Average Error: 10.0 → 0.1
Time: 2.6s
Precision: 64
\[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.96989924315931461 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{z}, x \cdot \frac{y}{z}\right) - x\\ \mathbf{elif}\;z \le 4098704.9989726637:\\ \;\;\;\;\frac{x}{z} \cdot \left(\left(y - z\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \end{array}\]
\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.96989924315931461 \cdot 10^{-11}:\\
\;\;\;\;\mathsf{fma}\left(1, \frac{x}{z}, x \cdot \frac{y}{z}\right) - x\\

\mathbf{elif}\;z \le 4098704.9989726637:\\
\;\;\;\;\frac{x}{z} \cdot \left(\left(y - z\right) + 1\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\

\end{array}
double code(double x, double y, double z) {
	return ((x * ((y - z) + 1.0)) / z);
}
↓
double code(double x, double y, double z) {
	double temp;
	if ((z <= -1.9698992431593146e-11)) {
		temp = (fma(1.0, (x / z), (x * (y / z))) - x);
	} else {
		double temp_1;
		if ((z <= 4098704.9989726637)) {
			temp_1 = ((x / z) * ((y - z) + 1.0));
		} else {
			temp_1 = (x / (z / ((y - z) + 1.0)));
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original10.0
Target0.4
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;x \lt -2.7148310671343599 \cdot 10^{-162}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x \lt 3.87410881643954616 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -1.9698992431593146e-11

    1. Initial program 16.6

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\]
    2. Using strategy rm

    3. Applied associate-/l*0.1

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{\left(y - z\right) + 1}}}\]

    4. Taylor expanded around 0 5.3

      \[\leadsto \color{blue}{\left(\frac{x \cdot y}{z} + 1 \cdot \frac{x}{z}\right) - x}\]

    5. Simplified5.3

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

    7. Applied *-un-lft-identity5.3

      \[\leadsto \mathsf{fma}\left(1, \frac{x}{z}, \frac{x \cdot y}{\color{blue}{1 \cdot z}}\right) - x\]

    8. Applied times-frac0.1

      \[\leadsto \mathsf{fma}\left(1, \frac{x}{z}, \color{blue}{\frac{x}{1} \cdot \frac{y}{z}}\right) - x\]

    9. Simplified0.1

      \[\leadsto \mathsf{fma}\left(1, \frac{x}{z}, \color{blue}{x} \cdot \frac{y}{z}\right) - x\]

    if -1.9698992431593146e-11 < z < 4098704.9989726637

    1. Initial program 0.1

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\]
    2. Using strategy rm

    3. Applied associate-/l*7.7

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{\left(y - z\right) + 1}}}\]
    4. Using strategy rm

    5. Applied associate-/r/0.1

      \[\leadsto \color{blue}{\frac{x}{z} \cdot \left(\left(y - z\right) + 1\right)}\]

    if 4098704.9989726637 < z

    1. Initial program 16.8

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\]
    2. Using strategy rm

    3. Applied associate-/l*0.1

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{\left(y - z\right) + 1}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.96989924315931461 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{z}, x \cdot \frac{y}{z}\right) - x\\ \mathbf{elif}\;z \le 4098704.9989726637:\\ \;\;\;\;\frac{x}{z} \cdot \left(\left(y - z\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020066 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.TwoD.Segment.Bernstein:evaluateBernstein from diagrams-lib-1.3.0.3"
  :precision binary64

  :herbie-target
  (if (< x -2.71483106713436e-162) (- (* (+ 1 y) (/ x z)) x) (if (< x 3.874108816439546e-197) (* (* x (+ (- y z) 1)) (/ 1 z)) (- (* (+ 1 y) (/ x z)) x)))

  (/ (* x (+ (- y z) 1)) z))