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

↓

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

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 r322808 = x;
        double r322809 = y;
        double r322810 = r322809 - r322808;
        double r322811 = z;
        double r322812 = r322810 * r322811;
        double r322813 = t;
        double r322814 = r322812 / r322813;
        double r322815 = r322808 + r322814;
        return r322815;
}

↓

double f(double x, double y, double z, double t) {
        double r322816 = x;
        double r322817 = y;
        double r322818 = r322817 - r322816;
        double r322819 = t;
        double r322820 = z;
        double r322821 = r322819 / r322820;
        double r322822 = r322818 / r322821;
        double r322823 = r322816 + r322822;
        return r322823;
}

Error

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

Target

Original	6.3
Target	2.2
Herbie	2.2

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

Derivation

Initial program 6.3
\[x + \frac{\left(y - x\right) \cdot z}{t}\]
Using strategy rm
Applied associate-/l*2.2
\[\leadsto x + \color{blue}{\frac{y - x}{\frac{t}{z}}}\]
Final simplification2.2
\[\leadsto x + \frac{y - x}{\frac{t}{z}}\]

Reproduce

herbie shell --seed 2019323 
(FPCore (x y z t)
  :name "Numeric.Histogram:binBounds from Chart-1.5.3"
  :precision binary64

  :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)))

In
Out

Error

Try it out

Target

Derivation

Reproduce