Average Error: 10.4 → 0.3
Time: 2.6s
Precision: binary64
\[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -4.934159841637241 \cdot 10^{-149} \lor \neg \left(x \leq 2.1558912601142047 \cdot 10^{+23}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, \frac{x}{z}\right) - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y + 1\right)}{z} - x\\ \end{array} \]
\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -4.934159841637241 \cdot 10^{-149} \lor \neg \left(x \leq 2.1558912601142047 \cdot 10^{+23}\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, \frac{x}{z}\right) - 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 (or (<= x -4.934159841637241e-149) (not (<= x 2.1558912601142047e+23)))
   (- (fma (/ x z) y (/ x z)) 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 ((x <= -4.934159841637241e-149) || !(x <= 2.1558912601142047e+23)) {
		tmp = fma((x / z), y, (x / z)) - x;
	} else {
		tmp = ((x * (y + 1.0)) / z) - x;
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original10.4
Target0.5
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 < -4.93415984163724069e-149 or 2.1558912601142047e23 < x

    1. Initial program 20.2

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

    2. Simplified20.2

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

    3. Taylor expanded in y around 0 6.5

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

    4. Simplified0.4

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

    if -4.93415984163724069e-149 < x < 2.1558912601142047e23

    1. Initial program 0.2

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

    2. Simplified0.2

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

    3. Taylor expanded in y around 0 0.1

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

    4. Simplified3.0

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

    5. Taylor expanded in x around 0 5.6

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

    6. Taylor expanded in z around 0 0.1

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.934159841637241 \cdot 10^{-149} \lor \neg \left(x \leq 2.1558912601142047 \cdot 10^{+23}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, \frac{x}{z}\right) - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y + 1\right)}{z} - x\\ \end{array} \]

Reproduce

herbie shell --seed 2021313 
(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))