Average Error: 10.6 → 0.9
Time: 4.7s
Precision: 64
\[x + \frac{\left(y - z\right) \cdot t}{a - z}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\left(y - z\right) \cdot t}{a - z} \le 4.5977456110666884 \cdot 10^{-7} \lor \neg \left(\frac{\left(y - z\right) \cdot t}{a - z} \le 3.5007108546600527 \cdot 10^{236}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a - z}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \end{array}\]
x + \frac{\left(y - z\right) \cdot t}{a - z}
↓
\begin{array}{l}
\mathbf{if}\;\frac{\left(y - z\right) \cdot t}{a - z} \le 4.5977456110666884 \cdot 10^{-7} \lor \neg \left(\frac{\left(y - z\right) \cdot t}{a - z} \le 3.5007108546600527 \cdot 10^{236}\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 r543674 = x;
        double r543675 = y;
        double r543676 = z;
        double r543677 = r543675 - r543676;
        double r543678 = t;
        double r543679 = r543677 * r543678;
        double r543680 = a;
        double r543681 = r543680 - r543676;
        double r543682 = r543679 / r543681;
        double r543683 = r543674 + r543682;
        return r543683;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r543684 = y;
        double r543685 = z;
        double r543686 = r543684 - r543685;
        double r543687 = t;
        double r543688 = r543686 * r543687;
        double r543689 = a;
        double r543690 = r543689 - r543685;
        double r543691 = r543688 / r543690;
        double r543692 = 4.5977456110666884e-07;
        bool r543693 = r543691 <= r543692;
        double r543694 = 3.5007108546600527e+236;
        bool r543695 = r543691 <= r543694;
        double r543696 = !r543695;
        bool r543697 = r543693 || r543696;
        double r543698 = r543686 / r543690;
        double r543699 = x;
        double r543700 = fma(r543698, r543687, r543699);
        double r543701 = r543699 + r543691;
        double r543702 = r543697 ? r543700 : r543701;
        return r543702;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original10.6
Target0.5
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;t \lt -1.0682974490174067 \cdot 10^{-39}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \mathbf{elif}\;t \lt 3.9110949887586375 \cdot 10^{-141}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (* (- y z) t) (- a z)) < 4.5977456110666884e-07 or 3.5007108546600527e+236 < (/ (* (- y z) t) (- a z))

    1. Initial program 12.2

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

    2. Simplified1.0

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

    if 4.5977456110666884e-07 < (/ (* (- y z) t) (- a z)) < 3.5007108546600527e+236

    1. Initial program 0.1

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(y - z\right) \cdot t}{a - z} \le 4.5977456110666884 \cdot 10^{-7} \lor \neg \left(\frac{\left(y - z\right) \cdot t}{a - z} \le 3.5007108546600527 \cdot 10^{236}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a - z}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 +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))))