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

Average Error: 0.5 → 0.6

Time: 10.4s

Precision: binary64

Cost: 832

Math

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

↓

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

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 (/ (/ -1.0 (- y t)) (- 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 * ((-1.0 / (y - t)) / (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 * (((-1.0d0) / (y - t)) / (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 * ((-1.0 / (y - t)) / (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 * ((-1.0 / (y - t)) / (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(x * Float64(Float64(-1.0 / Float64(y - t)) / 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 * ((-1.0 / (y - t)) / (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[(x * N[(N[(-1.0 / N[(y - t), $MachinePrecision]), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.5

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

2. Simplified 1.1

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

   [Start] 0.5	\[ 1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
   associate-/r* [=>] 1.1	\[ 1 - \color{blue}{\frac{\frac{x}{y - z}}{y - t}} \]

3. Applied egg-rr 0.6

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

4. Final simplification 0.6

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

Alternatives

Alternative 1
Error	11.7
Cost	1236

\[\begin{array}{l} t_1 := 1 - \frac{\frac{x}{y}}{y}\\ t_2 := 1 - \frac{x}{z \cdot \left(t - y\right)}\\ \mathbf{if}\;z \leq -3.1 \cdot 10^{-13}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -8 \cdot 10^{-69}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -2 \cdot 10^{-94}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -3 \cdot 10^{-219}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq -3.5 \cdot 10^{-302}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{x}{t}}{z - y}\\ \end{array} \]

Alternative 2
Error	8.1
Cost	968

\[\begin{array}{l} \mathbf{if}\;z \leq -1.9:\\ \;\;\;\;1 - \frac{\frac{x}{z}}{t - y}\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-188}:\\ \;\;\;\;1 - \frac{\frac{x}{y - t}}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \frac{1}{t \cdot \left(y - z\right)}\\ \end{array} \]

Alternative 3
Error	9.6
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{-131}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{-54}:\\ \;\;\;\;1 + x \cdot \frac{\frac{-1}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4
Error	7.6
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq -2.6:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{+22}:\\ \;\;\;\;1 - \frac{x}{z \cdot \left(t - y\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5
Error	8.8
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -22:\\ \;\;\;\;1 - \frac{x}{z \cdot \left(t - y\right)}\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{-188}:\\ \;\;\;\;1 - \frac{\frac{x}{y}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{x}{t}}{z - y}\\ \end{array} \]

Alternative 6
Error	8.9
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -3000:\\ \;\;\;\;1 - \frac{\frac{x}{z}}{t - y}\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{-188}:\\ \;\;\;\;1 - \frac{\frac{x}{y}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{x}{t}}{z - y}\\ \end{array} \]

Alternative 7
Error	8.4
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -7.8:\\ \;\;\;\;1 - \frac{\frac{x}{z}}{t - y}\\ \mathbf{elif}\;z \leq 7.6 \cdot 10^{-188}:\\ \;\;\;\;1 - \frac{\frac{x}{y - t}}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{x}{t}}{z - y}\\ \end{array} \]

Alternative 8
Error	0.6
Cost	832

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

Alternative 9
Error	9.6
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{-131}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 9.6 \cdot 10^{-54}:\\ \;\;\;\;1 - \frac{x}{t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 10
Error	0.5
Cost	704

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

Alternative 11
Error	13.1
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023060 
(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)))))