Average Error: 7.7 → 0.2
Time: 9.4s
Precision: binary64
\[\frac{x + y}{1 - \frac{y}{z}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -5.551876445681659 \cdot 10^{-269} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x}{y} \cdot \left(z + \frac{z}{\frac{y}{z}}\right)\right) - \left(z + \frac{z}{\frac{y}{z}}\right)\\ \end{array}\]
\frac{x + y}{1 - \frac{y}{z}}
↓
\begin{array}{l}
\mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -5.551876445681659 \cdot 10^{-269} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 0\right):\\
\;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\

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

\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))) -5.551876445681659e-269)
         (not (<= (/ (+ x y) (- 1.0 (/ y z))) 0.0)))
   (/ (+ x y) (- 1.0 (/ y z)))
   (- (- (* (/ x y) (+ z (/ z (/ y z))))) (+ z (/ z (/ y z))))))
double code(double x, double y, double z) {
	return (x + y) / (1.0 - (y / z));
}
↓
double code(double x, double y, double z) {
	double tmp;
	if ((((x + y) / (1.0 - (y / z))) <= -5.551876445681659e-269) || !(((x + y) / (1.0 - (y / z))) <= 0.0)) {
		tmp = (x + y) / (1.0 - (y / z));
	} else {
		tmp = -((x / y) * (z + (z / (y / z)))) - (z + (z / (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

Original7.7
Target4.3
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;y < -3.7429310762689856 \cdot 10^{+171}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \mathbf{elif}\;y < 3.5534662456086734 \cdot 10^{+168}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -5.55187644568165897e-269 or -0.0 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z)))

    1. Initial program 0.1

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

    if -5.55187644568165897e-269 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -0.0

    1. Initial program 57.0

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

    3. Applied clear-num_binary64_1610457.0

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

    4. Simplified57.0

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

    5. Taylor expanded around 0 1.4

      \[\leadsto \color{blue}{-\left(\frac{x \cdot {z}^{2}}{{y}^{2}} + \left(\frac{x \cdot z}{y} + \left(z + \frac{{z}^{2}}{y}\right)\right)\right)}\]

    6. Simplified0.7

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -5.551876445681659 \cdot 10^{-269} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x}{y} \cdot \left(z + \frac{z}{\frac{y}{z}}\right)\right) - \left(z + \frac{z}{\frac{y}{z}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021098 
(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))))