Average Error: 10.8 → 0.6
Time: 23.3s
Precision: 64
\[x + \frac{\left(y - z\right) \cdot t}{a - z}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -7.122674290888581465936789302887738189388 \cdot 10^{-47} \lor \neg \left(t \le 2.662174497494423670876869896388531842882 \cdot 10^{-151}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a - z}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \end{array}\]
x + \frac{\left(y - z\right) \cdot t}{a - z}
↓
\begin{array}{l}
\mathbf{if}\;t \le -7.122674290888581465936789302887738189388 \cdot 10^{-47} \lor \neg \left(t \le 2.662174497494423670876869896388531842882 \cdot 10^{-151}\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 r494378 = x;
        double r494379 = y;
        double r494380 = z;
        double r494381 = r494379 - r494380;
        double r494382 = t;
        double r494383 = r494381 * r494382;
        double r494384 = a;
        double r494385 = r494384 - r494380;
        double r494386 = r494383 / r494385;
        double r494387 = r494378 + r494386;
        return r494387;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r494388 = t;
        double r494389 = -7.122674290888581e-47;
        bool r494390 = r494388 <= r494389;
        double r494391 = 2.6621744974944237e-151;
        bool r494392 = r494388 <= r494391;
        double r494393 = !r494392;
        bool r494394 = r494390 || r494393;
        double r494395 = y;
        double r494396 = z;
        double r494397 = r494395 - r494396;
        double r494398 = a;
        double r494399 = r494398 - r494396;
        double r494400 = r494397 / r494399;
        double r494401 = x;
        double r494402 = fma(r494400, r494388, r494401);
        double r494403 = r494397 * r494388;
        double r494404 = r494403 / r494399;
        double r494405 = r494401 + r494404;
        double r494406 = r494394 ? r494402 : r494405;
        return r494406;
}

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
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;t \lt -1.068297449017406694366747246993994850729 \cdot 10^{-39}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \mathbf{elif}\;t \lt 3.911094988758637497591020599238553861375 \cdot 10^{-141}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if t < -7.122674290888581e-47 or 2.6621744974944237e-151 < t

    1. Initial program 16.9

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

    2. Simplified0.7

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

    if -7.122674290888581e-47 < t < 2.6621744974944237e-151

    1. Initial program 0.4

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -7.122674290888581465936789302887738189388 \cdot 10^{-47} \lor \neg \left(t \le 2.662174497494423670876869896388531842882 \cdot 10^{-151}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a - z}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 +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))))