Average Error: 26.5 → 0.8
Time: 11.9s
Precision: 64
\[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -3.5558868413567101 \cdot 10^{22} \lor \neg \left(x \le 1.1453649274389147 \cdot 10^{36}\right):\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922227999964, \frac{y}{{x}^{2}}\right) - 110.113924298481081\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \left(\left(\frac{\left({\left(\mathsf{fma}\left(x, 4.16438922227999964, 78.6994924154000017\right) \cdot x\right)}^{3} + {137.51941641600001}^{3}\right) \cdot x}{\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right) \cdot \left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right) + \left(137.51941641600001 \cdot 137.51941641600001 - \left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right) \cdot 137.51941641600001\right)} + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\\ \end{array}\]
\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 -3.5558868413567101 \cdot 10^{22} \lor \neg \left(x \le 1.1453649274389147 \cdot 10^{36}\right):\\
\;\;\;\;\mathsf{fma}\left(x, 4.16438922227999964, \frac{y}{{x}^{2}}\right) - 110.113924298481081\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(x - 2\right) \cdot \left(\left(\frac{\left({\left(\mathsf{fma}\left(x, 4.16438922227999964, 78.6994924154000017\right) \cdot x\right)}^{3} + {137.51941641600001}^{3}\right) \cdot x}{\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right) \cdot \left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right) + \left(137.51941641600001 \cdot 137.51941641600001 - \left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right) \cdot 137.51941641600001\right)} + 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 r364029 = x;
        double r364030 = 2.0;
        double r364031 = r364029 - r364030;
        double r364032 = 4.16438922228;
        double r364033 = r364029 * r364032;
        double r364034 = 78.6994924154;
        double r364035 = r364033 + r364034;
        double r364036 = r364035 * r364029;
        double r364037 = 137.519416416;
        double r364038 = r364036 + r364037;
        double r364039 = r364038 * r364029;
        double r364040 = y;
        double r364041 = r364039 + r364040;
        double r364042 = r364041 * r364029;
        double r364043 = z;
        double r364044 = r364042 + r364043;
        double r364045 = r364031 * r364044;
        double r364046 = 43.3400022514;
        double r364047 = r364029 + r364046;
        double r364048 = r364047 * r364029;
        double r364049 = 263.505074721;
        double r364050 = r364048 + r364049;
        double r364051 = r364050 * r364029;
        double r364052 = 313.399215894;
        double r364053 = r364051 + r364052;
        double r364054 = r364053 * r364029;
        double r364055 = 47.066876606;
        double r364056 = r364054 + r364055;
        double r364057 = r364045 / r364056;
        return r364057;
}
↓
double f(double x, double y, double z) {
        double r364058 = x;
        double r364059 = -3.55588684135671e+22;
        bool r364060 = r364058 <= r364059;
        double r364061 = 1.1453649274389147e+36;
        bool r364062 = r364058 <= r364061;
        double r364063 = !r364062;
        bool r364064 = r364060 || r364063;
        double r364065 = 4.16438922228;
        double r364066 = y;
        double r364067 = 2.0;
        double r364068 = pow(r364058, r364067);
        double r364069 = r364066 / r364068;
        double r364070 = fma(r364058, r364065, r364069);
        double r364071 = 110.11392429848108;
        double r364072 = r364070 - r364071;
        double r364073 = 2.0;
        double r364074 = r364058 - r364073;
        double r364075 = 78.6994924154;
        double r364076 = fma(r364058, r364065, r364075);
        double r364077 = r364076 * r364058;
        double r364078 = 3.0;
        double r364079 = pow(r364077, r364078);
        double r364080 = 137.519416416;
        double r364081 = pow(r364080, r364078);
        double r364082 = r364079 + r364081;
        double r364083 = r364082 * r364058;
        double r364084 = r364058 * r364065;
        double r364085 = r364084 + r364075;
        double r364086 = r364085 * r364058;
        double r364087 = r364086 * r364086;
        double r364088 = r364080 * r364080;
        double r364089 = r364086 * r364080;
        double r364090 = r364088 - r364089;
        double r364091 = r364087 + r364090;
        double r364092 = r364083 / r364091;
        double r364093 = r364092 + r364066;
        double r364094 = r364093 * r364058;
        double r364095 = z;
        double r364096 = r364094 + r364095;
        double r364097 = r364074 * r364096;
        double r364098 = 43.3400022514;
        double r364099 = r364058 + r364098;
        double r364100 = r364099 * r364058;
        double r364101 = 263.505074721;
        double r364102 = r364100 + r364101;
        double r364103 = r364102 * r364058;
        double r364104 = 313.399215894;
        double r364105 = r364103 + r364104;
        double r364106 = r364105 * r364058;
        double r364107 = 47.066876606;
        double r364108 = r364106 + r364107;
        double r364109 = r364097 / r364108;
        double r364110 = r364064 ? r364072 : r364109;
        return r364110;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original26.5
Target0.4
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;x \lt -3.3261287258700048 \cdot 10^{62}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999964 \cdot x\right) - 110.11392429848109\\ \mathbf{elif}\;x \lt 9.4299917145546727 \cdot 10^{55}:\\ \;\;\;\;\frac{x - 2}{1} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z}{\left(\left(263.50507472100003 \cdot x + \left(43.3400022514000014 \cdot \left(x \cdot x\right) + x \cdot \left(x \cdot x\right)\right)\right) + 313.399215894\right) \cdot x + 47.066876606000001}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999964 \cdot x\right) - 110.11392429848109\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -3.55588684135671e+22 or 1.1453649274389147e+36 < 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. Using strategy rm

    3. Applied flip--58.3

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

    4. Applied associate-*l/60.6

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

    5. Applied associate-/l/60.6

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

    6. Simplified60.6

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

    7. Taylor expanded around inf 1.1

      \[\leadsto \color{blue}{\left(\frac{y}{{x}^{2}} + 4.16438922227999964 \cdot x\right) - 110.113924298481081}\]

    8. Simplified1.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 4.16438922227999964, \frac{y}{{x}^{2}}\right) - 110.113924298481081}\]

    if -3.55588684135671e+22 < x < 1.1453649274389147e+36

    1. Initial program 0.6

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

    3. Applied flip3-+0.6

      \[\leadsto \frac{\left(x - 2\right) \cdot \left(\left(\color{blue}{\frac{{\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right)}^{3} + {137.51941641600001}^{3}}{\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right) \cdot \left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right) + \left(137.51941641600001 \cdot 137.51941641600001 - \left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right) \cdot 137.51941641600001\right)}} \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\]

    4. Applied associate-*l/0.6

      \[\leadsto \frac{\left(x - 2\right) \cdot \left(\left(\color{blue}{\frac{\left({\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right)}^{3} + {137.51941641600001}^{3}\right) \cdot x}{\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right) \cdot \left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right) + \left(137.51941641600001 \cdot 137.51941641600001 - \left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right) \cdot 137.51941641600001\right)}} + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\]

    5. Simplified0.6

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.5558868413567101 \cdot 10^{22} \lor \neg \left(x \le 1.1453649274389147 \cdot 10^{36}\right):\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922227999964, \frac{y}{{x}^{2}}\right) - 110.113924298481081\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \left(\left(\frac{\left({\left(\mathsf{fma}\left(x, 4.16438922227999964, 78.6994924154000017\right) \cdot x\right)}^{3} + {137.51941641600001}^{3}\right) \cdot x}{\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right) \cdot \left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right) + \left(137.51941641600001 \cdot 137.51941641600001 - \left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right) \cdot 137.51941641600001\right)} + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 +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)))