Average Error: 6.6 → 2.1
Time: 2.8s
Precision: 64
\[x + \frac{\left(y - x\right) \cdot z}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le 3.6000243022547191 \cdot 10^{-81} \lor \neg \left(z \le 1.640658992443321 \cdot 10^{237}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{t}, z, x\right)\\ \end{array}\]
x + \frac{\left(y - x\right) \cdot z}{t}
↓
\begin{array}{l}
\mathbf{if}\;z \le 3.6000243022547191 \cdot 10^{-81} \lor \neg \left(z \le 1.640658992443321 \cdot 10^{237}\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y - x, x\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{y - x}{t}, z, x\right)\\

\end{array}
double code(double x, double y, double z, double t) {
	return ((double) (x + ((double) (((double) (((double) (y - x)) * z)) / t))));
}
↓
double code(double x, double y, double z, double t) {
	double VAR;
	if (((z <= 3.600024302254719e-81) || !(z <= 1.640658992443321e+237))) {
		VAR = ((double) fma(((double) (z / t)), ((double) (y - x)), x));
	} else {
		VAR = ((double) fma(((double) (((double) (y - x)) / t)), z, x));
	}
	return VAR;
}

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

Original6.6
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. Split input into 2 regimes

  2. if z < 3.600024302254719e-81 or 1.640658992443321e+237 < z

    1. Initial program 6.1

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

    2. Simplified7.6

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

    4. Applied clear-num7.7

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

    6. Applied fma-udef7.7

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

    7. Simplified7.0

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

    9. Applied associate-/r/2.1

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

    10. Applied fma-def2.1

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

    if 3.600024302254719e-81 < z < 1.640658992443321e+237

    1. Initial program 8.1

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

    2. Simplified2.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - x}{t}, z, x\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le 3.6000243022547191 \cdot 10^{-81} \lor \neg \left(z \le 1.640658992443321 \cdot 10^{237}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{t}, z, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020123 +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)))