Average Error: 26.1 → 0.7
Time: 20.7s
Precision: 64
\[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -6.944319865754395156979684186439296035944 \cdot 10^{68} \lor \neg \left(x \le 2.184683867268628449643800778858061665375 \cdot 10^{74}\right):\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922227999963610045597306452691555, \frac{y}{{x}^{2}}\right) - 110.1139242984810948655649553984403610229\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, x, 2 \cdot \left(x + 2\right)\right)} \cdot \frac{{x}^{3} - {2}^{3}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x + 43.3400022514000013984514225739985704422, x, 263.5050747210000281484099105000495910645\right), x, 313.3992158940000081202015280723571777344\right), x, 47.06687660600000100430406746454536914825\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 4.16438922227999963610045597306452691555, 78.69949241540000173245061887428164482117\right), x, 137.5194164160000127594685181975364685059\right), x, y\right), x, z\right)}}\\ \end{array}\]
\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}
↓
\begin{array}{l}
\mathbf{if}\;x \le -6.944319865754395156979684186439296035944 \cdot 10^{68} \lor \neg \left(x \le 2.184683867268628449643800778858061665375 \cdot 10^{74}\right):\\
\;\;\;\;\mathsf{fma}\left(x, 4.16438922227999963610045597306452691555, \frac{y}{{x}^{2}}\right) - 110.1139242984810948655649553984403610229\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\mathsf{fma}\left(x, x, 2 \cdot \left(x + 2\right)\right)} \cdot \frac{{x}^{3} - {2}^{3}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x + 43.3400022514000013984514225739985704422, x, 263.5050747210000281484099105000495910645\right), x, 313.3992158940000081202015280723571777344\right), x, 47.06687660600000100430406746454536914825\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 4.16438922227999963610045597306452691555, 78.69949241540000173245061887428164482117\right), x, 137.5194164160000127594685181975364685059\right), x, y\right), x, z\right)}}\\

\end{array}
double f(double x, double y, double z) {
        double r461789 = x;
        double r461790 = 2.0;
        double r461791 = r461789 - r461790;
        double r461792 = 4.16438922228;
        double r461793 = r461789 * r461792;
        double r461794 = 78.6994924154;
        double r461795 = r461793 + r461794;
        double r461796 = r461795 * r461789;
        double r461797 = 137.519416416;
        double r461798 = r461796 + r461797;
        double r461799 = r461798 * r461789;
        double r461800 = y;
        double r461801 = r461799 + r461800;
        double r461802 = r461801 * r461789;
        double r461803 = z;
        double r461804 = r461802 + r461803;
        double r461805 = r461791 * r461804;
        double r461806 = 43.3400022514;
        double r461807 = r461789 + r461806;
        double r461808 = r461807 * r461789;
        double r461809 = 263.505074721;
        double r461810 = r461808 + r461809;
        double r461811 = r461810 * r461789;
        double r461812 = 313.399215894;
        double r461813 = r461811 + r461812;
        double r461814 = r461813 * r461789;
        double r461815 = 47.066876606;
        double r461816 = r461814 + r461815;
        double r461817 = r461805 / r461816;
        return r461817;
}
↓
double f(double x, double y, double z) {
        double r461818 = x;
        double r461819 = -6.944319865754395e+68;
        bool r461820 = r461818 <= r461819;
        double r461821 = 2.1846838672686284e+74;
        bool r461822 = r461818 <= r461821;
        double r461823 = !r461822;
        bool r461824 = r461820 || r461823;
        double r461825 = 4.16438922228;
        double r461826 = y;
        double r461827 = 2.0;
        double r461828 = pow(r461818, r461827);
        double r461829 = r461826 / r461828;
        double r461830 = fma(r461818, r461825, r461829);
        double r461831 = 110.1139242984811;
        double r461832 = r461830 - r461831;
        double r461833 = 1.0;
        double r461834 = 2.0;
        double r461835 = r461818 + r461834;
        double r461836 = r461834 * r461835;
        double r461837 = fma(r461818, r461818, r461836);
        double r461838 = r461833 / r461837;
        double r461839 = 3.0;
        double r461840 = pow(r461818, r461839);
        double r461841 = pow(r461834, r461839);
        double r461842 = r461840 - r461841;
        double r461843 = 43.3400022514;
        double r461844 = r461818 + r461843;
        double r461845 = 263.505074721;
        double r461846 = fma(r461844, r461818, r461845);
        double r461847 = 313.399215894;
        double r461848 = fma(r461846, r461818, r461847);
        double r461849 = 47.066876606;
        double r461850 = fma(r461848, r461818, r461849);
        double r461851 = 78.6994924154;
        double r461852 = fma(r461818, r461825, r461851);
        double r461853 = 137.519416416;
        double r461854 = fma(r461852, r461818, r461853);
        double r461855 = fma(r461854, r461818, r461826);
        double r461856 = z;
        double r461857 = fma(r461855, r461818, r461856);
        double r461858 = r461850 / r461857;
        double r461859 = r461842 / r461858;
        double r461860 = r461838 * r461859;
        double r461861 = r461824 ? r461832 : r461860;
        return r461861;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original26.1
Target0.4
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;x \lt -3.326128725870004842699683658678411714981 \cdot 10^{62}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229\\ \mathbf{elif}\;x \lt 9.429991714554672672712552870340896976735 \cdot 10^{55}:\\ \;\;\;\;\frac{x - 2}{1} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z}{\left(\left(263.5050747210000281484099105000495910645 \cdot x + \left(43.3400022514000013984514225739985704422 \cdot \left(x \cdot x\right) + x \cdot \left(x \cdot x\right)\right)\right) + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -6.944319865754395e+68 or 2.1846838672686284e+74 < x

    1. Initial program 64.0

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\]

    2. Simplified62.8

      \[\leadsto \color{blue}{\frac{x - 2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x + 43.3400022514000013984514225739985704422, x, 263.5050747210000281484099105000495910645\right), x, 313.3992158940000081202015280723571777344\right), x, 47.06687660600000100430406746454536914825\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 4.16438922227999963610045597306452691555, 78.69949241540000173245061887428164482117\right), x, 137.5194164160000127594685181975364685059\right), x, y\right), x, z\right)}}}\]

    3. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{\left(\frac{y}{{x}^{2}} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229}\]

    4. Simplified0.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 4.16438922227999963610045597306452691555, \frac{y}{{x}^{2}}\right) - 110.1139242984810948655649553984403610229}\]

    if -6.944319865754395e+68 < x < 2.1846838672686284e+74

    1. Initial program 3.2

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\]

    2. Simplified1.1

      \[\leadsto \color{blue}{\frac{x - 2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x + 43.3400022514000013984514225739985704422, x, 263.5050747210000281484099105000495910645\right), x, 313.3992158940000081202015280723571777344\right), x, 47.06687660600000100430406746454536914825\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 4.16438922227999963610045597306452691555, 78.69949241540000173245061887428164482117\right), x, 137.5194164160000127594685181975364685059\right), x, y\right), x, z\right)}}}\]
    3. Using strategy rm

    4. Applied flip3--1.1

      \[\leadsto \frac{\color{blue}{\frac{{x}^{3} - {2}^{3}}{x \cdot x + \left(2 \cdot 2 + x \cdot 2\right)}}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x + 43.3400022514000013984514225739985704422, x, 263.5050747210000281484099105000495910645\right), x, 313.3992158940000081202015280723571777344\right), x, 47.06687660600000100430406746454536914825\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 4.16438922227999963610045597306452691555, 78.69949241540000173245061887428164482117\right), x, 137.5194164160000127594685181975364685059\right), x, y\right), x, z\right)}}\]

    5. Applied associate-/l/1.1

      \[\leadsto \color{blue}{\frac{{x}^{3} - {2}^{3}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x + 43.3400022514000013984514225739985704422, x, 263.5050747210000281484099105000495910645\right), x, 313.3992158940000081202015280723571777344\right), x, 47.06687660600000100430406746454536914825\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 4.16438922227999963610045597306452691555, 78.69949241540000173245061887428164482117\right), x, 137.5194164160000127594685181975364685059\right), x, y\right), x, z\right)} \cdot \left(x \cdot x + \left(2 \cdot 2 + x \cdot 2\right)\right)}}\]

    6. Simplified1.1

      \[\leadsto \frac{{x}^{3} - {2}^{3}}{\color{blue}{\mathsf{fma}\left(x, x, 2 \cdot \left(x + 2\right)\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x + 43.3400022514000013984514225739985704422, x, 263.5050747210000281484099105000495910645\right), x, 313.3992158940000081202015280723571777344\right), x, 47.06687660600000100430406746454536914825\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 4.16438922227999963610045597306452691555, 78.69949241540000173245061887428164482117\right), x, 137.5194164160000127594685181975364685059\right), x, y\right), x, z\right)}}}\]
    7. Using strategy rm

    8. Applied *-un-lft-identity1.1

      \[\leadsto \frac{\color{blue}{1 \cdot \left({x}^{3} - {2}^{3}\right)}}{\mathsf{fma}\left(x, x, 2 \cdot \left(x + 2\right)\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x + 43.3400022514000013984514225739985704422, x, 263.5050747210000281484099105000495910645\right), x, 313.3992158940000081202015280723571777344\right), x, 47.06687660600000100430406746454536914825\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 4.16438922227999963610045597306452691555, 78.69949241540000173245061887428164482117\right), x, 137.5194164160000127594685181975364685059\right), x, y\right), x, z\right)}}\]

    9. Applied times-frac1.1

      \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(x, x, 2 \cdot \left(x + 2\right)\right)} \cdot \frac{{x}^{3} - {2}^{3}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x + 43.3400022514000013984514225739985704422, x, 263.5050747210000281484099105000495910645\right), x, 313.3992158940000081202015280723571777344\right), x, 47.06687660600000100430406746454536914825\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 4.16438922227999963610045597306452691555, 78.69949241540000173245061887428164482117\right), x, 137.5194164160000127594685181975364685059\right), x, y\right), x, z\right)}}}\]
    10. Using strategy rm

    11. Applied *-un-lft-identity1.1

      \[\leadsto \frac{1}{\mathsf{fma}\left(x, x, 2 \cdot \left(x + 2\right)\right)} \cdot \color{blue}{\left(1 \cdot \frac{{x}^{3} - {2}^{3}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x + 43.3400022514000013984514225739985704422, x, 263.5050747210000281484099105000495910645\right), x, 313.3992158940000081202015280723571777344\right), x, 47.06687660600000100430406746454536914825\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 4.16438922227999963610045597306452691555, 78.69949241540000173245061887428164482117\right), x, 137.5194164160000127594685181975364685059\right), x, y\right), x, z\right)}}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -6.944319865754395156979684186439296035944 \cdot 10^{68} \lor \neg \left(x \le 2.184683867268628449643800778858061665375 \cdot 10^{74}\right):\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922227999963610045597306452691555, \frac{y}{{x}^{2}}\right) - 110.1139242984810948655649553984403610229\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, x, 2 \cdot \left(x + 2\right)\right)} \cdot \frac{{x}^{3} - {2}^{3}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x + 43.3400022514000013984514225739985704422, x, 263.5050747210000281484099105000495910645\right), x, 313.3992158940000081202015280723571777344\right), x, 47.06687660600000100430406746454536914825\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 4.16438922227999963610045597306452691555, 78.69949241540000173245061887428164482117\right), x, 137.5194164160000127594685181975364685059\right), x, y\right), x, z\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019351 +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)))