Average Error: 10.2 → 0.1
Time: 4.7s
Precision: binary64
\[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
↓
\[\begin{array}{l} \mathbf{if}\;z \leq -18726034.17619243:\\ \;\;\;\;x \cdot \frac{1 + y}{z} - x\\ \mathbf{elif}\;z \leq 2.389419874305158 \cdot 10^{-56}:\\ \;\;\;\;\frac{x \cdot \left(1 + \left(y - z\right)\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-1 + \frac{1}{\frac{z}{1 + y}}\right)\\ \end{array} \]
\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \leq -18726034.17619243:\\
\;\;\;\;x \cdot \frac{1 + y}{z} - x\\

\mathbf{elif}\;z \leq 2.389419874305158 \cdot 10^{-56}:\\
\;\;\;\;\frac{x \cdot \left(1 + \left(y - z\right)\right)}{z}\\

\mathbf{else}:\\
\;\;\;\;x \cdot \left(-1 + \frac{1}{\frac{z}{1 + y}}\right)\\


\end{array}

(FPCore (x y z) :precision binary64 (/ (* x (+ (- y z) 1.0)) z))
↓
(FPCore (x y z)
 :precision binary64
 (if (<= z -18726034.17619243)
   (- (* x (/ (+ 1.0 y) z)) x)
   (if (<= z 2.389419874305158e-56)
     (/ (* x (+ 1.0 (- y z))) z)
     (* x (+ -1.0 (/ 1.0 (/ z (+ 1.0 y))))))))
double code(double x, double y, double z) {
	return (x * ((y - z) + 1.0)) / z;
}
↓
double code(double x, double y, double z) {
	double tmp;
	if (z <= -18726034.17619243) {
		tmp = (x * ((1.0 + y) / z)) - x;
	} else if (z <= 2.389419874305158e-56) {
		tmp = (x * (1.0 + (y - z))) / z;
	} else {
		tmp = x * (-1.0 + (1.0 / (z / (1.0 + 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

Original10.2
Target0.5
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;x < -2.71483106713436 \cdot 10^{-162}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x < 3.874108816439546 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if z < -18726034.1761924289

    1. Initial program 17.2

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

    2. Applied *-un-lft-identity_binary6417.2

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

    3. Applied times-frac_binary640.1

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

    4. Simplified0.1

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

    5. Simplified0.1

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

    6. Applied div-sub_binary640.1

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

    7. Simplified0.1

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

    8. Applied sub-neg_binary640.1

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

    9. Applied distribute-lft-in_binary640.1

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

    if -18726034.1761924289 < z < 2.3894198743051582e-56

    1. Initial program 0.1

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

    if 2.3894198743051582e-56 < z

    1. Initial program 14.2

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

    2. Applied *-un-lft-identity_binary6414.2

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

    3. Applied times-frac_binary640.2

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

    4. Simplified0.2

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

    5. Simplified0.2

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

    6. Applied div-sub_binary640.2

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

    7. Simplified0.2

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

    8. Applied clear-num_binary640.2

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

    9. Simplified0.2

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -18726034.17619243:\\ \;\;\;\;x \cdot \frac{1 + y}{z} - x\\ \mathbf{elif}\;z \leq 2.389419874305158 \cdot 10^{-56}:\\ \;\;\;\;\frac{x \cdot \left(1 + \left(y - z\right)\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-1 + \frac{1}{\frac{z}{1 + y}}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022024 
(FPCore (x y z)
  :name "Diagrams.TwoD.Segment.Bernstein:evaluateBernstein from diagrams-lib-1.3.0.3"
  :precision binary64

  :herbie-target
  (if (< x -2.71483106713436e-162) (- (* (+ 1.0 y) (/ x z)) x) (if (< x 3.874108816439546e-197) (* (* x (+ (- y z) 1.0)) (/ 1.0 z)) (- (* (+ 1.0 y) (/ x z)) x)))

  (/ (* x (+ (- y z) 1.0)) z))