Average Error: 4.1 → 1.4
Time: 23.7s
Precision: 64
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
↓
\[\frac{x}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(\frac{\frac{2}{t}}{3} - \left(a + \frac{5}{6}\right), b - c, \frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{t}} \cdot \left(\frac{\sqrt[3]{z}}{\sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}}\right)\right)\right)}, x\right)}\]
\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{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{t}} \cdot \left(\frac{\sqrt[3]{z}}{\sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}}\right)\right)\right)}, x\right)}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r62071 = x;
        double r62072 = y;
        double r62073 = 2.0;
        double r62074 = z;
        double r62075 = t;
        double r62076 = a;
        double r62077 = r62075 + r62076;
        double r62078 = sqrt(r62077);
        double r62079 = r62074 * r62078;
        double r62080 = r62079 / r62075;
        double r62081 = b;
        double r62082 = c;
        double r62083 = r62081 - r62082;
        double r62084 = 5.0;
        double r62085 = 6.0;
        double r62086 = r62084 / r62085;
        double r62087 = r62076 + r62086;
        double r62088 = 3.0;
        double r62089 = r62075 * r62088;
        double r62090 = r62073 / r62089;
        double r62091 = r62087 - r62090;
        double r62092 = r62083 * r62091;
        double r62093 = r62080 - r62092;
        double r62094 = r62073 * r62093;
        double r62095 = exp(r62094);
        double r62096 = r62072 * r62095;
        double r62097 = r62071 + r62096;
        double r62098 = r62071 / r62097;
        return r62098;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r62099 = x;
        double r62100 = y;
        double r62101 = 2.0;
        double r62102 = exp(r62101);
        double r62103 = t;
        double r62104 = r62101 / r62103;
        double r62105 = 3.0;
        double r62106 = r62104 / r62105;
        double r62107 = a;
        double r62108 = 5.0;
        double r62109 = 6.0;
        double r62110 = r62108 / r62109;
        double r62111 = r62107 + r62110;
        double r62112 = r62106 - r62111;
        double r62113 = b;
        double r62114 = c;
        double r62115 = r62113 - r62114;
        double r62116 = z;
        double r62117 = cbrt(r62116);
        double r62118 = r62117 * r62117;
        double r62119 = cbrt(r62103);
        double r62120 = r62118 / r62119;
        double r62121 = r62117 / r62119;
        double r62122 = r62103 + r62107;
        double r62123 = sqrt(r62122);
        double r62124 = r62123 / r62119;
        double r62125 = r62121 * r62124;
        double r62126 = r62120 * r62125;
        double r62127 = fma(r62112, r62115, r62126);
        double r62128 = pow(r62102, r62127);
        double r62129 = fma(r62100, r62128, r62099);
        double r62130 = r62099 / r62129;
        return r62130;
}

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 4.1

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

  2. Simplified2.6

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

  4. Applied add-cube-cbrt2.6

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

  5. Applied times-frac1.5

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

  7. Applied add-cube-cbrt1.5

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

  8. Applied times-frac1.5

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

  9. Applied associate-*l*1.4

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

  10. Final simplification1.4

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

Reproduce

herbie shell --seed 2019325 +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)))))))))))