Numeric.Histogram:binBounds from Chart-1.5.3

Average Error: 6.3 → 1.4

Time: 7.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -5.121129492313333 \cdot 10^{35}:\\ \;\;\;\;x + \left(\sqrt[3]{\frac{y - x}{\sqrt[3]{t} \cdot \sqrt[3]{t}}} \cdot \sqrt[3]{\frac{y - x}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}\right) \cdot \left(\sqrt[3]{\frac{y - x}{\sqrt[3]{t} \cdot \sqrt[3]{t}}} \cdot \frac{z}{\sqrt[3]{t}}\right)\\ \mathbf{elif}\;t \le 1.0394980955783228 \cdot 10^{74}:\\ \;\;\;\;x + \frac{1}{\frac{t}{\left(y - x\right) \cdot z}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\sqrt[3]{\frac{y - x}{\sqrt[3]{t} \cdot \sqrt[3]{t}}} \cdot \sqrt[3]{\frac{y - x}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}\right) \cdot \left(\left(\left(\sqrt[3]{\sqrt[3]{\frac{y - x}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}} \cdot \sqrt[3]{\sqrt[3]{\frac{y - x}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}}\right) \cdot \sqrt[3]{\sqrt[3]{\frac{y - x}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}}\right) \cdot \frac{z}{\sqrt[3]{t}}\right)\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t) {
	double VAR;
	if ((t <= -5.121129492313333e+35)) {
		VAR = (x + ((cbrt(((y - x) / (cbrt(t) * cbrt(t)))) * cbrt(((y - x) / (cbrt(t) * cbrt(t))))) * (cbrt(((y - x) / (cbrt(t) * cbrt(t)))) * (z / cbrt(t)))));
	} else {
		double VAR_1;
		if ((t <= 1.0394980955783228e+74)) {
			VAR_1 = (x + (1.0 / (t / ((y - x) * z))));
		} else {
			VAR_1 = (x + ((cbrt(((y - x) / (cbrt(t) * cbrt(t)))) * cbrt(((y - x) / (cbrt(t) * cbrt(t))))) * (((cbrt(cbrt(((y - x) / (cbrt(t) * cbrt(t))))) * cbrt(cbrt(((y - x) / (cbrt(t) * cbrt(t)))))) * cbrt(cbrt(((y - x) / (cbrt(t) * cbrt(t)))))) * (z / cbrt(t)))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	6.3
Target	2.0
Herbie	1.4

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

Derivation

1. Split input into 3 regimes

2. if t < -5.121129492313333e+35

   1. Initial program 10.4

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

   3. Applied add-cube-cbrt 10.6

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

   4. Applied times-frac 0.8

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

   6. Applied add-cube-cbrt 0.9

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

   7. Applied associate-*l* 0.9

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

   if -5.121129492313333e+35 < t < 1.0394980955783228e+74

   1. Initial program 1.7

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

   3. Applied clear-num 1.7

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

   if 1.0394980955783228e+74 < t

   1. Initial program 10.7

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

   3. Applied add-cube-cbrt 10.9

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

   4. Applied times-frac 1.0

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

   6. Applied add-cube-cbrt 1.1

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

   7. Applied associate-*l* 1.1

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

   9. Applied add-cube-cbrt 1.2

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

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -5.121129492313333 \cdot 10^{35}:\\ \;\;\;\;x + \left(\sqrt[3]{\frac{y - x}{\sqrt[3]{t} \cdot \sqrt[3]{t}}} \cdot \sqrt[3]{\frac{y - x}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}\right) \cdot \left(\sqrt[3]{\frac{y - x}{\sqrt[3]{t} \cdot \sqrt[3]{t}}} \cdot \frac{z}{\sqrt[3]{t}}\right)\\ \mathbf{elif}\;t \le 1.0394980955783228 \cdot 10^{74}:\\ \;\;\;\;x + \frac{1}{\frac{t}{\left(y - x\right) \cdot z}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\sqrt[3]{\frac{y - x}{\sqrt[3]{t} \cdot \sqrt[3]{t}}} \cdot \sqrt[3]{\frac{y - x}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}\right) \cdot \left(\left(\left(\sqrt[3]{\sqrt[3]{\frac{y - x}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}} \cdot \sqrt[3]{\sqrt[3]{\frac{y - x}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}}\right) \cdot \sqrt[3]{\sqrt[3]{\frac{y - x}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}}\right) \cdot \frac{z}{\sqrt[3]{t}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 
(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)))