Average Error: 4.1 → 1.4
Time: 12.4s
Precision: 64
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
↓
\[\frac{x}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(\frac{\frac{2}{3}}{t} - \left(a + \frac{5}{6}\right), b - c, \frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}}\right)\right)}, x\right)}\]
\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}{3}}{t} - \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 r91222 = x;
        double r91223 = y;
        double r91224 = 2.0;
        double r91225 = z;
        double r91226 = t;
        double r91227 = a;
        double r91228 = r91226 + r91227;
        double r91229 = sqrt(r91228);
        double r91230 = r91225 * r91229;
        double r91231 = r91230 / r91226;
        double r91232 = b;
        double r91233 = c;
        double r91234 = r91232 - r91233;
        double r91235 = 5.0;
        double r91236 = 6.0;
        double r91237 = r91235 / r91236;
        double r91238 = r91227 + r91237;
        double r91239 = 3.0;
        double r91240 = r91226 * r91239;
        double r91241 = r91224 / r91240;
        double r91242 = r91238 - r91241;
        double r91243 = r91234 * r91242;
        double r91244 = r91231 - r91243;
        double r91245 = r91224 * r91244;
        double r91246 = exp(r91245);
        double r91247 = r91223 * r91246;
        double r91248 = r91222 + r91247;
        double r91249 = r91222 / r91248;
        return r91249;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r91250 = x;
        double r91251 = y;
        double r91252 = 2.0;
        double r91253 = exp(r91252);
        double r91254 = 3.0;
        double r91255 = r91252 / r91254;
        double r91256 = t;
        double r91257 = r91255 / r91256;
        double r91258 = a;
        double r91259 = 5.0;
        double r91260 = 6.0;
        double r91261 = r91259 / r91260;
        double r91262 = r91258 + r91261;
        double r91263 = r91257 - r91262;
        double r91264 = b;
        double r91265 = c;
        double r91266 = r91264 - r91265;
        double r91267 = z;
        double r91268 = cbrt(r91256);
        double r91269 = r91268 * r91268;
        double r91270 = r91267 / r91269;
        double r91271 = r91256 + r91258;
        double r91272 = sqrt(r91271);
        double r91273 = r91272 / r91268;
        double r91274 = r91270 * r91273;
        double r91275 = fma(r91263, r91266, r91274);
        double r91276 = pow(r91253, r91275);
        double r91277 = fma(r91251, r91276, r91250);
        double r91278 = r91250 / r91277;
        return r91278;
}

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 4.1

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

  2. Simplified2.8

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

  4. Applied add-cube-cbrt2.8

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

  5. Applied times-frac1.4

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

  6. Final simplification1.4

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

Reproduce

herbie shell --seed 2020042 +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)))))))))))