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

Average Error: 5.9 → 4.1

Time: 31.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le 39120322101589186043319138230730752:\\ \;\;\;\;\left(0.9189385332046700050057097541866824030876 + \left(\sqrt{\left(\log x \cdot \left(\sqrt{x} - \sqrt{0.5}\right)\right) \cdot \left(\sqrt{x} + \sqrt{0.5}\right)} \cdot \sqrt{\left(x - 0.5\right) \cdot \log x} - x\right)\right) + \frac{0.08333333333333299564049667651488562114537 + \left(\left(7.936500793651000149400709382518925849581 \cdot 10^{-4} + y\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(7.936500793651000149400709382518925849581 \cdot 10^{-4} + y\right) \cdot \frac{z \cdot z}{x} - \frac{0.002777777777777800001512975569539776188321 \cdot z}{x}\right) + \left(\left(\left(\log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right) - x\right) + \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right)\right) + 0.9189385332046700050057097541866824030876\right)\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r24142545 = x;
        double r24142546 = 0.5;
        double r24142547 = r24142545 - r24142546;
        double r24142548 = log(r24142545);
        double r24142549 = r24142547 * r24142548;
        double r24142550 = r24142549 - r24142545;
        double r24142551 = 0.91893853320467;
        double r24142552 = r24142550 + r24142551;
        double r24142553 = y;
        double r24142554 = 0.0007936500793651;
        double r24142555 = r24142553 + r24142554;
        double r24142556 = z;
        double r24142557 = r24142555 * r24142556;
        double r24142558 = 0.0027777777777778;
        double r24142559 = r24142557 - r24142558;
        double r24142560 = r24142559 * r24142556;
        double r24142561 = 0.083333333333333;
        double r24142562 = r24142560 + r24142561;
        double r24142563 = r24142562 / r24142545;
        double r24142564 = r24142552 + r24142563;
        return r24142564;
}

↓

double f(double x, double y, double z) {
        double r24142565 = x;
        double r24142566 = 3.9120322101589186e+34;
        bool r24142567 = r24142565 <= r24142566;
        double r24142568 = 0.91893853320467;
        double r24142569 = log(r24142565);
        double r24142570 = sqrt(r24142565);
        double r24142571 = 0.5;
        double r24142572 = sqrt(r24142571);
        double r24142573 = r24142570 - r24142572;
        double r24142574 = r24142569 * r24142573;
        double r24142575 = r24142570 + r24142572;
        double r24142576 = r24142574 * r24142575;
        double r24142577 = sqrt(r24142576);
        double r24142578 = r24142565 - r24142571;
        double r24142579 = r24142578 * r24142569;
        double r24142580 = sqrt(r24142579);
        double r24142581 = r24142577 * r24142580;
        double r24142582 = r24142581 - r24142565;
        double r24142583 = r24142568 + r24142582;
        double r24142584 = 0.083333333333333;
        double r24142585 = 0.0007936500793651;
        double r24142586 = y;
        double r24142587 = r24142585 + r24142586;
        double r24142588 = z;
        double r24142589 = r24142587 * r24142588;
        double r24142590 = 0.0027777777777778;
        double r24142591 = r24142589 - r24142590;
        double r24142592 = r24142591 * r24142588;
        double r24142593 = r24142584 + r24142592;
        double r24142594 = r24142593 / r24142565;
        double r24142595 = r24142583 + r24142594;
        double r24142596 = r24142588 * r24142588;
        double r24142597 = r24142596 / r24142565;
        double r24142598 = r24142587 * r24142597;
        double r24142599 = r24142590 * r24142588;
        double r24142600 = r24142599 / r24142565;
        double r24142601 = r24142598 - r24142600;
        double r24142602 = cbrt(r24142565);
        double r24142603 = log(r24142602);
        double r24142604 = r24142603 * r24142578;
        double r24142605 = r24142604 - r24142565;
        double r24142606 = r24142602 * r24142602;
        double r24142607 = log(r24142606);
        double r24142608 = r24142607 * r24142578;
        double r24142609 = r24142605 + r24142608;
        double r24142610 = r24142609 + r24142568;
        double r24142611 = r24142601 + r24142610;
        double r24142612 = r24142567 ? r24142595 : r24142611;
        return r24142612;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	5.9
Target	1.4
Herbie	4.1

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

Derivation

1. Split input into 2 regimes

2. if x < 3.9120322101589186e+34

   1. Initial program 0.3

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

   3. Applied add-sqr-sqrt 0.4

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

   5. Applied add-sqr-sqrt 0.4

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

   6. Applied add-sqr-sqrt 0.4

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

   7. Applied difference-of-squares 0.4

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

   8. Applied associate-*l* 0.4

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

   if 3.9120322101589186e+34 < x

   1. Initial program 10.8

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

   3. Applied add-cube-cbrt 10.8

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

   4. Applied log-prod 10.9

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

   5. Applied distribute-lft-in 10.9

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

   6. Applied associate--l+ 10.8

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

   7. Taylor expanded around inf 11.0

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

   8. Simplified 7.3

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

4. Final simplification 4.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 39120322101589186043319138230730752:\\ \;\;\;\;\left(0.9189385332046700050057097541866824030876 + \left(\sqrt{\left(\log x \cdot \left(\sqrt{x} - \sqrt{0.5}\right)\right) \cdot \left(\sqrt{x} + \sqrt{0.5}\right)} \cdot \sqrt{\left(x - 0.5\right) \cdot \log x} - x\right)\right) + \frac{0.08333333333333299564049667651488562114537 + \left(\left(7.936500793651000149400709382518925849581 \cdot 10^{-4} + y\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(7.936500793651000149400709382518925849581 \cdot 10^{-4} + y\right) \cdot \frac{z \cdot z}{x} - \frac{0.002777777777777800001512975569539776188321 \cdot z}{x}\right) + \left(\left(\left(\log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right) - x\right) + \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right)\right) + 0.9189385332046700050057097541866824030876\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 
(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)))