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

Average Error: 0.6 → 1.0

Time: 11.6s

Precision: binary64

Cost: 704

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

Math

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

↓

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

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)) (- t y))))

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)) / (t - y));
}

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

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)) / (t - y))

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

Derivation

1. Initial program 0.6

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

2. Simplified 1.0

   \[\leadsto \color{blue}{1 + \frac{\frac{x}{y - z}}{t - y}} \]
   Proof
   (+.f64 1 (/.f64 (/.f64 x (-.f64 y z)) (-.f64 t y))): 0 points increase in error, 0 points decrease in error
   (+.f64 1 (/.f64 (/.f64 x (-.f64 y z)) (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (-.f64 t y)))))): 0 points increase in error, 0 points decrease in error
   (+.f64 1 (/.f64 (/.f64 x (-.f64 y z)) (neg.f64 (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 t y)))))): 0 points increase in error, 0 points decrease in error
   (+.f64 1 (/.f64 (/.f64 x (-.f64 y z)) (neg.f64 (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 0 t) y))))): 0 points increase in error, 0 points decrease in error
   (+.f64 1 (/.f64 (/.f64 x (-.f64 y z)) (neg.f64 (+.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 t)) y)))): 0 points increase in error, 0 points decrease in error
   (+.f64 1 (/.f64 (/.f64 x (-.f64 y z)) (neg.f64 (Rewrite<= +-commutative_binary64 (+.f64 y (neg.f64 t)))))): 0 points increase in error, 0 points decrease in error
   (+.f64 1 (/.f64 (/.f64 x (-.f64 y z)) (neg.f64 (Rewrite<= sub-neg_binary64 (-.f64 y t))))): 0 points increase in error, 0 points decrease in error
   (+.f64 1 (/.f64 (/.f64 x (-.f64 y z)) (Rewrite=> neg-mul-1_binary64 (*.f64 -1 (-.f64 y t))))): 0 points increase in error, 0 points decrease in error
   (+.f64 1 (/.f64 (/.f64 x (-.f64 y z)) (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 y t) -1)))): 0 points increase in error, 0 points decrease in error
   (+.f64 1 (Rewrite<= *-lft-identity_binary64 (*.f64 1 (/.f64 (/.f64 x (-.f64 y z)) (*.f64 (-.f64 y t) -1))))): 0 points increase in error, 0 points decrease in error
   (+.f64 1 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 1 (/.f64 x (-.f64 y z))) (*.f64 (-.f64 y t) -1)))): 0 points increase in error, 0 points decrease in error
   (+.f64 1 (/.f64 (*.f64 1 (/.f64 x (-.f64 y z))) (Rewrite=> *-commutative_binary64 (*.f64 -1 (-.f64 y t))))): 0 points increase in error, 0 points decrease in error
   (+.f64 1 (Rewrite=> times-frac_binary64 (*.f64 (/.f64 1 -1) (/.f64 (/.f64 x (-.f64 y z)) (-.f64 y t))))): 0 points increase in error, 0 points decrease in error
   (+.f64 1 (*.f64 (Rewrite=> metadata-eval -1) (/.f64 (/.f64 x (-.f64 y z)) (-.f64 y t)))): 0 points increase in error, 0 points decrease in error
   (+.f64 1 (*.f64 -1 (Rewrite<= associate-/r*_binary64 (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t)))))): 12 points increase in error, 9 points decrease in error
   (+.f64 1 (Rewrite<= neg-mul-1_binary64 (neg.f64 (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t)))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= sub-neg_binary64 (-.f64 1 (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))))): 0 points increase in error, 0 points decrease in error

3. Final simplification 1.0

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

Alternatives

Alternative 1
Error	16.5
Cost	1372

\[\begin{array}{l} t_1 := 1 + \frac{x}{y \cdot t}\\ t_2 := 1 - \frac{\frac{x}{t}}{z}\\ \mathbf{if}\;t \leq -1.95177832248065 \cdot 10^{+31}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-305}:\\ \;\;\;\;1 + \frac{x}{y \cdot z}\\ \mathbf{elif}\;t \leq 1.0681571990490262 \cdot 10^{-146}:\\ \;\;\;\;1 - \frac{\frac{x}{y}}{y}\\ \mathbf{elif}\;t \leq 1.1063921579305679 \cdot 10^{-125}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 1.5388256252053993 \cdot 10^{-38}:\\ \;\;\;\;1 - \frac{x}{y \cdot y}\\ \mathbf{elif}\;t \leq 9.001297347930891 \cdot 10^{+127}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.976186754053298 \cdot 10^{+218}:\\ \;\;\;\;1 - \frac{\frac{x}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	12.3
Cost	976

\[\begin{array}{l} t_1 := 1 - \frac{\frac{x}{y}}{y}\\ \mathbf{if}\;y \leq -2.81533444712011 \cdot 10^{-37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 10^{-157}:\\ \;\;\;\;1 - \frac{\frac{x}{z}}{t}\\ \mathbf{elif}\;y \leq 10^{-80}:\\ \;\;\;\;1 + \frac{x}{y \cdot t}\\ \mathbf{elif}\;y \leq 1.7711442963601074 \cdot 10^{-48}:\\ \;\;\;\;1 - \frac{\frac{x}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	10.8
Cost	976

\[\begin{array}{l} \mathbf{if}\;t \leq -1.95177832248065 \cdot 10^{+31}:\\ \;\;\;\;1 - \frac{\frac{x}{t}}{z}\\ \mathbf{elif}\;t \leq 6.974251998134922 \cdot 10^{-50}:\\ \;\;\;\;1 - \frac{x}{y \cdot \left(y - z\right)}\\ \mathbf{elif}\;t \leq 9.001297347930891 \cdot 10^{+127}:\\ \;\;\;\;1 - \frac{\frac{x}{y - t}}{y}\\ \mathbf{elif}\;t \leq 2.976186754053298 \cdot 10^{+218}:\\ \;\;\;\;1 - \frac{\frac{x}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y \cdot t}\\ \end{array} \]

Alternative 4
Error	9.6
Cost	972

\[\begin{array}{l} \mathbf{if}\;z \leq -1.3133227694107761 \cdot 10^{+258}:\\ \;\;\;\;1 - \frac{\frac{x}{t}}{z}\\ \mathbf{elif}\;z \leq -2.8156376890390167 \cdot 10^{-105}:\\ \;\;\;\;1 - \frac{x}{y \cdot \left(y - z\right)}\\ \mathbf{elif}\;z \leq 1.5607187827801354 \cdot 10^{-156}:\\ \;\;\;\;1 - \frac{\frac{x}{y - t}}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{x}{y - z}}{t}\\ \end{array} \]

Alternative 5
Error	9.5
Cost	840

\[\begin{array}{l} t_1 := 1 - \frac{\frac{x}{y - t}}{y}\\ \mathbf{if}\;y \leq -2.81533444712011 \cdot 10^{-37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 10^{-157}:\\ \;\;\;\;1 - \frac{\frac{x}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	5.6
Cost	840

\[\begin{array}{l} t_1 := \frac{x}{y - z}\\ t_2 := 1 + \frac{t_1}{t}\\ \mathbf{if}\;t \leq -4.188286125881617 \cdot 10^{-40}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 1.5388256252053993 \cdot 10^{-38}:\\ \;\;\;\;1 - \frac{t_1}{y}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Error	4.2
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -1.296468876758125 \cdot 10^{-81}:\\ \;\;\;\;1 + \frac{x}{z \cdot \left(y - t\right)}\\ \mathbf{elif}\;z \leq 1.5607187827801354 \cdot 10^{-156}:\\ \;\;\;\;1 - \frac{\frac{x}{y - t}}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{x}{y - z}}{t}\\ \end{array} \]

Alternative 8
Error	19.2
Cost	712

\[\begin{array}{l} t_1 := 1 + \frac{x}{y \cdot t}\\ \mathbf{if}\;y \leq -2.81533444712011 \cdot 10^{-37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 10^{-157}:\\ \;\;\;\;1 - \frac{\frac{x}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	1.0
Cost	704

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

Alternative 10
Error	25.7
Cost	448

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

Alternative 11
Error	25.7
Cost	448

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

Reproduce

herbie shell --seed 2022311 
(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)))))