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

↓

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

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 r25288823 = x;
        double r25288824 = y;
        double r25288825 = r25288824 - r25288823;
        double r25288826 = z;
        double r25288827 = r25288825 * r25288826;
        double r25288828 = t;
        double r25288829 = r25288827 / r25288828;
        double r25288830 = r25288823 + r25288829;
        return r25288830;
}

↓

double f(double x, double y, double z, double t) {
        double r25288831 = x;
        double r25288832 = y;
        double r25288833 = r25288832 - r25288831;
        double r25288834 = t;
        double r25288835 = z;
        double r25288836 = r25288834 / r25288835;
        double r25288837 = r25288833 / r25288836;
        double r25288838 = r25288831 + r25288837;
        return r25288838;
}

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
\[x + \frac{\left(y - x\right) \cdot z}{t}\]
Using strategy rm
Applied associate-/l*1.8
\[\leadsto x + \color{blue}{\frac{y - x}{\frac{t}{z}}}\]
Final simplification1.8
\[\leadsto x + \frac{y - x}{\frac{t}{z}}\]

Reproduce

herbie shell --seed 2019163 
(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)))

In
Out

Error

Try it out

Target

Derivation

Reproduce