Data.Random.Distribution.Triangular:triangularCDF from random-fu-0.2.6.2, B

Average Error: 7.5 → 2.1

Time: 8.8s

Precision: binary64

Cost: 1346

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \leq -1.0627057217112072 \cdot 10^{-284}:\\ \;\;\;\;\frac{1}{y - z} \cdot \frac{x}{t - z}\\ \mathbf{elif}\;z \leq 1.8551606080189914 \cdot 10^{-68}:\\ \;\;\;\;\frac{x}{\left(y - z\right) \cdot \left(t - z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{y - z}{x}}}{t - z}\\ \end{array}\]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.0627057217112072e-284)
   (* (/ 1.0 (- y z)) (/ x (- t z)))
   (if (<= z 1.8551606080189914e-68)
     (/ x (* (- y z) (- t z)))
     (/ (/ 1.0 (/ (- y z) x)) (- t z)))))

C

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

↓

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.0627057217112072e-284) {
		tmp = (1.0 / (y - z)) * (x / (t - z));
	} else if (z <= 1.8551606080189914e-68) {
		tmp = x / ((y - z) * (t - z));
	} else {
		tmp = (1.0 / ((y - z) / x)) / (t - z);
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	7.5
Target	8.2
Herbie	2.1

\[\begin{array}{l} \mathbf{if}\;\frac{x}{\left(y - z\right) \cdot \left(t - z\right)} < 0:\\ \;\;\;\;\frac{\frac{x}{y - z}}{t - z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{\left(y - z\right) \cdot \left(t - z\right)}\\ \end{array}\]

Alternatives

Alternative 1
Error	1.1
Cost	39744

\[\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}} \cdot \frac{\frac{\sqrt[3]{x}}{\sqrt[3]{y - z}}}{t - z}\]

Alternative 2
Error	1.0
Cost	2114

\[\begin{array}{l} \mathbf{if}\;\left(y - z\right) \cdot \left(t - z\right) \leq -1.4490301069605532 \cdot 10^{+290}:\\ \;\;\;\;\frac{1}{\frac{y - z}{\frac{x}{t - z}}}\\ \mathbf{elif}\;\left(y - z\right) \cdot \left(t - z\right) \leq 2.710281579753108 \cdot 10^{+205}:\\ \;\;\;\;\frac{x}{\left(y - z\right) \cdot \left(t - z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y - z} \cdot x}{t - z}\\ \end{array}\]

Alternative 3
Error	0.9
Cost	2114

\[\begin{array}{l} \mathbf{if}\;\left(y - z\right) \cdot \left(t - z\right) \leq -\infty:\\ \;\;\;\;\frac{\frac{1}{\frac{y - z}{x}}}{t - z}\\ \mathbf{elif}\;\left(y - z\right) \cdot \left(t - z\right) \leq 2.710281579753108 \cdot 10^{+205}:\\ \;\;\;\;\frac{x}{\left(y - z\right) \cdot \left(t - z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y - z} \cdot x}{t - z}\\ \end{array}\]

Alternative 4
Error	1.0
Cost	1800

\[\begin{array}{l} \mathbf{if}\;\left(y - z\right) \cdot \left(t - z\right) \leq -\infty \lor \neg \left(\left(y - z\right) \cdot \left(t - z\right) \leq 2.710281579753108 \cdot 10^{+205}\right):\\ \;\;\;\;\frac{\frac{1}{\frac{y - z}{x}}}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - z\right) \cdot \left(t - z\right)}\\ \end{array}\]

Alternative 5
Error	0.9
Cost	1672

\[\begin{array}{l} \mathbf{if}\;\left(y - z\right) \cdot \left(t - z\right) \leq -1.4490301069605532 \cdot 10^{+290} \lor \neg \left(\left(y - z\right) \cdot \left(t - z\right) \leq 3.6081242556375764 \cdot 10^{+188}\right):\\ \;\;\;\;\frac{\frac{x}{y - z}}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - z\right) \cdot \left(t - z\right)}\\ \end{array}\]

Alternative 6
Error	2.1
Cost	576

\[\frac{\frac{x}{y - z}}{t - z}\]

Alternative 7
Error	13.6
Cost	1796

\[\begin{array}{l} \mathbf{if}\;t \leq -90.41343743129991:\\ \;\;\;\;\frac{x}{\left(y - z\right) \cdot t}\\ \mathbf{elif}\;t \leq -9.865164912019664 \cdot 10^{-88}:\\ \;\;\;\;\frac{\frac{-x}{y - z}}{z}\\ \mathbf{elif}\;t \leq -2.4414779933159523 \cdot 10^{-164}:\\ \;\;\;\;\frac{\frac{x}{y}}{t - z}\\ \mathbf{elif}\;t \leq 7.866291426314132 \cdot 10^{-73}:\\ \;\;\;\;\frac{\frac{-x}{y - z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - z\right) \cdot t}\\ \end{array}\]

Alternative 8
Error	14.8
Cost	1796

\[\begin{array}{l} \mathbf{if}\;y \leq -2.1051889596342455 \cdot 10^{-158}:\\ \;\;\;\;\frac{\frac{x}{y}}{t - z}\\ \mathbf{elif}\;y \leq 4.9549839596378684 \cdot 10^{-95}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t - z}\\ \mathbf{elif}\;y \leq 6.797466776022858 \cdot 10^{-59}:\\ \;\;\;\;\frac{x}{y \cdot \left(t - z\right)}\\ \mathbf{elif}\;y \leq 1.7343185145153442 \cdot 10^{+64}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t - z}\\ \end{array}\]

Alternative 9
Error	16.1
Cost	776

\[\begin{array}{l} \mathbf{if}\;y \leq -2.1051889596342455 \cdot 10^{-158} \lor \neg \left(y \leq 6.967413386655891 \cdot 10^{-116}\right):\\ \;\;\;\;\frac{\frac{x}{y}}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot \left(z - t\right)}\\ \end{array}\]

Alternative 10
Error	17.7
Cost	776

\[\begin{array}{l} \mathbf{if}\;z \leq -4.5430164082148206 \cdot 10^{-89} \lor \neg \left(z \leq 1.2825980939759143 \cdot 10^{-65}\right):\\ \;\;\;\;\frac{x}{z \cdot \left(z - y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t - z}\\ \end{array}\]

Alternative 11
Error	21.3
Cost	776

\[\begin{array}{l} \mathbf{if}\;y \leq -9.349038285353448 \cdot 10^{-159} \lor \neg \left(y \leq 3.1003388053934303 \cdot 10^{-140}\right):\\ \;\;\;\;\frac{\frac{x}{y}}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot z}\\ \end{array}\]

Alternative 12
Error	27.0
Cost	962

\[\begin{array}{l} \mathbf{if}\;z \leq -4.5430164082148206 \cdot 10^{-89}:\\ \;\;\;\;\frac{x}{z \cdot z}\\ \mathbf{elif}\;z \leq 1.1581699379218829 \cdot 10^{-66}:\\ \;\;\;\;\frac{x}{y \cdot t}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Alternative 13
Error	33.3
Cost	962

\[\begin{array}{l} \mathbf{if}\;t \leq -9.13718258547169 \cdot 10^{+15}:\\ \;\;\;\;0\\ \mathbf{elif}\;t \leq 2.4085052544920162 \cdot 10^{+112}:\\ \;\;\;\;\frac{x}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Alternative 14
Error	36.9
Cost	64

\[0\]

Alternative 15
Error	61.8
Cost	64

\[1\]

Derivation

1. Split input into 3 regimes

2. if z < -1.06270572171120721e-284

   1. Initial program 7.3

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

   3. Applied *-un-lft-identity_binary64_22243 7.3

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

   4. Applied times-frac_binary64_22249 1.8

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

   5. Simplified 1.8

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

   if -1.06270572171120721e-284 < z < 1.8551606080189914e-68

   1. Initial program 5.5

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

   2. Simplified 5.5

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

   if 1.8551606080189914e-68 < z

   1. Initial program 8.9

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

   3. Applied associate-/r*_binary64_22187 0.4

      \[\leadsto \color{blue}{\frac{\frac{x}{y - z}}{t - z}}\]
   4. Using strategy rm

   5. Applied clear-num_binary64_22242 0.5

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

   6. Simplified 0.5

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

4. Final simplification 2.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.0627057217112072 \cdot 10^{-284}:\\ \;\;\;\;\frac{1}{y - z} \cdot \frac{x}{t - z}\\ \mathbf{elif}\;z \leq 1.8551606080189914 \cdot 10^{-68}:\\ \;\;\;\;\frac{x}{\left(y - z\right) \cdot \left(t - z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{y - z}{x}}}{t - z}\\ \end{array}\]

Reproduce

herbie shell --seed 2021044 
(FPCore (x y z t)
  :name "Data.Random.Distribution.Triangular:triangularCDF from random-fu-0.2.6.2, B"
  :precision binary64

  :herbie-target
  (if (< (/ x (* (- y z) (- t z))) 0.0) (/ (/ x (- y z)) (- t z)) (* x (/ 1.0 (* (- y z) (- t z)))))

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