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

↓

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

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 r20360101 = x;
        double r20360102 = y;
        double r20360103 = r20360102 - r20360101;
        double r20360104 = z;
        double r20360105 = r20360103 * r20360104;
        double r20360106 = t;
        double r20360107 = r20360105 / r20360106;
        double r20360108 = r20360101 + r20360107;
        return r20360108;
}

↓

double f(double x, double y, double z, double t) {
        double r20360109 = y;
        double r20360110 = x;
        double r20360111 = r20360109 - r20360110;
        double r20360112 = z;
        double r20360113 = t;
        double r20360114 = r20360112 / r20360113;
        double r20360115 = fma(r20360111, r20360114, r20360110);
        return r20360115;
}

Error

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

Target

Original	6.4
Target	1.9
Herbie	2.0

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

Derivation

Initial program 6.4
\[x + \frac{\left(y - x\right) \cdot z}{t}\]
Simplified2.0
\[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)}\]
Final simplification2.0
\[\leadsto \mathsf{fma}\left(y - x, \frac{z}{t}, x\right)\]

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)))