Average Error: 7.4 → 0.7
Time: 3.3s
Precision: binary64
\[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
↓
\[\begin{array}{l} \mathbf{if}\;z \leq -8.970176524587994 \cdot 10^{+114}:\\ \;\;\;\;x \cdot \frac{\left(1 + y\right) - z}{z}\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_0 := 1 + \left(y - z\right)\\ \mathbf{if}\;z \leq 5.246984068361077 \cdot 10^{-119}:\\ \;\;\;\;\frac{x}{z} \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{t_0}}\\ \end{array}\\ \end{array} \]
\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \leq -8.970176524587994 \cdot 10^{+114}:\\
\;\;\;\;x \cdot \frac{\left(1 + y\right) - z}{z}\\

\mathbf{else}:\\
\;\;\;\;\begin{array}{l}
t_0 := 1 + \left(y - z\right)\\
\mathbf{if}\;z \leq 5.246984068361077 \cdot 10^{-119}:\\
\;\;\;\;\frac{x}{z} \cdot t_0\\

\mathbf{else}:\\
\;\;\;\;\frac{x}{\frac{z}{t_0}}\\


\end{array}\\


\end{array}

(FPCore (x y z) :precision binary64 (/ (* x (+ (- y z) 1.0)) z))
↓
(FPCore (x y z)
 :precision binary64
 (if (<= z -8.970176524587994e+114)
   (* x (/ (- (+ 1.0 y) z) z))
   (let* ((t_0 (+ 1.0 (- y z))))
     (if (<= z 5.246984068361077e-119) (* (/ x z) t_0) (/ x (/ z t_0))))))
double code(double x, double y, double z) {
	return (x * ((y - z) + 1.0)) / z;
}
↓
double code(double x, double y, double z) {
	double tmp;
	if (z <= -8.970176524587994e+114) {
		tmp = x * (((1.0 + y) - z) / z);
	} else {
		double t_0 = 1.0 + (y - z);
		double tmp_1;
		if (z <= 5.246984068361077e-119) {
			tmp_1 = (x / z) * t_0;
		} else {
			tmp_1 = x / (z / t_0);
		}
		tmp = tmp_1;
	}
	return tmp;
}

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

Original7.4
Target0.4
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;x < -2.71483106713436 \cdot 10^{-162}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x < 3.874108816439546 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if z < -8.970176524587994e114

    1. Initial program 20.1

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

    2. Applied *-un-lft-identity_binary6420.1

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

    3. Applied times-frac_binary640.0

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

    4. Simplified0.0

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

    5. Simplified0.0

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

    if -8.970176524587994e114 < z < 5.24698406836107669e-119

    1. Initial program 1.1

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

    2. Applied associate-/l*_binary644.4

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

    3. Applied associate-/r/_binary641.1

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

    if 5.24698406836107669e-119 < z

    1. Initial program 10.7

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

    2. Applied associate-/l*_binary640.6

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.970176524587994 \cdot 10^{+114}:\\ \;\;\;\;x \cdot \frac{\left(1 + y\right) - z}{z}\\ \mathbf{elif}\;z \leq 5.246984068361077 \cdot 10^{-119}:\\ \;\;\;\;\frac{x}{z} \cdot \left(1 + \left(y - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{1 + \left(y - z\right)}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022068 
(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) (- (* (+ 1.0 y) (/ x z)) x) (if (< x 3.874108816439546e-197) (* (* x (+ (- y z) 1.0)) (/ 1.0 z)) (- (* (+ 1.0 y) (/ x z)) x)))

  (/ (* x (+ (- y z) 1.0)) z))