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

↓

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

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 r480450 = x;
        double r480451 = y;
        double r480452 = r480451 - r480450;
        double r480453 = z;
        double r480454 = r480452 * r480453;
        double r480455 = t;
        double r480456 = r480454 / r480455;
        double r480457 = r480450 + r480456;
        return r480457;
}

↓

double f(double x, double y, double z, double t) {
        double r480458 = x;
        double r480459 = y;
        double r480460 = r480459 - r480458;
        double r480461 = t;
        double r480462 = z;
        double r480463 = r480461 / r480462;
        double r480464 = r480460 / r480463;
        double r480465 = r480458 + r480464;
        return r480465;
}

Error

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

Target

Original	6.5
Target	2.1
Herbie	1.9

\[\begin{array}{l} \mathbf{if}\;x \lt -9.0255111955330046 \cdot 10^{-135}:\\ \;\;\;\;x - \frac{z}{t} \cdot \left(x - y\right)\\ \mathbf{elif}\;x \lt 4.2750321637007147 \cdot 10^{-250}:\\ \;\;\;\;x + \frac{y - x}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array}\]

Derivation

Initial program 6.5
\[x + \frac{\left(y - x\right) \cdot z}{t}\]
Using strategy rm
Applied associate-/l*1.9
\[\leadsto x + \color{blue}{\frac{y - x}{\frac{t}{z}}}\]
Final simplification1.9
\[\leadsto x + \frac{y - x}{\frac{t}{z}}\]

Reproduce

herbie shell --seed 2020024 +o rules:numerics
(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