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

↓

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

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

↓

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

double f(double x, double y, double z, double t) {
        double r19957153 = x;
        double r19957154 = y;
        double r19957155 = r19957154 - r19957153;
        double r19957156 = z;
        double r19957157 = r19957155 * r19957156;
        double r19957158 = t;
        double r19957159 = r19957157 / r19957158;
        double r19957160 = r19957153 + r19957159;
        return r19957160;
}

↓

double f(double x, double y, double z, double t) {
        double r19957161 = y;
        double r19957162 = x;
        double r19957163 = r19957161 - r19957162;
        double r19957164 = z;
        double r19957165 = t;
        double r19957166 = r19957164 / r19957165;
        double r19957167 = fma(r19957163, r19957166, r19957162);
        return r19957167;
}

Error

The X axis uses an exponential scale — Bits error versus `x`

Target

Original	6.4
Target	2.0
Herbie	1.8

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

Derivation

Initial program 6.4
\[x + \frac{\left(y - x\right) \cdot z}{t}\]
Simplified1.8
\[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)}\]
Final simplification1.8
\[\leadsto \mathsf{fma}\left(y - x, \frac{z}{t}, x\right)\]

Reproduce

herbie shell --seed 2019171 +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)))