Average Error: 6.5 → 2.1
Time: 4.4s
Precision: 64
\[x + \frac{\left(y - x\right) \cdot z}{t}\]
↓
\[\frac{\frac{z}{t}}{\frac{1}{y - x}} + x\]
x + \frac{\left(y - x\right) \cdot z}{t}
↓
\frac{\frac{z}{t}}{\frac{1}{y - x}} + x
double f(double x, double y, double z, double t) {
        double r594396 = x;
        double r594397 = y;
        double r594398 = r594397 - r594396;
        double r594399 = z;
        double r594400 = r594398 * r594399;
        double r594401 = t;
        double r594402 = r594400 / r594401;
        double r594403 = r594396 + r594402;
        return r594403;
}
↓
double f(double x, double y, double z, double t) {
        double r594404 = z;
        double r594405 = t;
        double r594406 = r594404 / r594405;
        double r594407 = 1.0;
        double r594408 = y;
        double r594409 = x;
        double r594410 = r594408 - r594409;
        double r594411 = r594407 / r594410;
        double r594412 = r594406 / r594411;
        double r594413 = r594412 + r594409;
        return r594413;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original6.5
Target2.0
Herbie2.1
\[\begin{array}{l} \mathbf{if}\;x \lt -9.0255111955330046 \cdot 10^{-135}:\\ \;\;\;\;x - \frac{z}{t} \cdot \left(x - y\right)\\ \mathbf{elif}\;x \lt 4.2750321637007147 \cdot 10^{-250}:\\ \;\;\;\;x + \frac{y - x}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array}\]

Derivation

  1. Initial program 6.5

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

  2. Simplified6.3

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - x}{t}, z, x\right)}\]
  3. Using strategy rm

  4. Applied clear-num6.4

    \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\frac{t}{y - x}}}, z, x\right)\]
  5. Using strategy rm

  6. Applied fma-udef6.4

    \[\leadsto \color{blue}{\frac{1}{\frac{t}{y - x}} \cdot z + x}\]

  7. Simplified5.8

    \[\leadsto \color{blue}{\frac{z}{\frac{t}{y - x}}} + x\]
  8. Using strategy rm

  9. Applied div-inv5.8

    \[\leadsto \frac{z}{\color{blue}{t \cdot \frac{1}{y - x}}} + x\]

  10. Applied associate-/r*2.1

    \[\leadsto \color{blue}{\frac{\frac{z}{t}}{\frac{1}{y - x}}} + x\]

  11. Final simplification2.1

    \[\leadsto \frac{\frac{z}{t}}{\frac{1}{y - x}} + x\]

Reproduce

herbie shell --seed 2020057 +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)))