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

Average Error: 7.6 → 1.2

Time: 13.4s

Precision: binary64

Cost: 1608

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

Math

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

↓

\[\begin{array}{l} t_1 := \left(y - z\right) \cdot \left(t - z\right)\\ \mathbf{if}\;t_1 \leq 2 \cdot 10^{-161}:\\ \;\;\;\;\frac{\frac{x}{z - y}}{z - t}\\ \mathbf{elif}\;t_1 \leq 10^{+157}:\\ \;\;\;\;\frac{x}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z - t}}{z - y}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- y z) (- t z))))
   (if (<= t_1 2e-161)
     (/ (/ x (- z y)) (- z t))
     (if (<= t_1 1e+157) (/ x t_1) (/ (/ 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) {
	double t_1 = (y - z) * (t - z);
	double tmp;
	if (t_1 <= 2e-161) {
		tmp = (x / (z - y)) / (z - t);
	} else if (t_1 <= 1e+157) {
		tmp = x / t_1;
	} else {
		tmp = (x / (z - t)) / (z - y);
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (y - z) * (t - z)
    if (t_1 <= 2d-161) then
        tmp = (x / (z - y)) / (z - t)
    else if (t_1 <= 1d+157) then
        tmp = x / t_1
    else
        tmp = (x / (z - t)) / (z - y)
    end if
    code = tmp
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) {
	double t_1 = (y - z) * (t - z);
	double tmp;
	if (t_1 <= 2e-161) {
		tmp = (x / (z - y)) / (z - t);
	} else if (t_1 <= 1e+157) {
		tmp = x / t_1;
	} else {
		tmp = (x / (z - t)) / (z - y);
	}
	return tmp;
}

Python

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

↓

def code(x, y, z, t):
	t_1 = (y - z) * (t - z)
	tmp = 0
	if t_1 <= 2e-161:
		tmp = (x / (z - y)) / (z - t)
	elif t_1 <= 1e+157:
		tmp = x / t_1
	else:
		tmp = (x / (z - t)) / (z - y)
	return tmp

Julia

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

↓

function code(x, y, z, t)
	t_1 = Float64(Float64(y - z) * Float64(t - z))
	tmp = 0.0
	if (t_1 <= 2e-161)
		tmp = Float64(Float64(x / Float64(z - y)) / Float64(z - t));
	elseif (t_1 <= 1e+157)
		tmp = Float64(x / t_1);
	else
		tmp = Float64(Float64(x / Float64(z - t)) / Float64(z - y));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y, z, t)
	t_1 = (y - z) * (t - z);
	tmp = 0.0;
	if (t_1 <= 2e-161)
		tmp = (x / (z - y)) / (z - t);
	elseif (t_1 <= 1e+157)
		tmp = x / t_1;
	else
		tmp = (x / (z - t)) / (z - y);
	end
	tmp_2 = tmp;
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_] := Block[{t$95$1 = N[(N[(y - z), $MachinePrecision] * N[(t - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-161], N[(N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision] / N[(z - t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+157], N[(x / t$95$1), $MachinePrecision], N[(N[(x / N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	7.6
Target	8.2
Herbie	1.2

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 y z) (-.f64 t z)) < 2.00000000000000006e-161

   1. Initial program 7.6

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

   2. Simplified 7.6

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

      [Start] 7.6	\[ \frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \]
      sub-neg [=>] 7.6	\[ \frac{x}{\color{blue}{\left(y + \left(-z\right)\right)} \cdot \left(t - z\right)} \]
      +-commutative [=>] 7.6	\[ \frac{x}{\color{blue}{\left(\left(-z\right) + y\right)} \cdot \left(t - z\right)} \]
      neg-sub0 [=>] 7.6	\[ \frac{x}{\left(\color{blue}{\left(0 - z\right)} + y\right) \cdot \left(t - z\right)} \]
      associate-+l- [=>] 7.6	\[ \frac{x}{\color{blue}{\left(0 - \left(z - y\right)\right)} \cdot \left(t - z\right)} \]
      sub0-neg [=>] 7.6	\[ \frac{x}{\color{blue}{\left(-\left(z - y\right)\right)} \cdot \left(t - z\right)} \]
      distribute-lft-neg-out [=>] 7.6	\[ \frac{x}{\color{blue}{-\left(z - y\right) \cdot \left(t - z\right)}} \]
      distribute-rgt-neg-in [=>] 7.6	\[ \frac{x}{\color{blue}{\left(z - y\right) \cdot \left(-\left(t - z\right)\right)}} \]
      neg-sub0 [=>] 7.6	\[ \frac{x}{\left(z - y\right) \cdot \color{blue}{\left(0 - \left(t - z\right)\right)}} \]
      associate-+l- [<=] 7.6	\[ \frac{x}{\left(z - y\right) \cdot \color{blue}{\left(\left(0 - t\right) + z\right)}} \]
      neg-sub0 [<=] 7.6	\[ \frac{x}{\left(z - y\right) \cdot \left(\color{blue}{\left(-t\right)} + z\right)} \]
      +-commutative [<=] 7.6	\[ \frac{x}{\left(z - y\right) \cdot \color{blue}{\left(z + \left(-t\right)\right)}} \]
      sub-neg [<=] 7.6	\[ \frac{x}{\left(z - y\right) \cdot \color{blue}{\left(z - t\right)}} \]

   3. Taylor expanded in x around 0 7.6

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

   4. Simplified 2.7

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

      [Start] 7.6	\[ \frac{x}{\left(z - t\right) \cdot \left(z - y\right)} \]
      associate-/l/ [<=] 2.7	\[ \color{blue}{\frac{\frac{x}{z - y}}{z - t}} \]

   if 2.00000000000000006e-161 < (*.f64 (-.f64 y z) (-.f64 t z)) < 9.99999999999999983e156

   1. Initial program 0.2

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

   if 9.99999999999999983e156 < (*.f64 (-.f64 y z) (-.f64 t z))

   1. Initial program 10.7

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

   2. Simplified 0.3

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

      [Start] 10.7	\[ \frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \]
      sub-neg [=>] 10.7	\[ \frac{x}{\color{blue}{\left(y + \left(-z\right)\right)} \cdot \left(t - z\right)} \]
      +-commutative [=>] 10.7	\[ \frac{x}{\color{blue}{\left(\left(-z\right) + y\right)} \cdot \left(t - z\right)} \]
      neg-sub0 [=>] 10.7	\[ \frac{x}{\left(\color{blue}{\left(0 - z\right)} + y\right) \cdot \left(t - z\right)} \]
      associate-+l- [=>] 10.7	\[ \frac{x}{\color{blue}{\left(0 - \left(z - y\right)\right)} \cdot \left(t - z\right)} \]
      sub0-neg [=>] 10.7	\[ \frac{x}{\color{blue}{\left(-\left(z - y\right)\right)} \cdot \left(t - z\right)} \]
      distribute-lft-neg-out [=>] 10.7	\[ \frac{x}{\color{blue}{-\left(z - y\right) \cdot \left(t - z\right)}} \]
      distribute-rgt-neg-in [=>] 10.7	\[ \frac{x}{\color{blue}{\left(z - y\right) \cdot \left(-\left(t - z\right)\right)}} \]
      neg-sub0 [=>] 10.7	\[ \frac{x}{\left(z - y\right) \cdot \color{blue}{\left(0 - \left(t - z\right)\right)}} \]
      associate-+l- [<=] 10.7	\[ \frac{x}{\left(z - y\right) \cdot \color{blue}{\left(\left(0 - t\right) + z\right)}} \]
      neg-sub0 [<=] 10.7	\[ \frac{x}{\left(z - y\right) \cdot \left(\color{blue}{\left(-t\right)} + z\right)} \]
      +-commutative [<=] 10.7	\[ \frac{x}{\left(z - y\right) \cdot \color{blue}{\left(z + \left(-t\right)\right)}} \]
      sub-neg [<=] 10.7	\[ \frac{x}{\left(z - y\right) \cdot \color{blue}{\left(z - t\right)}} \]
      associate-/l/ [<=] 0.3	\[ \color{blue}{\frac{\frac{x}{z - t}}{z - y}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 1.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y - z\right) \cdot \left(t - z\right) \leq 2 \cdot 10^{-161}:\\ \;\;\;\;\frac{\frac{x}{z - y}}{z - t}\\ \mathbf{elif}\;\left(y - z\right) \cdot \left(t - z\right) \leq 10^{+157}:\\ \;\;\;\;\frac{x}{\left(y - z\right) \cdot \left(t - z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z - t}}{z - y}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.3
Cost	1609

\[\begin{array}{l} t_1 := \left(y - z\right) \cdot \left(t - z\right)\\ \mathbf{if}\;t_1 \leq 0 \lor \neg \left(t_1 \leq 2 \cdot 10^{+143}\right):\\ \;\;\;\;\frac{\frac{x}{z - t}}{z - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t_1}\\ \end{array} \]

Alternative 2
Error	5.6
Cost	1104

\[\begin{array}{l} t_1 := \frac{x}{\left(y - z\right) \cdot \left(t - z\right)}\\ \mathbf{if}\;t \leq -1.55 \cdot 10^{+149}:\\ \;\;\;\;\frac{\frac{-x}{y}}{z - t}\\ \mathbf{elif}\;t \leq -1.25 \cdot 10^{-211}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{-241}:\\ \;\;\;\;\frac{\frac{x}{z}}{z - y}\\ \mathbf{elif}\;t \leq 6 \cdot 10^{+177}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{t}}{z - y}\\ \end{array} \]

Alternative 3
Error	5.8
Cost	1104

\[\begin{array}{l} t_1 := \frac{x}{\left(y - z\right) \cdot \left(t - z\right)}\\ \mathbf{if}\;t \leq -1.55 \cdot 10^{+149}:\\ \;\;\;\;\frac{\frac{-x}{y}}{z - t}\\ \mathbf{elif}\;t \leq -1.4 \cdot 10^{-210}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.18 \cdot 10^{-240}:\\ \;\;\;\;\frac{\frac{x}{z}}{z - y}\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{+196}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{t}}{\frac{z - y}{x}}\\ \end{array} \]

Alternative 4
Error	14.1
Cost	776

\[\begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{+21}:\\ \;\;\;\;\frac{\frac{x}{z}}{z - t}\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{+55}:\\ \;\;\;\;\frac{\frac{-x}{t}}{z - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{z - y}\\ \end{array} \]

Alternative 5
Error	11.5
Cost	776

\[\begin{array}{l} \mathbf{if}\;y \leq -2050000000000:\\ \;\;\;\;\frac{\frac{-x}{y}}{z - t}\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{-160}:\\ \;\;\;\;\frac{\frac{x}{z}}{z - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{t}}{z - y}\\ \end{array} \]

Alternative 6
Error	13.7
Cost	776

\[\begin{array}{l} \mathbf{if}\;z \leq -3.7 \cdot 10^{+20}:\\ \;\;\;\;\frac{\frac{x}{z}}{z - t}\\ \mathbf{elif}\;z \leq 10^{+55}:\\ \;\;\;\;\frac{\frac{-x}{z - y}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{z - y}\\ \end{array} \]

Alternative 7
Error	17.6
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{+24} \lor \neg \left(z \leq 2.2 \cdot 10^{+79}\right):\\ \;\;\;\;\frac{\frac{x}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t \cdot \left(y - z\right)}\\ \end{array} \]

Alternative 8
Error	14.0
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+18} \lor \neg \left(z \leq 3.1 \cdot 10^{-45}\right):\\ \;\;\;\;\frac{\frac{x}{z}}{z - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t \cdot \left(y - z\right)}\\ \end{array} \]

Alternative 9
Error	14.1
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{+18}:\\ \;\;\;\;\frac{\frac{x}{z}}{z - t}\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{-41}:\\ \;\;\;\;\frac{x}{t \cdot \left(y - z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{z - y}\\ \end{array} \]

Alternative 10
Error	2.0
Cost	704

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

Alternative 11
Error	35.2
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+47} \lor \neg \left(z \leq 9.2 \cdot 10^{+54}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t \cdot y}\\ \end{array} \]

Alternative 12
Error	25.0
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{+20} \lor \neg \left(z \leq 2.5 \cdot 10^{+54}\right):\\ \;\;\;\;\frac{x}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t \cdot y}\\ \end{array} \]

Alternative 13
Error	24.1
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{+21} \lor \neg \left(z \leq 2.2 \cdot 10^{+79}\right):\\ \;\;\;\;\frac{x}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{y}\\ \end{array} \]

Alternative 14
Error	21.8
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{+20} \lor \neg \left(z \leq 2.4 \cdot 10^{+79}\right):\\ \;\;\;\;\frac{\frac{x}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{y}\\ \end{array} \]

Alternative 15
Error	35.2
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{+50}:\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{+73}:\\ \;\;\;\;\frac{x}{t \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot y}\\ \end{array} \]

Alternative 16
Error	50.7
Cost	320

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

Reproduce

herbie shell --seed 2023031 
(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))))