Average Error: 7.6 → 0.5
Time: 6.6s
Precision: binary64
\[\frac{x + y}{1 - \frac{y}{z}} \]
↓
\[\begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t_0 \leq -1.4564281761318297 \cdot 10^{-249} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-\left(\frac{x \cdot {z}^{2}}{{y}^{2}} + \left(\frac{{z}^{2}}{y} + \left(\frac{x \cdot z}{y} + \left(z + \frac{{z}^{3}}{{y}^{2}}\right)\right)\right)\right)\\ \end{array} \]
\frac{x + y}{1 - \frac{y}{z}}
↓
\begin{array}{l}
t_0 := \frac{x + y}{1 - \frac{y}{z}}\\
\mathbf{if}\;t_0 \leq -1.4564281761318297 \cdot 10^{-249} \lor \neg \left(t_0 \leq 0\right):\\
\;\;\;\;t_0\\

\mathbf{else}:\\
\;\;\;\;-\left(\frac{x \cdot {z}^{2}}{{y}^{2}} + \left(\frac{{z}^{2}}{y} + \left(\frac{x \cdot z}{y} + \left(z + \frac{{z}^{3}}{{y}^{2}}\right)\right)\right)\right)\\


\end{array}

(FPCore (x y z) :precision binary64 (/ (+ x y) (- 1.0 (/ y z))))
↓
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (or (<= t_0 -1.4564281761318297e-249) (not (<= t_0 0.0)))
     t_0
     (-
      (+
       (/ (* x (pow z 2.0)) (pow y 2.0))
       (+
        (/ (pow z 2.0) y)
        (+ (/ (* x z) y) (+ z (/ (pow z 3.0) (pow y 2.0))))))))))
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 = (x + y) / (1.0 - (y / z));
	double tmp;
	if ((t_0 <= -1.4564281761318297e-249) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = -(((x * pow(z, 2.0)) / pow(y, 2.0)) + ((pow(z, 2.0) / y) + (((x * z) / y) + (z + (pow(z, 3.0) / pow(y, 2.0))))));
	}
	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

Original7.6
Target3.9
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;y < -3.7429310762689856 \cdot 10^{+171}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \mathbf{elif}\;y < 3.5534662456086734 \cdot 10^{+168}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \end{array} \]

Derivation

  1. Split input into 2 regimes

  2. if (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -1.4564281761318297e-249 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 -1.4564281761318297e-249 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -0.0

    1. Initial program 54.9

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

    2. Taylor expanded in y around inf 2.8

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -1.4564281761318297 \cdot 10^{-249} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;-\left(\frac{x \cdot {z}^{2}}{{y}^{2}} + \left(\frac{{z}^{2}}{y} + \left(\frac{x \cdot z}{y} + \left(z + \frac{{z}^{3}}{{y}^{2}}\right)\right)\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021307 
(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))))