Average Error: 7.6 → 6.5
Time: 3.5s
Precision: binary64
\[\frac{x + y}{1 - \frac{y}{z}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \le -1.2682815818638479 \cdot 10^{-282}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{elif}\;\frac{x + y}{1 - \frac{y}{z}} \le -0.0:\\ \;\;\;\;\frac{1}{\frac{\sqrt{1} + \frac{\sqrt{y}}{\sqrt{z}}}{\sqrt{x + y}} \cdot \frac{\sqrt{1} - \frac{\sqrt{y}}{\sqrt{z}}}{\sqrt{x + y}}}\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) \cdot \frac{1}{1 - \frac{y}{z}}\\ \end{array}\]
\frac{x + y}{1 - \frac{y}{z}}
↓
\begin{array}{l}
\mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \le -1.2682815818638479 \cdot 10^{-282}:\\
\;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\

\mathbf{elif}\;\frac{x + y}{1 - \frac{y}{z}} \le -0.0:\\
\;\;\;\;\frac{1}{\frac{\sqrt{1} + \frac{\sqrt{y}}{\sqrt{z}}}{\sqrt{x + y}} \cdot \frac{\sqrt{1} - \frac{\sqrt{y}}{\sqrt{z}}}{\sqrt{x + y}}}\\

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

\end{array}
double code(double x, double y, double z) {
	return ((double) (((double) (x + y)) / ((double) (1.0 - ((double) (y / z))))));
}
↓
double code(double x, double y, double z) {
	double VAR;
	if ((((double) (((double) (x + y)) / ((double) (1.0 - ((double) (y / z)))))) <= -1.268281581863848e-282)) {
		VAR = ((double) (((double) (x + y)) / ((double) (1.0 - ((double) (y / z))))));
	} else {
		double VAR_1;
		if ((((double) (((double) (x + y)) / ((double) (1.0 - ((double) (y / z)))))) <= -0.0)) {
			VAR_1 = ((double) (1.0 / ((double) (((double) (((double) (((double) sqrt(1.0)) + ((double) (((double) sqrt(y)) / ((double) sqrt(z)))))) / ((double) sqrt(((double) (x + y)))))) * ((double) (((double) (((double) sqrt(1.0)) - ((double) (((double) sqrt(y)) / ((double) sqrt(z)))))) / ((double) sqrt(((double) (x + y))))))))));
		} else {
			VAR_1 = ((double) (((double) (x + y)) * ((double) (1.0 / ((double) (1.0 - ((double) (y / z))))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

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

Original7.6
Target3.8
Herbie6.5
\[\begin{array}{l} \mathbf{if}\;y \lt -3.74293107626898565 \cdot 10^{171}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \mathbf{elif}\;y \lt 3.55346624560867344 \cdot 10^{168}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (/ (+ x y) (- 1.0 (/ y z))) < -1.2682815818638479e-282

    1. Initial program 0.1

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

    if -1.2682815818638479e-282 < (/ (+ x y) (- 1.0 (/ y z))) < -0.0

    1. Initial program 58.4

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

    3. Applied clear-num58.4

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

    5. Applied add-sqr-sqrt61.4

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

    6. Applied add-sqr-sqrt62.4

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

    7. Applied add-sqr-sqrt62.4

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

    8. Applied times-frac62.4

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

    9. Applied add-sqr-sqrt62.4

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

    10. Applied difference-of-squares62.4

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

    11. Applied times-frac49.8

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

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

    1. Initial program 0.1

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

    3. Applied div-inv0.2

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

  4. Final simplification6.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \le -1.2682815818638479 \cdot 10^{-282}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{elif}\;\frac{x + y}{1 - \frac{y}{z}} \le -0.0:\\ \;\;\;\;\frac{1}{\frac{\sqrt{1} + \frac{\sqrt{y}}{\sqrt{z}}}{\sqrt{x + y}} \cdot \frac{\sqrt{1} - \frac{\sqrt{y}}{\sqrt{z}}}{\sqrt{x + y}}}\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) \cdot \frac{1}{1 - \frac{y}{z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020149 
(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) (neg y)) z) (if (< y 3.5534662456086734e+168) (/ (+ x y) (- 1.0 (/ y z))) (* (/ (+ y x) (neg y)) z)))

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