Average Error: 3.7 → 2.5
Time: 26.2s
Precision: 64
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
↓
\[\frac{x}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(\frac{\frac{2}{t}}{3} - \left(a + \frac{5}{6}\right), b - c, \frac{z \cdot \sqrt{t + a}}{t}\right)\right)}, x\right)}\]
\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 \cdot \sqrt{t + a}}{t}\right)\right)}, x\right)}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r64204 = x;
        double r64205 = y;
        double r64206 = 2.0;
        double r64207 = z;
        double r64208 = t;
        double r64209 = a;
        double r64210 = r64208 + r64209;
        double r64211 = sqrt(r64210);
        double r64212 = r64207 * r64211;
        double r64213 = r64212 / r64208;
        double r64214 = b;
        double r64215 = c;
        double r64216 = r64214 - r64215;
        double r64217 = 5.0;
        double r64218 = 6.0;
        double r64219 = r64217 / r64218;
        double r64220 = r64209 + r64219;
        double r64221 = 3.0;
        double r64222 = r64208 * r64221;
        double r64223 = r64206 / r64222;
        double r64224 = r64220 - r64223;
        double r64225 = r64216 * r64224;
        double r64226 = r64213 - r64225;
        double r64227 = r64206 * r64226;
        double r64228 = exp(r64227);
        double r64229 = r64205 * r64228;
        double r64230 = r64204 + r64229;
        double r64231 = r64204 / r64230;
        return r64231;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r64232 = x;
        double r64233 = y;
        double r64234 = 2.0;
        double r64235 = exp(r64234);
        double r64236 = t;
        double r64237 = r64234 / r64236;
        double r64238 = 3.0;
        double r64239 = r64237 / r64238;
        double r64240 = a;
        double r64241 = 5.0;
        double r64242 = 6.0;
        double r64243 = r64241 / r64242;
        double r64244 = r64240 + r64243;
        double r64245 = r64239 - r64244;
        double r64246 = b;
        double r64247 = c;
        double r64248 = r64246 - r64247;
        double r64249 = z;
        double r64250 = r64236 + r64240;
        double r64251 = sqrt(r64250);
        double r64252 = r64249 * r64251;
        double r64253 = r64252 / r64236;
        double r64254 = fma(r64245, r64248, r64253);
        double r64255 = pow(r64235, r64254);
        double r64256 = fma(r64233, r64255, r64232);
        double r64257 = r64232 / r64256;
        return r64257;
}

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. Simplified2.5

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

  3. Final simplification2.5

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

Reproduce

herbie shell --seed 2019306 +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)))))))))))