Average Error: 3.9 → 1.5
Time: 34.3s
Precision: 64
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
↓
\[\frac{x}{\mathsf{fma}\left(y, e^{2 \cdot \mathsf{fma}\left(c - b, \left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}, \frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}}\right)}, x\right)}\]
\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, \left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}, \frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}}\right)}, x\right)}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r6080230 = x;
        double r6080231 = y;
        double r6080232 = 2.0;
        double r6080233 = z;
        double r6080234 = t;
        double r6080235 = a;
        double r6080236 = r6080234 + r6080235;
        double r6080237 = sqrt(r6080236);
        double r6080238 = r6080233 * r6080237;
        double r6080239 = r6080238 / r6080234;
        double r6080240 = b;
        double r6080241 = c;
        double r6080242 = r6080240 - r6080241;
        double r6080243 = 5.0;
        double r6080244 = 6.0;
        double r6080245 = r6080243 / r6080244;
        double r6080246 = r6080235 + r6080245;
        double r6080247 = 3.0;
        double r6080248 = r6080234 * r6080247;
        double r6080249 = r6080232 / r6080248;
        double r6080250 = r6080246 - r6080249;
        double r6080251 = r6080242 * r6080250;
        double r6080252 = r6080239 - r6080251;
        double r6080253 = r6080232 * r6080252;
        double r6080254 = exp(r6080253);
        double r6080255 = r6080231 * r6080254;
        double r6080256 = r6080230 + r6080255;
        double r6080257 = r6080230 / r6080256;
        return r6080257;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r6080258 = x;
        double r6080259 = y;
        double r6080260 = 2.0;
        double r6080261 = c;
        double r6080262 = b;
        double r6080263 = r6080261 - r6080262;
        double r6080264 = a;
        double r6080265 = 5.0;
        double r6080266 = 6.0;
        double r6080267 = r6080265 / r6080266;
        double r6080268 = r6080264 + r6080267;
        double r6080269 = t;
        double r6080270 = 3.0;
        double r6080271 = r6080269 * r6080270;
        double r6080272 = r6080260 / r6080271;
        double r6080273 = r6080268 - r6080272;
        double r6080274 = z;
        double r6080275 = cbrt(r6080269);
        double r6080276 = r6080275 * r6080275;
        double r6080277 = r6080274 / r6080276;
        double r6080278 = r6080269 + r6080264;
        double r6080279 = sqrt(r6080278);
        double r6080280 = r6080279 / r6080275;
        double r6080281 = r6080277 * r6080280;
        double r6080282 = fma(r6080263, r6080273, r6080281);
        double r6080283 = r6080260 * r6080282;
        double r6080284 = exp(r6080283);
        double r6080285 = fma(r6080259, r6080284, r6080258);
        double r6080286 = r6080258 / r6080285;
        return r6080286;
}

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 3.9

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

  2. Simplified2.5

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

  4. Applied add-cube-cbrt2.5

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

  5. Applied times-frac1.5

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

  6. Final simplification1.5

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

Reproduce

herbie shell --seed 2019173 +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)))))))))))