Average Error: 25.7 → 0.9
Time: 47.8s
Precision: binary64
\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
↓
\[\begin{array}{l} \mathbf{if}\;z \leq -6.974765525849115 \cdot 10^{+19}:\\ \;\;\;\;\left(\left(\frac{y}{z \cdot z} \cdot \left(457.9610022158428 + \frac{a}{z}\right) + \mathsf{fma}\left(3.13060547623, y, x\right)\right) + \frac{y}{z} \cdot \left(\frac{t}{z} - \frac{5864.8025282699045}{z \cdot z}\right)\right) - \mathsf{fma}\left(36.52704169880642, \frac{y}{z}, 15.234687407 \cdot \left(t \cdot \frac{y}{{z}^{3}}\right)\right)\\ \mathbf{elif}\;z \leq 6.433678393573731 \cdot 10^{+39}:\\ \;\;\;\;x + y \cdot \left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 3.13060547623, 11.1667541262\right), t\right), a\right), b\right) \cdot \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \left(\frac{457.9610022158428}{z \cdot z} + \frac{t}{z \cdot z}\right) + \left(3.13060547623 - \frac{36.52704169880642}{z}\right), x\right)\\ \end{array} \]
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 -6.974765525849115 \cdot 10^{+19}:\\
\;\;\;\;\left(\left(\frac{y}{z \cdot z} \cdot \left(457.9610022158428 + \frac{a}{z}\right) + \mathsf{fma}\left(3.13060547623, y, x\right)\right) + \frac{y}{z} \cdot \left(\frac{t}{z} - \frac{5864.8025282699045}{z \cdot z}\right)\right) - \mathsf{fma}\left(36.52704169880642, \frac{y}{z}, 15.234687407 \cdot \left(t \cdot \frac{y}{{z}^{3}}\right)\right)\\

\mathbf{elif}\;z \leq 6.433678393573731 \cdot 10^{+39}:\\
\;\;\;\;x + y \cdot \left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 3.13060547623, 11.1667541262\right), t\right), a\right), b\right) \cdot \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(y, \left(\frac{457.9610022158428}{z \cdot z} + \frac{t}{z \cdot z}\right) + \left(3.13060547623 - \frac{36.52704169880642}{z}\right), x\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 -6.974765525849115e+19)
   (-
    (+
     (+
      (* (/ y (* z z)) (+ 457.9610022158428 (/ a z)))
      (fma 3.13060547623 y x))
     (* (/ y z) (- (/ t z) (/ 5864.8025282699045 (* z z)))))
    (fma 36.52704169880642 (/ y z) (* 15.234687407 (* t (/ y (pow z 3.0))))))
   (if (<= z 6.433678393573731e+39)
     (+
      x
      (*
       y
       (*
        (fma z (fma z (fma z (fma z 3.13060547623 11.1667541262) t) a) b)
        (/
         1.0
         (fma
          z
          (fma z (fma z (+ z 15.234687407) 31.4690115749) 11.9400905721)
          0.607771387771)))))
     (fma
      y
      (+
       (+ (/ 457.9610022158428 (* z z)) (/ t (* z z)))
       (- 3.13060547623 (/ 36.52704169880642 z)))
      x))))
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 <= -6.974765525849115e+19) {
		tmp = ((((y / (z * z)) * (457.9610022158428 + (a / z))) + fma(3.13060547623, y, x)) + ((y / z) * ((t / z) - (5864.8025282699045 / (z * z))))) - fma(36.52704169880642, (y / z), (15.234687407 * (t * (y / pow(z, 3.0)))));
	} else if (z <= 6.433678393573731e+39) {
		tmp = x + (y * (fma(z, fma(z, fma(z, fma(z, 3.13060547623, 11.1667541262), t), a), b) * (1.0 / fma(z, fma(z, fma(z, (z + 15.234687407), 31.4690115749), 11.9400905721), 0.607771387771))));
	} else {
		tmp = fma(y, (((457.9610022158428 / (z * z)) + (t / (z * z))) + (3.13060547623 - (36.52704169880642 / z))), x);
	}
	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

Target

Original25.7
Target0.9
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;z < -6.499344996252632 \cdot 10^{+53}:\\ \;\;\;\;x + \left(\left(3.13060547623 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \mathbf{elif}\;z < 7.066965436914287 \cdot 10^{+59}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}{\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(3.13060547623 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if z < -69747655258491150300

    1. Initial program 55.7

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

    2. Simplified53.2

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

    3. Taylor expanded in z around inf 12.8

      \[\leadsto \color{blue}{\left(\frac{y \cdot t}{{z}^{2}} + \left(3.13060547623 \cdot y + \left(\frac{y \cdot a}{{z}^{3}} + \left(457.9610022158428 \cdot \frac{y}{{z}^{2}} + x\right)\right)\right)\right) - \left(5864.8025282699045 \cdot \frac{y}{{z}^{3}} + \left(36.52704169880642 \cdot \frac{y}{z} + 15.234687407 \cdot \frac{y \cdot t}{{z}^{3}}\right)\right)} \]

    4. Simplified1.4

      \[\leadsto \color{blue}{\left(\left(\frac{y}{z \cdot z} \cdot \left(457.9610022158428 + \frac{a}{z}\right) + \mathsf{fma}\left(3.13060547623, y, x\right)\right) + \frac{y}{z} \cdot \left(\frac{t}{z} - \frac{5864.8025282699045}{z \cdot z}\right)\right) - \mathsf{fma}\left(36.52704169880642, \frac{y}{z}, 15.234687407 \cdot \left(\frac{y}{{z}^{3}} \cdot t\right)\right)} \]

    if -69747655258491150300 < z < 6.4336783935737311e39

    1. Initial program 0.9

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

    2. Simplified0.5

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

    3. Applied fma-udef_binary640.5

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

    4. Applied div-inv_binary640.5

      \[\leadsto y \cdot \color{blue}{\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 3.13060547623, 11.1667541262\right), t\right), a\right), b\right) \cdot \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}\right)} + x \]

    if 6.4336783935737311e39 < z

    1. Initial program 59.3

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

    2. Simplified56.9

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

    3. Taylor expanded in z around inf 1.3

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(3.13060547623 + \left(457.9610022158428 \cdot \frac{1}{{z}^{2}} + \frac{t}{{z}^{2}}\right)\right) - 36.52704169880642 \cdot \frac{1}{z}}, x\right) \]

    4. Simplified1.3

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(\frac{457.9610022158428}{z \cdot z} + \frac{t}{z \cdot z}\right) + \left(3.13060547623 - \frac{36.52704169880642}{z}\right)}, x\right) \]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.974765525849115 \cdot 10^{+19}:\\ \;\;\;\;\left(\left(\frac{y}{z \cdot z} \cdot \left(457.9610022158428 + \frac{a}{z}\right) + \mathsf{fma}\left(3.13060547623, y, x\right)\right) + \frac{y}{z} \cdot \left(\frac{t}{z} - \frac{5864.8025282699045}{z \cdot z}\right)\right) - \mathsf{fma}\left(36.52704169880642, \frac{y}{z}, 15.234687407 \cdot \left(t \cdot \frac{y}{{z}^{3}}\right)\right)\\ \mathbf{elif}\;z \leq 6.433678393573731 \cdot 10^{+39}:\\ \;\;\;\;x + y \cdot \left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 3.13060547623, 11.1667541262\right), t\right), a\right), b\right) \cdot \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \left(\frac{457.9610022158428}{z \cdot z} + \frac{t}{z \cdot z}\right) + \left(3.13060547623 - \frac{36.52704169880642}{z}\right), x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022068 
(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))))