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

Average Error: 12.02% → 1.76%

Time: 13.8s

Precision: binary64

Cost: 1737

\[ \begin{array}{c}[y, t] = \mathsf{sort}([y, t])\\ \end{array} \]

Math

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

↓

\[\begin{array}{l} t_1 := \left(y - z\right) \cdot \left(t - z\right)\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq -1 \cdot 10^{-96}\right):\\ \;\;\;\;\frac{\frac{x}{z - t}}{z - y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{1}{z - y}}{z - t}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- y z) (- t z))))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 -1e-96)))
     (/ (/ x (- z t)) (- z y))
     (* x (/ (/ 1.0 (- z y)) (- z t))))))

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 t_1 = (y - z) * (t - z);
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= -1e-96)) {
		tmp = (x / (z - t)) / (z - y);
	} else {
		tmp = x * ((1.0 / (z - y)) / (z - t));
	}
	return tmp;
}

Java

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

↓

public static double code(double x, double y, double z, double t) {
	double t_1 = (y - z) * (t - z);
	double tmp;
	if ((t_1 <= -Double.POSITIVE_INFINITY) || !(t_1 <= -1e-96)) {
		tmp = (x / (z - t)) / (z - y);
	} else {
		tmp = x * ((1.0 / (z - y)) / (z - t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	return x / ((y - z) * (t - z))

↓

def code(x, y, z, t):
	t_1 = (y - z) * (t - z)
	tmp = 0
	if (t_1 <= -math.inf) or not (t_1 <= -1e-96):
		tmp = (x / (z - t)) / (z - y)
	else:
		tmp = x * ((1.0 / (z - y)) / (z - t))
	return tmp

Julia

function code(x, y, z, t)
	return Float64(x / Float64(Float64(y - z) * Float64(t - z)))
end

↓

function code(x, y, z, t)
	t_1 = Float64(Float64(y - z) * Float64(t - z))
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= -1e-96))
		tmp = Float64(Float64(x / Float64(z - t)) / Float64(z - y));
	else
		tmp = Float64(x * Float64(Float64(1.0 / Float64(z - y)) / Float64(z - t)));
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x / ((y - z) * (t - z));
end

↓

function tmp_2 = code(x, y, z, t)
	t_1 = (y - z) * (t - z);
	tmp = 0.0;
	if ((t_1 <= -Inf) || ~((t_1 <= -1e-96)))
		tmp = (x / (z - t)) / (z - y);
	else
		tmp = x * ((1.0 / (z - y)) / (z - t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := N[(x / N[(N[(y - z), $MachinePrecision] * N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y - z), $MachinePrecision] * N[(t - z), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, -1e-96]], $MachinePrecision]], N[(N[(x / N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(1.0 / N[(z - y), $MachinePrecision]), $MachinePrecision] / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	12.02%
Target	13.15%
Herbie	1.76%

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 y z) (-.f64 t z)) < -inf.0 or -9.9999999999999991e-97 < (*.f64 (-.f64 y z) (-.f64 t z))

   1. Initial program 15.04

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

   2. Simplified 2.1

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

      [Start] 15.04	\[ \frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \]
      sub-neg [=>] 15.04	\[ \frac{x}{\color{blue}{\left(y + \left(-z\right)\right)} \cdot \left(t - z\right)} \]
      +-commutative [=>] 15.04	\[ \frac{x}{\color{blue}{\left(\left(-z\right) + y\right)} \cdot \left(t - z\right)} \]
      neg-sub0 [=>] 15.04	\[ \frac{x}{\left(\color{blue}{\left(0 - z\right)} + y\right) \cdot \left(t - z\right)} \]
      associate-+l- [=>] 15.04	\[ \frac{x}{\color{blue}{\left(0 - \left(z - y\right)\right)} \cdot \left(t - z\right)} \]
      sub0-neg [=>] 15.04	\[ \frac{x}{\color{blue}{\left(-\left(z - y\right)\right)} \cdot \left(t - z\right)} \]
      distribute-lft-neg-out [=>] 15.04	\[ \frac{x}{\color{blue}{-\left(z - y\right) \cdot \left(t - z\right)}} \]
      distribute-rgt-neg-in [=>] 15.04	\[ \frac{x}{\color{blue}{\left(z - y\right) \cdot \left(-\left(t - z\right)\right)}} \]
      neg-sub0 [=>] 15.04	\[ \frac{x}{\left(z - y\right) \cdot \color{blue}{\left(0 - \left(t - z\right)\right)}} \]
      associate-+l- [<=] 15.04	\[ \frac{x}{\left(z - y\right) \cdot \color{blue}{\left(\left(0 - t\right) + z\right)}} \]
      neg-sub0 [<=] 15.04	\[ \frac{x}{\left(z - y\right) \cdot \left(\color{blue}{\left(-t\right)} + z\right)} \]
      +-commutative [<=] 15.04	\[ \frac{x}{\left(z - y\right) \cdot \color{blue}{\left(z + \left(-t\right)\right)}} \]
      sub-neg [<=] 15.04	\[ \frac{x}{\left(z - y\right) \cdot \color{blue}{\left(z - t\right)}} \]
      associate-/l/ [<=] 2.1	\[ \color{blue}{\frac{\frac{x}{z - t}}{z - y}} \]

   if -inf.0 < (*.f64 (-.f64 y z) (-.f64 t z)) < -9.9999999999999991e-97

   1. Initial program 0.33

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

   2. Simplified 8.06

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

      [Start] 0.33	\[ \frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \]
      sub-neg [=>] 0.33	\[ \frac{x}{\color{blue}{\left(y + \left(-z\right)\right)} \cdot \left(t - z\right)} \]
      +-commutative [=>] 0.33	\[ \frac{x}{\color{blue}{\left(\left(-z\right) + y\right)} \cdot \left(t - z\right)} \]
      neg-sub0 [=>] 0.33	\[ \frac{x}{\left(\color{blue}{\left(0 - z\right)} + y\right) \cdot \left(t - z\right)} \]
      associate-+l- [=>] 0.33	\[ \frac{x}{\color{blue}{\left(0 - \left(z - y\right)\right)} \cdot \left(t - z\right)} \]
      sub0-neg [=>] 0.33	\[ \frac{x}{\color{blue}{\left(-\left(z - y\right)\right)} \cdot \left(t - z\right)} \]
      distribute-lft-neg-out [=>] 0.33	\[ \frac{x}{\color{blue}{-\left(z - y\right) \cdot \left(t - z\right)}} \]
      distribute-rgt-neg-in [=>] 0.33	\[ \frac{x}{\color{blue}{\left(z - y\right) \cdot \left(-\left(t - z\right)\right)}} \]
      neg-sub0 [=>] 0.33	\[ \frac{x}{\left(z - y\right) \cdot \color{blue}{\left(0 - \left(t - z\right)\right)}} \]
      associate-+l- [<=] 0.33	\[ \frac{x}{\left(z - y\right) \cdot \color{blue}{\left(\left(0 - t\right) + z\right)}} \]
      neg-sub0 [<=] 0.33	\[ \frac{x}{\left(z - y\right) \cdot \left(\color{blue}{\left(-t\right)} + z\right)} \]
      +-commutative [<=] 0.33	\[ \frac{x}{\left(z - y\right) \cdot \color{blue}{\left(z + \left(-t\right)\right)}} \]
      sub-neg [<=] 0.33	\[ \frac{x}{\left(z - y\right) \cdot \color{blue}{\left(z - t\right)}} \]
      associate-/l/ [<=] 8.06	\[ \color{blue}{\frac{\frac{x}{z - t}}{z - y}} \]

   3. Applied egg-rr 0.43

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

4. Final simplification 1.76

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y - z\right) \cdot \left(t - z\right) \leq -\infty \lor \neg \left(\left(y - z\right) \cdot \left(t - z\right) \leq -1 \cdot 10^{-96}\right):\\ \;\;\;\;\frac{\frac{x}{z - t}}{z - y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{1}{z - y}}{z - t}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.6%
Cost	1609

\[\begin{array}{l} t_1 := \left(y - z\right) \cdot \left(t - z\right)\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+248} \lor \neg \left(t_1 \leq -5 \cdot 10^{-282}\right):\\ \;\;\;\;\frac{\frac{x}{z - t}}{z - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t_1}\\ \end{array} \]

Alternative 2
Error	44.11%
Cost	1308

\[\begin{array}{l} t_1 := \frac{\frac{x}{y}}{t}\\ t_2 := \frac{\frac{x}{z}}{z}\\ \mathbf{if}\;y \leq -3.5 \cdot 10^{+261}:\\ \;\;\;\;\frac{\frac{-x}{z}}{y}\\ \mathbf{elif}\;y \leq -7.6 \cdot 10^{+121}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -2.25 \cdot 10^{+81}:\\ \;\;\;\;\frac{-x}{y \cdot z}\\ \mathbf{elif}\;y \leq -1.8 \cdot 10^{-59}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -1.05 \cdot 10^{-67}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-267}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-29}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	43.6%
Cost	1308

\[\begin{array}{l} t_1 := \frac{\frac{x}{y}}{t}\\ t_2 := \frac{\frac{x}{z}}{z}\\ \mathbf{if}\;y \leq -3.9 \cdot 10^{+260}:\\ \;\;\;\;\frac{\frac{-x}{y}}{z}\\ \mathbf{elif}\;y \leq -1.5 \cdot 10^{+122}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.85 \cdot 10^{+81}:\\ \;\;\;\;\frac{-x}{y \cdot z}\\ \mathbf{elif}\;y \leq -1.95 \cdot 10^{-59}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -1.05 \cdot 10^{-67}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-267}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-27}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	43.36%
Cost	1308

\[\begin{array}{l} t_1 := \frac{\frac{x}{y}}{t}\\ t_2 := \frac{\frac{x}{z}}{z}\\ \mathbf{if}\;y \leq -7.8 \cdot 10^{+259}:\\ \;\;\;\;\frac{\frac{-x}{y}}{z}\\ \mathbf{elif}\;y \leq -6.5 \cdot 10^{+122}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -6.5 \cdot 10^{+87}:\\ \;\;\;\;\frac{-x}{y \cdot z}\\ \mathbf{elif}\;y \leq -1.8 \cdot 10^{-59}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -1 \cdot 10^{-70}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{-265}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-33}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{y}\\ \end{array} \]

Alternative 5
Error	43.69%
Cost	1176

\[\begin{array}{l} t_1 := \frac{\frac{x}{z}}{z}\\ t_2 := \frac{\frac{x}{y}}{t}\\ \mathbf{if}\;y \leq -1.45 \cdot 10^{+122}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -1.15 \cdot 10^{+87}:\\ \;\;\;\;\frac{-x}{y \cdot z}\\ \mathbf{elif}\;y \leq -1.95 \cdot 10^{-59}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -8.2 \cdot 10^{-68}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 1.22 \cdot 10^{-265}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{-29}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 6
Error	44.05%
Cost	1044

\[\begin{array}{l} t_1 := \frac{\frac{x}{y}}{t}\\ t_2 := \frac{\frac{x}{z}}{z}\\ \mathbf{if}\;y \leq -6 \cdot 10^{+107}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.8 \cdot 10^{-59}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -5.2 \cdot 10^{-70}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{-265}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{-31}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	35.62%
Cost	976

\[\begin{array}{l} t_1 := \frac{\frac{x}{y}}{t}\\ \mathbf{if}\;y \leq -9.5 \cdot 10^{+260}:\\ \;\;\;\;\frac{\frac{-x}{y}}{z}\\ \mathbf{elif}\;y \leq -6.8 \cdot 10^{+121}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.35 \cdot 10^{+81}:\\ \;\;\;\;\frac{-x}{y \cdot z}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-28}:\\ \;\;\;\;\frac{x}{z \cdot \left(z - t\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	33.84%
Cost	976

\[\begin{array}{l} t_1 := \frac{\frac{x}{y}}{t}\\ \mathbf{if}\;y \leq -7.2 \cdot 10^{+260}:\\ \;\;\;\;\frac{\frac{-x}{y}}{z}\\ \mathbf{elif}\;y \leq -1.9 \cdot 10^{+122}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.65 \cdot 10^{-101}:\\ \;\;\;\;\frac{x}{z \cdot \left(z - y\right)}\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{-26}:\\ \;\;\;\;\frac{x}{z \cdot \left(z - t\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	31.29%
Cost	976

\[\begin{array}{l} t_1 := \frac{\frac{x}{y}}{t}\\ \mathbf{if}\;y \leq -1.42 \cdot 10^{+261}:\\ \;\;\;\;\frac{\frac{-x}{y}}{z}\\ \mathbf{elif}\;y \leq -3.2 \cdot 10^{+121}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -3.1 \cdot 10^{+73}:\\ \;\;\;\;\frac{x}{z \cdot \left(z - y\right)}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-26}:\\ \;\;\;\;\frac{\frac{x}{z}}{z - t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	20.7%
Cost	976

\[\begin{array}{l} t_1 := \frac{\frac{x}{z - y}}{z}\\ t_2 := \frac{-x}{y \cdot \left(z - t\right)}\\ \mathbf{if}\;t \leq -6.2 \cdot 10^{-142}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{-125}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{-94}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 6 \cdot 10^{+62}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y - z}}{t}\\ \end{array} \]

Alternative 11
Error	30.13%
Cost	844

\[\begin{array}{l} \mathbf{if}\;t \leq -3.6 \cdot 10^{-96}:\\ \;\;\;\;\frac{x}{y \cdot t}\\ \mathbf{elif}\;t \leq 1.26 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{x}{z}}{z - y}\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{+167}:\\ \;\;\;\;\frac{\frac{x}{z}}{z - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{y}\\ \end{array} \]

Alternative 12
Error	20.83%
Cost	713

\[\begin{array}{l} \mathbf{if}\;t \leq -5.6 \cdot 10^{-96} \lor \neg \left(t \leq 4.8 \cdot 10^{+62}\right):\\ \;\;\;\;\frac{\frac{x}{y - z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{z - y}\\ \end{array} \]

Alternative 13
Error	20.38%
Cost	713

\[\begin{array}{l} \mathbf{if}\;t \leq -4.9 \cdot 10^{-96} \lor \neg \left(t \leq 5.2 \cdot 10^{+62}\right):\\ \;\;\;\;\frac{\frac{x}{y - z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z - y}}{z}\\ \end{array} \]

Alternative 14
Error	10.33%
Cost	708

\[\begin{array}{l} \mathbf{if}\;t \leq 3.7 \cdot 10^{+133}:\\ \;\;\;\;\frac{x}{\left(y - z\right) \cdot \left(t - z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y - z}}{t}\\ \end{array} \]

Alternative 15
Error	55.42%
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{+71} \lor \neg \left(z \leq 2.15 \cdot 10^{+68}\right):\\ \;\;\;\;\frac{x}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot t}\\ \end{array} \]

Alternative 16
Error	39.96%
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -16000000000 \lor \neg \left(z \leq 1.45 \cdot 10^{-87}\right):\\ \;\;\;\;\frac{x}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot t}\\ \end{array} \]

Alternative 17
Error	39.76%
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{+93} \lor \neg \left(z \leq 1.45 \cdot 10^{-87}\right):\\ \;\;\;\;\frac{x}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{y}\\ \end{array} \]

Alternative 18
Error	39.77%
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -8.2 \cdot 10^{+91} \lor \neg \left(z \leq 1.45 \cdot 10^{-87}\right):\\ \;\;\;\;\frac{x}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t}\\ \end{array} \]

Alternative 19
Error	36.19%
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -8.2 \cdot 10^{+91} \lor \neg \left(z \leq 1.45 \cdot 10^{-87}\right):\\ \;\;\;\;\frac{\frac{x}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t}\\ \end{array} \]

Alternative 20
Error	62.65%
Cost	320

\[\frac{x}{y \cdot t} \]

Reproduce

herbie shell --seed 2023121 
(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))))