Diagrams.TwoD.Segment.Bernstein:evaluateBernstein from diagrams-lib-1.3.0.3

Average Error: 10.4 → 0.1

Time: 6.5s

Precision: binary64

Cost: 904

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \leq -2.025964934226397 \cdot 10^{-05} \lor \neg \left(z \leq 2033294.3313340866\right):\\ \;\;\;\;x \cdot \frac{1 + \left(y - z\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + x \cdot \left(y - z\right)}{z}\\ \end{array}\]

FPCore

(FPCore (x y z) :precision binary64 (/ (* x (+ (- y z) 1.0)) z))

↓

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2.025964934226397e-05) (not (<= z 2033294.3313340866)))
   (* x (/ (+ 1.0 (- y z)) z))
   (/ (+ x (* x (- y z))) z)))

C

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 <= -2.025964934226397e-05) || !(z <= 2033294.3313340866)) {
		tmp = x * ((1.0 + (y - z)) / z);
	} else {
		tmp = (x + (x * (y - z))) / z;
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	10.4
Target	0.4
Herbie	0.1

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

Alternatives

Alternative 1
Error	1.7
Cost	576

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

Alternative 2
Error	0.1
Cost	904

\[\begin{array}{l} \mathbf{if}\;z \leq -4.7891591680294165 \cdot 10^{-14} \lor \neg \left(z \leq 6756302.354297653\right):\\ \;\;\;\;\frac{x}{\frac{z}{1 + \left(y - z\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + x \cdot \left(y - z\right)}{z}\\ \end{array}\]

Alternative 3
Error	1.8
Cost	904

\[\begin{array}{l} \mathbf{if}\;z \leq -4.8064786163120255 \cdot 10^{+60} \lor \neg \left(z \leq 14597444746214.799\right):\\ \;\;\;\;y \cdot \frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x + x \cdot \left(y - z\right)}{z}\\ \end{array}\]

Alternative 4
Error	2.4
Cost	776

\[\begin{array}{l} \mathbf{if}\;y \leq -17.382468885251733 \lor \neg \left(y \leq 0.5827356549752174\right):\\ \;\;\;\;y \cdot \frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array}\]

Alternative 5
Error	9.7
Cost	1090

\[\begin{array}{l} \mathbf{if}\;z \leq -2.5710013950639173 \cdot 10^{+76}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 8.354933333947367 \cdot 10^{-10}:\\ \;\;\;\;\frac{x}{z} \cdot \left(1 + y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array}\]

Alternative 6
Error	9.5
Cost	1090

\[\begin{array}{l} \mathbf{if}\;z \leq -4.575836686361241 \cdot 10^{+70}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 2.2224369997613503 \cdot 10^{-10}:\\ \;\;\;\;\frac{x + x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array}\]

Alternative 7
Error	11.8
Cost	648

\[\begin{array}{l} \mathbf{if}\;y \leq -2.1834251543746128 \cdot 10^{+24} \lor \neg \left(y \leq 3877231641.368535\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array}\]

Alternative 8
Error	24.3
Cost	648

\[\begin{array}{l} \mathbf{if}\;z \leq -5.7014279737727145 \cdot 10^{+75} \lor \neg \left(z \leq 193563.38636236975\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array}\]

Alternative 9
Error	24.1
Cost	648

\[\begin{array}{l} \mathbf{if}\;z \leq -4.575836686361241 \cdot 10^{+70} \lor \neg \left(z \leq 1880735.2428012835\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array}\]

Alternative 10
Error	32.8
Cost	128

\[-x\]

Alternative 11
Error	61.8
Cost	64

\[1\]

Derivation

1. Split input into 2 regimes

2. if z < -2.025964934226397e-5 or 2033294.3313340866 < z

   1. Initial program 17.0

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\]
   2. Using strategy rm

   3. Applied *-un-lft-identity_binary64_19174 17.0

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

   4. Applied times-frac_binary64_19180 0.1

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

   5. Simplified 0.1

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

   6. Simplified 0.1

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

   if -2.025964934226397e-5 < z < 2033294.3313340866

   1. Initial program 0.1

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\]
   2. Using strategy rm

   3. Applied distribute-rgt-in_binary64_19124 0.1

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

   4. Simplified 0.1

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

   5. Simplified 0.1

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

   6. Simplified 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.025964934226397 \cdot 10^{-05} \lor \neg \left(z \leq 2033294.3313340866\right):\\ \;\;\;\;x \cdot \frac{1 + \left(y - z\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + x \cdot \left(y - z\right)}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2021044 
(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))