Average Error: 7.2 → 3.2
Time: 21.7s
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 r30250125 = x;
        double r30250126 = y;
        double r30250127 = z;
        double r30250128 = r30250126 * r30250127;
        double r30250129 = r30250128 - r30250125;
        double r30250130 = t;
        double r30250131 = r30250130 * r30250127;
        double r30250132 = r30250131 - r30250125;
        double r30250133 = r30250129 / r30250132;
        double r30250134 = r30250125 + r30250133;
        double r30250135 = 1.0;
        double r30250136 = r30250125 + r30250135;
        double r30250137 = r30250134 / r30250136;
        return r30250137;
}
↓
double f(double x, double y, double z, double t) {
        double r30250138 = z;
        double r30250139 = -7.259731899519289e-62;
        bool r30250140 = r30250138 <= r30250139;
        double r30250141 = 1.0;
        double r30250142 = t;
        double r30250143 = r30250142 * r30250138;
        double r30250144 = x;
        double r30250145 = r30250143 - r30250144;
        double r30250146 = y;
        double r30250147 = r30250145 / r30250146;
        double r30250148 = r30250141 / r30250147;
        double r30250149 = fma(r30250148, r30250138, r30250144);
        double r30250150 = r30250144 / r30250145;
        double r30250151 = r30250149 - r30250150;
        double r30250152 = 1.0;
        double r30250153 = r30250144 + r30250152;
        double r30250154 = r30250151 / r30250153;
        double r30250155 = 1.415907658888434e+57;
        bool r30250156 = r30250138 <= r30250155;
        double r30250157 = r30250138 * r30250146;
        double r30250158 = r30250157 - r30250144;
        double r30250159 = r30250141 / r30250145;
        double r30250160 = r30250158 * r30250159;
        double r30250161 = r30250160 + r30250144;
        double r30250162 = r30250161 / r30250153;
        double r30250163 = r30250146 / r30250142;
        double r30250164 = r30250163 + r30250144;
        double r30250165 = r30250164 / r30250153;
        double r30250166 = r30250156 ? r30250162 : r30250165;
        double r30250167 = r30250140 ? r30250154 : r30250166;
        return r30250167;
}

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)))