Average Error: 6.3 → 2.2
Time: 2.2s
Precision: 64
\[x + \frac{\left(y - x\right) \cdot z}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.33529079990606012 \cdot 10^{-79} \lor \neg \left(z \le 2.779801418179183 \cdot 10^{-143}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t} - \frac{x}{t}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \end{array}\]
x + \frac{\left(y - x\right) \cdot z}{t}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.33529079990606012 \cdot 10^{-79} \lor \neg \left(z \le 2.779801418179183 \cdot 10^{-143}\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{t} - \frac{x}{t}, z, x\right)\\

\mathbf{else}:\\
\;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\

\end{array}
double code(double x, double y, double z, double t) {
	return (x + (((y - x) * z) / t));
}
↓
double code(double x, double y, double z, double t) {
	double VAR;
	if (((z <= -1.3352907999060601e-79) || !(z <= 2.779801418179183e-143))) {
		VAR = fma(((y / t) - (x / t)), z, x);
	} else {
		VAR = (x + (((y - x) * z) / t));
	}
	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

Original6.3
Target2.0
Herbie2.2
\[\begin{array}{l} \mathbf{if}\;x \lt -9.0255111955330046 \cdot 10^{-135}:\\ \;\;\;\;x - \frac{z}{t} \cdot \left(x - y\right)\\ \mathbf{elif}\;x \lt 4.2750321637007147 \cdot 10^{-250}:\\ \;\;\;\;x + \frac{y - x}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -1.3352907999060601e-79 or 2.779801418179183e-143 < z

    1. Initial program 10.0

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

    2. Simplified3.0

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

    4. Applied div-sub3.0

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

    if -1.3352907999060601e-79 < z < 2.779801418179183e-143

    1. Initial program 1.1

      \[x + \frac{\left(y - x\right) \cdot z}{t}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.33529079990606012 \cdot 10^{-79} \lor \neg \left(z \le 2.779801418179183 \cdot 10^{-143}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t} - \frac{x}{t}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 +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)))