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

↓

\[x + \frac{y - x}{\frac{t}{z}}\]

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

↓

x + \frac{y - x}{\frac{t}{z}}

double f(double x, double y, double z, double t) {
        double r29880 = x;
        double r29881 = y;
        double r29882 = r29881 - r29880;
        double r29883 = z;
        double r29884 = r29882 * r29883;
        double r29885 = t;
        double r29886 = r29884 / r29885;
        double r29887 = r29880 + r29886;
        return r29887;
}

↓

double f(double x, double y, double z, double t) {
        double r29888 = x;
        double r29889 = y;
        double r29890 = r29889 - r29888;
        double r29891 = t;
        double r29892 = z;
        double r29893 = r29891 / r29892;
        double r29894 = r29890 / r29893;
        double r29895 = r29888 + r29894;
        return r29895;
}

Error

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

Target

Original	6.3
Target	2.0
Herbie	1.6

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

Derivation

Initial program 6.3
\[x + \frac{\left(y - x\right) \cdot z}{t}\]
Using strategy rm
Applied associate-/l*1.6
\[\leadsto x + \color{blue}{\frac{y - x}{\frac{t}{z}}}\]
Final simplification1.6
\[\leadsto x + \frac{y - x}{\frac{t}{z}}\]

Reproduce

herbie shell --seed 2019315 
(FPCore (x y z t)
  :name "Numeric.Histogram:binBounds from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (if (< x -9.0255111955330046e-135) (- x (* (/ z t) (- x y))) (if (< x 4.2750321637007147e-250) (+ x (* (/ (- y x) t) z)) (+ x (/ (- y x) (/ t z)))))

  (+ x (/ (* (- y x) z) t)))

In
Out

Error

Try it out

Target

Derivation

Reproduce