Average Error: 6.4 → 1.8
Time: 16.2s
Precision: 64
\[x + \frac{\left(y - x\right) \cdot z}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.47437460621894122515597907358401131123 \cdot 10^{-73}:\\ \;\;\;\;\frac{y - x}{t} \cdot z + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)\\ \end{array}\]
x + \frac{\left(y - x\right) \cdot z}{t}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.47437460621894122515597907358401131123 \cdot 10^{-73}:\\
\;\;\;\;\frac{y - x}{t} \cdot z + x\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r431830 = x;
        double r431831 = y;
        double r431832 = r431831 - r431830;
        double r431833 = z;
        double r431834 = r431832 * r431833;
        double r431835 = t;
        double r431836 = r431834 / r431835;
        double r431837 = r431830 + r431836;
        return r431837;
}
↓
double f(double x, double y, double z, double t) {
        double r431838 = z;
        double r431839 = -1.4743746062189412e-73;
        bool r431840 = r431838 <= r431839;
        double r431841 = y;
        double r431842 = x;
        double r431843 = r431841 - r431842;
        double r431844 = t;
        double r431845 = r431843 / r431844;
        double r431846 = r431845 * r431838;
        double r431847 = r431846 + r431842;
        double r431848 = r431838 / r431844;
        double r431849 = fma(r431843, r431848, r431842);
        double r431850 = r431840 ? r431847 : r431849;
        return r431850;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original6.4
Target1.9
Herbie1.8
\[\begin{array}{l} \mathbf{if}\;x \lt -9.025511195533004570453352523209034680317 \cdot 10^{-135}:\\ \;\;\;\;x - \frac{z}{t} \cdot \left(x - y\right)\\ \mathbf{elif}\;x \lt 4.275032163700714748507147332551979944314 \cdot 10^{-250}:\\ \;\;\;\;x + \frac{y - x}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -1.4743746062189412e-73

    1. Initial program 11.2

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

    2. Simplified2.7

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

    4. Applied fma-udef2.7

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

    5. Simplified2.1

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

    if -1.4743746062189412e-73 < z

    1. Initial program 4.8

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

    2. Simplified1.8

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

  4. Final simplification1.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.47437460621894122515597907358401131123 \cdot 10^{-73}:\\ \;\;\;\;\frac{y - x}{t} \cdot z + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 +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)))