Average Error: 3.7 → 1.3
Time: 29.2s
Precision: 64
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
↓
\[\frac{x}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(\frac{\frac{2}{t}}{3} - \left(a + \frac{5}{6}\right), b - c, \frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}}\right)\right)}, x\right)}\]
\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, {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(\frac{\frac{2}{t}}{3} - \left(a + \frac{5}{6}\right), b - c, \frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}}\right)\right)}, x\right)}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r85248 = x;
        double r85249 = y;
        double r85250 = 2.0;
        double r85251 = z;
        double r85252 = t;
        double r85253 = a;
        double r85254 = r85252 + r85253;
        double r85255 = sqrt(r85254);
        double r85256 = r85251 * r85255;
        double r85257 = r85256 / r85252;
        double r85258 = b;
        double r85259 = c;
        double r85260 = r85258 - r85259;
        double r85261 = 5.0;
        double r85262 = 6.0;
        double r85263 = r85261 / r85262;
        double r85264 = r85253 + r85263;
        double r85265 = 3.0;
        double r85266 = r85252 * r85265;
        double r85267 = r85250 / r85266;
        double r85268 = r85264 - r85267;
        double r85269 = r85260 * r85268;
        double r85270 = r85257 - r85269;
        double r85271 = r85250 * r85270;
        double r85272 = exp(r85271);
        double r85273 = r85249 * r85272;
        double r85274 = r85248 + r85273;
        double r85275 = r85248 / r85274;
        return r85275;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r85276 = x;
        double r85277 = y;
        double r85278 = 2.0;
        double r85279 = exp(r85278);
        double r85280 = t;
        double r85281 = r85278 / r85280;
        double r85282 = 3.0;
        double r85283 = r85281 / r85282;
        double r85284 = a;
        double r85285 = 5.0;
        double r85286 = 6.0;
        double r85287 = r85285 / r85286;
        double r85288 = r85284 + r85287;
        double r85289 = r85283 - r85288;
        double r85290 = b;
        double r85291 = c;
        double r85292 = r85290 - r85291;
        double r85293 = z;
        double r85294 = cbrt(r85280);
        double r85295 = r85294 * r85294;
        double r85296 = r85293 / r85295;
        double r85297 = r85280 + r85284;
        double r85298 = sqrt(r85297);
        double r85299 = r85298 / r85294;
        double r85300 = r85296 * r85299;
        double r85301 = fma(r85289, r85292, r85300);
        double r85302 = pow(r85279, r85301);
        double r85303 = fma(r85277, r85302, r85276);
        double r85304 = r85276 / r85303;
        return r85304;
}

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 3.7

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

  2. Simplified2.4

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

  4. Applied add-cube-cbrt2.4

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

  5. Applied times-frac1.3

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

  6. Final simplification1.3

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

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(FPCore (x y z t a b c)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2"
  :precision binary64
  (/ x (+ x (* y (exp (* 2 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5 6)) (/ 2 (* t 3)))))))))))