Average Error: 29.3 → 1.1
Time: 8.3s
Precision: binary64
\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \leq -1.5826630768290659 \cdot 10^{+59}:\\ \;\;\;\;x + \left(y \cdot 3.13060547623 + \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(\left(\frac{t}{z} + -36.527041698806414\right) \cdot \frac{\sqrt[3]{y}}{z}\right)\right)\\ \mathbf{elif}\;z \leq 2.6725087956259304 \cdot 10^{+43}:\\ \;\;\;\;x + \frac{y}{\frac{z \cdot \left(z \cdot \left(z \cdot \left(z + 15.234687407\right) + 31.4690115749\right) + 11.9400905721\right) + 0.607771387771}{z \cdot \left(z \cdot \left(t + z \cdot \left(z \cdot 3.13060547623 + 11.1667541262\right)\right) + a\right) + b}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot \left(3.13060547623 + \frac{t}{z \cdot z}\right) + -36.527041698806414 \cdot \frac{y}{z}\right)\\ \end{array}\]
x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}
↓
\begin{array}{l}
\mathbf{if}\;z \leq -1.5826630768290659 \cdot 10^{+59}:\\
\;\;\;\;x + \left(y \cdot 3.13060547623 + \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(\left(\frac{t}{z} + -36.527041698806414\right) \cdot \frac{\sqrt[3]{y}}{z}\right)\right)\\

\mathbf{elif}\;z \leq 2.6725087956259304 \cdot 10^{+43}:\\
\;\;\;\;x + \frac{y}{\frac{z \cdot \left(z \cdot \left(z \cdot \left(z + 15.234687407\right) + 31.4690115749\right) + 11.9400905721\right) + 0.607771387771}{z \cdot \left(z \cdot \left(t + z \cdot \left(z \cdot 3.13060547623 + 11.1667541262\right)\right) + a\right) + b}}\\

\mathbf{else}:\\
\;\;\;\;x + \left(y \cdot \left(3.13060547623 + \frac{t}{z \cdot z}\right) + -36.527041698806414 \cdot \frac{y}{z}\right)\\

\end{array}
(FPCore (x y z t a b)
 :precision binary64
 (+
  x
  (/
   (*
    y
    (+ (* (+ (* (+ (* (+ (* z 3.13060547623) 11.1667541262) z) t) z) a) z) b))
   (+
    (* (+ (* (+ (* (+ z 15.234687407) z) 31.4690115749) z) 11.9400905721) z)
    0.607771387771))))
↓
(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -1.5826630768290659e+59)
   (+
    x
    (+
     (* y 3.13060547623)
     (*
      (* (cbrt y) (cbrt y))
      (* (+ (/ t z) -36.527041698806414) (/ (cbrt y) z)))))
   (if (<= z 2.6725087956259304e+43)
     (+
      x
      (/
       y
       (/
        (+
         (*
          z
          (+ (* z (+ (* z (+ z 15.234687407)) 31.4690115749)) 11.9400905721))
         0.607771387771)
        (+
         (* z (+ (* z (+ t (* z (+ (* z 3.13060547623) 11.1667541262)))) a))
         b))))
     (+
      x
      (+
       (* y (+ 3.13060547623 (/ t (* z z))))
       (* -36.527041698806414 (/ y z)))))))
double code(double x, double y, double z, double t, double a, double b) {
	return x + ((y * ((((((((z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / (((((((z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771));
}
↓
double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -1.5826630768290659e+59) {
		tmp = x + ((y * 3.13060547623) + ((cbrt(y) * cbrt(y)) * (((t / z) + -36.527041698806414) * (cbrt(y) / z))));
	} else if (z <= 2.6725087956259304e+43) {
		tmp = x + (y / (((z * ((z * ((z * (z + 15.234687407)) + 31.4690115749)) + 11.9400905721)) + 0.607771387771) / ((z * ((z * (t + (z * ((z * 3.13060547623) + 11.1667541262)))) + a)) + b)));
	} else {
		tmp = x + ((y * (3.13060547623 + (t / (z * z)))) + (-36.527041698806414 * (y / z)));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.3
Target1.0
Herbie1.1
\[\begin{array}{l} \mathbf{if}\;z < -6.499344996252632 \cdot 10^{+53}:\\ \;\;\;\;x + \left(\left(3.13060547623 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \mathbf{elif}\;z < 7.066965436914287 \cdot 10^{+59}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}{\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(3.13060547623 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -1.58266307682907e59

    1. Initial program 62.1

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}\]

    2. Taylor expanded around inf 8.0

      \[\leadsto x + \color{blue}{\left(\left(3.13060547623 \cdot y + \frac{t \cdot y}{{z}^{2}}\right) - 36.527041698806414 \cdot \frac{y}{z}\right)}\]

    3. Simplified0.8

      \[\leadsto x + \color{blue}{\left(y \cdot 3.13060547623 + \frac{y}{z} \cdot \left(\frac{t}{z} + -36.527041698806414\right)\right)}\]
    4. Using strategy rm

    5. Applied *-un-lft-identity_binary64_109900.8

      \[\leadsto x + \left(y \cdot 3.13060547623 + \frac{y}{\color{blue}{1 \cdot z}} \cdot \left(\frac{t}{z} + -36.527041698806414\right)\right)\]

    6. Applied add-cube-cbrt_binary64_110250.8

      \[\leadsto x + \left(y \cdot 3.13060547623 + \frac{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}{1 \cdot z} \cdot \left(\frac{t}{z} + -36.527041698806414\right)\right)\]

    7. Applied times-frac_binary64_109960.8

      \[\leadsto x + \left(y \cdot 3.13060547623 + \color{blue}{\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{1} \cdot \frac{\sqrt[3]{y}}{z}\right)} \cdot \left(\frac{t}{z} + -36.527041698806414\right)\right)\]

    8. Applied associate-*l*_binary64_109310.6

      \[\leadsto x + \left(y \cdot 3.13060547623 + \color{blue}{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{1} \cdot \left(\frac{\sqrt[3]{y}}{z} \cdot \left(\frac{t}{z} + -36.527041698806414\right)\right)}\right)\]

    9. Simplified0.6

      \[\leadsto x + \left(y \cdot 3.13060547623 + \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{1} \cdot \color{blue}{\left(\left(\frac{t}{z} + -36.527041698806414\right) \cdot \frac{\sqrt[3]{y}}{z}\right)}\right)\]
    10. Using strategy rm

    11. Applied div-inv_binary64_109870.6

      \[\leadsto x + \left(y \cdot 3.13060547623 + \color{blue}{\left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \frac{1}{1}\right)} \cdot \left(\left(\frac{t}{z} + -36.527041698806414\right) \cdot \frac{\sqrt[3]{y}}{z}\right)\right)\]

    12. Applied associate-*l*_binary64_109310.6

      \[\leadsto x + \left(y \cdot 3.13060547623 + \color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(\frac{1}{1} \cdot \left(\left(\frac{t}{z} + -36.527041698806414\right) \cdot \frac{\sqrt[3]{y}}{z}\right)\right)}\right)\]

    13. Simplified0.6

      \[\leadsto x + \left(y \cdot 3.13060547623 + \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \color{blue}{\left(\left(\frac{t}{z} + -36.527041698806414\right) \cdot \frac{\sqrt[3]{y}}{z}\right)}\right)\]

    if -1.58266307682907e59 < z < 2.67250879562593e43

    1. Initial program 2.7

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}\]
    2. Using strategy rm

    3. Applied associate-/l*_binary64_109351.2

      \[\leadsto x + \color{blue}{\frac{y}{\frac{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}{\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}}\]

    4. Simplified1.2

      \[\leadsto x + \frac{y}{\color{blue}{\frac{z \cdot \left(z \cdot \left(z \cdot \left(z + 15.234687407\right) + 31.4690115749\right) + 11.9400905721\right) + 0.607771387771}{z \cdot \left(z \cdot \left(z \cdot \left(z \cdot 3.13060547623 + 11.1667541262\right) + t\right) + a\right) + b}}}\]

    if 2.67250879562593e43 < z

    1. Initial program 60.1

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}\]

    2. Taylor expanded around inf 8.5

      \[\leadsto x + \color{blue}{\left(\left(3.13060547623 \cdot y + \frac{t \cdot y}{{z}^{2}}\right) - 36.527041698806414 \cdot \frac{y}{z}\right)}\]

    3. Simplified1.2

      \[\leadsto x + \color{blue}{\left(y \cdot 3.13060547623 + \frac{y}{z} \cdot \left(\frac{t}{z} + -36.527041698806414\right)\right)}\]
    4. Using strategy rm

    5. Applied distribute-rgt-in_binary64_109401.2

      \[\leadsto x + \left(y \cdot 3.13060547623 + \color{blue}{\left(\frac{t}{z} \cdot \frac{y}{z} + -36.527041698806414 \cdot \frac{y}{z}\right)}\right)\]

    6. Applied associate-+r+_binary64_109221.2

      \[\leadsto x + \color{blue}{\left(\left(y \cdot 3.13060547623 + \frac{t}{z} \cdot \frac{y}{z}\right) + -36.527041698806414 \cdot \frac{y}{z}\right)}\]

    7. Simplified1.1

      \[\leadsto x + \left(\color{blue}{y \cdot \left(\frac{t}{z \cdot z} + 3.13060547623\right)} + -36.527041698806414 \cdot \frac{y}{z}\right)\]
  3. Recombined 3 regimes into one program.

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.5826630768290659 \cdot 10^{+59}:\\ \;\;\;\;x + \left(y \cdot 3.13060547623 + \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(\left(\frac{t}{z} + -36.527041698806414\right) \cdot \frac{\sqrt[3]{y}}{z}\right)\right)\\ \mathbf{elif}\;z \leq 2.6725087956259304 \cdot 10^{+43}:\\ \;\;\;\;x + \frac{y}{\frac{z \cdot \left(z \cdot \left(z \cdot \left(z + 15.234687407\right) + 31.4690115749\right) + 11.9400905721\right) + 0.607771387771}{z \cdot \left(z \cdot \left(t + z \cdot \left(z \cdot 3.13060547623 + 11.1667541262\right)\right) + a\right) + b}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot \left(3.13060547623 + \frac{t}{z \cdot z}\right) + -36.527041698806414 \cdot \frac{y}{z}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020355 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, D"
  :precision binary64

  :herbie-target
  (if (< z -6.499344996252632e+53) (+ x (* (+ (- 3.13060547623 (/ 36.527041698806414 z)) (/ t (* z z))) (/ y 1.0))) (if (< z 7.066965436914287e+59) (+ x (/ y (/ (+ (* (+ (* (+ (* (+ z 15.234687407) z) 31.4690115749) z) 11.9400905721) z) 0.607771387771) (+ (* (+ (* (+ (* (+ (* z 3.13060547623) 11.1667541262) z) t) z) a) z) b)))) (+ x (* (+ (- 3.13060547623 (/ 36.527041698806414 z)) (/ t (* z z))) (/ y 1.0)))))

  (+ x (/ (* y (+ (* (+ (* (+ (* (+ (* z 3.13060547623) 11.1667541262) z) t) z) a) z) b)) (+ (* (+ (* (+ (* (+ z 15.234687407) z) 31.4690115749) z) 11.9400905721) z) 0.607771387771))))