Average Error: 3.7 → 1.4
Time: 46.1s
Precision: 64
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
↓
\[\frac{x}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(\frac{\frac{2}{t}}{3} - \left(a + \frac{5}{6}\right), b - c, \frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}}\right)\right)}, x\right)}\]
\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}{t}}{3} - \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 r62103 = x;
        double r62104 = y;
        double r62105 = 2.0;
        double r62106 = z;
        double r62107 = t;
        double r62108 = a;
        double r62109 = r62107 + r62108;
        double r62110 = sqrt(r62109);
        double r62111 = r62106 * r62110;
        double r62112 = r62111 / r62107;
        double r62113 = b;
        double r62114 = c;
        double r62115 = r62113 - r62114;
        double r62116 = 5.0;
        double r62117 = 6.0;
        double r62118 = r62116 / r62117;
        double r62119 = r62108 + r62118;
        double r62120 = 3.0;
        double r62121 = r62107 * r62120;
        double r62122 = r62105 / r62121;
        double r62123 = r62119 - r62122;
        double r62124 = r62115 * r62123;
        double r62125 = r62112 - r62124;
        double r62126 = r62105 * r62125;
        double r62127 = exp(r62126);
        double r62128 = r62104 * r62127;
        double r62129 = r62103 + r62128;
        double r62130 = r62103 / r62129;
        return r62130;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r62131 = x;
        double r62132 = y;
        double r62133 = 2.0;
        double r62134 = exp(r62133);
        double r62135 = t;
        double r62136 = r62133 / r62135;
        double r62137 = 3.0;
        double r62138 = r62136 / r62137;
        double r62139 = a;
        double r62140 = 5.0;
        double r62141 = 6.0;
        double r62142 = r62140 / r62141;
        double r62143 = r62139 + r62142;
        double r62144 = r62138 - r62143;
        double r62145 = b;
        double r62146 = c;
        double r62147 = r62145 - r62146;
        double r62148 = z;
        double r62149 = cbrt(r62135);
        double r62150 = r62149 * r62149;
        double r62151 = r62148 / r62150;
        double r62152 = r62135 + r62139;
        double r62153 = sqrt(r62152);
        double r62154 = r62153 / r62149;
        double r62155 = r62151 * r62154;
        double r62156 = fma(r62144, r62147, r62155);
        double r62157 = pow(r62134, r62156);
        double r62158 = fma(r62132, r62157, r62131);
        double r62159 = r62131 / r62158;
        return r62159;
}

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. Using strategy rm

  4. Applied add-cube-cbrt2.5

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

  5. Applied times-frac1.4

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

  6. Final simplification1.4

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

Reproduce

herbie shell --seed 2019199 +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)))))))))))