Numeric.Histogram:binBounds from Chart-1.5.3

Average Error: 6.8 → 0.9

Time: 3.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot z}{t} = -inf.0:\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{z}{t}\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot z}{t} \le 6.64996797919708278 \cdot 10^{294}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{z}{\frac{t}{y}} - \frac{x}{\frac{t}{z}}\right)\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t) {
	double VAR;
	if ((((double) (x + ((double) (((double) (((double) (y - x)) * z)) / t)))) <= -inf.0)) {
		VAR = ((double) (x + ((double) (((double) (y - x)) * ((double) (z / t))))));
	} else {
		double VAR_1;
		if ((((double) (x + ((double) (((double) (((double) (y - x)) * z)) / t)))) <= 6.649967979197083e+294)) {
			VAR_1 = ((double) (x + ((double) (((double) (((double) (y - x)) * z)) / t))));
		} else {
			VAR_1 = ((double) (x + ((double) (((double) (z / ((double) (t / y)))) - ((double) (x / ((double) (t / z))))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	6.8
Target	2.1
Herbie	0.9

\[\begin{array}{l} \mathbf{if}\;x \lt -9.0255111955330046 \cdot 10^{-135}:\\ \;\;\;\;x - \frac{z}{t} \cdot \left(x - y\right)\\ \mathbf{elif}\;x \lt 4.2750321637007147 \cdot 10^{-250}:\\ \;\;\;\;x + \frac{y - x}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if (+ x (/ (* (- y x) z) t)) < -inf.0

   1. Initial program 64.0

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

   2. Simplified 0.2

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

   if -inf.0 < (+ x (/ (* (- y x) z) t)) < 6.64996797919708278e294

   1. Initial program 0.8

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

   if 6.64996797919708278e294 < (+ x (/ (* (- y x) z) t))

   1. Initial program 52.4

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

   2. Simplified 0.7

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

   4. Applied add-cube-cbrt 1.4

      \[\leadsto x + \left(y - x\right) \cdot \frac{z}{\color{blue}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}}\]

   5. Applied *-un-lft-identity 1.4

      \[\leadsto x + \left(y - x\right) \cdot \frac{\color{blue}{1 \cdot z}}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}\]

   6. Applied times-frac 1.4

      \[\leadsto x + \left(y - x\right) \cdot \color{blue}{\left(\frac{1}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}}\right)}\]

   7. Applied associate-*r* 4.7

      \[\leadsto x + \color{blue}{\left(\left(y - x\right) \cdot \frac{1}{\sqrt[3]{t} \cdot \sqrt[3]{t}}\right) \cdot \frac{z}{\sqrt[3]{t}}}\]

   8. Simplified 4.7

      \[\leadsto x + \color{blue}{\frac{y - x}{\sqrt[3]{t} \cdot \sqrt[3]{t}}} \cdot \frac{z}{\sqrt[3]{t}}\]
   9. Using strategy rm

   10. Applied add-cube-cbrt 4.8

       \[\leadsto x + \frac{y - x}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{\color{blue}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}}}\]

   11. Applied cbrt-prod 4.8

       \[\leadsto x + \frac{y - x}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z}{\color{blue}{\sqrt[3]{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \sqrt[3]{\sqrt[3]{t}}}}\]

   12. Applied add-cube-cbrt 5.0

       \[\leadsto x + \frac{y - x}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}}{\sqrt[3]{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \sqrt[3]{\sqrt[3]{t}}}\]

   13. Applied times-frac 5.0

       \[\leadsto x + \frac{y - x}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \color{blue}{\left(\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{\sqrt[3]{t} \cdot \sqrt[3]{t}}} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{\sqrt[3]{t}}}\right)}\]

   14. Applied associate-*r* 5.0

       \[\leadsto x + \color{blue}{\left(\frac{y - x}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{\sqrt[3]{t} \cdot \sqrt[3]{t}}}\right) \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{\sqrt[3]{t}}}}\]

   15. Simplified 5.0

       \[\leadsto x + \color{blue}{\left(\frac{y - x}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \left(\sqrt[3]{z} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{\sqrt[3]{t} \cdot \sqrt[3]{t}}}\right)\right)} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{\sqrt[3]{t}}}\]

   16. Taylor expanded around 0 52.4

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

   17. Simplified 2.6

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

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot z}{t} = -inf.0:\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{z}{t}\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot z}{t} \le 6.64996797919708278 \cdot 10^{294}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{z}{\frac{t}{y}} - \frac{x}{\frac{t}{z}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020184 
(FPCore (x y z t)
  :name "Numeric.Histogram:binBounds from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (if (< x -9.025511195533005e-135) (- x (* (/ z t) (- x y))) (if (< x 4.275032163700715e-250) (+ x (* (/ (- y x) t) z)) (+ x (/ (- y x) (/ t z)))))

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