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

Average Accuracy: 88.2% → 96.6%

Time: 12.2s

Precision: binary64

Cost: 576

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

Math

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

↓

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

FPCore

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

↓

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

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) {
	return (x / (z - t)) / (z - 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 = x / ((y - z) * (t - z))
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 = (x / (z - t)) / (z - y)
end function

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) {
	return (x / (z - t)) / (z - y);
}

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

function tmp = code(x, y, z, t)
	tmp = (x / (z - t)) / (z - y);
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_] := N[(N[(x / N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]

Target

Original	88.2%
Target	87.2%
Herbie	96.6%

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

Derivation

1. Initial program 88.2%

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

2. Simplified 96.6%

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

   [Start] 88.2	\[ \frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \]
   sub-neg [=>] 88.2	\[ \frac{x}{\color{blue}{\left(y + \left(-z\right)\right)} \cdot \left(t - z\right)} \]
   +-commutative [=>] 88.2	\[ \frac{x}{\color{blue}{\left(\left(-z\right) + y\right)} \cdot \left(t - z\right)} \]
   neg-sub0 [=>] 88.2	\[ \frac{x}{\left(\color{blue}{\left(0 - z\right)} + y\right) \cdot \left(t - z\right)} \]
   associate-+l- [=>] 88.2	\[ \frac{x}{\color{blue}{\left(0 - \left(z - y\right)\right)} \cdot \left(t - z\right)} \]
   sub0-neg [=>] 88.2	\[ \frac{x}{\color{blue}{\left(-\left(z - y\right)\right)} \cdot \left(t - z\right)} \]
   distribute-lft-neg-out [=>] 88.2	\[ \frac{x}{\color{blue}{-\left(z - y\right) \cdot \left(t - z\right)}} \]
   distribute-rgt-neg-in [=>] 88.2	\[ \frac{x}{\color{blue}{\left(z - y\right) \cdot \left(-\left(t - z\right)\right)}} \]
   neg-sub0 [=>] 88.2	\[ \frac{x}{\left(z - y\right) \cdot \color{blue}{\left(0 - \left(t - z\right)\right)}} \]
   associate-+l- [<=] 88.2	\[ \frac{x}{\left(z - y\right) \cdot \color{blue}{\left(\left(0 - t\right) + z\right)}} \]
   neg-sub0 [<=] 88.2	\[ \frac{x}{\left(z - y\right) \cdot \left(\color{blue}{\left(-t\right)} + z\right)} \]
   +-commutative [<=] 88.2	\[ \frac{x}{\left(z - y\right) \cdot \color{blue}{\left(z + \left(-t\right)\right)}} \]
   sub-neg [<=] 88.2	\[ \frac{x}{\left(z - y\right) \cdot \color{blue}{\left(z - t\right)}} \]
   associate-/l/ [<=] 96.6	\[ \color{blue}{\frac{\frac{x}{z - t}}{z - y}} \]

3. Final simplification 96.6%

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

Alternatives

Alternative 1
Accuracy	91.8%
Cost	1104

\[\begin{array}{l} t_1 := \frac{x}{\left(y - z\right) \cdot \left(t - z\right)}\\ \mathbf{if}\;z \leq -1.3 \cdot 10^{+131}:\\ \;\;\;\;\frac{\frac{x}{z}}{z - t}\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{-306}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{-212}:\\ \;\;\;\;\frac{\frac{x}{t}}{y - z}\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{+43}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z - y} \cdot \frac{1}{z}\\ \end{array} \]

Alternative 2
Accuracy	68.1%
Cost	844

\[\begin{array}{l} \mathbf{if}\;t \leq -6 \cdot 10^{-161}:\\ \;\;\;\;\frac{\frac{x}{t}}{y}\\ \mathbf{elif}\;t \leq 5.3 \cdot 10^{-204}:\\ \;\;\;\;\frac{\frac{-x}{z}}{y}\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{-68}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t \cdot \left(y - z\right)}\\ \end{array} \]

Alternative 3
Accuracy	76.8%
Cost	713

\[\begin{array}{l} \mathbf{if}\;t \leq -6 \cdot 10^{-161} \lor \neg \left(t \leq 7.5 \cdot 10^{-68}\right):\\ \;\;\;\;\frac{\frac{x}{t}}{y - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot \left(z - y\right)}\\ \end{array} \]

Alternative 4
Accuracy	78.7%
Cost	713

\[\begin{array}{l} \mathbf{if}\;t \leq -6 \cdot 10^{-161} \lor \neg \left(t \leq 2 \cdot 10^{-64}\right):\\ \;\;\;\;\frac{\frac{x}{t}}{y - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{z - y}\\ \end{array} \]

Alternative 5
Accuracy	66.3%
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -2000000000:\\ \;\;\;\;\frac{\frac{x}{z}}{z}\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{+42}:\\ \;\;\;\;\frac{\frac{x}{t}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{1}{z}\\ \end{array} \]

Alternative 6
Accuracy	74.8%
Cost	712

\[\begin{array}{l} \mathbf{if}\;t \leq -6 \cdot 10^{-161}:\\ \;\;\;\;\frac{\frac{x}{t}}{y}\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{-69}:\\ \;\;\;\;\frac{x}{z \cdot \left(z - y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t \cdot \left(y - z\right)}\\ \end{array} \]

Alternative 7
Accuracy	44.3%
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{+144} \lor \neg \left(z \leq 2.8 \cdot 10^{+95}\right):\\ \;\;\;\;\frac{x}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t \cdot y}\\ \end{array} \]

Alternative 8
Accuracy	61.0%
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -216000 \lor \neg \left(z \leq 1.12 \cdot 10^{+14}\right):\\ \;\;\;\;\frac{x}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t \cdot y}\\ \end{array} \]

Alternative 9
Accuracy	62.3%
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -23000000000 \lor \neg \left(z \leq 2.65 \cdot 10^{+42}\right):\\ \;\;\;\;\frac{x}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{y}\\ \end{array} \]

Alternative 10
Accuracy	66.4%
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -27000000000 \lor \neg \left(z \leq 1.25 \cdot 10^{+41}\right):\\ \;\;\;\;\frac{\frac{x}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{y}\\ \end{array} \]

Alternative 11
Accuracy	37.4%
Cost	320

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

Reproduce

herbie shell --seed 2023138 
(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))))