Average Error: 7.6 → 4.7
Time: 9.7s
Precision: 64
\[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]
↓
\[\frac{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right)}{\left(x + 1\right) \cdot 1} - \frac{\frac{x}{t \cdot z - x}}{x + 1}\]
\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}
↓
\frac{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right)}{\left(x + 1\right) \cdot 1} - \frac{\frac{x}{t \cdot z - x}}{x + 1}
double f(double x, double y, double z, double t) {
        double r3897 = x;
        double r3898 = y;
        double r3899 = z;
        double r3900 = r3898 * r3899;
        double r3901 = r3900 - r3897;
        double r3902 = t;
        double r3903 = r3902 * r3899;
        double r3904 = r3903 - r3897;
        double r3905 = r3901 / r3904;
        double r3906 = r3897 + r3905;
        double r3907 = 1.0;
        double r3908 = r3897 + r3907;
        double r3909 = r3906 / r3908;
        return r3909;
}
↓
double f(double x, double y, double z, double t) {
        double r3910 = y;
        double r3911 = t;
        double r3912 = z;
        double r3913 = r3911 * r3912;
        double r3914 = x;
        double r3915 = r3913 - r3914;
        double r3916 = r3910 / r3915;
        double r3917 = fma(r3916, r3912, r3914);
        double r3918 = 1.0;
        double r3919 = r3914 + r3918;
        double r3920 = 1.0;
        double r3921 = r3919 * r3920;
        double r3922 = r3917 / r3921;
        double r3923 = r3914 / r3915;
        double r3924 = r3923 / r3919;
        double r3925 = r3922 - r3924;
        return r3925;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original7.6
Target0.4
Herbie4.7
\[\frac{x + \left(\frac{y}{t - \frac{x}{z}} - \frac{x}{t \cdot z - x}\right)}{x + 1}\]

Derivation

  1. Initial program 7.6

    \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]
  2. Using strategy rm

  3. Applied div-sub7.6

    \[\leadsto \frac{x + \color{blue}{\left(\frac{y \cdot z}{t \cdot z - x} - \frac{x}{t \cdot z - x}\right)}}{x + 1}\]

  4. Applied associate-+r-7.6

    \[\leadsto \frac{\color{blue}{\left(x + \frac{y \cdot z}{t \cdot z - x}\right) - \frac{x}{t \cdot z - x}}}{x + 1}\]

  5. Applied div-sub7.6

    \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z}{t \cdot z - x}}{x + 1} - \frac{\frac{x}{t \cdot z - x}}{x + 1}}\]

  6. Simplified4.7

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right)}{\left(x + 1\right) \cdot 1}} - \frac{\frac{x}{t \cdot z - x}}{x + 1}\]

  7. Final simplification4.7

    \[\leadsto \frac{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right)}{\left(x + 1\right) \cdot 1} - \frac{\frac{x}{t \cdot z - x}}{x + 1}\]

Reproduce

herbie shell --seed 2020025 +o rules:numerics
(FPCore (x y z t)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, A"
  :precision binary64

  :herbie-target
  (/ (+ x (- (/ y (- t (/ x z))) (/ x (- (* t z) x)))) (+ x 1))

  (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1)))