Average Error: 7.7 → 6.3
Time: 5.5s
Precision: 64
\[\frac{x + y}{1 - \frac{y}{z}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \le -1.94587273222047501 \cdot 10^{-303}:\\ \;\;\;\;\left(x + y\right) \cdot \frac{1}{1 - \frac{y}{z}}\\ \mathbf{elif}\;\frac{x + y}{1 - \frac{y}{z}} \le -0.0:\\ \;\;\;\;\frac{x + y}{\sqrt{1} + \frac{\sqrt{y}}{\sqrt{z}}} \cdot \frac{\sqrt{1}}{\sqrt{1} - \frac{\sqrt{y}}{\sqrt{z}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \end{array}\]
\frac{x + y}{1 - \frac{y}{z}}
↓
\begin{array}{l}
\mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \le -1.94587273222047501 \cdot 10^{-303}:\\
\;\;\;\;\left(x + y\right) \cdot \frac{1}{1 - \frac{y}{z}}\\

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

\mathbf{else}:\\
\;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\

\end{array}
double code(double x, double y, double z) {
	return ((x + y) / (1.0 - (y / z)));
}
↓
double code(double x, double y, double z) {
	double temp;
	if ((((x + y) / (1.0 - (y / z))) <= -1.945872732220475e-303)) {
		temp = ((x + y) * (1.0 / (1.0 - (y / z))));
	} else {
		double temp_1;
		if ((((x + y) / (1.0 - (y / z))) <= -0.0)) {
			temp_1 = (((x + y) / (sqrt(1.0) + (sqrt(y) / sqrt(z)))) * (sqrt(1.0) / (sqrt(1.0) - (sqrt(y) / sqrt(z)))));
		} else {
			temp_1 = ((x + y) / (1.0 - (y / z)));
		}
		temp = temp_1;
	}
	return temp;
}

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
Target4.0
Herbie6.3
\[\begin{array}{l} \mathbf{if}\;y \lt -3.74293107626898565 \cdot 10^{171}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \mathbf{elif}\;y \lt 3.55346624560867344 \cdot 10^{168}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (/ (+ x y) (- 1.0 (/ y z))) < -1.945872732220475e-303

    1. Initial program 0.1

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

    3. Applied div-inv0.1

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

    if -1.945872732220475e-303 < (/ (+ x y) (- 1.0 (/ y z))) < -0.0

    1. Initial program 59.9

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

    3. Applied div-inv59.9

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

    5. Applied add-sqr-sqrt61.9

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

    6. Applied add-sqr-sqrt63.0

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

    7. Applied times-frac63.0

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

    8. Applied add-sqr-sqrt63.0

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

    9. Applied difference-of-squares63.0

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

    10. Applied add-sqr-sqrt63.0

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

    11. Applied times-frac62.1

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

    12. Applied associate-*r*48.7

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

    13. Simplified48.7

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

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

    1. Initial program 0.1

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

  4. Final simplification6.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \le -1.94587273222047501 \cdot 10^{-303}:\\ \;\;\;\;\left(x + y\right) \cdot \frac{1}{1 - \frac{y}{z}}\\ \mathbf{elif}\;\frac{x + y}{1 - \frac{y}{z}} \le -0.0:\\ \;\;\;\;\frac{x + y}{\sqrt{1} + \frac{\sqrt{y}}{\sqrt{z}}} \cdot \frac{\sqrt{1}}{\sqrt{1} - \frac{\sqrt{y}}{\sqrt{z}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 +o rules:numerics
(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 (/ y z))) (* (/ (+ y x) (- y)) z)))

  (/ (+ x y) (- 1 (/ y z))))