Average Error: 26.5 → 0.7
Time: 11.3s
Precision: 64
\[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -5.029347387268419 \cdot 10^{53} \lor \neg \left(x \le 3.9187971556134169 \cdot 10^{49}\right):\\ \;\;\;\;\left(\frac{y}{{x}^{2}} + 4.16438922227999964 \cdot x\right) - 110.11392429848109\\ \mathbf{else}:\\ \;\;\;\;\frac{x - 2}{\sqrt{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z}{\sqrt{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}}\\ \end{array}\]
\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 -5.029347387268419 \cdot 10^{53} \lor \neg \left(x \le 3.9187971556134169 \cdot 10^{49}\right):\\
\;\;\;\;\left(\frac{y}{{x}^{2}} + 4.16438922227999964 \cdot x\right) - 110.11392429848109\\

\mathbf{else}:\\
\;\;\;\;\frac{x - 2}{\sqrt{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z}{\sqrt{\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 r372190 = x;
        double r372191 = 2.0;
        double r372192 = r372190 - r372191;
        double r372193 = 4.16438922228;
        double r372194 = r372190 * r372193;
        double r372195 = 78.6994924154;
        double r372196 = r372194 + r372195;
        double r372197 = r372196 * r372190;
        double r372198 = 137.519416416;
        double r372199 = r372197 + r372198;
        double r372200 = r372199 * r372190;
        double r372201 = y;
        double r372202 = r372200 + r372201;
        double r372203 = r372202 * r372190;
        double r372204 = z;
        double r372205 = r372203 + r372204;
        double r372206 = r372192 * r372205;
        double r372207 = 43.3400022514;
        double r372208 = r372190 + r372207;
        double r372209 = r372208 * r372190;
        double r372210 = 263.505074721;
        double r372211 = r372209 + r372210;
        double r372212 = r372211 * r372190;
        double r372213 = 313.399215894;
        double r372214 = r372212 + r372213;
        double r372215 = r372214 * r372190;
        double r372216 = 47.066876606;
        double r372217 = r372215 + r372216;
        double r372218 = r372206 / r372217;
        return r372218;
}
↓
double f(double x, double y, double z) {
        double r372219 = x;
        double r372220 = -5.029347387268419e+53;
        bool r372221 = r372219 <= r372220;
        double r372222 = 3.918797155613417e+49;
        bool r372223 = r372219 <= r372222;
        double r372224 = !r372223;
        bool r372225 = r372221 || r372224;
        double r372226 = y;
        double r372227 = 2.0;
        double r372228 = pow(r372219, r372227);
        double r372229 = r372226 / r372228;
        double r372230 = 4.16438922228;
        double r372231 = r372230 * r372219;
        double r372232 = r372229 + r372231;
        double r372233 = 110.1139242984811;
        double r372234 = r372232 - r372233;
        double r372235 = 2.0;
        double r372236 = r372219 - r372235;
        double r372237 = 43.3400022514;
        double r372238 = r372219 + r372237;
        double r372239 = r372238 * r372219;
        double r372240 = 263.505074721;
        double r372241 = r372239 + r372240;
        double r372242 = r372241 * r372219;
        double r372243 = 313.399215894;
        double r372244 = r372242 + r372243;
        double r372245 = r372244 * r372219;
        double r372246 = 47.066876606;
        double r372247 = r372245 + r372246;
        double r372248 = sqrt(r372247);
        double r372249 = r372236 / r372248;
        double r372250 = r372219 * r372230;
        double r372251 = 78.6994924154;
        double r372252 = r372250 + r372251;
        double r372253 = r372252 * r372219;
        double r372254 = 137.519416416;
        double r372255 = r372253 + r372254;
        double r372256 = r372255 * r372219;
        double r372257 = r372256 + r372226;
        double r372258 = r372257 * r372219;
        double r372259 = z;
        double r372260 = r372258 + r372259;
        double r372261 = r372260 / r372248;
        double r372262 = r372249 * r372261;
        double r372263 = r372225 ? r372234 : r372262;
        return r372263;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original26.5
Target0.5
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;x \lt -3.3261287258700048 \cdot 10^{62}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999964 \cdot x\right) - 110.11392429848109\\ \mathbf{elif}\;x \lt 9.4299917145546727 \cdot 10^{55}:\\ \;\;\;\;\frac{x - 2}{1} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z}{\left(\left(263.50507472100003 \cdot x + \left(43.3400022514000014 \cdot \left(x \cdot x\right) + x \cdot \left(x \cdot x\right)\right)\right) + 313.399215894\right) \cdot x + 47.066876606000001}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999964 \cdot x\right) - 110.11392429848109\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -5.029347387268419e+53 or 3.918797155613417e+49 < x

    1. Initial program 62.4

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

    2. Taylor expanded around inf 0.3

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

    if -5.029347387268419e+53 < x < 3.918797155613417e+49

    1. Initial program 1.3

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

    3. Applied add-sqr-sqrt1.4

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

    4. Applied times-frac0.9

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -5.029347387268419 \cdot 10^{53} \lor \neg \left(x \le 3.9187971556134169 \cdot 10^{49}\right):\\ \;\;\;\;\left(\frac{y}{{x}^{2}} + 4.16438922227999964 \cdot x\right) - 110.11392429848109\\ \mathbf{else}:\\ \;\;\;\;\frac{x - 2}{\sqrt{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z}{\sqrt{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 
(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)))