Average Error: 26.9 → 1.1
Time: 9.9s
Precision: 64
\[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.5660234670732025 \cdot 10^{61} \lor \neg \left(x \le 27074635474235478000\right):\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922227999964, \frac{y}{{x}^{2}} - 110.11392429848109\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\\ \end{array}\]
\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 -1.5660234670732025 \cdot 10^{61} \lor \neg \left(x \le 27074635474235478000\right):\\
\;\;\;\;\mathsf{fma}\left(x, 4.16438922227999964, \frac{y}{{x}^{2}} - 110.11392429848109\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 r412035 = x;
        double r412036 = 2.0;
        double r412037 = r412035 - r412036;
        double r412038 = 4.16438922228;
        double r412039 = r412035 * r412038;
        double r412040 = 78.6994924154;
        double r412041 = r412039 + r412040;
        double r412042 = r412041 * r412035;
        double r412043 = 137.519416416;
        double r412044 = r412042 + r412043;
        double r412045 = r412044 * r412035;
        double r412046 = y;
        double r412047 = r412045 + r412046;
        double r412048 = r412047 * r412035;
        double r412049 = z;
        double r412050 = r412048 + r412049;
        double r412051 = r412037 * r412050;
        double r412052 = 43.3400022514;
        double r412053 = r412035 + r412052;
        double r412054 = r412053 * r412035;
        double r412055 = 263.505074721;
        double r412056 = r412054 + r412055;
        double r412057 = r412056 * r412035;
        double r412058 = 313.399215894;
        double r412059 = r412057 + r412058;
        double r412060 = r412059 * r412035;
        double r412061 = 47.066876606;
        double r412062 = r412060 + r412061;
        double r412063 = r412051 / r412062;
        return r412063;
}
↓
double f(double x, double y, double z) {
        double r412064 = x;
        double r412065 = -1.5660234670732025e+61;
        bool r412066 = r412064 <= r412065;
        double r412067 = 2.707463547423548e+19;
        bool r412068 = r412064 <= r412067;
        double r412069 = !r412068;
        bool r412070 = r412066 || r412069;
        double r412071 = 4.16438922228;
        double r412072 = y;
        double r412073 = 2.0;
        double r412074 = pow(r412064, r412073);
        double r412075 = r412072 / r412074;
        double r412076 = 110.1139242984811;
        double r412077 = r412075 - r412076;
        double r412078 = fma(r412064, r412071, r412077);
        double r412079 = 2.0;
        double r412080 = r412064 - r412079;
        double r412081 = r412064 * r412071;
        double r412082 = 78.6994924154;
        double r412083 = r412081 + r412082;
        double r412084 = r412083 * r412064;
        double r412085 = 137.519416416;
        double r412086 = r412084 + r412085;
        double r412087 = r412086 * r412064;
        double r412088 = r412087 + r412072;
        double r412089 = r412088 * r412064;
        double r412090 = z;
        double r412091 = r412089 + r412090;
        double r412092 = r412080 * r412091;
        double r412093 = 43.3400022514;
        double r412094 = r412064 + r412093;
        double r412095 = r412094 * r412064;
        double r412096 = 263.505074721;
        double r412097 = r412095 + r412096;
        double r412098 = r412097 * r412064;
        double r412099 = 313.399215894;
        double r412100 = r412098 + r412099;
        double r412101 = r412100 * r412064;
        double r412102 = 47.066876606;
        double r412103 = r412101 + r412102;
        double r412104 = r412092 / r412103;
        double r412105 = r412070 ? r412078 : r412104;
        return r412105;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original26.9
Target0.5
Herbie1.1
\[\begin{array}{l} \mathbf{if}\;x \lt -3.3261287258700048 \cdot 10^{62}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999964 \cdot x\right) - 110.11392429848109\\ \mathbf{elif}\;x \lt 9.4299917145546727 \cdot 10^{55}:\\ \;\;\;\;\frac{x - 2}{1} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z}{\left(\left(263.50507472100003 \cdot x + \left(43.3400022514000014 \cdot \left(x \cdot x\right) + x \cdot \left(x \cdot x\right)\right)\right) + 313.399215894\right) \cdot x + 47.066876606000001}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999964 \cdot x\right) - 110.11392429848109\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -1.5660234670732025e+61 or 2.707463547423548e+19 < x

    1. Initial program 59.9

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

    2. Simplified55.8

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

    3. Taylor expanded around inf 1.1

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

    4. Simplified1.1

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

    if -1.5660234670732025e+61 < x < 2.707463547423548e+19

    1. Initial program 1.1

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.5660234670732025 \cdot 10^{61} \lor \neg \left(x \le 27074635474235478000\right):\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922227999964, \frac{y}{{x}^{2}} - 110.11392429848109\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 +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)))