Average Error: 4.0 → 1.6
Time: 32.8s
Precision: 64
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
↓
\[\frac{x}{\mathsf{fma}\left(y, e^{2 \cdot \mathsf{fma}\left(c - b, \frac{5}{6} + \left(a - \frac{\frac{2}{t}}{3}\right), \frac{z}{\sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t} \cdot \sqrt[3]{t}}\right)}, x\right)}\]
\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, \frac{5}{6} + \left(a - \frac{\frac{2}{t}}{3}\right), \frac{z}{\sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t} \cdot \sqrt[3]{t}}\right)}, x\right)}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r2810199 = x;
        double r2810200 = y;
        double r2810201 = 2.0;
        double r2810202 = z;
        double r2810203 = t;
        double r2810204 = a;
        double r2810205 = r2810203 + r2810204;
        double r2810206 = sqrt(r2810205);
        double r2810207 = r2810202 * r2810206;
        double r2810208 = r2810207 / r2810203;
        double r2810209 = b;
        double r2810210 = c;
        double r2810211 = r2810209 - r2810210;
        double r2810212 = 5.0;
        double r2810213 = 6.0;
        double r2810214 = r2810212 / r2810213;
        double r2810215 = r2810204 + r2810214;
        double r2810216 = 3.0;
        double r2810217 = r2810203 * r2810216;
        double r2810218 = r2810201 / r2810217;
        double r2810219 = r2810215 - r2810218;
        double r2810220 = r2810211 * r2810219;
        double r2810221 = r2810208 - r2810220;
        double r2810222 = r2810201 * r2810221;
        double r2810223 = exp(r2810222);
        double r2810224 = r2810200 * r2810223;
        double r2810225 = r2810199 + r2810224;
        double r2810226 = r2810199 / r2810225;
        return r2810226;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r2810227 = x;
        double r2810228 = y;
        double r2810229 = 2.0;
        double r2810230 = c;
        double r2810231 = b;
        double r2810232 = r2810230 - r2810231;
        double r2810233 = 5.0;
        double r2810234 = 6.0;
        double r2810235 = r2810233 / r2810234;
        double r2810236 = a;
        double r2810237 = t;
        double r2810238 = r2810229 / r2810237;
        double r2810239 = 3.0;
        double r2810240 = r2810238 / r2810239;
        double r2810241 = r2810236 - r2810240;
        double r2810242 = r2810235 + r2810241;
        double r2810243 = z;
        double r2810244 = cbrt(r2810237);
        double r2810245 = r2810243 / r2810244;
        double r2810246 = r2810237 + r2810236;
        double r2810247 = sqrt(r2810246);
        double r2810248 = r2810244 * r2810244;
        double r2810249 = r2810247 / r2810248;
        double r2810250 = r2810245 * r2810249;
        double r2810251 = fma(r2810232, r2810242, r2810250);
        double r2810252 = r2810229 * r2810251;
        double r2810253 = exp(r2810252);
        double r2810254 = fma(r2810228, r2810253, r2810227);
        double r2810255 = r2810227 / r2810254;
        return r2810255;
}

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 4.0

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

  2. Simplified1.8

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

  4. Applied add-cube-cbrt1.8

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

  5. Applied *-un-lft-identity1.8

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

  6. Applied times-frac1.8

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

  7. Applied associate-*r*1.6

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

  8. Simplified1.6

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

  9. Final simplification1.6

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

Reproduce

herbie shell --seed 2019172 +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)))))))))))