Average Error: 6.4 → 1.5
Time: 8.8s
Precision: 64
\[x + \frac{\left(y - x\right) \cdot z}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.6809054761397501 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y - x}{t}, x\right)\\ \mathbf{elif}\;t \le 2.3137374841890364 \cdot 10^{-267}:\\ \;\;\;\;x + \frac{z \cdot \left(y - x\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - x}{\frac{t}{z}} + x\\ \end{array}\]
x + \frac{\left(y - x\right) \cdot z}{t}
↓
\begin{array}{l}
\mathbf{if}\;t \le -1.6809054761397501 \cdot 10^{-10}:\\
\;\;\;\;\mathsf{fma}\left(z, \frac{y - x}{t}, x\right)\\

\mathbf{elif}\;t \le 2.3137374841890364 \cdot 10^{-267}:\\
\;\;\;\;x + \frac{z \cdot \left(y - x\right)}{t}\\

\mathbf{else}:\\
\;\;\;\;\frac{y - x}{\frac{t}{z}} + x\\

\end{array}
double f(double x, double y, double z, double t) {
        double r8095331 = x;
        double r8095332 = y;
        double r8095333 = r8095332 - r8095331;
        double r8095334 = z;
        double r8095335 = r8095333 * r8095334;
        double r8095336 = t;
        double r8095337 = r8095335 / r8095336;
        double r8095338 = r8095331 + r8095337;
        return r8095338;
}
↓
double f(double x, double y, double z, double t) {
        double r8095339 = t;
        double r8095340 = -1.6809054761397501e-10;
        bool r8095341 = r8095339 <= r8095340;
        double r8095342 = z;
        double r8095343 = y;
        double r8095344 = x;
        double r8095345 = r8095343 - r8095344;
        double r8095346 = r8095345 / r8095339;
        double r8095347 = fma(r8095342, r8095346, r8095344);
        double r8095348 = 2.3137374841890364e-267;
        bool r8095349 = r8095339 <= r8095348;
        double r8095350 = r8095342 * r8095345;
        double r8095351 = r8095350 / r8095339;
        double r8095352 = r8095344 + r8095351;
        double r8095353 = r8095339 / r8095342;
        double r8095354 = r8095345 / r8095353;
        double r8095355 = r8095354 + r8095344;
        double r8095356 = r8095349 ? r8095352 : r8095355;
        double r8095357 = r8095341 ? r8095347 : r8095356;
        return r8095357;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original6.4
Target1.9
Herbie1.5
\[\begin{array}{l} \mathbf{if}\;x \lt -9.025511195533005 \cdot 10^{-135}:\\ \;\;\;\;x - \frac{z}{t} \cdot \left(x - y\right)\\ \mathbf{elif}\;x \lt 4.275032163700715 \cdot 10^{-250}:\\ \;\;\;\;x + \frac{y - x}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if t < -1.6809054761397501e-10

    1. Initial program 9.8

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

    2. Simplified0.8

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

    if -1.6809054761397501e-10 < t < 2.3137374841890364e-267

    1. Initial program 2.0

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

    if 2.3137374841890364e-267 < t

    1. Initial program 6.0

      \[x + \frac{\left(y - x\right) \cdot z}{t}\]
    2. Using strategy rm

    3. Applied associate-/l*1.7

      \[\leadsto x + \color{blue}{\frac{y - x}{\frac{t}{z}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification1.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.6809054761397501 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y - x}{t}, x\right)\\ \mathbf{elif}\;t \le 2.3137374841890364 \cdot 10^{-267}:\\ \;\;\;\;x + \frac{z \cdot \left(y - x\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - x}{\frac{t}{z}} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 +o rules:numerics
(FPCore (x y z t)
  :name "Numeric.Histogram:binBounds from Chart-1.5.3"

  :herbie-target
  (if (< x -9.025511195533005e-135) (- x (* (/ z t) (- x y))) (if (< x 4.275032163700715e-250) (+ x (* (/ (- y x) t) z)) (+ x (/ (- y x) (/ t z)))))

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