333;
        double r371463 = r371461 + r371462;
        double r371464 = r371463 / r371446;
        double r371465 = r371453 + r371464;
        return r371465;
}

↓

double f(double x, double y, double z) {
        double r371466 = x;
        double r371467 = 29348759.76926812;
        bool r371468 = r371466 <= r371467;
        double r371469 = z;
        double r371470 = -r371469;
        double r371471 = 0.0027777777777778;
        double r371472 = r371470 * r371471;
        double r371473 = 0.0007936500793651;
        double r371474 = y;
        double r371475 = r371473 + r371474;
        double r371476 = r371469 * r371469;
        double r371477 = r371475 * r371476;
        double r371478 = r371472 + r371477;
        double r371479 = 0.083333333333333;
        double r371480 = r371478 + r371479;
        double r371481 = r371480 / r371466;
        double r371482 = 0.91893853320467;
        double r371483 = r371481 + r371482;
        double r371484 = 0.5;
        double r371485 = r371466 - r371484;
        double r371486 = log(r371466);
        double r371487 = r371485 * r371486;
        double r371488 = r371487 - r371466;
        double r371489 = r371483 + r371488;
        double r371490 = r371466 / r371469;
        double r371491 = r371469 / r371490;
        double r371492 = r371475 * r371491;
        double r371493 = r371471 * r371469;
        double r371494 = r371493 / r371466;
        double r371495 = r371492 - r371494;
        double r371496 = r371482 + r371495;
        double r371497 = cbrt(r371466);
        double r371498 = r371497 * r371497;
        double r371499 = log(r371498);
        double r371500 = r371485 * r371499;
        double r371501 = log(r371497);
        double r371502 = r371501 * r371485;
        double r371503 = r371502 - r371466;
        double r371504 = r371500 + r371503;
        double r371505 = r371496 + r371504;
        double r371506 = r371468 ? r371489 : r371505;
        return r371506;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	5.9
Target	1.2
Herbie	0.3

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

Derivation

1. Split input into 2 regimes

2. if x < 29348759.76926812

   1. Initial program 0.1

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

   2. Simplified 0.1

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

   4. Applied sub-neg 0.1

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

   5. Applied distribute-lft-in 0.1

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

   6. Simplified 0.1

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

   7. Simplified 0.1

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

   if 29348759.76926812 < x

   1. Initial program 10.0

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

   2. Simplified 10.0

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

   4. Applied add-cube-cbrt 10.0

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

   5. Applied log-prod 10.1

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

   6. Applied distribute-lft-in 10.1

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

   7. Applied associate--l+ 10.0

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

   8. Simplified 10.0

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

   9. Taylor expanded around inf 10.2

      \[\leadsto \left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \left(\log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right) - x\right)\right) + \left(0.9189385332046700050057097541866824030876 + \color{blue}{\left(\left(7.936500793651000149400709382518925849581 \cdot 10^{-4} \cdot \frac{{z}^{2}}{x} + \frac{{z}^{2} \cdot y}{x}\right) - 0.002777777777777800001512975569539776188321 \cdot \frac{z}{x}\right)}\right)\]

   10. Simplified 0.5

       \[\leadsto \left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \left(\log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right) - x\right)\right) + \left(0.9189385332046700050057097541866824030876 + \color{blue}{\left(\frac{z}{\frac{x}{z}} \cdot \left(7.936500793651000149400709382518925849581 \cdot 10^{-4} + y\right) - \frac{z \cdot 0.002777777777777800001512975569539776188321}{x}\right)}\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 29348759.7692681215703487396240234375:\\ \;\;\;\;\left(\frac{\left(\left(-z\right) \cdot 0.002777777777777800001512975569539776188321 + \left(7.936500793651000149400709382518925849581 \cdot 10^{-4} + y\right) \cdot \left(z \cdot z\right)\right) + 0.08333333333333299564049667651488562114537}{x} + 0.9189385332046700050057097541866824030876\right) + \left(\left(x - 0.5\right) \cdot \log x - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.9189385332046700050057097541866824030876 + \left(\left(7.936500793651000149400709382518925849581 \cdot 10^{-4} + y\right) \cdot \frac{z}{\frac{x}{z}} - \frac{0.002777777777777800001512975569539776188321 \cdot z}{x}\right)\right) + \left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \left(\log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right) - x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 
(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)))