Average Error: 9.8 → 0.3
Time: 2.6s
Precision: binary64
\[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -3.9126001633745477 \cdot 10^{-11} \lor \neg \left(x \leq 2.944504131166995 \cdot 10^{-212}\right):\\ \;\;\;\;\left(\frac{x}{z} + \frac{x}{z} \cdot y\right) - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x + x \cdot \left(y - z\right)}{z}\\ \end{array}\]
\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -3.9126001633745477 \cdot 10^{-11} \lor \neg \left(x \leq 2.944504131166995 \cdot 10^{-212}\right):\\
\;\;\;\;\left(\frac{x}{z} + \frac{x}{z} \cdot y\right) - x\\

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

\end{array}
(FPCore (x y z) :precision binary64 (/ (* x (+ (- y z) 1.0)) z))
↓
(FPCore (x y z)
 :precision binary64
 (if (or (<= x -3.9126001633745477e-11) (not (<= x 2.944504131166995e-212)))
   (- (+ (/ x z) (* (/ x z) y)) x)
   (/ (+ x (* x (- y z))) z)))
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 ((x <= -3.9126001633745477e-11) || !(x <= 2.944504131166995e-212)) {
		tmp = ((x / z) + ((x / z) * y)) - x;
	} else {
		tmp = (x + (x * (y - z))) / 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

Original9.8
Target0.4
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;x < -2.71483106713436 \cdot 10^{-162}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x < 3.874108816439546 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -3.91260016337454765e-11 or 2.94450413116699479e-212 < x

    1. Initial program 16.0

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

    2. Taylor expanded around 0 5.6

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

    3. Simplified0.5

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

    if -3.91260016337454765e-11 < x < 2.94450413116699479e-212

    1. Initial program 0.2

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

    3. Applied distribute-rgt-in_binary64_111500.2

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

    4. Simplified0.2

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

    5. Simplified0.2

      \[\leadsto \frac{x \cdot \left(y - z\right) + \color{blue}{x}}{z}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.9126001633745477 \cdot 10^{-11} \lor \neg \left(x \leq 2.944504131166995 \cdot 10^{-212}\right):\\ \;\;\;\;\left(\frac{x}{z} + \frac{x}{z} \cdot y\right) - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x + x \cdot \left(y - z\right)}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020281 
(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))