Average Error: 7.7 → 0.1
Time: 8.5s
Precision: binary64
\[\frac{x + y}{1 - \frac{y}{z}} \]
↓
\[\begin{array}{l} t_0 := 1 - \frac{y}{z}\\ t_1 := \frac{x + y}{t_0}\\ \mathbf{if}\;t_1 \leq -1.74949953938085 \cdot 10^{-302}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;-\frac{\left(x + y\right) \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t_0} + \frac{y}{t_0}\\ \end{array} \]
\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 -1.74949953938085 \cdot 10^{-302}:\\
\;\;\;\;t_1\\

\mathbf{elif}\;t_1 \leq 0:\\
\;\;\;\;-\frac{\left(x + y\right) \cdot z}{y}\\

\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 -1.74949953938085e-302)
     t_1
     (if (<= t_1 0.0) (- (/ (* (+ x y) z) y)) (+ (/ 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 <= -1.74949953938085e-302) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = -(((x + y) * z) / y);
	} 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

Original7.7
Target4.1
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;y < -3.7429310762689856 \cdot 10^{+171}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \mathbf{elif}\;y < 3.5534662456086734 \cdot 10^{+168}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -1.7494995393808499e-302

    1. Initial program 0.1

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

    if -1.7494995393808499e-302 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < 0.0

    1. Initial program 59.9

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

    2. Taylor expanded in z around 0 0.3

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

    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 simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -1.74949953938085 \cdot 10^{-302}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{elif}\;\frac{x + y}{1 - \frac{y}{z}} \leq 0:\\ \;\;\;\;-\frac{\left(x + y\right) \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}} + \frac{y}{1 - \frac{y}{z}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022068 
(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))))