Average Error: 10.8 → 0.3
Time: 14.5s
Precision: 64
\[x + \frac{\left(y - z\right) \cdot t}{a - z}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\left(y - z\right) \cdot t}{a - z} = -\infty \lor \neg \left(\frac{\left(y - z\right) \cdot t}{a - z} \le 4.80072190399451909 \cdot 10^{294}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a - z}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \end{array}\]
x + \frac{\left(y - z\right) \cdot t}{a - z}
↓
\begin{array}{l}
\mathbf{if}\;\frac{\left(y - z\right) \cdot t}{a - z} = -\infty \lor \neg \left(\frac{\left(y - z\right) \cdot t}{a - z} \le 4.80072190399451909 \cdot 10^{294}\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{y - z}{a - z}, t, x\right)\\

\mathbf{else}:\\
\;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r515122 = x;
        double r515123 = y;
        double r515124 = z;
        double r515125 = r515123 - r515124;
        double r515126 = t;
        double r515127 = r515125 * r515126;
        double r515128 = a;
        double r515129 = r515128 - r515124;
        double r515130 = r515127 / r515129;
        double r515131 = r515122 + r515130;
        return r515131;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r515132 = y;
        double r515133 = z;
        double r515134 = r515132 - r515133;
        double r515135 = t;
        double r515136 = r515134 * r515135;
        double r515137 = a;
        double r515138 = r515137 - r515133;
        double r515139 = r515136 / r515138;
        double r515140 = -inf.0;
        bool r515141 = r515139 <= r515140;
        double r515142 = 4.800721903994519e+294;
        bool r515143 = r515139 <= r515142;
        double r515144 = !r515143;
        bool r515145 = r515141 || r515144;
        double r515146 = r515134 / r515138;
        double r515147 = x;
        double r515148 = fma(r515146, r515135, r515147);
        double r515149 = r515147 + r515139;
        double r515150 = r515145 ? r515148 : r515149;
        return r515150;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original10.8
Target0.6
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;t \lt -1.0682974490174067 \cdot 10^{-39}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \mathbf{elif}\;t \lt 3.9110949887586375 \cdot 10^{-141}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (* (- y z) t) (- a z)) < -inf.0 or 4.800721903994519e+294 < (/ (* (- y z) t) (- a z))

    1. Initial program 62.7

      \[x + \frac{\left(y - z\right) \cdot t}{a - z}\]

    2. Simplified0.2

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

    if -inf.0 < (/ (* (- y z) t) (- a z)) < 4.800721903994519e+294

    1. Initial program 0.3

      \[x + \frac{\left(y - z\right) \cdot t}{a - z}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(y - z\right) \cdot t}{a - z} = -\infty \lor \neg \left(\frac{\left(y - z\right) \cdot t}{a - z} \le 4.80072190399451909 \cdot 10^{294}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a - z}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, A"
  :precision binary64

  :herbie-target
  (if (< t -1.0682974490174067e-39) (+ x (* (/ (- y z) (- a z)) t)) (if (< t 3.9110949887586375e-141) (+ x (/ (* (- y z) t) (- a z))) (+ x (* (/ (- y z) (- a z)) t))))

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