Average Error: 9.9 → 1.7
Time: 3.1s
Precision: binary64
\[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
↓
\[\begin{array}{l} \mathbf{if}\;z \leq -6.09357792742113 \cdot 10^{-51}:\\ \;\;\;\;\frac{x}{\frac{z}{y + 1}} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y + 1\right)}{z} - x\\ \end{array} \]
\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \leq -6.09357792742113 \cdot 10^{-51}:\\
\;\;\;\;\frac{x}{\frac{z}{y + 1}} - x\\

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


\end{array}

(FPCore (x y z) :precision binary64 (/ (* x (+ (- y z) 1.0)) z))
↓
(FPCore (x y z)
 :precision binary64
 (if (<= z -6.09357792742113e-51)
   (- (/ x (/ z (+ y 1.0))) x)
   (- (/ (* x (+ y 1.0)) 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 (z <= -6.09357792742113e-51) {
		tmp = (x / (z / (y + 1.0))) - x;
	} else {
		tmp = ((x * (y + 1.0)) / 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

Original9.9
Target0.4
Herbie1.7
\[\begin{array}{l} \mathbf{if}\;x < -2.71483106713436 \cdot 10^{-162}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x < 3.874108816439546 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \end{array} \]

Derivation

  1. Split input into 2 regimes

  2. if z < -6.0935779274211301e-51

    1. Initial program 14.7

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

    2. Simplified14.7

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, y - z, x\right)}{z}} \]

    3. Taylor expanded in y around 0 5.0

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

    4. Simplified2.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{z}, y, \frac{x}{z}\right) - x} \]

    5. Taylor expanded in z around 0 5.0

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

    6. Applied *-un-lft-identity_binary645.0

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

    7. Applied distribute-rgt-out_binary645.0

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

    8. Applied associate-/l*_binary640.2

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

    if -6.0935779274211301e-51 < z

    1. Initial program 7.4

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

    2. Simplified7.4

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, y - z, x\right)}{z}} \]

    3. Taylor expanded in y around 0 2.5

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

    4. Simplified1.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{z}, y, \frac{x}{z}\right) - x} \]

    5. Taylor expanded in z around 0 2.5

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

    6. Taylor expanded in x around 0 2.5

      \[\leadsto \frac{\color{blue}{\left(1 + y\right) \cdot x}}{z} - x \]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.09357792742113 \cdot 10^{-51}:\\ \;\;\;\;\frac{x}{\frac{z}{y + 1}} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y + 1\right)}{z} - x\\ \end{array} \]

Reproduce

herbie shell --seed 2022067 
(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))