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

↓

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

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

↓

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

double f(double x, double y, double z, double t) {
        double r20256397 = x;
        double r20256398 = y;
        double r20256399 = r20256398 - r20256397;
        double r20256400 = z;
        double r20256401 = r20256399 * r20256400;
        double r20256402 = t;
        double r20256403 = r20256401 / r20256402;
        double r20256404 = r20256397 + r20256403;
        return r20256404;
}

↓

double f(double x, double y, double z, double t) {
        double r20256405 = z;
        double r20256406 = t;
        double r20256407 = r20256405 / r20256406;
        double r20256408 = y;
        double r20256409 = x;
        double r20256410 = r20256408 - r20256409;
        double r20256411 = fma(r20256407, r20256410, r20256409);
        return r20256411;
}

Error

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

Target

Original	6.1
Target	1.7
Herbie	1.8

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

Derivation

Initial program 6.1
\[x + \frac{\left(y - x\right) \cdot z}{t}\]
Simplified1.8
\[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{t}, y - x, x\right)}\]
Final simplification1.8
\[\leadsto \mathsf{fma}\left(\frac{z}{t}, y - x, x\right)\]

Reproduce

herbie shell --seed 2019163 +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)))