Average Error: 7.4 → 0.6
Time: 5.0s
Precision: binary64
\[\frac{x + y}{1 - \frac{y}{z}} \]
↓
\[\begin{array}{l} t_0 := 1 - \frac{y}{z}\\ t_1 := \frac{x + y}{t_0}\\ \mathbf{if}\;t_1 \leq -6.827332753572353 \cdot 10^{-230}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{1}{\frac{1}{x} - \frac{y}{x \cdot z}} - z\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t_0} + \frac{y}{t_0}\\ \end{array} \]
\frac{x + y}{1 - \frac{y}{z}}
↓
\begin{array}{l}
t_0 := 1 - \frac{y}{z}\\
t_1 := \frac{x + y}{t_0}\\
\mathbf{if}\;t_1 \leq -6.827332753572353 \cdot 10^{-230}:\\
\;\;\;\;t_1\\

\mathbf{elif}\;t_1 \leq 0:\\
\;\;\;\;\frac{1}{\frac{1}{x} - \frac{y}{x \cdot z}} - z\\

\mathbf{else}:\\
\;\;\;\;\frac{x}{t_0} + \frac{y}{t_0}\\


\end{array}

(FPCore (x y z) :precision binary64 (/ (+ x y) (- 1.0 (/ y z))))
↓
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ y z))) (t_1 (/ (+ x y) t_0)))
   (if (<= t_1 -6.827332753572353e-230)
     t_1
     (if (<= t_1 0.0)
       (- (/ 1.0 (- (/ 1.0 x) (/ y (* x z)))) z)
       (+ (/ x t_0) (/ y t_0))))))
double code(double x, double y, double z) {
	return (x + y) / (1.0 - (y / z));
}
↓
double code(double x, double y, double z) {
	double t_0 = 1.0 - (y / z);
	double t_1 = (x + y) / t_0;
	double tmp;
	if (t_1 <= -6.827332753572353e-230) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = (1.0 / ((1.0 / x) - (y / (x * z)))) - z;
	} else {
		tmp = (x / t_0) + (y / t_0);
	}
	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
Target3.6
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;y < -3.7429310762689856 \cdot 10^{+171}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \mathbf{elif}\;y < 3.5534662456086734 \cdot 10^{+168}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -6.82733275357235265e-230

    1. Initial program 0.1

      \[\frac{x + y}{1 - \frac{y}{z}} \]

    2. Applied *-un-lft-identity_binary640.1

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

    3. Applied associate-/r*_binary640.1

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

    if -6.82733275357235265e-230 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -0.0

    1. Initial program 51.9

      \[\frac{x + y}{1 - \frac{y}{z}} \]

    2. Taylor expanded in x around 0 51.9

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

    3. Applied clear-num_binary6451.9

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

    4. Taylor expanded in y around 0 45.4

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

    5. Taylor expanded in y around inf 4.0

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

    if -0.0 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z)))

    1. Initial program 0.1

      \[\frac{x + y}{1 - \frac{y}{z}} \]

    2. Taylor expanded in x around 0 0.1

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -6.827332753572353 \cdot 10^{-230}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{elif}\;\frac{x + y}{1 - \frac{y}{z}} \leq 0:\\ \;\;\;\;\frac{1}{\frac{1}{x} - \frac{y}{x \cdot z}} - z\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}} + \frac{y}{1 - \frac{y}{z}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022077 
(FPCore (x y z)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, A"
  :precision binary64

  :herbie-target
  (if (< y -3.7429310762689856e+171) (* (/ (+ y x) (- y)) z) (if (< y 3.5534662456086734e+168) (/ (+ x y) (- 1.0 (/ y z))) (* (/ (+ y x) (- y)) z)))

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