Diagrams.Trail:splitAtParam from diagrams-lib-1.3.0.3, A

Average Error: 7.5 → 3.6

Time: 4.4s

Precision: 64

Math

\[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.862397195409269 \cdot 10^{-252} \lor \neg \left(x \le 1.3799272751964713 \cdot 10^{-126}\right):\\ \;\;\;\;\left(\frac{y}{t \cdot z - x} \cdot \frac{z}{x + 1} + \frac{1}{x + 1} \cdot x\right) - \frac{\frac{x}{t \cdot z - x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y \cdot z}{\left(x + 1\right) \cdot \left(t \cdot z - x\right)} + \frac{1}{x + 1} \cdot x\right) - \frac{\frac{x}{t \cdot z - x}}{x + 1}\\ \end{array}\]

C

double code(double x, double y, double z, double t) {
	return ((x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0));
}

↓

double code(double x, double y, double z, double t) {
	double temp;
	if (((x <= -1.862397195409269e-252) || !(x <= 1.3799272751964713e-126))) {
		temp = ((((y / ((t * z) - x)) * (z / (x + 1.0))) + ((1.0 / (x + 1.0)) * x)) - ((x / ((t * z) - x)) / (x + 1.0)));
	} else {
		temp = ((((y * z) / ((x + 1.0) * ((t * z) - x))) + ((1.0 / (x + 1.0)) * x)) - ((x / ((t * z) - x)) / (x + 1.0)));
	}
	return temp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	7.5
Target	0.4
Herbie	3.6

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

Derivation

1. Split input into 2 regimes

2. if x < -1.862397195409269e-252 or 1.3799272751964713e-126 < x

   1. Initial program 7.5

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]
   2. Using strategy rm

   3. Applied div-sub 7.5

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

   4. Applied associate-+r- 7.5

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

   5. Applied div-sub 7.5

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

   6. Simplified 3.1

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right)}{\left(x + 1\right) \cdot 1}} - \frac{\frac{x}{t \cdot z - x}}{x + 1}\]
   7. Using strategy rm

   8. Applied *-un-lft-identity 3.1

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

   9. Applied times-frac 3.2

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

   10. Simplified 3.2

       \[\leadsto \frac{1}{x + 1} \cdot \color{blue}{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right)} - \frac{\frac{x}{t \cdot z - x}}{x + 1}\]
   11. Using strategy rm

   12. Applied fma-udef 3.2

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

   13. Applied distribute-lft-in 3.2

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

   14. Simplified 3.2

       \[\leadsto \left(\color{blue}{\frac{\frac{y}{t \cdot z - x} \cdot z}{x + 1}} + \frac{1}{x + 1} \cdot x\right) - \frac{\frac{x}{t \cdot z - x}}{x + 1}\]
   15. Using strategy rm

   16. Applied *-un-lft-identity 3.2

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

   17. Applied times-frac 2.7

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

   18. Simplified 2.7

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

   if -1.862397195409269e-252 < x < 1.3799272751964713e-126

   1. Initial program 7.8

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]
   2. Using strategy rm

   3. Applied div-sub 7.8

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

   4. Applied associate-+r- 7.8

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

   5. Applied div-sub 7.8

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

   6. Simplified 13.1

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right)}{\left(x + 1\right) \cdot 1}} - \frac{\frac{x}{t \cdot z - x}}{x + 1}\]
   7. Using strategy rm

   8. Applied *-un-lft-identity 13.1

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

   9. Applied times-frac 13.1

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

   10. Simplified 13.1

       \[\leadsto \frac{1}{x + 1} \cdot \color{blue}{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right)} - \frac{\frac{x}{t \cdot z - x}}{x + 1}\]
   11. Using strategy rm

   12. Applied fma-udef 13.1

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

   13. Applied distribute-lft-in 13.1

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

   14. Simplified 13.1

       \[\leadsto \left(\color{blue}{\frac{\frac{y}{t \cdot z - x} \cdot z}{x + 1}} + \frac{1}{x + 1} \cdot x\right) - \frac{\frac{x}{t \cdot z - x}}{x + 1}\]
   15. Using strategy rm

   16. Applied associate-*l/ 7.8

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

   17. Applied associate-/l/ 7.8

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

4. Final simplification 3.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.862397195409269 \cdot 10^{-252} \lor \neg \left(x \le 1.3799272751964713 \cdot 10^{-126}\right):\\ \;\;\;\;\left(\frac{y}{t \cdot z - x} \cdot \frac{z}{x + 1} + \frac{1}{x + 1} \cdot x\right) - \frac{\frac{x}{t \cdot z - x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y \cdot z}{\left(x + 1\right) \cdot \left(t \cdot z - x\right)} + \frac{1}{x + 1} \cdot x\right) - \frac{\frac{x}{t \cdot z - x}}{x + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(FPCore (x y z t)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, A"
  :precision binary64

  :herbie-target
  (/ (+ x (- (/ y (- t (/ x z))) (/ x (- (* t z) x)))) (+ x 1))

  (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1)))