Average Error: 7.5 → 0.7
Time: 3.4s
Precision: binary64
\[\frac{x + y}{1 - \frac{y}{z}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -1.6757501527623584 \cdot 10^{-239} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{x + y} - \frac{y}{\left(x + y\right) \cdot z}}\\ \end{array}\]
\frac{x + y}{1 - \frac{y}{z}}
↓
\begin{array}{l}
\mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -1.6757501527623584 \cdot 10^{-239} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 0\right):\\
\;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\

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

\end{array}
(FPCore (x y z) :precision binary64 (/ (+ x y) (- 1.0 (/ y z))))
↓
(FPCore (x y z)
 :precision binary64
 (if (or (<= (/ (+ x y) (- 1.0 (/ y z))) -1.6757501527623584e-239)
         (not (<= (/ (+ x y) (- 1.0 (/ y z))) 0.0)))
   (/ (+ x y) (- 1.0 (/ y z)))
   (/ 1.0 (- (/ 1.0 (+ x y)) (/ y (* (+ x y) z))))))
double code(double x, double y, double z) {
	return (((double) (x + y)) / ((double) (1.0 - (y / z))));
}
↓
double code(double x, double y, double z) {
	double tmp;
	if ((((((double) (x + y)) / ((double) (1.0 - (y / z)))) <= -1.6757501527623584e-239) || !((((double) (x + y)) / ((double) (1.0 - (y / z)))) <= 0.0))) {
		tmp = (((double) (x + y)) / ((double) (1.0 - (y / z))));
	} else {
		tmp = (1.0 / ((double) ((1.0 / ((double) (x + y))) - (y / ((double) (((double) (x + y)) * z))))));
	}
	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

Original7.5
Target4.0
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;y < -3.7429310762689856 \cdot 10^{+171}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \mathbf{elif}\;y < 3.5534662456086734 \cdot 10^{+168}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (+ x y) (- 1.0 (/ y z))) < -1.6757501527623584e-239 or 0.0 < (/ (+ x y) (- 1.0 (/ y z)))

    1. Initial program Error: 0.1 bits

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

    if -1.6757501527623584e-239 < (/ (+ x y) (- 1.0 (/ y z))) < 0.0

    1. Initial program Error: 52.4 bits

      \[\frac{x + y}{1 - \frac{y}{z}}\]
    2. Using strategy rm

    3. Applied clear-numError: 52.4 bits

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

    5. Applied div-subError: 52.4 bits

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

    6. SimplifiedError: 52.4 bits

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

    7. SimplifiedError: 4.3 bits

      \[\leadsto \frac{1}{\frac{1}{y + x} - \color{blue}{\frac{y}{z \cdot \left(y + x\right)}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplificationError: 0.7 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -1.6757501527623584 \cdot 10^{-239} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{x + y} - \frac{y}{\left(x + y\right) \cdot z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020204 
(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))))