Average Error: 7.5 → 0.4
Time: 3.1s
Precision: binary64
\[\frac{x + y}{1 - \frac{y}{z}} \]
↓
\[\begin{array}{l} t_0 := 1 - \frac{y}{z}\\ t_1 := \frac{x + y}{t_0}\\ \mathbf{if}\;t_1 \leq -3.1159828002826525 \cdot 10^{-282}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 2.2399534121698782 \cdot 10^{-277}:\\ \;\;\;\;\frac{z \cdot \left(-1 + \frac{x}{y} \cdot \frac{x}{y}\right)}{1 - \frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t_0} + \frac{y}{t_0}\\ \end{array} \]
\frac{x + y}{1 - \frac{y}{z}}
↓
\begin{array}{l}
t_0 := 1 - \frac{y}{z}\\
t_1 := \frac{x + y}{t_0}\\
\mathbf{if}\;t_1 \leq -3.1159828002826525 \cdot 10^{-282}:\\
\;\;\;\;t_1\\

\mathbf{elif}\;t_1 \leq 2.2399534121698782 \cdot 10^{-277}:\\
\;\;\;\;\frac{z \cdot \left(-1 + \frac{x}{y} \cdot \frac{x}{y}\right)}{1 - \frac{x}{y}}\\

\mathbf{else}:\\
\;\;\;\;\frac{x}{t_0} + \frac{y}{t_0}\\


\end{array}

(FPCore (x y z) :precision binary64 (/ (+ x y) (- 1.0 (/ y z))))
↓
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ y z))) (t_1 (/ (+ x y) t_0)))
   (if (<= t_1 -3.1159828002826525e-282)
     t_1
     (if (<= t_1 2.2399534121698782e-277)
       (/ (* z (+ -1.0 (* (/ x y) (/ x y)))) (- 1.0 (/ x y)))
       (+ (/ x t_0) (/ y t_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 = 1.0 - (y / z);
	double t_1 = (x + y) / t_0;
	double tmp;
	if (t_1 <= -3.1159828002826525e-282) {
		tmp = t_1;
	} else if (t_1 <= 2.2399534121698782e-277) {
		tmp = (z * (-1.0 + ((x / y) * (x / y)))) / (1.0 - (x / y));
	} else {
		tmp = (x / t_0) + (y / t_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

Original7.5
Target4.1
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;y < -3.7429310762689856 \cdot 10^{+171}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \mathbf{elif}\;y < 3.5534662456086734 \cdot 10^{+168}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -3.11598280028265255e-282

    1. Initial program 0.1

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

    if -3.11598280028265255e-282 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < 2.23995341216987818e-277

    1. Initial program 55.5

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

    2. Taylor expanded in z around 0 2.3

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

    3. Simplified10.6

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

    4. Taylor expanded in z around 0 1.2

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

    5. Applied flip-+_binary642.4

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

    6. Applied associate-*r/_binary642.5

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

    7. Simplified2.5

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

    if 2.23995341216987818e-277 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z)))

    1. Initial program 0.1

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

    2. Taylor expanded in x around 0 0.1

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -3.1159828002826525 \cdot 10^{-282}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{elif}\;\frac{x + y}{1 - \frac{y}{z}} \leq 2.2399534121698782 \cdot 10^{-277}:\\ \;\;\;\;\frac{z \cdot \left(-1 + \frac{x}{y} \cdot \frac{x}{y}\right)}{1 - \frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}} + \frac{y}{1 - \frac{y}{z}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022081 
(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))))