Jmat.Real.gamma, branch z less than 0.5

Average Error: 1.8 → 0.4

Time: 5.8m

Precision: 64

• could not determine a ground truth for program body

  1. z = -1.5889599578209953e+300

Math

\[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099298181841732002794742584 + \frac{676.5203681218850988443591631948947906494}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.139216722402807135949842631816864014}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531346025876700878143310547}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621405870046146446838974952698}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.50734327868690520801919774385169148445}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.1385710952657201178173096423051902092993}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019571583242346146658263705831 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.505632735149311617592788074479481785772 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]

↓

\[\left(\left(\left(\left(\left(\left(\frac{676.5203681218850988443591631948947906494}{1 - z} + \frac{-1259.139216722402807135949842631816864014}{\left(-z\right) + 2}\right) + \frac{771.3234287776531346025876700878143310547}{\left(-z\right) + 3}\right) + 0.9999999999998099298181841732002794742584\right) + \frac{12.50734327868690520801919774385169148445}{\left(-z\right) + 5}\right) + \left(\frac{-176.6150291621405870046146446838974952698}{\left(-z\right) + 4} + \left(\left(\frac{-0.1385710952657201178173096423051902092993}{\left(-z\right) + 6} + \frac{1.505632735149311617592788074479481785772 \cdot 10^{-7}}{\left(-z\right) + 8}\right) + \frac{9.984369578019571583242346146658263705831 \cdot 10^{-6}}{7 + \left(-z\right)}\right)\right)\right) \cdot \sqrt{2 \cdot \pi}\right) \cdot \left(\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \frac{{\left(\sqrt[3]{7 + \left(\left(-z\right) + 0.5\right)} \cdot \sqrt[3]{7 + \left(\left(-z\right) + 0.5\right)}\right)}^{\left(\left(-z\right) + 0.5\right)}}{\frac{e^{7 + \left(\left(-z\right) + 0.5\right)}}{{\left(\sqrt[3]{7 + \left(\left(-z\right) + 0.5\right)}\right)}^{\left(\left(-z\right) + 0.5\right)}}}\right)\]

C

double f(double z) {
        double r8204150 = atan2(1.0, 0.0);
        double r8204151 = z;
        double r8204152 = r8204150 * r8204151;
        double r8204153 = sin(r8204152);
        double r8204154 = r8204150 / r8204153;
        double r8204155 = 2.0;
        double r8204156 = r8204150 * r8204155;
        double r8204157 = sqrt(r8204156);
        double r8204158 = 1.0;
        double r8204159 = r8204158 - r8204151;
        double r8204160 = r8204159 - r8204158;
        double r8204161 = 7.0;
        double r8204162 = r8204160 + r8204161;
        double r8204163 = 0.5;
        double r8204164 = r8204162 + r8204163;
        double r8204165 = r8204160 + r8204163;
        double r8204166 = pow(r8204164, r8204165);
        double r8204167 = r8204157 * r8204166;
        double r8204168 = -r8204164;
        double r8204169 = exp(r8204168);
        double r8204170 = r8204167 * r8204169;
        double r8204171 = 0.9999999999998099;
        double r8204172 = 676.5203681218851;
        double r8204173 = r8204160 + r8204158;
        double r8204174 = r8204172 / r8204173;
        double r8204175 = r8204171 + r8204174;
        double r8204176 = -1259.1392167224028;
        double r8204177 = r8204160 + r8204155;
        double r8204178 = r8204176 / r8204177;
        double r8204179 = r8204175 + r8204178;
        double r8204180 = 771.3234287776531;
        double r8204181 = 3.0;
        double r8204182 = r8204160 + r8204181;
        double r8204183 = r8204180 / r8204182;
        double r8204184 = r8204179 + r8204183;
        double r8204185 = -176.6150291621406;
        double r8204186 = 4.0;
        double r8204187 = r8204160 + r8204186;
        double r8204188 = r8204185 / r8204187;
        double r8204189 = r8204184 + r8204188;
        double r8204190 = 12.507343278686905;
        double r8204191 = 5.0;
        double r8204192 = r8204160 + r8204191;
        double r8204193 = r8204190 / r8204192;
        double r8204194 = r8204189 + r8204193;
        double r8204195 = -0.13857109526572012;
        double r8204196 = 6.0;
        double r8204197 = r8204160 + r8204196;
        double r8204198 = r8204195 / r8204197;
        double r8204199 = r8204194 + r8204198;
        double r8204200 = 9.984369578019572e-06;
        double r8204201 = r8204200 / r8204162;
        double r8204202 = r8204199 + r8204201;
        double r8204203 = 1.5056327351493116e-07;
        double r8204204 = 8.0;
        double r8204205 = r8204160 + r8204204;
        double r8204206 = r8204203 / r8204205;
        double r8204207 = r8204202 + r8204206;
        double r8204208 = r8204170 * r8204207;
        double r8204209 = r8204154 * r8204208;
        return r8204209;
}

↓

double f(double z) {
        double r8204210 = 676.5203681218851;
        double r8204211 = 1.0;
        double r8204212 = z;
        double r8204213 = r8204211 - r8204212;
        double r8204214 = r8204210 / r8204213;
        double r8204215 = -1259.1392167224028;
        double r8204216 = -r8204212;
        double r8204217 = 2.0;
        double r8204218 = r8204216 + r8204217;
        double r8204219 = r8204215 / r8204218;
        double r8204220 = r8204214 + r8204219;
        double r8204221 = 771.3234287776531;
        double r8204222 = 3.0;
        double r8204223 = r8204216 + r8204222;
        double r8204224 = r8204221 / r8204223;
        double r8204225 = r8204220 + r8204224;
        double r8204226 = 0.9999999999998099;
        double r8204227 = r8204225 + r8204226;
        double r8204228 = 12.507343278686905;
        double r8204229 = 5.0;
        double r8204230 = r8204216 + r8204229;
        double r8204231 = r8204228 / r8204230;
        double r8204232 = r8204227 + r8204231;
        double r8204233 = -176.6150291621406;
        double r8204234 = 4.0;
        double r8204235 = r8204216 + r8204234;
        double r8204236 = r8204233 / r8204235;
        double r8204237 = -0.13857109526572012;
        double r8204238 = 6.0;
        double r8204239 = r8204216 + r8204238;
        double r8204240 = r8204237 / r8204239;
        double r8204241 = 1.5056327351493116e-07;
        double r8204242 = 8.0;
        double r8204243 = r8204216 + r8204242;
        double r8204244 = r8204241 / r8204243;
        double r8204245 = r8204240 + r8204244;
        double r8204246 = 9.984369578019572e-06;
        double r8204247 = 7.0;
        double r8204248 = r8204247 + r8204216;
        double r8204249 = r8204246 / r8204248;
        double r8204250 = r8204245 + r8204249;
        double r8204251 = r8204236 + r8204250;
        double r8204252 = r8204232 + r8204251;
        double r8204253 = atan2(1.0, 0.0);
        double r8204254 = r8204217 * r8204253;
        double r8204255 = sqrt(r8204254);
        double r8204256 = r8204252 * r8204255;
        double r8204257 = r8204212 * r8204253;
        double r8204258 = sin(r8204257);
        double r8204259 = r8204253 / r8204258;
        double r8204260 = 0.5;
        double r8204261 = r8204216 + r8204260;
        double r8204262 = r8204247 + r8204261;
        double r8204263 = cbrt(r8204262);
        double r8204264 = r8204263 * r8204263;
        double r8204265 = pow(r8204264, r8204261);
        double r8204266 = exp(r8204262);
        double r8204267 = pow(r8204263, r8204261);
        double r8204268 = r8204266 / r8204267;
        double r8204269 = r8204265 / r8204268;
        double r8204270 = r8204259 * r8204269;
        double r8204271 = r8204256 * r8204270;
        return r8204271;
}

Error

Bits error versus z

Derivation

1. Initial program 1.8

   \[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099298181841732002794742584 + \frac{676.5203681218850988443591631948947906494}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.139216722402807135949842631816864014}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531346025876700878143310547}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621405870046146446838974952698}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.50734327868690520801919774385169148445}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.1385710952657201178173096423051902092993}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019571583242346146658263705831 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.505632735149311617592788074479481785772 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]

2. Simplified 0.8

   \[\leadsto \color{blue}{\left(\left(\left(\left(\left(\left(\frac{676.5203681218850988443591631948947906494}{1 - z} + \frac{-1259.139216722402807135949842631816864014}{2 + \left(-z\right)}\right) + \frac{771.3234287776531346025876700878143310547}{3 + \left(-z\right)}\right) + 0.9999999999998099298181841732002794742584\right) + \frac{12.50734327868690520801919774385169148445}{\left(-z\right) + 5}\right) + \left(\frac{-176.6150291621405870046146446838974952698}{\left(-z\right) + 4} + \left(\left(\frac{-0.1385710952657201178173096423051902092993}{6 + \left(-z\right)} + \frac{1.505632735149311617592788074479481785772 \cdot 10^{-7}}{8 + \left(-z\right)}\right) + \frac{9.984369578019571583242346146658263705831 \cdot 10^{-6}}{7 + \left(-z\right)}\right)\right)\right) \cdot \sqrt{2 \cdot \pi}\right) \cdot \left(\frac{{\left(7 + \left(0.5 + \left(-z\right)\right)\right)}^{\left(0.5 + \left(-z\right)\right)}}{e^{7 + \left(0.5 + \left(-z\right)\right)}} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right)}\]
3. Using strategy rm

4. Applied add-cube-cbrt 0.8

   \[\leadsto \left(\left(\left(\left(\left(\left(\frac{676.5203681218850988443591631948947906494}{1 - z} + \frac{-1259.139216722402807135949842631816864014}{2 + \left(-z\right)}\right) + \frac{771.3234287776531346025876700878143310547}{3 + \left(-z\right)}\right) + 0.9999999999998099298181841732002794742584\right) + \frac{12.50734327868690520801919774385169148445}{\left(-z\right) + 5}\right) + \left(\frac{-176.6150291621405870046146446838974952698}{\left(-z\right) + 4} + \left(\left(\frac{-0.1385710952657201178173096423051902092993}{6 + \left(-z\right)} + \frac{1.505632735149311617592788074479481785772 \cdot 10^{-7}}{8 + \left(-z\right)}\right) + \frac{9.984369578019571583242346146658263705831 \cdot 10^{-6}}{7 + \left(-z\right)}\right)\right)\right) \cdot \sqrt{2 \cdot \pi}\right) \cdot \left(\frac{{\color{blue}{\left(\left(\sqrt[3]{7 + \left(0.5 + \left(-z\right)\right)} \cdot \sqrt[3]{7 + \left(0.5 + \left(-z\right)\right)}\right) \cdot \sqrt[3]{7 + \left(0.5 + \left(-z\right)\right)}\right)}}^{\left(0.5 + \left(-z\right)\right)}}{e^{7 + \left(0.5 + \left(-z\right)\right)}} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right)\]

5. Applied unpow-prod-down 0.8

   \[\leadsto \left(\left(\left(\left(\left(\left(\frac{676.5203681218850988443591631948947906494}{1 - z} + \frac{-1259.139216722402807135949842631816864014}{2 + \left(-z\right)}\right) + \frac{771.3234287776531346025876700878143310547}{3 + \left(-z\right)}\right) + 0.9999999999998099298181841732002794742584\right) + \frac{12.50734327868690520801919774385169148445}{\left(-z\right) + 5}\right) + \left(\frac{-176.6150291621405870046146446838974952698}{\left(-z\right) + 4} + \left(\left(\frac{-0.1385710952657201178173096423051902092993}{6 + \left(-z\right)} + \frac{1.505632735149311617592788074479481785772 \cdot 10^{-7}}{8 + \left(-z\right)}\right) + \frac{9.984369578019571583242346146658263705831 \cdot 10^{-6}}{7 + \left(-z\right)}\right)\right)\right) \cdot \sqrt{2 \cdot \pi}\right) \cdot \left(\frac{\color{blue}{{\left(\sqrt[3]{7 + \left(0.5 + \left(-z\right)\right)} \cdot \sqrt[3]{7 + \left(0.5 + \left(-z\right)\right)}\right)}^{\left(0.5 + \left(-z\right)\right)} \cdot {\left(\sqrt[3]{7 + \left(0.5 + \left(-z\right)\right)}\right)}^{\left(0.5 + \left(-z\right)\right)}}}{e^{7 + \left(0.5 + \left(-z\right)\right)}} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right)\]

6. Applied associate-/l* 0.4

   \[\leadsto \left(\left(\left(\left(\left(\left(\frac{676.5203681218850988443591631948947906494}{1 - z} + \frac{-1259.139216722402807135949842631816864014}{2 + \left(-z\right)}\right) + \frac{771.3234287776531346025876700878143310547}{3 + \left(-z\right)}\right) + 0.9999999999998099298181841732002794742584\right) + \frac{12.50734327868690520801919774385169148445}{\left(-z\right) + 5}\right) + \left(\frac{-176.6150291621405870046146446838974952698}{\left(-z\right) + 4} + \left(\left(\frac{-0.1385710952657201178173096423051902092993}{6 + \left(-z\right)} + \frac{1.505632735149311617592788074479481785772 \cdot 10^{-7}}{8 + \left(-z\right)}\right) + \frac{9.984369578019571583242346146658263705831 \cdot 10^{-6}}{7 + \left(-z\right)}\right)\right)\right) \cdot \sqrt{2 \cdot \pi}\right) \cdot \left(\color{blue}{\frac{{\left(\sqrt[3]{7 + \left(0.5 + \left(-z\right)\right)} \cdot \sqrt[3]{7 + \left(0.5 + \left(-z\right)\right)}\right)}^{\left(0.5 + \left(-z\right)\right)}}{\frac{e^{7 + \left(0.5 + \left(-z\right)\right)}}{{\left(\sqrt[3]{7 + \left(0.5 + \left(-z\right)\right)}\right)}^{\left(0.5 + \left(-z\right)\right)}}}} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right)\]

7. Final simplification 0.4

   \[\leadsto \left(\left(\left(\left(\left(\left(\frac{676.5203681218850988443591631948947906494}{1 - z} + \frac{-1259.139216722402807135949842631816864014}{\left(-z\right) + 2}\right) + \frac{771.3234287776531346025876700878143310547}{\left(-z\right) + 3}\right) + 0.9999999999998099298181841732002794742584\right) + \frac{12.50734327868690520801919774385169148445}{\left(-z\right) + 5}\right) + \left(\frac{-176.6150291621405870046146446838974952698}{\left(-z\right) + 4} + \left(\left(\frac{-0.1385710952657201178173096423051902092993}{\left(-z\right) + 6} + \frac{1.505632735149311617592788074479481785772 \cdot 10^{-7}}{\left(-z\right) + 8}\right) + \frac{9.984369578019571583242346146658263705831 \cdot 10^{-6}}{7 + \left(-z\right)}\right)\right)\right) \cdot \sqrt{2 \cdot \pi}\right) \cdot \left(\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \frac{{\left(\sqrt[3]{7 + \left(\left(-z\right) + 0.5\right)} \cdot \sqrt[3]{7 + \left(\left(-z\right) + 0.5\right)}\right)}^{\left(\left(-z\right) + 0.5\right)}}{\frac{e^{7 + \left(\left(-z\right) + 0.5\right)}}{{\left(\sqrt[3]{7 + \left(\left(-z\right) + 0.5\right)}\right)}^{\left(\left(-z\right) + 0.5\right)}}}\right)\]

Reproduce

herbie shell --seed 2019179 
(FPCore (z)
  :name "Jmat.Real.gamma, branch z less than 0.5"
  (* (/ PI (sin (* PI z))) (* (* (* (sqrt (* PI 2.0)) (pow (+ (+ (- (- 1.0 z) 1.0) 7.0) 0.5) (+ (- (- 1.0 z) 1.0) 0.5))) (exp (- (+ (+ (- (- 1.0 z) 1.0) 7.0) 0.5)))) (+ (+ (+ (+ (+ (+ (+ (+ 0.9999999999998099 (/ 676.5203681218851 (+ (- (- 1.0 z) 1.0) 1.0))) (/ -1259.1392167224028 (+ (- (- 1.0 z) 1.0) 2.0))) (/ 771.3234287776531 (+ (- (- 1.0 z) 1.0) 3.0))) (/ -176.6150291621406 (+ (- (- 1.0 z) 1.0) 4.0))) (/ 12.507343278686905 (+ (- (- 1.0 z) 1.0) 5.0))) (/ -0.13857109526572012 (+ (- (- 1.0 z) 1.0) 6.0))) (/ 9.984369578019572e-06 (+ (- (- 1.0 z) 1.0) 7.0))) (/ 1.5056327351493116e-07 (+ (- (- 1.0 z) 1.0) 8.0))))))