Average Error: 4.0 → 1.4
Time: 19.6s
Precision: 64
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
↓
\[\frac{x}{\mathsf{fma}\left(y, e^{2 \cdot \mathsf{fma}\left(c - b, \frac{5}{6} + \left(a - \frac{\frac{2}{t}}{3}\right), \left(\sqrt{t + a} \cdot \frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{t} \cdot \sqrt[3]{t}}\right) \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{t}}\right)}, x\right)}\]
\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), \left(\sqrt{t + a} \cdot \frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{t} \cdot \sqrt[3]{t}}\right) \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{t}}\right)}, x\right)}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r54103 = x;
        double r54104 = y;
        double r54105 = 2.0;
        double r54106 = z;
        double r54107 = t;
        double r54108 = a;
        double r54109 = r54107 + r54108;
        double r54110 = sqrt(r54109);
        double r54111 = r54106 * r54110;
        double r54112 = r54111 / r54107;
        double r54113 = b;
        double r54114 = c;
        double r54115 = r54113 - r54114;
        double r54116 = 5.0;
        double r54117 = 6.0;
        double r54118 = r54116 / r54117;
        double r54119 = r54108 + r54118;
        double r54120 = 3.0;
        double r54121 = r54107 * r54120;
        double r54122 = r54105 / r54121;
        double r54123 = r54119 - r54122;
        double r54124 = r54115 * r54123;
        double r54125 = r54112 - r54124;
        double r54126 = r54105 * r54125;
        double r54127 = exp(r54126);
        double r54128 = r54104 * r54127;
        double r54129 = r54103 + r54128;
        double r54130 = r54103 / r54129;
        return r54130;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r54131 = x;
        double r54132 = y;
        double r54133 = 2.0;
        double r54134 = c;
        double r54135 = b;
        double r54136 = r54134 - r54135;
        double r54137 = 5.0;
        double r54138 = 6.0;
        double r54139 = r54137 / r54138;
        double r54140 = a;
        double r54141 = t;
        double r54142 = r54133 / r54141;
        double r54143 = 3.0;
        double r54144 = r54142 / r54143;
        double r54145 = r54140 - r54144;
        double r54146 = r54139 + r54145;
        double r54147 = r54141 + r54140;
        double r54148 = sqrt(r54147);
        double r54149 = z;
        double r54150 = cbrt(r54149);
        double r54151 = r54150 * r54150;
        double r54152 = cbrt(r54141);
        double r54153 = r54152 * r54152;
        double r54154 = r54151 / r54153;
        double r54155 = r54148 * r54154;
        double r54156 = r54150 / r54152;
        double r54157 = r54155 * r54156;
        double r54158 = fma(r54136, r54146, r54157);
        double r54159 = r54133 * r54158;
        double r54160 = exp(r54159);
        double r54161 = fma(r54132, r54160, r54131);
        double r54162 = r54131 / r54161;
        return r54162;
}

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 add-cube-cbrt1.8

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

  6. Applied times-frac1.8

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

  7. Applied associate-*r*1.4

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

  8. Simplified1.4

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

  9. Final simplification1.4

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

Reproduce

herbie shell --seed 2019179 +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)))))))))))