Average Error: 7.8 → 6.3
Time: 3.5s
Precision: binary64
\[\frac{x + y}{1 - \frac{y}{z}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -1.4483950108988509 \cdot 10^{-297} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{1 + \frac{\sqrt{y}}{\sqrt{z}}}}{\frac{1 - \frac{\sqrt{y}}{\sqrt{z}}}{x + y}}\\ \end{array}\]
\frac{x + y}{1 - \frac{y}{z}}
↓
\begin{array}{l}
\mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -1.4483950108988509 \cdot 10^{-297} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 0\right):\\
\;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{1}{1 + \frac{\sqrt{y}}{\sqrt{z}}}}{\frac{1 - \frac{\sqrt{y}}{\sqrt{z}}}{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))) -1.4483950108988509e-297)
         (not (<= (/ (+ x y) (- 1.0 (/ y z))) 0.0)))
   (/ (+ x y) (- 1.0 (/ y z)))
   (/
    (/ 1.0 (+ 1.0 (/ (sqrt y) (sqrt z))))
    (/ (- 1.0 (/ (sqrt y) (sqrt z))) (+ 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))) <= -1.4483950108988509e-297) || !(((x + y) / (1.0 - (y / z))) <= 0.0)) {
		tmp = (x + y) / (1.0 - (y / z));
	} else {
		tmp = (1.0 / (1.0 + (sqrt(y) / sqrt(z)))) / ((1.0 - (sqrt(y) / sqrt(z))) / (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

Original7.8
Target4.2
Herbie6.3
\[\begin{array}{l} \mathbf{if}\;y < -3.7429310762689856 \cdot 10^{+171}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \mathbf{elif}\;y < 3.5534662456086734 \cdot 10^{+168}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -1.44839501089885086e-297 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 -1.44839501089885086e-297 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < 0.0

    1. Initial program 59.4

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

    3. Applied clear-num_binary64_984959.4

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

    5. Applied *-un-lft-identity_binary64_985059.4

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

    6. Applied add-sqr-sqrt_binary64_987161.4

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

    7. Applied add-sqr-sqrt_binary64_987162.7

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

    8. Applied times-frac_binary64_985662.7

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

    9. Applied *-un-lft-identity_binary64_985062.7

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

    10. Applied difference-of-squares_binary64_981962.7

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

    11. Applied times-frac_binary64_985647.7

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

    12. Applied associate-/r*_binary64_979647.6

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

    13. Simplified47.6

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

  4. Final simplification6.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -1.4483950108988509 \cdot 10^{-297} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{1 + \frac{\sqrt{y}}{\sqrt{z}}}}{\frac{1 - \frac{\sqrt{y}}{\sqrt{z}}}{x + y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020281 
(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))))