Average Error: 10.1 → 0.1
Time: 3.1s
Precision: binary64
\[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
↓
\[\begin{array}{l} \mathbf{if}\;z \leq -2.236168945046539 \cdot 10^{-18}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x, \frac{x}{z}\right) - x\\ \mathbf{elif}\;z \leq 0.0007078159893443644:\\ \;\;\;\;\frac{x}{z} \cdot \left(\left(y - z\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(\frac{y}{z} + \frac{1}{z}\right) - 1\right)\\ \end{array} \]
\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \leq -2.236168945046539 \cdot 10^{-18}:\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x, \frac{x}{z}\right) - x\\

\mathbf{elif}\;z \leq 0.0007078159893443644:\\
\;\;\;\;\frac{x}{z} \cdot \left(\left(y - z\right) + 1\right)\\

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


\end{array}

(FPCore (x y z) :precision binary64 (/ (* x (+ (- y z) 1.0)) z))
↓
(FPCore (x y z)
 :precision binary64
 (if (<= z -2.236168945046539e-18)
   (- (fma (/ y z) x (/ x z)) x)
   (if (<= z 0.0007078159893443644)
     (* (/ x z) (+ (- y z) 1.0))
     (* x (- (+ (/ y z) (/ 1.0 z)) 1.0)))))
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 <= -2.236168945046539e-18) {
		tmp = fma((y / z), x, (x / z)) - x;
	} else if (z <= 0.0007078159893443644) {
		tmp = (x / z) * ((y - z) + 1.0);
	} else {
		tmp = x * (((y / z) + (1.0 / z)) - 1.0);
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original10.1
Target0.4
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;x < -2.71483106713436 \cdot 10^{-162}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x < 3.874108816439546 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if z < -2.236168945046539e-18

    1. Initial program 15.8

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

    2. Applied associate-/l*_binary640.1

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

    3. Taylor expanded in z around 0 5.8

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

    4. Simplified0.1

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

    if -2.236168945046539e-18 < z < 7.07815989344364447e-4

    1. Initial program 0.1

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

    2. Applied associate-/l*_binary648.0

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

    3. Applied associate-/r/_binary640.1

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

    if 7.07815989344364447e-4 < z

    1. Initial program 16.4

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

    2. Applied associate-/l*_binary640.1

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

    3. Taylor expanded in z around 0 5.4

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

    4. Simplified0.1

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

    5. Taylor expanded in x around 0 0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.236168945046539 \cdot 10^{-18}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x, \frac{x}{z}\right) - x\\ \mathbf{elif}\;z \leq 0.0007078159893443644:\\ \;\;\;\;\frac{x}{z} \cdot \left(\left(y - z\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(\frac{y}{z} + \frac{1}{z}\right) - 1\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022020 
(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))