Average Error: 7.4 → 7.0
Time: 3.8s
Precision: binary64
\[\frac{x + y}{1 - \frac{y}{z}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -3.637592698794902 \cdot 10^{-218} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{x + y}}{1 + \frac{\sqrt{y}}{\sqrt{z}}}}{\frac{1 - \frac{\sqrt{y}}{\sqrt{z}}}{\sqrt{x + y}}}\\ \end{array}\]
\frac{x + y}{1 - \frac{y}{z}}
↓
\begin{array}{l}
\mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -3.637592698794902 \cdot 10^{-218} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 0\right):\\
\;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\

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

\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))) -3.637592698794902e-218)
         (not (<= (/ (+ x y) (- 1.0 (/ y z))) 0.0)))
   (/ (+ x y) (- 1.0 (/ y z)))
   (/
    (/ (sqrt (+ x y)) (+ 1.0 (/ (sqrt y) (sqrt z))))
    (/ (- 1.0 (/ (sqrt y) (sqrt z))) (sqrt (+ x y))))))
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))) <= -3.637592698794902e-218) || !(((x + y) / (1.0 - (y / z))) <= 0.0)) {
		tmp = (x + y) / (1.0 - (y / z));
	} else {
		tmp = (sqrt(x + y) / (1.0 + (sqrt(y) / sqrt(z)))) / ((1.0 - (sqrt(y) / sqrt(z))) / sqrt(x + y));
	}
	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

Original7.4
Target4.2
Herbie7.0
\[\begin{array}{l} \mathbf{if}\;y < -3.7429310762689856 \cdot 10^{+171}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \mathbf{elif}\;y < 3.5534662456086734 \cdot 10^{+168}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -3.6375926987949019e-218 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 -3.6375926987949019e-218 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < 0.0

    1. Initial program 49.8

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

    3. Applied clear-num_binary6449.8

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

    5. Applied add-sqr-sqrt_binary6458.7

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

    6. Applied add-sqr-sqrt_binary6459.9

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

    7. Applied add-sqr-sqrt_binary6459.9

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

    8. Applied times-frac_binary6459.9

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

    9. Applied *-un-lft-identity_binary6459.9

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

    10. Applied difference-of-squares_binary6459.9

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

    11. Applied times-frac_binary6447.2

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

    12. Applied associate-/r*_binary6447.2

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

    13. Simplified47.2

      \[\leadsto \frac{\color{blue}{\frac{\sqrt{y + x}}{1 + \frac{\sqrt{y}}{\sqrt{z}}}}}{\frac{1 - \frac{\sqrt{y}}{\sqrt{z}}}{\sqrt{x + y}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification7.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -3.637592698794902 \cdot 10^{-218} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{x + y}}{1 + \frac{\sqrt{y}}{\sqrt{z}}}}{\frac{1 - \frac{\sqrt{y}}{\sqrt{z}}}{\sqrt{x + y}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020231 
(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))))