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

Average Error: 10.2 → 0.3

Time: 3.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -4.471222327351869 \cdot 10^{+87}:\\ \;\;\;\;x \cdot \frac{y + 1}{z} - x\\ \mathbf{elif}\;x \leq 220123720629973600:\\ \;\;\;\;\frac{x \cdot \left(y + 1\right)}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{y + 1}{\frac{z}{x}} - x\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= x -4.471222327351869e+87)
   (- (* x (/ (+ y 1.0) z)) x)
   (if (<= x 220123720629973600.0)
     (- (/ (* x (+ y 1.0)) z) x)
     (- (/ (+ y 1.0) (/ z x)) x))))

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 (x <= -4.471222327351869e+87) {
		tmp = (x * ((y + 1.0) / z)) - x;
	} else if (x <= 220123720629973600.0) {
		tmp = ((x * (y + 1.0)) / z) - x;
	} else {
		tmp = ((y + 1.0) / (z / x)) - x;
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	10.2
Target	0.5
Herbie	0.3

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

Derivation

1. Split input into 3 regimes

2. if x < -4.4712223273518688e87

   1. Initial program 35.6

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

   2. Simplified 35.6

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

   3. Taylor expanded in y around 0 12.1

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

   4. Simplified 0.1

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

   5. Taylor expanded in z around 0 12.1

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

   6. Applied *-un-lft-identity_binary64 12.1

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

   7. Applied *-un-lft-identity_binary64 12.1

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

   8. Applied distribute-rgt-out_binary64 12.1

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

   9. Applied times-frac_binary64 0.1

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

   10. Simplified 0.1

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

   if -4.4712223273518688e87 < x < 220123720629973600

   1. Initial program 0.9

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

   2. Simplified 0.9

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

   3. Taylor expanded in y around 0 0.4

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

   4. Simplified 2.4

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

   5. Taylor expanded in z around 0 0.4

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

   6. Applied *-un-lft-identity_binary64 0.4

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

   7. Applied distribute-rgt-out_binary64 0.4

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

   if 220123720629973600 < x

   1. Initial program 26.9

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

   2. Simplified 26.9

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

   3. Taylor expanded in y around 0 10.1

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

   4. Simplified 0.1

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

   5. Taylor expanded in z around 0 10.1

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

   6. Applied distribute-lft1-in_binary64 10.1

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

   7. Applied associate-/l*_binary64 0.1

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.471222327351869 \cdot 10^{+87}:\\ \;\;\;\;x \cdot \frac{y + 1}{z} - x\\ \mathbf{elif}\;x \leq 220123720629973600:\\ \;\;\;\;\frac{x \cdot \left(y + 1\right)}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{y + 1}{\frac{z}{x}} - x\\ \end{array} \]

Reproduce

herbie shell --seed 2022097 
(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))