Average Error: 10.8 → 0.3
Time: 11.1s
Precision: 64
\[x + \frac{\left(y - z\right) \cdot t}{a - z}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\left(y - z\right) \cdot t}{a - z} = -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(y - z\right) \cdot \frac{1}{a - z}, t, x\right)\\ \mathbf{elif}\;\frac{\left(y - z\right) \cdot t}{a - z} \le 4.80072190399451909 \cdot 10^{294}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a - z}, t, x\right)\\ \end{array}\]
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:\\
\;\;\;\;\mathsf{fma}\left(\left(y - z\right) \cdot \frac{1}{a - z}, t, x\right)\\

\mathbf{elif}\;\frac{\left(y - z\right) \cdot t}{a - z} \le 4.80072190399451909 \cdot 10^{294}:\\
\;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{y - z}{a - z}, t, x\right)\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r571034 = x;
        double r571035 = y;
        double r571036 = z;
        double r571037 = r571035 - r571036;
        double r571038 = t;
        double r571039 = r571037 * r571038;
        double r571040 = a;
        double r571041 = r571040 - r571036;
        double r571042 = r571039 / r571041;
        double r571043 = r571034 + r571042;
        return r571043;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r571044 = y;
        double r571045 = z;
        double r571046 = r571044 - r571045;
        double r571047 = t;
        double r571048 = r571046 * r571047;
        double r571049 = a;
        double r571050 = r571049 - r571045;
        double r571051 = r571048 / r571050;
        double r571052 = -inf.0;
        bool r571053 = r571051 <= r571052;
        double r571054 = 1.0;
        double r571055 = r571054 / r571050;
        double r571056 = r571046 * r571055;
        double r571057 = x;
        double r571058 = fma(r571056, r571047, r571057);
        double r571059 = 4.800721903994519e+294;
        bool r571060 = r571051 <= r571059;
        double r571061 = r571057 + r571051;
        double r571062 = r571046 / r571050;
        double r571063 = fma(r571062, r571047, r571057);
        double r571064 = r571060 ? r571061 : r571063;
        double r571065 = r571053 ? r571058 : r571064;
        return r571065;
}

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 3 regimes

  2. if (/ (* (- y z) t) (- a z)) < -inf.0

    1. Initial program 64.0

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

    2. Simplified0.1

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

    4. Applied div-inv0.2

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

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

    1. Initial program 0.3

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

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

    1. Initial program 61.4

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

    2. Simplified0.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t, x\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(y - z\right) \cdot t}{a - z} = -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(y - z\right) \cdot \frac{1}{a - z}, t, x\right)\\ \mathbf{elif}\;\frac{\left(y - z\right) \cdot t}{a - z} \le 4.80072190399451909 \cdot 10^{294}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a - z}, t, x\right)\\ \end{array}\]

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))))