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

Average Error: 1.15% → 1.75%

Time: 9.9s

Precision: binary64

Cost: 704

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

Math

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

↓

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

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 z)) (- y t))))

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 - z)) / (y - t));
}

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 - z)) / (y - t))
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 - z)) / (y - t));
}

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 - z)) / (y - t))

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 - z)) / Float64(y - t)))
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 - z)) / (y - t));
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 - z), $MachinePrecision]), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 1.15

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

2. Simplified 1.75

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

   [Start] 1.15	\[ 1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
   associate-/r* [=>] 1.75	\[ 1 - \color{blue}{\frac{\frac{x}{y - z}}{y - t}} \]

3. Final simplification 1.75

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

Alternatives

Alternative 1
Error	22.64%
Cost	1240

\[\begin{array}{l} t_1 := 1 + \frac{x}{y \cdot z}\\ t_2 := 1 - \frac{\frac{x}{t}}{z}\\ \mathbf{if}\;t \leq -1.5 \cdot 10^{-80}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -3.1 \cdot 10^{-307}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2 \cdot 10^{-234}:\\ \;\;\;\;1 - \frac{\frac{x}{y}}{y}\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{-190}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{-24}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 3 \cdot 10^{+59}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 2
Error	22.62%
Cost	1240

\[\begin{array}{l} t_1 := 1 + \frac{x}{y \cdot z}\\ \mathbf{if}\;t \leq -1.1 \cdot 10^{-80}:\\ \;\;\;\;1 - \frac{\frac{x}{t}}{z}\\ \mathbf{elif}\;t \leq 3 \cdot 10^{-308}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{-232}:\\ \;\;\;\;1 - \frac{\frac{x}{y}}{y}\\ \mathbf{elif}\;t \leq 6.6 \cdot 10^{-189}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{-24}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{+59}:\\ \;\;\;\;1 - \frac{\frac{x}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 3
Error	16.8%
Cost	844

\[\begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-80}:\\ \;\;\;\;1 - \frac{\frac{x}{t}}{z}\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{-24}:\\ \;\;\;\;1 - \frac{x}{y \cdot \left(y - z\right)}\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{+59}:\\ \;\;\;\;1 - \frac{\frac{x}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4
Error	8.41%
Cost	841

\[\begin{array}{l} \mathbf{if}\;t \leq -1.8 \cdot 10^{-80} \lor \neg \left(t \leq 3.2 \cdot 10^{-29}\right):\\ \;\;\;\;1 - \frac{\frac{x}{t}}{z - y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y \cdot \left(y - z\right)}\\ \end{array} \]

Alternative 5
Error	7.83%
Cost	841

\[\begin{array}{l} \mathbf{if}\;t \leq -1.1 \cdot 10^{-80} \lor \neg \left(t \leq 3 \cdot 10^{-33}\right):\\ \;\;\;\;1 - \frac{\frac{x}{t}}{z - y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{x}{y - z}}{y}\\ \end{array} \]

Alternative 6
Error	6.33%
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -8.6 \cdot 10^{-38}:\\ \;\;\;\;1 + \frac{x}{z \cdot \left(y - t\right)}\\ \mathbf{elif}\;z \leq 5.7 \cdot 10^{-145}:\\ \;\;\;\;1 - \frac{\frac{x}{y - t}}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{x}{t}}{z - y}\\ \end{array} \]

Alternative 7
Error	15.11%
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -2.55 \cdot 10^{-85}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{-113}:\\ \;\;\;\;1 - \frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8
Error	1.15%
Cost	704

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

Alternative 9
Error	21.05%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023090 
(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)))))