Average Error: 26.3 → 1.0
Time: 8.7s
Precision: 64
\[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -26739016327637995500 \lor \neg \left(x \le 3.6330534504476709 \cdot 10^{40}\right):\\ \;\;\;\;\left(x - 2\right) \cdot \left(\left(\frac{y}{{x}^{3}} + 4.16438922227999964\right) - 101.785145853921094 \cdot \frac{1}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\\ \end{array}\]
\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}
↓
\begin{array}{l}
\mathbf{if}\;x \le -26739016327637995500 \lor \neg \left(x \le 3.6330534504476709 \cdot 10^{40}\right):\\
\;\;\;\;\left(x - 2\right) \cdot \left(\left(\frac{y}{{x}^{3}} + 4.16438922227999964\right) - 101.785145853921094 \cdot \frac{1}{x}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\\

\end{array}
double f(double x, double y, double z) {
        double r440034 = x;
        double r440035 = 2.0;
        double r440036 = r440034 - r440035;
        double r440037 = 4.16438922228;
        double r440038 = r440034 * r440037;
        double r440039 = 78.6994924154;
        double r440040 = r440038 + r440039;
        double r440041 = r440040 * r440034;
        double r440042 = 137.519416416;
        double r440043 = r440041 + r440042;
        double r440044 = r440043 * r440034;
        double r440045 = y;
        double r440046 = r440044 + r440045;
        double r440047 = r440046 * r440034;
        double r440048 = z;
        double r440049 = r440047 + r440048;
        double r440050 = r440036 * r440049;
        double r440051 = 43.3400022514;
        double r440052 = r440034 + r440051;
        double r440053 = r440052 * r440034;
        double r440054 = 263.505074721;
        double r440055 = r440053 + r440054;
        double r440056 = r440055 * r440034;
        double r440057 = 313.399215894;
        double r440058 = r440056 + r440057;
        double r440059 = r440058 * r440034;
        double r440060 = 47.066876606;
        double r440061 = r440059 + r440060;
        double r440062 = r440050 / r440061;
        return r440062;
}
↓
double f(double x, double y, double z) {
        double r440063 = x;
        double r440064 = -2.6739016327637996e+19;
        bool r440065 = r440063 <= r440064;
        double r440066 = 3.633053450447671e+40;
        bool r440067 = r440063 <= r440066;
        double r440068 = !r440067;
        bool r440069 = r440065 || r440068;
        double r440070 = 2.0;
        double r440071 = r440063 - r440070;
        double r440072 = y;
        double r440073 = 3.0;
        double r440074 = pow(r440063, r440073);
        double r440075 = r440072 / r440074;
        double r440076 = 4.16438922228;
        double r440077 = r440075 + r440076;
        double r440078 = 101.7851458539211;
        double r440079 = 1.0;
        double r440080 = r440079 / r440063;
        double r440081 = r440078 * r440080;
        double r440082 = r440077 - r440081;
        double r440083 = r440071 * r440082;
        double r440084 = r440063 * r440076;
        double r440085 = 78.6994924154;
        double r440086 = r440084 + r440085;
        double r440087 = r440086 * r440063;
        double r440088 = 137.519416416;
        double r440089 = r440087 + r440088;
        double r440090 = r440089 * r440063;
        double r440091 = r440090 + r440072;
        double r440092 = r440091 * r440063;
        double r440093 = z;
        double r440094 = r440092 + r440093;
        double r440095 = r440071 * r440094;
        double r440096 = 43.3400022514;
        double r440097 = r440063 + r440096;
        double r440098 = r440097 * r440063;
        double r440099 = 263.505074721;
        double r440100 = r440098 + r440099;
        double r440101 = r440100 * r440063;
        double r440102 = 313.399215894;
        double r440103 = r440101 + r440102;
        double r440104 = r440103 * r440063;
        double r440105 = 47.066876606;
        double r440106 = r440104 + r440105;
        double r440107 = r440095 / r440106;
        double r440108 = r440069 ? r440083 : r440107;
        return r440108;
}

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

Original26.3
Target0.5
Herbie1.0
\[\begin{array}{l} \mathbf{if}\;x \lt -3.3261287258700048 \cdot 10^{62}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999964 \cdot x\right) - 110.11392429848109\\ \mathbf{elif}\;x \lt 9.4299917145546727 \cdot 10^{55}:\\ \;\;\;\;\frac{x - 2}{1} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z}{\left(\left(263.50507472100003 \cdot x + \left(43.3400022514000014 \cdot \left(x \cdot x\right) + x \cdot \left(x \cdot x\right)\right)\right) + 313.399215894\right) \cdot x + 47.066876606000001}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999964 \cdot x\right) - 110.11392429848109\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -2.6739016327637996e+19 or 3.633053450447671e+40 < x

    1. Initial program 58.3

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\]

    2. Simplified54.3

      \[\leadsto \color{blue}{\frac{x - 2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x + 43.3400022514000014, x, 263.50507472100003\right), x, 313.399215894\right), x, 47.066876606000001\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 4.16438922227999964, 78.6994924154000017\right), x, 137.51941641600001\right), x, y\right), x, z\right)}}}\]
    3. Using strategy rm

    4. Applied div-inv54.3

      \[\leadsto \color{blue}{\left(x - 2\right) \cdot \frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x + 43.3400022514000014, x, 263.50507472100003\right), x, 313.399215894\right), x, 47.066876606000001\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 4.16438922227999964, 78.6994924154000017\right), x, 137.51941641600001\right), x, y\right), x, z\right)}}}\]

    5. Taylor expanded around inf 1.5

      \[\leadsto \left(x - 2\right) \cdot \color{blue}{\left(\left(\frac{y}{{x}^{3}} + 4.16438922227999964\right) - 101.785145853921094 \cdot \frac{1}{x}\right)}\]

    if -2.6739016327637996e+19 < x < 3.633053450447671e+40

    1. Initial program 0.6

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -26739016327637995500 \lor \neg \left(x \le 3.6330534504476709 \cdot 10^{40}\right):\\ \;\;\;\;\left(x - 2\right) \cdot \left(\left(\frac{y}{{x}^{3}} + 4.16438922227999964\right) - 101.785145853921094 \cdot \frac{1}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\\ \end{array}\]

Reproduce

herbie shell --seed 2020024 +o rules:numerics
(FPCore (x y z)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, C"
  :precision binary64

  :herbie-target
  (if (< x -3.326128725870005e+62) (- (+ (/ y (* x x)) (* 4.16438922228 x)) 110.1139242984811) (if (< x 9.429991714554673e+55) (* (/ (- x 2) 1) (/ (+ (* (+ (* (+ (* (+ (* x 4.16438922228) 78.6994924154) x) 137.519416416) x) y) x) z) (+ (* (+ (+ (* 263.505074721 x) (+ (* 43.3400022514 (* x x)) (* x (* x x)))) 313.399215894) x) 47.066876606))) (- (+ (/ y (* x x)) (* 4.16438922228 x)) 110.1139242984811)))

  (/ (* (- x 2) (+ (* (+ (* (+ (* (+ (* x 4.16438922228) 78.6994924154) x) 137.519416416) x) y) x) z)) (+ (* (+ (* (+ (* (+ x 43.3400022514) x) 263.505074721) x) 313.399215894) x) 47.066876606)))