Average Error: 3.8 → 1.7
Time: 23.9s
Precision: 64
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
↓
\[\frac{x}{\mathsf{fma}\left(y, e^{2 \cdot \mathsf{fma}\left(c - b, \frac{5}{6} + \left(a - \frac{\frac{2}{t}}{3}\right), \left(\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt{t + a}\right) \cdot \frac{\sqrt[3]{z}}{t}\right)}, x\right)}\]
\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}
↓
\frac{x}{\mathsf{fma}\left(y, e^{2 \cdot \mathsf{fma}\left(c - b, \frac{5}{6} + \left(a - \frac{\frac{2}{t}}{3}\right), \left(\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt{t + a}\right) \cdot \frac{\sqrt[3]{z}}{t}\right)}, x\right)}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r62228 = x;
        double r62229 = y;
        double r62230 = 2.0;
        double r62231 = z;
        double r62232 = t;
        double r62233 = a;
        double r62234 = r62232 + r62233;
        double r62235 = sqrt(r62234);
        double r62236 = r62231 * r62235;
        double r62237 = r62236 / r62232;
        double r62238 = b;
        double r62239 = c;
        double r62240 = r62238 - r62239;
        double r62241 = 5.0;
        double r62242 = 6.0;
        double r62243 = r62241 / r62242;
        double r62244 = r62233 + r62243;
        double r62245 = 3.0;
        double r62246 = r62232 * r62245;
        double r62247 = r62230 / r62246;
        double r62248 = r62244 - r62247;
        double r62249 = r62240 * r62248;
        double r62250 = r62237 - r62249;
        double r62251 = r62230 * r62250;
        double r62252 = exp(r62251);
        double r62253 = r62229 * r62252;
        double r62254 = r62228 + r62253;
        double r62255 = r62228 / r62254;
        return r62255;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r62256 = x;
        double r62257 = y;
        double r62258 = 2.0;
        double r62259 = c;
        double r62260 = b;
        double r62261 = r62259 - r62260;
        double r62262 = 5.0;
        double r62263 = 6.0;
        double r62264 = r62262 / r62263;
        double r62265 = a;
        double r62266 = t;
        double r62267 = r62258 / r62266;
        double r62268 = 3.0;
        double r62269 = r62267 / r62268;
        double r62270 = r62265 - r62269;
        double r62271 = r62264 + r62270;
        double r62272 = z;
        double r62273 = cbrt(r62272);
        double r62274 = r62273 * r62273;
        double r62275 = r62266 + r62265;
        double r62276 = sqrt(r62275);
        double r62277 = r62274 * r62276;
        double r62278 = r62273 / r62266;
        double r62279 = r62277 * r62278;
        double r62280 = fma(r62261, r62271, r62279);
        double r62281 = r62258 * r62280;
        double r62282 = exp(r62281);
        double r62283 = fma(r62257, r62282, r62256);
        double r62284 = r62256 / r62283;
        return r62284;
}

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

Bits error versus c

Derivation

  1. Initial program 3.8

    \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]

  2. Simplified1.9

    \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, e^{2 \cdot \mathsf{fma}\left(c - b, \frac{5}{6} + \left(a - \frac{\frac{2}{t}}{3}\right), \sqrt{a + t} \cdot \frac{z}{t}\right)}, x\right)}}\]
  3. Using strategy rm

  4. Applied *-un-lft-identity1.9

    \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2 \cdot \mathsf{fma}\left(c - b, \frac{5}{6} + \left(a - \frac{\frac{2}{t}}{3}\right), \sqrt{a + t} \cdot \frac{z}{\color{blue}{1 \cdot t}}\right)}, x\right)}\]

  5. Applied add-cube-cbrt1.9

    \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2 \cdot \mathsf{fma}\left(c - b, \frac{5}{6} + \left(a - \frac{\frac{2}{t}}{3}\right), \sqrt{a + t} \cdot \frac{\color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}}{1 \cdot t}\right)}, x\right)}\]

  6. Applied times-frac1.9

    \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2 \cdot \mathsf{fma}\left(c - b, \frac{5}{6} + \left(a - \frac{\frac{2}{t}}{3}\right), \sqrt{a + t} \cdot \color{blue}{\left(\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{1} \cdot \frac{\sqrt[3]{z}}{t}\right)}\right)}, x\right)}\]

  7. Applied associate-*r*1.7

    \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2 \cdot \mathsf{fma}\left(c - b, \frac{5}{6} + \left(a - \frac{\frac{2}{t}}{3}\right), \color{blue}{\left(\sqrt{a + t} \cdot \frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{1}\right) \cdot \frac{\sqrt[3]{z}}{t}}\right)}, x\right)}\]

  8. Simplified1.7

    \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2 \cdot \mathsf{fma}\left(c - b, \frac{5}{6} + \left(a - \frac{\frac{2}{t}}{3}\right), \color{blue}{\left(\sqrt{t + a} \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right)} \cdot \frac{\sqrt[3]{z}}{t}\right)}, x\right)}\]

  9. Final simplification1.7

    \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2 \cdot \mathsf{fma}\left(c - b, \frac{5}{6} + \left(a - \frac{\frac{2}{t}}{3}\right), \left(\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt{t + a}\right) \cdot \frac{\sqrt[3]{z}}{t}\right)}, x\right)}\]

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (x y z t a b c)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2"
  (/ x (+ x (* y (exp (* 2.0 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0)))))))))))