Average Error: 6.1 → 0.4
Time: 13.9s
Precision: binary64
\[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\]
↓
\[\left(\left(\left(\left(\log \left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right) \cdot \left(x + -0.5\right)\right) \cdot 2 + \left(2 \cdot \left(\left(x + -0.5\right) \cdot \log \left(\sqrt[3]{\sqrt[3]{x}}\right)\right) + \left(x + -0.5\right) \cdot \log \left(\sqrt[3]{x}\right)\right)\right) - x\right) + 0.91893853320467\right) + \left(\left(\frac{0.083333333333333}{x} + \left(z \cdot \frac{z}{x}\right) \cdot \left(0.0007936500793651 + y\right)\right) + \frac{z}{x} \cdot -0.0027777777777778\right)\]
\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}
↓
\left(\left(\left(\left(\log \left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right) \cdot \left(x + -0.5\right)\right) \cdot 2 + \left(2 \cdot \left(\left(x + -0.5\right) \cdot \log \left(\sqrt[3]{\sqrt[3]{x}}\right)\right) + \left(x + -0.5\right) \cdot \log \left(\sqrt[3]{x}\right)\right)\right) - x\right) + 0.91893853320467\right) + \left(\left(\frac{0.083333333333333}{x} + \left(z \cdot \frac{z}{x}\right) \cdot \left(0.0007936500793651 + y\right)\right) + \frac{z}{x} \cdot -0.0027777777777778\right)
(FPCore (x y z)
 :precision binary64
 (+
  (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
  (/
   (+
    (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
    0.083333333333333)
   x)))
↓
(FPCore (x y z)
 :precision binary64
 (+
  (+
   (-
    (+
     (* (* (log (cbrt (* (cbrt x) (cbrt x)))) (+ x -0.5)) 2.0)
     (+
      (* 2.0 (* (+ x -0.5) (log (cbrt (cbrt x)))))
      (* (+ x -0.5) (log (cbrt x)))))
    x)
   0.91893853320467)
  (+
   (+ (/ 0.083333333333333 x) (* (* z (/ z x)) (+ 0.0007936500793651 y)))
   (* (/ z x) -0.0027777777777778))))
double code(double x, double y, double z) {
	return ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
}
↓
double code(double x, double y, double z) {
	return (((((log(cbrt(cbrt(x) * cbrt(x))) * (x + -0.5)) * 2.0) + ((2.0 * ((x + -0.5) * log(cbrt(cbrt(x))))) + ((x + -0.5) * log(cbrt(x))))) - x) + 0.91893853320467) + (((0.083333333333333 / x) + ((z * (z / x)) * (0.0007936500793651 + y))) + ((z / x) * -0.0027777777777778));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original6.1
Target1.2
Herbie0.4
\[\left(\left(\left(x - 0.5\right) \cdot \log x + \left(0.91893853320467 - x\right)\right) + \frac{0.083333333333333}{x}\right) + \frac{z}{x} \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)\]

Derivation

  1. Initial program 6.1

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

  2. Taylor expanded around 0 6.3

    \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\left(\left(0.0007936500793651 \cdot \frac{{z}^{2}}{x} + \left(0.083333333333333 \cdot \frac{1}{x} + \frac{{z}^{2} \cdot y}{x}\right)\right) - 0.0027777777777778 \cdot \frac{z}{x}\right)}\]

  3. Simplified4.2

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

  5. Applied *-un-lft-identity_binary64_147414.2

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

  6. Applied times-frac_binary64_147470.4

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

  7. Simplified0.4

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

  9. Applied add-cube-cbrt_binary64_147760.4

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

  10. Applied log-prod_binary64_148270.4

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

  11. Applied distribute-rgt-in_binary64_146910.4

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

  12. Simplified0.4

    \[\leadsto \left(\left(\left(\color{blue}{2 \cdot \left(\left(x + -0.5\right) \cdot \log \left(\sqrt[3]{x}\right)\right)} + \log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right)\right) - x\right) + 0.91893853320467\right) + \left(\left(\frac{0.083333333333333}{x} + \left(z \cdot \frac{z}{x}\right) \cdot \left(0.0007936500793651 + y\right)\right) + \frac{z}{x} \cdot -0.0027777777777778\right)\]

  13. Simplified0.4

    \[\leadsto \left(\left(\left(2 \cdot \left(\left(x + -0.5\right) \cdot \log \left(\sqrt[3]{x}\right)\right) + \color{blue}{\left(x + -0.5\right) \cdot \log \left(\sqrt[3]{x}\right)}\right) - x\right) + 0.91893853320467\right) + \left(\left(\frac{0.083333333333333}{x} + \left(z \cdot \frac{z}{x}\right) \cdot \left(0.0007936500793651 + y\right)\right) + \frac{z}{x} \cdot -0.0027777777777778\right)\]
  14. Using strategy rm

  15. Applied add-cube-cbrt_binary64_147760.4

    \[\leadsto \left(\left(\left(2 \cdot \left(\left(x + -0.5\right) \cdot \log \left(\sqrt[3]{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}\right)\right) + \left(x + -0.5\right) \cdot \log \left(\sqrt[3]{x}\right)\right) - x\right) + 0.91893853320467\right) + \left(\left(\frac{0.083333333333333}{x} + \left(z \cdot \frac{z}{x}\right) \cdot \left(0.0007936500793651 + y\right)\right) + \frac{z}{x} \cdot -0.0027777777777778\right)\]

  16. Applied cbrt-prod_binary64_147720.4

    \[\leadsto \left(\left(\left(2 \cdot \left(\left(x + -0.5\right) \cdot \log \color{blue}{\left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)}\right) + \left(x + -0.5\right) \cdot \log \left(\sqrt[3]{x}\right)\right) - x\right) + 0.91893853320467\right) + \left(\left(\frac{0.083333333333333}{x} + \left(z \cdot \frac{z}{x}\right) \cdot \left(0.0007936500793651 + y\right)\right) + \frac{z}{x} \cdot -0.0027777777777778\right)\]

  17. Applied log-prod_binary64_148270.4

    \[\leadsto \left(\left(\left(2 \cdot \left(\left(x + -0.5\right) \cdot \color{blue}{\left(\log \left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right) + \log \left(\sqrt[3]{\sqrt[3]{x}}\right)\right)}\right) + \left(x + -0.5\right) \cdot \log \left(\sqrt[3]{x}\right)\right) - x\right) + 0.91893853320467\right) + \left(\left(\frac{0.083333333333333}{x} + \left(z \cdot \frac{z}{x}\right) \cdot \left(0.0007936500793651 + y\right)\right) + \frac{z}{x} \cdot -0.0027777777777778\right)\]

  18. Applied distribute-rgt-in_binary64_146910.4

    \[\leadsto \left(\left(\left(2 \cdot \color{blue}{\left(\log \left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right) \cdot \left(x + -0.5\right) + \log \left(\sqrt[3]{\sqrt[3]{x}}\right) \cdot \left(x + -0.5\right)\right)} + \left(x + -0.5\right) \cdot \log \left(\sqrt[3]{x}\right)\right) - x\right) + 0.91893853320467\right) + \left(\left(\frac{0.083333333333333}{x} + \left(z \cdot \frac{z}{x}\right) \cdot \left(0.0007936500793651 + y\right)\right) + \frac{z}{x} \cdot -0.0027777777777778\right)\]

  19. Applied distribute-rgt-in_binary64_146910.4

    \[\leadsto \left(\left(\left(\color{blue}{\left(\left(\log \left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right) \cdot \left(x + -0.5\right)\right) \cdot 2 + \left(\log \left(\sqrt[3]{\sqrt[3]{x}}\right) \cdot \left(x + -0.5\right)\right) \cdot 2\right)} + \left(x + -0.5\right) \cdot \log \left(\sqrt[3]{x}\right)\right) - x\right) + 0.91893853320467\right) + \left(\left(\frac{0.083333333333333}{x} + \left(z \cdot \frac{z}{x}\right) \cdot \left(0.0007936500793651 + y\right)\right) + \frac{z}{x} \cdot -0.0027777777777778\right)\]

  20. Applied associate-+l+_binary64_146740.4

    \[\leadsto \left(\left(\color{blue}{\left(\left(\log \left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right) \cdot \left(x + -0.5\right)\right) \cdot 2 + \left(\left(\log \left(\sqrt[3]{\sqrt[3]{x}}\right) \cdot \left(x + -0.5\right)\right) \cdot 2 + \left(x + -0.5\right) \cdot \log \left(\sqrt[3]{x}\right)\right)\right)} - x\right) + 0.91893853320467\right) + \left(\left(\frac{0.083333333333333}{x} + \left(z \cdot \frac{z}{x}\right) \cdot \left(0.0007936500793651 + y\right)\right) + \frac{z}{x} \cdot -0.0027777777777778\right)\]

  21. Final simplification0.4

    \[\leadsto \left(\left(\left(\left(\log \left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right) \cdot \left(x + -0.5\right)\right) \cdot 2 + \left(2 \cdot \left(\left(x + -0.5\right) \cdot \log \left(\sqrt[3]{\sqrt[3]{x}}\right)\right) + \left(x + -0.5\right) \cdot \log \left(\sqrt[3]{x}\right)\right)\right) - x\right) + 0.91893853320467\right) + \left(\left(\frac{0.083333333333333}{x} + \left(z \cdot \frac{z}{x}\right) \cdot \left(0.0007936500793651 + y\right)\right) + \frac{z}{x} \cdot -0.0027777777777778\right)\]

Reproduce

herbie shell --seed 2021098 
(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)) (- 0.91893853320467 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)))