Average Error: 7.2 → 3.2
Time: 19.8s
Precision: 64
\[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -7.259731899519289 \cdot 10^{-62}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{\frac{t \cdot z - x}{y}}, z, x\right) - \frac{x}{t \cdot z - x}}{x + 1.0}\\ \mathbf{elif}\;z \le 1.415907658888434 \cdot 10^{+57}:\\ \;\;\;\;\frac{\left(z \cdot y - x\right) \cdot \frac{1}{t \cdot z - x} + x}{x + 1.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x + 1.0}\\ \end{array}\]
\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1.0}
↓
\begin{array}{l}
\mathbf{if}\;z \le -7.259731899519289 \cdot 10^{-62}:\\
\;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{\frac{t \cdot z - x}{y}}, z, x\right) - \frac{x}{t \cdot z - x}}{x + 1.0}\\

\mathbf{elif}\;z \le 1.415907658888434 \cdot 10^{+57}:\\
\;\;\;\;\frac{\left(z \cdot y - x\right) \cdot \frac{1}{t \cdot z - x} + x}{x + 1.0}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{y}{t} + x}{x + 1.0}\\

\end{array}
double f(double x, double y, double z, double t) {
        double r30750054 = x;
        double r30750055 = y;
        double r30750056 = z;
        double r30750057 = r30750055 * r30750056;
        double r30750058 = r30750057 - r30750054;
        double r30750059 = t;
        double r30750060 = r30750059 * r30750056;
        double r30750061 = r30750060 - r30750054;
        double r30750062 = r30750058 / r30750061;
        double r30750063 = r30750054 + r30750062;
        double r30750064 = 1.0;
        double r30750065 = r30750054 + r30750064;
        double r30750066 = r30750063 / r30750065;
        return r30750066;
}
↓
double f(double x, double y, double z, double t) {
        double r30750067 = z;
        double r30750068 = -7.259731899519289e-62;
        bool r30750069 = r30750067 <= r30750068;
        double r30750070 = 1.0;
        double r30750071 = t;
        double r30750072 = r30750071 * r30750067;
        double r30750073 = x;
        double r30750074 = r30750072 - r30750073;
        double r30750075 = y;
        double r30750076 = r30750074 / r30750075;
        double r30750077 = r30750070 / r30750076;
        double r30750078 = fma(r30750077, r30750067, r30750073);
        double r30750079 = r30750073 / r30750074;
        double r30750080 = r30750078 - r30750079;
        double r30750081 = 1.0;
        double r30750082 = r30750073 + r30750081;
        double r30750083 = r30750080 / r30750082;
        double r30750084 = 1.415907658888434e+57;
        bool r30750085 = r30750067 <= r30750084;
        double r30750086 = r30750067 * r30750075;
        double r30750087 = r30750086 - r30750073;
        double r30750088 = r30750070 / r30750074;
        double r30750089 = r30750087 * r30750088;
        double r30750090 = r30750089 + r30750073;
        double r30750091 = r30750090 / r30750082;
        double r30750092 = r30750075 / r30750071;
        double r30750093 = r30750092 + r30750073;
        double r30750094 = r30750093 / r30750082;
        double r30750095 = r30750085 ? r30750091 : r30750094;
        double r30750096 = r30750069 ? r30750083 : r30750095;
        return r30750096;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original7.2
Target0.4
Herbie3.2
\[\frac{x + \left(\frac{y}{t - \frac{x}{z}} - \frac{x}{t \cdot z - x}\right)}{x + 1.0}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -7.259731899519289e-62

    1. Initial program 12.0

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1.0}\]
    2. Using strategy rm

    3. Applied div-sub12.0

      \[\leadsto \frac{x + \color{blue}{\left(\frac{y \cdot z}{t \cdot z - x} - \frac{x}{t \cdot z - x}\right)}}{x + 1.0}\]

    4. Applied associate-+r-12.0

      \[\leadsto \frac{\color{blue}{\left(x + \frac{y \cdot z}{t \cdot z - x}\right) - \frac{x}{t \cdot z - x}}}{x + 1.0}\]

    5. Simplified4.3

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right)} - \frac{x}{t \cdot z - x}}{x + 1.0}\]
    6. Using strategy rm

    7. Applied clear-num4.5

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{1}{\frac{t \cdot z - x}{y}}}, z, x\right) - \frac{x}{t \cdot z - x}}{x + 1.0}\]

    if -7.259731899519289e-62 < z < 1.415907658888434e+57

    1. Initial program 0.3

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1.0}\]
    2. Using strategy rm

    3. Applied div-inv0.4

      \[\leadsto \frac{x + \color{blue}{\left(y \cdot z - x\right) \cdot \frac{1}{t \cdot z - x}}}{x + 1.0}\]

    if 1.415907658888434e+57 < z

    1. Initial program 17.8

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1.0}\]

    2. Taylor expanded around inf 8.2

      \[\leadsto \frac{x + \color{blue}{\frac{y}{t}}}{x + 1.0}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification3.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -7.259731899519289 \cdot 10^{-62}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{\frac{t \cdot z - x}{y}}, z, x\right) - \frac{x}{t \cdot z - x}}{x + 1.0}\\ \mathbf{elif}\;z \le 1.415907658888434 \cdot 10^{+57}:\\ \;\;\;\;\frac{\left(z \cdot y - x\right) \cdot \frac{1}{t \cdot z - x} + x}{x + 1.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x + 1.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (x y z t)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, A"

  :herbie-target
  (/ (+ x (- (/ y (- t (/ x z))) (/ x (- (* t z) x)))) (+ x 1.0))

  (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)))