Average Error: 6.5 → 1.4
Time: 23.3s
Precision: 64
\[x + \frac{\left(y - x\right) \cdot z}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.242090268965133989846420111071056511087 \cdot 10^{-14} \lor \neg \left(t \le 1.904850199149619616791444194911664558784 \cdot 10^{52}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{t}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(y - x\right) \cdot z}{t} + x\\ \end{array}\]
x + \frac{\left(y - x\right) \cdot z}{t}
↓
\begin{array}{l}
\mathbf{if}\;t \le -1.242090268965133989846420111071056511087 \cdot 10^{-14} \lor \neg \left(t \le 1.904850199149619616791444194911664558784 \cdot 10^{52}\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{y - x}{t}, z, x\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(y - x\right) \cdot z}{t} + x\\

\end{array}
double f(double x, double y, double z, double t) {
        double r257051 = x;
        double r257052 = y;
        double r257053 = r257052 - r257051;
        double r257054 = z;
        double r257055 = r257053 * r257054;
        double r257056 = t;
        double r257057 = r257055 / r257056;
        double r257058 = r257051 + r257057;
        return r257058;
}
↓
double f(double x, double y, double z, double t) {
        double r257059 = t;
        double r257060 = -1.242090268965134e-14;
        bool r257061 = r257059 <= r257060;
        double r257062 = 1.9048501991496196e+52;
        bool r257063 = r257059 <= r257062;
        double r257064 = !r257063;
        bool r257065 = r257061 || r257064;
        double r257066 = y;
        double r257067 = x;
        double r257068 = r257066 - r257067;
        double r257069 = r257068 / r257059;
        double r257070 = z;
        double r257071 = fma(r257069, r257070, r257067);
        double r257072 = r257068 * r257070;
        double r257073 = r257072 / r257059;
        double r257074 = r257073 + r257067;
        double r257075 = r257065 ? r257071 : r257074;
        return r257075;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original6.5
Target2.0
Herbie1.4
\[\begin{array}{l} \mathbf{if}\;x \lt -9.025511195533004570453352523209034680317 \cdot 10^{-135}:\\ \;\;\;\;x - \frac{z}{t} \cdot \left(x - y\right)\\ \mathbf{elif}\;x \lt 4.275032163700714748507147332551979944314 \cdot 10^{-250}:\\ \;\;\;\;x + \frac{y - x}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if t < -1.242090268965134e-14 or 1.9048501991496196e+52 < t

    1. Initial program 9.7

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

    2. Simplified1.0

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

    if -1.242090268965134e-14 < t < 1.9048501991496196e+52

    1. Initial program 2.0

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

    2. Simplified14.6

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

    4. Applied fma-udef14.6

      \[\leadsto \color{blue}{\frac{y - x}{t} \cdot z + x}\]

    5. Simplified2.0

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

  4. Final simplification1.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.242090268965133989846420111071056511087 \cdot 10^{-14} \lor \neg \left(t \le 1.904850199149619616791444194911664558784 \cdot 10^{52}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{t}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(y - x\right) \cdot z}{t} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(FPCore (x y z t)
  :name "Numeric.Histogram:binBounds from Chart-1.5.3"
  :precision binary64

  :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)))