Average Error: 7.4 → 6.1
Time: 6.3s
Precision: binary64
\[\frac{x + y}{1 - \frac{y}{z}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -8.981432546252582 \cdot 10^{-303}:\\ \;\;\;\;\left(x + y\right) \cdot \frac{1}{1 - \frac{y}{z}}\\ \mathbf{elif}\;\frac{x + y}{1 - \frac{y}{z}} \leq 0:\\ \;\;\;\;\frac{1}{1 + \frac{\sqrt{y}}{\sqrt{z}}} \cdot \frac{x + y}{1 - \frac{\sqrt{y}}{\sqrt{z}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \end{array}\]
\frac{x + y}{1 - \frac{y}{z}}
↓
\begin{array}{l}
\mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -8.981432546252582 \cdot 10^{-303}:\\
\;\;\;\;\left(x + y\right) \cdot \frac{1}{1 - \frac{y}{z}}\\

\mathbf{elif}\;\frac{x + y}{1 - \frac{y}{z}} \leq 0:\\
\;\;\;\;\frac{1}{1 + \frac{\sqrt{y}}{\sqrt{z}}} \cdot \frac{x + y}{1 - \frac{\sqrt{y}}{\sqrt{z}}}\\

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

\end{array}
(FPCore (x y z) :precision binary64 (/ (+ x y) (- 1.0 (/ y z))))
↓
(FPCore (x y z)
 :precision binary64
 (if (<= (/ (+ x y) (- 1.0 (/ y z))) -8.981432546252582e-303)
   (* (+ x y) (/ 1.0 (- 1.0 (/ y z))))
   (if (<= (/ (+ x y) (- 1.0 (/ y z))) 0.0)
     (*
      (/ 1.0 (+ 1.0 (/ (sqrt y) (sqrt z))))
      (/ (+ x y) (- 1.0 (/ (sqrt y) (sqrt z)))))
     (/ (+ x y) (- 1.0 (/ 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))) <= -8.981432546252582e-303) {
		tmp = (x + y) * (1.0 / (1.0 - (y / z)));
	} else if (((x + y) / (1.0 - (y / z))) <= 0.0) {
		tmp = (1.0 / (1.0 + (sqrt(y) / sqrt(z)))) * ((x + y) / (1.0 - (sqrt(y) / sqrt(z))));
	} else {
		tmp = (x + y) / (1.0 - (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

Original7.4
Target4.1
Herbie6.1
\[\begin{array}{l} \mathbf{if}\;y < -3.7429310762689856 \cdot 10^{+171}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \mathbf{elif}\;y < 3.5534662456086734 \cdot 10^{+168}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -8.98143254625258169e-303

    1. Initial program 0.1

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

    3. Applied div-inv_binary64_96230.2

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

    if -8.98143254625258169e-303 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < 0.0

    1. Initial program 59.7

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

    3. Applied add-sqr-sqrt_binary64_964861.7

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

    4. Applied add-sqr-sqrt_binary64_964862.9

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

    5. Applied times-frac_binary64_963262.9

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

    6. Applied add-sqr-sqrt_binary64_964862.9

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

    7. Applied difference-of-squares_binary64_959562.9

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

    8. Applied *-un-lft-identity_binary64_962662.9

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

    9. Applied times-frac_binary64_963248.6

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

    10. Simplified48.6

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

    11. Simplified48.6

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

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

    1. Initial program 0.1

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

  4. Final simplification6.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -8.981432546252582 \cdot 10^{-303}:\\ \;\;\;\;\left(x + y\right) \cdot \frac{1}{1 - \frac{y}{z}}\\ \mathbf{elif}\;\frac{x + y}{1 - \frac{y}{z}} \leq 0:\\ \;\;\;\;\frac{1}{1 + \frac{\sqrt{y}}{\sqrt{z}}} \cdot \frac{x + y}{1 - \frac{\sqrt{y}}{\sqrt{z}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020354 
(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))))