Average Error: 3.8 → 2.7
Time: 7.1s
Precision: 64
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
↓
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\log \left(e^{\mathsf{fma}\left(z \cdot \sqrt{t + a}, \frac{1}{t}, -\left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right) \cdot \left(b - c\right)\right)}\right) + \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right) \cdot \left(\left(-\left(b - c\right)\right) + \left(b - c\right)\right)\right)}}\]
\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}{x + y \cdot e^{2 \cdot \left(\log \left(e^{\mathsf{fma}\left(z \cdot \sqrt{t + a}, \frac{1}{t}, -\left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right) \cdot \left(b - c\right)\right)}\right) + \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right) \cdot \left(\left(-\left(b - c\right)\right) + \left(b - c\right)\right)\right)}}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r55104 = x;
        double r55105 = y;
        double r55106 = 2.0;
        double r55107 = z;
        double r55108 = t;
        double r55109 = a;
        double r55110 = r55108 + r55109;
        double r55111 = sqrt(r55110);
        double r55112 = r55107 * r55111;
        double r55113 = r55112 / r55108;
        double r55114 = b;
        double r55115 = c;
        double r55116 = r55114 - r55115;
        double r55117 = 5.0;
        double r55118 = 6.0;
        double r55119 = r55117 / r55118;
        double r55120 = r55109 + r55119;
        double r55121 = 3.0;
        double r55122 = r55108 * r55121;
        double r55123 = r55106 / r55122;
        double r55124 = r55120 - r55123;
        double r55125 = r55116 * r55124;
        double r55126 = r55113 - r55125;
        double r55127 = r55106 * r55126;
        double r55128 = exp(r55127);
        double r55129 = r55105 * r55128;
        double r55130 = r55104 + r55129;
        double r55131 = r55104 / r55130;
        return r55131;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r55132 = x;
        double r55133 = y;
        double r55134 = 2.0;
        double r55135 = z;
        double r55136 = t;
        double r55137 = a;
        double r55138 = r55136 + r55137;
        double r55139 = sqrt(r55138);
        double r55140 = r55135 * r55139;
        double r55141 = 1.0;
        double r55142 = r55141 / r55136;
        double r55143 = 5.0;
        double r55144 = 6.0;
        double r55145 = r55143 / r55144;
        double r55146 = r55137 + r55145;
        double r55147 = 3.0;
        double r55148 = r55136 * r55147;
        double r55149 = r55134 / r55148;
        double r55150 = r55146 - r55149;
        double r55151 = b;
        double r55152 = c;
        double r55153 = r55151 - r55152;
        double r55154 = r55150 * r55153;
        double r55155 = -r55154;
        double r55156 = fma(r55140, r55142, r55155);
        double r55157 = exp(r55156);
        double r55158 = log(r55157);
        double r55159 = -r55153;
        double r55160 = r55159 + r55153;
        double r55161 = r55150 * r55160;
        double r55162 = r55158 + r55161;
        double r55163 = r55134 * r55162;
        double r55164 = exp(r55163);
        double r55165 = r55133 * r55164;
        double r55166 = r55132 + r55165;
        double r55167 = r55132 / r55166;
        return r55167;
}

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 3.8

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

  3. Applied div-inv3.8

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

  4. Applied prod-diff22.5

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

  5. Simplified2.7

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

  7. Applied add-log-exp2.7

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

  8. Final simplification2.7

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

Reproduce

herbie shell --seed 2020001 +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)))))))))))