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

Percentage Accurate: 99.1% → 98.4%

Time: 14.5s

Precision: binary64

Cost: 832

• Unsound rule application detected in e-graph. Results may not be sound.

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

Math

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

↓

\[1 + \frac{x}{y - t} \cdot \frac{-1}{y - z} \]

FPCore

(FPCore (x y z t) :precision binary64 (- 1.0 (/ x (* (- y z) (- y t)))))

↓

(FPCore (x y z t)
 :precision binary64
 (+ 1.0 (* (/ x (- y t)) (/ -1.0 (- y z)))))

C

double code(double x, double y, double z, double t) {
	return 1.0 - (x / ((y - z) * (y - t)));
}

↓

double code(double x, double y, double z, double t) {
	return 1.0 + ((x / (y - t)) * (-1.0 / (y - z)));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = 1.0d0 - (x / ((y - z) * (y - t)))
end function

↓

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = 1.0d0 + ((x / (y - t)) * ((-1.0d0) / (y - z)))
end function

Java

public static double code(double x, double y, double z, double t) {
	return 1.0 - (x / ((y - z) * (y - t)));
}

↓

public static double code(double x, double y, double z, double t) {
	return 1.0 + ((x / (y - t)) * (-1.0 / (y - z)));
}

Python

def code(x, y, z, t):
	return 1.0 - (x / ((y - z) * (y - t)))

↓

def code(x, y, z, t):
	return 1.0 + ((x / (y - t)) * (-1.0 / (y - z)))

Julia

function code(x, y, z, t)
	return Float64(1.0 - Float64(x / Float64(Float64(y - z) * Float64(y - t))))
end

↓

function code(x, y, z, t)
	return Float64(1.0 + Float64(Float64(x / Float64(y - t)) * Float64(-1.0 / Float64(y - z))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = 1.0 - (x / ((y - z) * (y - t)));
end

↓

function tmp = code(x, y, z, t)
	tmp = 1.0 + ((x / (y - t)) * (-1.0 / (y - z)));
end

Wolfram

code[x_, y_, z_, t_] := N[(1.0 - N[(x / N[(N[(y - z), $MachinePrecision] * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_] := N[(1.0 + N[(N[(x / N[(y - t), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.3%

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

2. Applied egg-rr 98.5%

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

   [Start] 98.3%	\[ 1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
   associate-/l/ [<=] 98.4%	\[ 1 - \color{blue}{\frac{\frac{x}{y - t}}{y - z}} \]
   div-inv [=>] 98.5%	\[ 1 - \color{blue}{\frac{x}{y - t} \cdot \frac{1}{y - z}} \]

3. Final simplification 98.5%

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

Alternatives

Alternative 1
Accuracy	98.4%
Cost	832

\[1 + \frac{x}{y - t} \cdot \frac{-1}{y - z} \]

Alternative 2
Accuracy	81.0%
Cost	841

\[\begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{-27} \lor \neg \left(y \leq 8.2 \cdot 10^{-17}\right):\\ \;\;\;\;1 - \frac{x}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{\frac{t}{\frac{x}{z}}}\\ \end{array} \]

Alternative 3
Accuracy	85.4%
Cost	841

\[\begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{-86} \lor \neg \left(y \leq 2.55 \cdot 10^{-72}\right):\\ \;\;\;\;1 - \frac{x}{y \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{z} \cdot \frac{-1}{t}\\ \end{array} \]

Alternative 4
Accuracy	84.9%
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{-75}:\\ \;\;\;\;1 - \frac{x}{y \cdot \left(y - t\right)}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-27}:\\ \;\;\;\;1 + \frac{-1}{\frac{t}{\frac{x}{z}}}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{x}{y}}{y - z}\\ \end{array} \]

Alternative 5
Accuracy	90.0%
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{-106}:\\ \;\;\;\;1 + \frac{\frac{x}{y - t}}{z}\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-122}:\\ \;\;\;\;1 - \frac{x}{y \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{z \cdot \frac{t}{x}}\\ \end{array} \]

Alternative 6
Accuracy	90.0%
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -4.5 \cdot 10^{-106}:\\ \;\;\;\;1 - \frac{x}{y - t} \cdot \frac{-1}{z}\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{-124}:\\ \;\;\;\;1 - \frac{x}{y \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{z \cdot \frac{t}{x}}\\ \end{array} \]

Alternative 7
Accuracy	66.5%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{-127} \lor \neg \left(z \leq 5.5 \cdot 10^{-126}\right):\\ \;\;\;\;1 - \frac{x}{t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y \cdot t}\\ \end{array} \]

Alternative 8
Accuracy	81.5%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{-23} \lor \neg \left(y \leq 8.6 \cdot 10^{-17}\right):\\ \;\;\;\;1 - \frac{x}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{t \cdot z}\\ \end{array} \]

Alternative 9
Accuracy	81.0%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-24} \lor \neg \left(y \leq 9.6 \cdot 10^{-17}\right):\\ \;\;\;\;1 - \frac{x}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{x}{z}}{t}\\ \end{array} \]

Alternative 10
Accuracy	99.1%
Cost	704

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

Alternative 11
Accuracy	98.4%
Cost	704

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

Alternative 12
Accuracy	60.9%
Cost	448

\[1 - \frac{x}{t \cdot z} \]

Reproduce

herbie shell --seed 2023272 
(FPCore (x y z t)
  :name "Data.Random.Distribution.Triangular:triangularCDF from random-fu-0.2.6.2, A"
  :precision binary64
  (- 1.0 (/ x (* (- y z) (- y t)))))