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

↓

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

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 r20872987 = x;
        double r20872988 = y;
        double r20872989 = r20872988 - r20872987;
        double r20872990 = z;
        double r20872991 = r20872989 * r20872990;
        double r20872992 = t;
        double r20872993 = r20872991 / r20872992;
        double r20872994 = r20872987 + r20872993;
        return r20872994;
}

↓

double f(double x, double y, double z, double t) {
        double r20872995 = x;
        double r20872996 = y;
        double r20872997 = r20872996 - r20872995;
        double r20872998 = t;
        double r20872999 = z;
        double r20873000 = r20872998 / r20872999;
        double r20873001 = r20872997 / r20873000;
        double r20873002 = r20872995 + r20873001;
        return r20873002;
}

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