Average Error: 26.9 → 0.8
Time: 1.1m
Precision: 64
\[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -5.30632711353810765 \cdot 10^{24} \lor \neg \left(x \le 3.111579507262274 \cdot 10^{42}\right):\\ \;\;\;\;\left(\frac{y}{{x}^{2}} + 4.16438922227999964 \cdot x\right) - 110.11392429848109\\ \mathbf{else}:\\ \;\;\;\;\frac{x - 2}{\frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 4.16438922227999964, 78.6994924154000017\right), x, 137.51941641600001\right), x, y\right), x, z\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x + 43.3400022514000014, x, 263.50507472100003\right), x, 313.399215894\right), x, 47.066876606000001\right)}}}\\ \end{array}\]
\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 -5.30632711353810765 \cdot 10^{24} \lor \neg \left(x \le 3.111579507262274 \cdot 10^{42}\right):\\
\;\;\;\;\left(\frac{y}{{x}^{2}} + 4.16438922227999964 \cdot x\right) - 110.11392429848109\\

\mathbf{else}:\\
\;\;\;\;\frac{x - 2}{\frac{1}{\frac{\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)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x + 43.3400022514000014, x, 263.50507472100003\right), x, 313.399215894\right), x, 47.066876606000001\right)}}}\\

\end{array}
double f(double x, double y, double z) {
        double r534029 = x;
        double r534030 = 2.0;
        double r534031 = r534029 - r534030;
        double r534032 = 4.16438922228;
        double r534033 = r534029 * r534032;
        double r534034 = 78.6994924154;
        double r534035 = r534033 + r534034;
        double r534036 = r534035 * r534029;
        double r534037 = 137.519416416;
        double r534038 = r534036 + r534037;
        double r534039 = r534038 * r534029;
        double r534040 = y;
        double r534041 = r534039 + r534040;
        double r534042 = r534041 * r534029;
        double r534043 = z;
        double r534044 = r534042 + r534043;
        double r534045 = r534031 * r534044;
        double r534046 = 43.3400022514;
        double r534047 = r534029 + r534046;
        double r534048 = r534047 * r534029;
        double r534049 = 263.505074721;
        double r534050 = r534048 + r534049;
        double r534051 = r534050 * r534029;
        double r534052 = 313.399215894;
        double r534053 = r534051 + r534052;
        double r534054 = r534053 * r534029;
        double r534055 = 47.066876606;
        double r534056 = r534054 + r534055;
        double r534057 = r534045 / r534056;
        return r534057;
}
↓
double f(double x, double y, double z) {
        double r534058 = x;
        double r534059 = -5.306327113538108e+24;
        bool r534060 = r534058 <= r534059;
        double r534061 = 3.111579507262274e+42;
        bool r534062 = r534058 <= r534061;
        double r534063 = !r534062;
        bool r534064 = r534060 || r534063;
        double r534065 = y;
        double r534066 = 2.0;
        double r534067 = pow(r534058, r534066);
        double r534068 = r534065 / r534067;
        double r534069 = 4.16438922228;
        double r534070 = r534069 * r534058;
        double r534071 = r534068 + r534070;
        double r534072 = 110.1139242984811;
        double r534073 = r534071 - r534072;
        double r534074 = 2.0;
        double r534075 = r534058 - r534074;
        double r534076 = 1.0;
        double r534077 = 78.6994924154;
        double r534078 = fma(r534058, r534069, r534077);
        double r534079 = 137.519416416;
        double r534080 = fma(r534078, r534058, r534079);
        double r534081 = fma(r534080, r534058, r534065);
        double r534082 = z;
        double r534083 = fma(r534081, r534058, r534082);
        double r534084 = 43.3400022514;
        double r534085 = r534058 + r534084;
        double r534086 = 263.505074721;
        double r534087 = fma(r534085, r534058, r534086);
        double r534088 = 313.399215894;
        double r534089 = fma(r534087, r534058, r534088);
        double r534090 = 47.066876606;
        double r534091 = fma(r534089, r534058, r534090);
        double r534092 = r534083 / r534091;
        double r534093 = r534076 / r534092;
        double r534094 = r534075 / r534093;
        double r534095 = r534064 ? r534073 : r534094;
        return r534095;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original26.9
Target0.5
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 < -5.306327113538108e+24 or 3.111579507262274e+42 < x

    1. Initial program 59.4

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

    2. Simplified55.4

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

    4. Applied clear-num55.4

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

    5. Taylor expanded around inf 1.3

      \[\leadsto \color{blue}{\left(\frac{y}{{x}^{2}} + 4.16438922227999964 \cdot x\right) - 110.11392429848109}\]

    if -5.306327113538108e+24 < x < 3.111579507262274e+42

    1. Initial program 0.7

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

    2. Simplified0.5

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

    4. Applied clear-num0.5

      \[\leadsto \frac{x - 2}{\color{blue}{\frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 4.16438922227999964, 78.6994924154000017\right), x, 137.51941641600001\right), x, y\right), x, z\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x + 43.3400022514000014, x, 263.50507472100003\right), x, 313.399215894\right), x, 47.066876606000001\right)}}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -5.30632711353810765 \cdot 10^{24} \lor \neg \left(x \le 3.111579507262274 \cdot 10^{42}\right):\\ \;\;\;\;\left(\frac{y}{{x}^{2}} + 4.16438922227999964 \cdot x\right) - 110.11392429848109\\ \mathbf{else}:\\ \;\;\;\;\frac{x - 2}{\frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 4.16438922227999964, 78.6994924154000017\right), x, 137.51941641600001\right), x, y\right), x, z\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x + 43.3400022514000014, x, 263.50507472100003\right), x, 313.399215894\right), x, 47.066876606000001\right)}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020034 +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)))