Average Error: 10.6 → 0.2
Time: 5.0s
Precision: binary64
\[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \leq -3.337197997942166 \cdot 10^{+64} \lor \neg \left(\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \leq 279321992.8220806\right):\\ \;\;\;\;\frac{x}{z} \cdot \left(y + 1\right) - x\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{z} + x \cdot \frac{y}{z}\right) - x\\ \end{array}\]
\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \leq -3.337197997942166 \cdot 10^{+64} \lor \neg \left(\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \leq 279321992.8220806\right):\\
\;\;\;\;\frac{x}{z} \cdot \left(y + 1\right) - x\\

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

\end{array}
(FPCore (x y z) :precision binary64 (/ (* x (+ (- y z) 1.0)) z))
↓
(FPCore (x y z)
 :precision binary64
 (if (or (<= (/ (* x (+ (- y z) 1.0)) z) -3.337197997942166e+64)
         (not (<= (/ (* x (+ (- y z) 1.0)) z) 279321992.8220806)))
   (- (* (/ x z) (+ y 1.0)) x)
   (- (+ (/ x z) (* x (/ y z))) x)))
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 * ((y - z) + 1.0)) / z) <= -3.337197997942166e+64) || !(((x * ((y - z) + 1.0)) / z) <= 279321992.8220806)) {
		tmp = ((x / z) * (y + 1.0)) - x;
	} else {
		tmp = ((x / z) + (x * (y / z))) - x;
	}
	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 2 regimes

  2. if (/.f64 (*.f64 x (+.f64 (-.f64 y z) 1)) z) < -3.33719799794216578e64 or 279321992.8220806 < (/.f64 (*.f64 x (+.f64 (-.f64 y z) 1)) z)

    1. Initial program 19.4

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

    2. Taylor expanded around 0 6.5

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

    3. Simplified0.1

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

    4. Taylor expanded around 0 6.2

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

    5. Simplified0.1

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

    if -3.33719799794216578e64 < (/.f64 (*.f64 x (+.f64 (-.f64 y z) 1)) z) < 279321992.8220806

    1. Initial program 0.1

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

    2. Taylor expanded around 0 0.1

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

    3. Simplified3.4

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

    5. Applied div-inv_binary64_181483.5

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

    6. Applied associate-*l*_binary64_180920.3

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

    7. Simplified0.3

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \leq -3.337197997942166 \cdot 10^{+64} \lor \neg \left(\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \leq 279321992.8220806\right):\\ \;\;\;\;\frac{x}{z} \cdot \left(y + 1\right) - x\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{z} + x \cdot \frac{y}{z}\right) - x\\ \end{array}\]

Reproduce

herbie shell --seed 2021043 
(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))