Average Error: 30.0 → 1.0
Time: 17.6s
Precision: 64
\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.604519949088867154891817928391541043497 \cdot 10^{47} \lor \neg \left(z \le 79071068415003800898868704359389069312\right):\\ \;\;\;\;\mathsf{fma}\left(y, 3.130605476229999961645944495103321969509 + \frac{1}{z} \cdot \frac{t}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.2346874069999991263557603815570473671, z, 31.46901157490000144889563671313226222992\right), z, 11.94009057210000079862766142468899488449\right), z, 0.6077713877710000378584709324059076607227\right)}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.130605476229999961645944495103321969509, 11.16675412620000074070958362426608800888\right), z, t\right), z, a\right), z, b\right), x\right)\\ \end{array}\]
x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.604519949088867154891817928391541043497 \cdot 10^{47} \lor \neg \left(z \le 79071068415003800898868704359389069312\right):\\
\;\;\;\;\mathsf{fma}\left(y, 3.130605476229999961645944495103321969509 + \frac{1}{z} \cdot \frac{t}{z}, x\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(y \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.2346874069999991263557603815570473671, z, 31.46901157490000144889563671313226222992\right), z, 11.94009057210000079862766142468899488449\right), z, 0.6077713877710000378584709324059076607227\right)}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.130605476229999961645944495103321969509, 11.16675412620000074070958362426608800888\right), z, t\right), z, a\right), z, b\right), x\right)\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r196063 = x;
        double r196064 = y;
        double r196065 = z;
        double r196066 = 3.13060547623;
        double r196067 = r196065 * r196066;
        double r196068 = 11.1667541262;
        double r196069 = r196067 + r196068;
        double r196070 = r196069 * r196065;
        double r196071 = t;
        double r196072 = r196070 + r196071;
        double r196073 = r196072 * r196065;
        double r196074 = a;
        double r196075 = r196073 + r196074;
        double r196076 = r196075 * r196065;
        double r196077 = b;
        double r196078 = r196076 + r196077;
        double r196079 = r196064 * r196078;
        double r196080 = 15.234687407;
        double r196081 = r196065 + r196080;
        double r196082 = r196081 * r196065;
        double r196083 = 31.4690115749;
        double r196084 = r196082 + r196083;
        double r196085 = r196084 * r196065;
        double r196086 = 11.9400905721;
        double r196087 = r196085 + r196086;
        double r196088 = r196087 * r196065;
        double r196089 = 0.607771387771;
        double r196090 = r196088 + r196089;
        double r196091 = r196079 / r196090;
        double r196092 = r196063 + r196091;
        return r196092;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r196093 = z;
        double r196094 = -1.6045199490888672e+47;
        bool r196095 = r196093 <= r196094;
        double r196096 = 7.90710684150038e+37;
        bool r196097 = r196093 <= r196096;
        double r196098 = !r196097;
        bool r196099 = r196095 || r196098;
        double r196100 = y;
        double r196101 = 3.13060547623;
        double r196102 = 1.0;
        double r196103 = r196102 / r196093;
        double r196104 = t;
        double r196105 = r196104 / r196093;
        double r196106 = r196103 * r196105;
        double r196107 = r196101 + r196106;
        double r196108 = x;
        double r196109 = fma(r196100, r196107, r196108);
        double r196110 = 15.234687407;
        double r196111 = r196093 + r196110;
        double r196112 = 31.4690115749;
        double r196113 = fma(r196111, r196093, r196112);
        double r196114 = 11.9400905721;
        double r196115 = fma(r196113, r196093, r196114);
        double r196116 = 0.607771387771;
        double r196117 = fma(r196115, r196093, r196116);
        double r196118 = r196102 / r196117;
        double r196119 = r196100 * r196118;
        double r196120 = 11.1667541262;
        double r196121 = fma(r196093, r196101, r196120);
        double r196122 = fma(r196121, r196093, r196104);
        double r196123 = a;
        double r196124 = fma(r196122, r196093, r196123);
        double r196125 = b;
        double r196126 = fma(r196124, r196093, r196125);
        double r196127 = fma(r196119, r196126, r196108);
        double r196128 = r196099 ? r196109 : r196127;
        return r196128;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Target

Original30.0
Target0.9
Herbie1.0
\[\begin{array}{l} \mathbf{if}\;z \lt -6.499344996252631754123144978817242590467 \cdot 10^{53}:\\ \;\;\;\;x + \left(\left(3.130605476229999961645944495103321969509 - \frac{36.52704169880641416057187598198652267456}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \mathbf{elif}\;z \lt 7.066965436914286795694558389038333165002 \cdot 10^{59}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}{\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(3.130605476229999961645944495103321969509 - \frac{36.52704169880641416057187598198652267456}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -1.6045199490888672e+47 or 7.90710684150038e+37 < z

    1. Initial program 60.6

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}\]

    2. Simplified58.7

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.2346874069999991263557603815570473671, z, 31.46901157490000144889563671313226222992\right), z, 11.94009057210000079862766142468899488449\right), z, 0.6077713877710000378584709324059076607227\right)}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.130605476229999961645944495103321969509, 11.16675412620000074070958362426608800888\right), z, t\right), z, a\right), z, b\right), x\right)}\]

    3. Taylor expanded around inf 8.7

      \[\leadsto \color{blue}{x + \left(\frac{t \cdot y}{{z}^{2}} + 3.130605476229999961645944495103321969509 \cdot y\right)}\]

    4. Simplified1.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 3.130605476229999961645944495103321969509 + \frac{t}{{z}^{2}}, x\right)}\]
    5. Using strategy rm

    6. Applied add-sqr-sqrt32.3

      \[\leadsto \mathsf{fma}\left(y, 3.130605476229999961645944495103321969509 + \frac{t}{{\color{blue}{\left(\sqrt{z} \cdot \sqrt{z}\right)}}^{2}}, x\right)\]

    7. Applied unpow-prod-down32.3

      \[\leadsto \mathsf{fma}\left(y, 3.130605476229999961645944495103321969509 + \frac{t}{\color{blue}{{\left(\sqrt{z}\right)}^{2} \cdot {\left(\sqrt{z}\right)}^{2}}}, x\right)\]

    8. Applied *-un-lft-identity32.3

      \[\leadsto \mathsf{fma}\left(y, 3.130605476229999961645944495103321969509 + \frac{\color{blue}{1 \cdot t}}{{\left(\sqrt{z}\right)}^{2} \cdot {\left(\sqrt{z}\right)}^{2}}, x\right)\]

    9. Applied times-frac32.3

      \[\leadsto \mathsf{fma}\left(y, 3.130605476229999961645944495103321969509 + \color{blue}{\frac{1}{{\left(\sqrt{z}\right)}^{2}} \cdot \frac{t}{{\left(\sqrt{z}\right)}^{2}}}, x\right)\]

    10. Simplified32.3

      \[\leadsto \mathsf{fma}\left(y, 3.130605476229999961645944495103321969509 + \color{blue}{\frac{1}{z}} \cdot \frac{t}{{\left(\sqrt{z}\right)}^{2}}, x\right)\]

    11. Simplified1.1

      \[\leadsto \mathsf{fma}\left(y, 3.130605476229999961645944495103321969509 + \frac{1}{z} \cdot \color{blue}{\frac{t}{z}}, x\right)\]

    if -1.6045199490888672e+47 < z < 7.90710684150038e+37

    1. Initial program 1.9

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}\]

    2. Simplified0.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.2346874069999991263557603815570473671, z, 31.46901157490000144889563671313226222992\right), z, 11.94009057210000079862766142468899488449\right), z, 0.6077713877710000378584709324059076607227\right)}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.130605476229999961645944495103321969509, 11.16675412620000074070958362426608800888\right), z, t\right), z, a\right), z, b\right), x\right)}\]
    3. Using strategy rm

    4. Applied div-inv0.9

      \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.2346874069999991263557603815570473671, z, 31.46901157490000144889563671313226222992\right), z, 11.94009057210000079862766142468899488449\right), z, 0.6077713877710000378584709324059076607227\right)}}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.130605476229999961645944495103321969509, 11.16675412620000074070958362426608800888\right), z, t\right), z, a\right), z, b\right), x\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.604519949088867154891817928391541043497 \cdot 10^{47} \lor \neg \left(z \le 79071068415003800898868704359389069312\right):\\ \;\;\;\;\mathsf{fma}\left(y, 3.130605476229999961645944495103321969509 + \frac{1}{z} \cdot \frac{t}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.2346874069999991263557603815570473671, z, 31.46901157490000144889563671313226222992\right), z, 11.94009057210000079862766142468899488449\right), z, 0.6077713877710000378584709324059076607227\right)}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.130605476229999961645944495103321969509, 11.16675412620000074070958362426608800888\right), z, t\right), z, a\right), z, b\right), x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019305 +o rules:numerics
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, D"
  :precision binary64

  :herbie-target
  (if (< z -6.4993449962526318e53) (+ x (* (+ (- 3.13060547622999996 (/ 36.527041698806414 z)) (/ t (* z z))) (/ y 1))) (if (< z 7.0669654369142868e59) (+ x (/ y (/ (+ (* (+ (* (+ (* (+ z 15.234687406999999) z) 31.469011574900001) z) 11.940090572100001) z) 0.60777138777100004) (+ (* (+ (* (+ (* (+ (* z 3.13060547622999996) 11.166754126200001) z) t) z) a) z) b)))) (+ x (* (+ (- 3.13060547622999996 (/ 36.527041698806414 z)) (/ t (* z z))) (/ y 1)))))

  (+ x (/ (* y (+ (* (+ (* (+ (* (+ (* z 3.13060547622999996) 11.166754126200001) z) t) z) a) z) b)) (+ (* (+ (* (+ (* (+ z 15.234687406999999) z) 31.469011574900001) z) 11.940090572100001) z) 0.60777138777100004))))