Average Error: 6.7 → 0.6
Time: 3.6s
Precision: binary64
\[x + \frac{y \cdot \left(z - x\right)}{t} \]
↓
\[\begin{array}{l} t_1 := x + \frac{y \cdot \left(z - x\right)}{t}\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 9.14241781620673 \cdot 10^{+304}\right):\\ \;\;\;\;x + \left(z - x\right) \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
x + \frac{y \cdot \left(z - x\right)}{t}
↓
\begin{array}{l}
t_1 := x + \frac{y \cdot \left(z - x\right)}{t}\\
\mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 9.14241781620673 \cdot 10^{+304}\right):\\
\;\;\;\;x + \left(z - x\right) \cdot \frac{y}{t}\\

\mathbf{else}:\\
\;\;\;\;t_1\\


\end{array}

(FPCore (x y z t) :precision binary64 (+ x (/ (* y (- z x)) t)))
↓
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ x (/ (* y (- z x)) t))))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 9.14241781620673e+304)))
     (+ x (* (- z x) (/ y t)))
     t_1)))
double code(double x, double y, double z, double t) {
	return x + ((y * (z - x)) / t);
}
↓
double code(double x, double y, double z, double t) {
	double t_1 = x + ((y * (z - x)) / t);
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 9.14241781620673e+304)) {
		tmp = x + ((z - x) * (y / t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

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

Original6.7
Target2.0
Herbie0.6
\[x - \left(x \cdot \frac{y}{t} + \left(-z\right) \cdot \frac{y}{t}\right) \]

Derivation

  1. Split input into 2 regimes

  2. if (+.f64 x (/.f64 (*.f64 y (-.f64 z x)) t)) < -inf.0 or 9.14241781620672955e304 < (+.f64 x (/.f64 (*.f64 y (-.f64 z x)) t))

    1. Initial program 61.6

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

    2. Simplified1.6

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

    3. Taylor expanded in y around 0 61.6

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

    4. Simplified0.3

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

    5. Applied fma-udef_binary640.3

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

    if -inf.0 < (+.f64 x (/.f64 (*.f64 y (-.f64 z x)) t)) < 9.14241781620672955e304

    1. Initial program 0.7

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{y \cdot \left(z - x\right)}{t} \leq -\infty \lor \neg \left(x + \frac{y \cdot \left(z - x\right)}{t} \leq 9.14241781620673 \cdot 10^{+304}\right):\\ \;\;\;\;x + \left(z - x\right) \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(z - x\right)}{t}\\ \end{array} \]

Reproduce

herbie shell --seed 2022081 
(FPCore (x y z t)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, D"
  :precision binary64

  :herbie-target
  (- x (+ (* x (/ y t)) (* (- z) (/ y t))))

  (+ x (/ (* y (- z x)) t)))