Numeric.Histogram:binBounds from Chart-1.5.3

Average Error: 6.2 → 1.7

Time: 5.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le 9.93422309972294151 \cdot 10^{-57}:\\ \;\;\;\;x + \sqrt[3]{\frac{\sqrt[3]{z}}{t}} \cdot \left(\left(y - x\right) \cdot \left({\left(\sqrt[3]{z}\right)}^{2} \cdot \left(\sqrt[3]{\frac{\sqrt[3]{z}}{t}} \cdot \sqrt[3]{\frac{\sqrt[3]{z}}{t}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}}\\ \end{array}\]

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 ((z <= 9.934223099722942e-57)) {
		VAR = ((double) (x + ((double) (((double) cbrt(((double) (((double) cbrt(z)) / t)))) * ((double) (((double) (y - x)) * ((double) (((double) pow(((double) cbrt(z)), 2.0)) * ((double) (((double) cbrt(((double) (((double) cbrt(z)) / t)))) * ((double) cbrt(((double) (((double) cbrt(z)) / t))))))))))))));
	} else {
		VAR = ((double) (x + ((double) (((double) (((double) (y - x)) / ((double) (((double) cbrt(t)) * ((double) cbrt(t)))))) * ((double) (z / ((double) cbrt(t))))))));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	6.2
Target	2.1
Herbie	1.7

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

Derivation

1. Split input into 2 regimes

2. if z < 9.93422309972294151e-57

   1. Initial program 4.6

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

   2. Simplified 1.8

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

   4. Applied *-un-lft-identity 1.8

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

   5. Applied add-cube-cbrt 2.3

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

   6. Applied times-frac 2.3

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

   7. Applied associate-*r* 3.4

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

   8. Simplified 3.4

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

   10. Applied add-cube-cbrt 3.5

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

   11. Applied associate-*r* 3.5

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

   12. Simplified 1.7

       \[\leadsto x + \color{blue}{\left(\left(y - x\right) \cdot \left({\left(\sqrt[3]{z}\right)}^{2} \cdot \left(\sqrt[3]{\frac{\sqrt[3]{z}}{t}} \cdot \sqrt[3]{\frac{\sqrt[3]{z}}{t}}\right)\right)\right)} \cdot \sqrt[3]{\frac{\sqrt[3]{z}}{t}}\]

   if 9.93422309972294151e-57 < z

   1. Initial program 11.6

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

   2. Simplified 3.1

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

   4. Applied add-cube-cbrt 3.8

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

   5. Applied *-un-lft-identity 3.8

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

   6. Applied times-frac 3.8

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

   7. Applied associate-*r* 1.7

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

   8. Simplified 1.7

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

4. Final simplification 1.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le 9.93422309972294151 \cdot 10^{-57}:\\ \;\;\;\;x + \sqrt[3]{\frac{\sqrt[3]{z}}{t}} \cdot \left(\left(y - x\right) \cdot \left({\left(\sqrt[3]{z}\right)}^{2} \cdot \left(\sqrt[3]{\frac{\sqrt[3]{z}}{t}} \cdot \sqrt[3]{\frac{\sqrt[3]{z}}{t}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020179 
(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)))