Average Error: 10.6 → 0.2
Time: 6.7s
Precision: binary64
\[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \leq -3.429579611887415 \cdot 10^{-48}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y + 1\right) - z}}\\ \mathbf{elif}\;z \leq 5.186717654845177 \cdot 10^{-57}:\\ \;\;\;\;\left(\frac{x}{z} + \frac{x \cdot y}{z}\right) - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y + 1}{z} - 1\right)\\ \end{array}\]
\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \leq -3.429579611887415 \cdot 10^{-48}:\\
\;\;\;\;\frac{x}{\frac{z}{\left(y + 1\right) - z}}\\

\mathbf{elif}\;z \leq 5.186717654845177 \cdot 10^{-57}:\\
\;\;\;\;\left(\frac{x}{z} + \frac{x \cdot y}{z}\right) - x\\

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

\end{array}
(FPCore (x y z) :precision binary64 (/ (* x (+ (- y z) 1.0)) z))
↓
(FPCore (x y z)
 :precision binary64
 (if (<= z -3.429579611887415e-48)
   (/ x (/ z (- (+ y 1.0) z)))
   (if (<= z 5.186717654845177e-57)
     (- (+ (/ x z) (/ (* x y) z)) x)
     (* x (- (/ (+ y 1.0) z) 1.0)))))
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 <= -3.429579611887415e-48) {
		tmp = x / (z / ((y + 1.0) - z));
	} else if (z <= 5.186717654845177e-57) {
		tmp = ((x / z) + ((x * y) / z)) - x;
	} else {
		tmp = x * (((y + 1.0) / z) - 1.0);
	}
	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

Original10.6
Target0.5
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;x < -2.71483106713436 \cdot 10^{-162}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x < 3.874108816439546 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -3.4295796118874147e-48

    1. Initial program 16.1

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\]
    2. Using strategy rm

    3. Applied associate-/l*_binary64_201420.2

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

    4. Simplified0.2

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

    if -3.4295796118874147e-48 < z < 5.18671765484517694e-57

    1. Initial program 0.1

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

    2. Taylor expanded around 0 0.1

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

    if 5.18671765484517694e-57 < z

    1. Initial program 14.4

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity_binary64_2019714.4

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

    4. Applied times-frac_binary64_202030.3

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

    5. Simplified0.3

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

    6. Simplified0.3

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

    8. Applied div-sub_binary64_202020.3

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

    9. Simplified0.3

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.429579611887415 \cdot 10^{-48}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y + 1\right) - z}}\\ \mathbf{elif}\;z \leq 5.186717654845177 \cdot 10^{-57}:\\ \;\;\;\;\left(\frac{x}{z} + \frac{x \cdot y}{z}\right) - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y + 1}{z} - 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021023 
(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))