Average Error: 3.8 → 1.8
Time: 15.4s
Precision: 64
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
↓
\[\frac{x}{x + y \cdot e^{2 \cdot \mathsf{fma}\left(\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}}, \frac{\sqrt{t + a}}{\sqrt[3]{t}}, -\left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
\frac{x}{x + y \cdot e^{2 \cdot \left(\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 \mathsf{fma}\left(\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}}, \frac{\sqrt{t + a}}{\sqrt[3]{t}}, -\left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r100062 = x;
        double r100063 = y;
        double r100064 = 2.0;
        double r100065 = z;
        double r100066 = t;
        double r100067 = a;
        double r100068 = r100066 + r100067;
        double r100069 = sqrt(r100068);
        double r100070 = r100065 * r100069;
        double r100071 = r100070 / r100066;
        double r100072 = b;
        double r100073 = c;
        double r100074 = r100072 - r100073;
        double r100075 = 5.0;
        double r100076 = 6.0;
        double r100077 = r100075 / r100076;
        double r100078 = r100067 + r100077;
        double r100079 = 3.0;
        double r100080 = r100066 * r100079;
        double r100081 = r100064 / r100080;
        double r100082 = r100078 - r100081;
        double r100083 = r100074 * r100082;
        double r100084 = r100071 - r100083;
        double r100085 = r100064 * r100084;
        double r100086 = exp(r100085);
        double r100087 = r100063 * r100086;
        double r100088 = r100062 + r100087;
        double r100089 = r100062 / r100088;
        return r100089;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r100090 = x;
        double r100091 = y;
        double r100092 = 2.0;
        double r100093 = z;
        double r100094 = t;
        double r100095 = cbrt(r100094);
        double r100096 = r100095 * r100095;
        double r100097 = r100093 / r100096;
        double r100098 = a;
        double r100099 = r100094 + r100098;
        double r100100 = sqrt(r100099);
        double r100101 = r100100 / r100095;
        double r100102 = b;
        double r100103 = c;
        double r100104 = r100102 - r100103;
        double r100105 = 5.0;
        double r100106 = 6.0;
        double r100107 = r100105 / r100106;
        double r100108 = r100098 + r100107;
        double r100109 = 3.0;
        double r100110 = r100094 * r100109;
        double r100111 = r100092 / r100110;
        double r100112 = r100108 - r100111;
        double r100113 = r100104 * r100112;
        double r100114 = -r100113;
        double r100115 = fma(r100097, r100101, r100114);
        double r100116 = r100092 * r100115;
        double r100117 = exp(r100116);
        double r100118 = r100091 * r100117;
        double r100119 = r100090 + r100118;
        double r100120 = r100090 / r100119;
        return r100120;
}

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 add-cube-cbrt3.8

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

  4. Applied times-frac2.6

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

  5. Applied fma-neg1.8

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

  6. Final simplification1.8

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

Reproduce

herbie shell --seed 2020060 +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)))))))))))