Average Error: 10.8 → 0.4
Time: 3.4s
Precision: 64
\[x + \frac{\left(y - z\right) \cdot t}{a - z}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\left(y - z\right) \cdot t}{a - z} \le -1.3990348901059605 \cdot 10^{295}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{a - z}{y - z}}, t, x\right)\\ \mathbf{elif}\;\frac{\left(y - z\right) \cdot t}{a - z} \le 1.11870472799551397 \cdot 10^{252}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a - z}, t, x\right)\\ \end{array}\]
x + \frac{\left(y - z\right) \cdot t}{a - z}
↓
\begin{array}{l}
\mathbf{if}\;\frac{\left(y - z\right) \cdot t}{a - z} \le -1.3990348901059605 \cdot 10^{295}:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{a - z}{y - z}}, t, x\right)\\

\mathbf{elif}\;\frac{\left(y - z\right) \cdot t}{a - z} \le 1.11870472799551397 \cdot 10^{252}:\\
\;\;\;\;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 r670517 = x;
        double r670518 = y;
        double r670519 = z;
        double r670520 = r670518 - r670519;
        double r670521 = t;
        double r670522 = r670520 * r670521;
        double r670523 = a;
        double r670524 = r670523 - r670519;
        double r670525 = r670522 / r670524;
        double r670526 = r670517 + r670525;
        return r670526;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r670527 = y;
        double r670528 = z;
        double r670529 = r670527 - r670528;
        double r670530 = t;
        double r670531 = r670529 * r670530;
        double r670532 = a;
        double r670533 = r670532 - r670528;
        double r670534 = r670531 / r670533;
        double r670535 = -1.3990348901059605e+295;
        bool r670536 = r670534 <= r670535;
        double r670537 = 1.0;
        double r670538 = r670533 / r670529;
        double r670539 = r670537 / r670538;
        double r670540 = x;
        double r670541 = fma(r670539, r670530, r670540);
        double r670542 = 1.118704727995514e+252;
        bool r670543 = r670534 <= r670542;
        double r670544 = r670540 + r670534;
        double r670545 = r670529 / r670533;
        double r670546 = fma(r670545, r670530, r670540);
        double r670547 = r670543 ? r670544 : r670546;
        double r670548 = r670536 ? r670541 : r670547;
        return r670548;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original10.8
Target0.7
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;t \lt -1.0682974490174067 \cdot 10^{-39}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \mathbf{elif}\;t \lt 3.9110949887586375 \cdot 10^{-141}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (/ (* (- y z) t) (- a z)) < -1.3990348901059605e+295

    1. Initial program 61.5

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

    2. Simplified0.4

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

    4. Applied clear-num0.4

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

    if -1.3990348901059605e+295 < (/ (* (- y z) t) (- a z)) < 1.118704727995514e+252

    1. Initial program 0.3

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

    if 1.118704727995514e+252 < (/ (* (- y z) t) (- a z))

    1. Initial program 54.8

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

    2. Simplified1.9

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(y - z\right) \cdot t}{a - z} \le -1.3990348901059605 \cdot 10^{295}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{a - z}{y - z}}, t, x\right)\\ \mathbf{elif}\;\frac{\left(y - z\right) \cdot t}{a - z} \le 1.11870472799551397 \cdot 10^{252}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a - z}, t, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 +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))))