Average Error: 10.3 → 0.2
Time: 3.5s
Precision: binary64
\[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -3.1371785873879285 \cdot 10^{+55}:\\ \;\;\;\;x \cdot \left(\frac{y + 1}{z} + -1\right)\\ \mathbf{elif}\;x \leq 1.9249634187461288 \cdot 10^{-34}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, x\right)}{z} - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} + \frac{1}{z}\right) - x\\ \end{array} \]
\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -3.1371785873879285 \cdot 10^{+55}:\\
\;\;\;\;x \cdot \left(\frac{y + 1}{z} + -1\right)\\

\mathbf{elif}\;x \leq 1.9249634187461288 \cdot 10^{-34}:\\
\;\;\;\;\frac{\mathsf{fma}\left(y, x, x\right)}{z} - x\\

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


\end{array}

(FPCore (x y z) :precision binary64 (/ (* x (+ (- y z) 1.0)) z))
↓
(FPCore (x y z)
 :precision binary64
 (if (<= x -3.1371785873879285e+55)
   (* x (+ (/ (+ y 1.0) z) -1.0))
   (if (<= x 1.9249634187461288e-34)
     (- (/ (fma y x x) z) x)
     (- (* x (+ (/ y z) (/ 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 <= -3.1371785873879285e+55) {
		tmp = x * (((y + 1.0) / z) + -1.0);
	} else if (x <= 1.9249634187461288e-34) {
		tmp = (fma(y, x, x) / z) - x;
	} else {
		tmp = (x * ((y / z) + (1.0 / z))) - x;
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original10.3
Target0.4
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 x < -3.1371785873879285e55

    1. Initial program 31.8

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

    2. Simplified31.8

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

    3. Taylor expanded in y around 0 9.6

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

    4. Simplified0.1

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

    5. Taylor expanded in x around 0 0.1

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

    6. Applied *-un-lft-identity_binary640.1

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

    7. Applied distribute-rgt-out--_binary640.1

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

    8. Simplified0.1

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

    if -3.1371785873879285e55 < x < 1.92496341874612876e-34

    1. Initial program 0.5

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

    2. Simplified0.5

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

    3. Taylor expanded in x around 0 0.5

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

    4. Simplified0.2

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

    if 1.92496341874612876e-34 < x

    1. Initial program 23.4

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

    2. Simplified23.4

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

    3. Taylor expanded in y around 0 7.6

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

    4. Simplified0.1

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

    5. Taylor expanded in x around 0 0.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.1371785873879285 \cdot 10^{+55}:\\ \;\;\;\;x \cdot \left(\frac{y + 1}{z} + -1\right)\\ \mathbf{elif}\;x \leq 1.9249634187461288 \cdot 10^{-34}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, x\right)}{z} - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} + \frac{1}{z}\right) - x\\ \end{array} \]

Reproduce

herbie shell --seed 2021275 
(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))