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

Percentage Accurate: 99.1% → 98.5%

Time: 6.7s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

(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)));
}

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

Java

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)))

Julia

function code(x, y, z, t)
	return Float64(1.0 - Float64(x / Float64(Float64(y - z) * Float64(y - t))))
end

MATLAB

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]

Initial Program: 99.1% accurate, 1.0× speedup

Math

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

FPCore

(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)));
}

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

Java

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)))

Julia

function code(x, y, z, t)
	return Float64(1.0 - Float64(x / Float64(Float64(y - z) * Float64(y - t))))
end

MATLAB

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]

Alternative 1: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ 1 - \frac{\frac{x}{y - t}}{y - z} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (- 1.0 (/ (/ x (- y t)) (- y z))))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	return 1.0 - ((x / (y - t)) / (y - z));
}

Fortran

NOTE: z and t should be sorted in increasing order before calling this 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 - t)) / (y - z))
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t) {
	return 1.0 - ((x / (y - t)) / (y - z));
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	return 1.0 - ((x / (y - t)) / (y - z))

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	return Float64(1.0 - Float64(Float64(x / Float64(y - t)) / Float64(y - z)))
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp = code(x, y, z, t)
	tmp = 1.0 - ((x / (y - t)) / (y - z));
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(1.0 - N[(N[(x / N[(y - t), $MachinePrecision]), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.6%

   \[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
2. Step-by-step derivation

   1. associate-/l/ 98.1%

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

3. Simplified 98.1%

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

4. Final simplification 98.1%

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

Alternative 2: 85.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-133} \lor \neg \left(y \leq 6 \cdot 10^{-69}\right):\\ \;\;\;\;1 - \frac{x}{y \cdot \left(y - z\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{x}{t}}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -4e-133) (not (<= y 6e-69)))
   (- 1.0 (/ x (* y (- y z))))
   (- 1.0 (/ (/ x t) z))))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -4e-133) || !(y <= 6e-69)) {
		tmp = 1.0 - (x / (y * (y - z)));
	} else {
		tmp = 1.0 - ((x / t) / z);
	}
	return tmp;
}

Fortran

NOTE: z and t should be sorted in increasing order before calling this 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) :: tmp
    if ((y <= (-4d-133)) .or. (.not. (y <= 6d-69))) then
        tmp = 1.0d0 - (x / (y * (y - z)))
    else
        tmp = 1.0d0 - ((x / t) / z)
    end if
    code = tmp
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -4e-133) || !(y <= 6e-69)) {
		tmp = 1.0 - (x / (y * (y - z)));
	} else {
		tmp = 1.0 - ((x / t) / z);
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	tmp = 0
	if (y <= -4e-133) or not (y <= 6e-69):
		tmp = 1.0 - (x / (y * (y - z)))
	else:
		tmp = 1.0 - ((x / t) / z)
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -4e-133) || !(y <= 6e-69))
		tmp = Float64(1.0 - Float64(x / Float64(y * Float64(y - z))));
	else
		tmp = Float64(1.0 - Float64(Float64(x / t) / z));
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y <= -4e-133) || ~((y <= 6e-69)))
		tmp = 1.0 - (x / (y * (y - z)));
	else
		tmp = 1.0 - ((x / t) / z);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[Or[LessEqual[y, -4e-133], N[Not[LessEqual[y, 6e-69]], $MachinePrecision]], N[(1.0 - N[(x / N[(y * N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(x / t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.0000000000000003e-133 or 5.99999999999999978e-69 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in t around 0 89.7%

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

   if -4.0000000000000003e-133 < y < 5.99999999999999978e-69

   1. Initial program 96.5%

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

   2. Taylor expanded in y around 0 79.7%

      \[\leadsto 1 - \color{blue}{\frac{x}{t \cdot z}} \]
   3. Step-by-step derivation

      1. associate-/r* 80.7%

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

   4. Simplified 80.7%

      \[\leadsto 1 - \color{blue}{\frac{\frac{x}{t}}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-133} \lor \neg \left(y \leq 6 \cdot 10^{-69}\right):\\ \;\;\;\;1 - \frac{x}{y \cdot \left(y - z\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{x}{t}}{z}\\ \end{array} \]

Alternative 3: 70.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1.42 \cdot 10^{-29} \lor \neg \left(y \leq 9 \cdot 10^{-34}\right):\\ \;\;\;\;1 - \frac{x}{y \cdot t}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{t \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -1.42e-29) (not (<= y 9e-34)))
   (- 1.0 (/ x (* y t)))
   (- 1.0 (/ x (* t z)))))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -1.42e-29) || !(y <= 9e-34)) {
		tmp = 1.0 - (x / (y * t));
	} else {
		tmp = 1.0 - (x / (t * z));
	}
	return tmp;
}

Fortran

NOTE: z and t should be sorted in increasing order before calling this 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) :: tmp
    if ((y <= (-1.42d-29)) .or. (.not. (y <= 9d-34))) then
        tmp = 1.0d0 - (x / (y * t))
    else
        tmp = 1.0d0 - (x / (t * z))
    end if
    code = tmp
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -1.42e-29) || !(y <= 9e-34)) {
		tmp = 1.0 - (x / (y * t));
	} else {
		tmp = 1.0 - (x / (t * z));
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	tmp = 0
	if (y <= -1.42e-29) or not (y <= 9e-34):
		tmp = 1.0 - (x / (y * t))
	else:
		tmp = 1.0 - (x / (t * z))
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -1.42e-29) || !(y <= 9e-34))
		tmp = Float64(1.0 - Float64(x / Float64(y * t)));
	else
		tmp = Float64(1.0 - Float64(x / Float64(t * z)));
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y <= -1.42e-29) || ~((y <= 9e-34)))
		tmp = 1.0 - (x / (y * t));
	else
		tmp = 1.0 - (x / (t * z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[Or[LessEqual[y, -1.42e-29], N[Not[LessEqual[y, 9e-34]], $MachinePrecision]], N[(1.0 - N[(x / N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(x / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.4200000000000001e-29 or 9.00000000000000085e-34 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in z around 0 96.2%

      \[\leadsto 1 - \color{blue}{\frac{x}{\left(y - t\right) \cdot y}} \]
   3. Step-by-step derivation

      1. associate-/l/ 96.2%

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

   4. Simplified 96.2%

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

   5. Taylor expanded in y around 0 75.3%

      \[\leadsto 1 - \color{blue}{-1 \cdot \frac{x}{y \cdot t}} \]
   6. Step-by-step derivation

      1. associate-*r/ 75.3%

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

      2. neg-mul-1 75.3%

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

   7. Simplified 75.3%

      \[\leadsto 1 - \color{blue}{\frac{-x}{y \cdot t}} \]
   8. Step-by-step derivation

      1. expm1-log1p-u 73.4%

         \[\leadsto 1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-x}{y \cdot t}\right)\right)} \]

      2. expm1-udef 73.4%

         \[\leadsto 1 - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{-x}{y \cdot t}\right)} - 1\right)} \]

      3. add-sqr-sqrt 42.6%

         \[\leadsto 1 - \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}}{y \cdot t}\right)} - 1\right) \]

      4. sqrt-unprod 65.5%

         \[\leadsto 1 - \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}}{y \cdot t}\right)} - 1\right) \]

      5. sqr-neg 65.5%

         \[\leadsto 1 - \left(e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{x \cdot x}}}{y \cdot t}\right)} - 1\right) \]

      6. sqrt-unprod 30.8%

         \[\leadsto 1 - \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{y \cdot t}\right)} - 1\right) \]

      7. add-sqr-sqrt 73.0%

         \[\leadsto 1 - \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{x}}{y \cdot t}\right)} - 1\right) \]

      8. *-commutative 73.0%

         \[\leadsto 1 - \left(e^{\mathsf{log1p}\left(\frac{x}{\color{blue}{t \cdot y}}\right)} - 1\right) \]

   9. Applied egg-rr 73.0%

      \[\leadsto 1 - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x}{t \cdot y}\right)} - 1\right)} \]
   10. Step-by-step derivation

       1. expm1-def 73.0%

          \[\leadsto 1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{t \cdot y}\right)\right)} \]

       2. expm1-log1p 73.5%

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

   11. Simplified 73.5%

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

   if -1.4200000000000001e-29 < y < 9.00000000000000085e-34

   1. Initial program 97.3%

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

   2. Taylor expanded in y around 0 76.0%

      \[\leadsto 1 - \color{blue}{\frac{x}{t \cdot z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.42 \cdot 10^{-29} \lor \neg \left(y \leq 9 \cdot 10^{-34}\right):\\ \;\;\;\;1 - \frac{x}{y \cdot t}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{t \cdot z}\\ \end{array} \]

Alternative 4: 81.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{-40} \lor \neg \left(y \leq 7.2 \cdot 10^{-15}\right):\\ \;\;\;\;1 - \frac{x}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{t \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -1.6e-40) (not (<= y 7.2e-15)))
   (- 1.0 (/ x (* y y)))
   (- 1.0 (/ x (* t z)))))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -1.6e-40) || !(y <= 7.2e-15)) {
		tmp = 1.0 - (x / (y * y));
	} else {
		tmp = 1.0 - (x / (t * z));
	}
	return tmp;
}

Fortran

NOTE: z and t should be sorted in increasing order before calling this 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) :: tmp
    if ((y <= (-1.6d-40)) .or. (.not. (y <= 7.2d-15))) then
        tmp = 1.0d0 - (x / (y * y))
    else
        tmp = 1.0d0 - (x / (t * z))
    end if
    code = tmp
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -1.6e-40) || !(y <= 7.2e-15)) {
		tmp = 1.0 - (x / (y * y));
	} else {
		tmp = 1.0 - (x / (t * z));
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	tmp = 0
	if (y <= -1.6e-40) or not (y <= 7.2e-15):
		tmp = 1.0 - (x / (y * y))
	else:
		tmp = 1.0 - (x / (t * z))
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -1.6e-40) || !(y <= 7.2e-15))
		tmp = Float64(1.0 - Float64(x / Float64(y * y)));
	else
		tmp = Float64(1.0 - Float64(x / Float64(t * z)));
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y <= -1.6e-40) || ~((y <= 7.2e-15)))
		tmp = 1.0 - (x / (y * y));
	else
		tmp = 1.0 - (x / (t * z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[Or[LessEqual[y, -1.6e-40], N[Not[LessEqual[y, 7.2e-15]], $MachinePrecision]], N[(1.0 - N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(x / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.60000000000000001e-40 or 7.2000000000000002e-15 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 91.6%

      \[\leadsto 1 - \color{blue}{\frac{x}{{y}^{2}}} \]
   3. Step-by-step derivation

      1. unpow2 91.6%

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

   4. Simplified 91.6%

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

   if -1.60000000000000001e-40 < y < 7.2000000000000002e-15

   1. Initial program 97.3%

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

   2. Taylor expanded in y around 0 75.2%

      \[\leadsto 1 - \color{blue}{\frac{x}{t \cdot z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{-40} \lor \neg \left(y \leq 7.2 \cdot 10^{-15}\right):\\ \;\;\;\;1 - \frac{x}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{t \cdot z}\\ \end{array} \]

Alternative 5: 81.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -9.6 \cdot 10^{-40} \lor \neg \left(y \leq 3.4 \cdot 10^{-15}\right):\\ \;\;\;\;1 - \frac{x}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{x}{t}}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -9.6e-40) (not (<= y 3.4e-15)))
   (- 1.0 (/ x (* y y)))
   (- 1.0 (/ (/ x t) z))))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -9.6e-40) || !(y <= 3.4e-15)) {
		tmp = 1.0 - (x / (y * y));
	} else {
		tmp = 1.0 - ((x / t) / z);
	}
	return tmp;
}

Fortran

NOTE: z and t should be sorted in increasing order before calling this 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) :: tmp
    if ((y <= (-9.6d-40)) .or. (.not. (y <= 3.4d-15))) then
        tmp = 1.0d0 - (x / (y * y))
    else
        tmp = 1.0d0 - ((x / t) / z)
    end if
    code = tmp
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -9.6e-40) || !(y <= 3.4e-15)) {
		tmp = 1.0 - (x / (y * y));
	} else {
		tmp = 1.0 - ((x / t) / z);
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	tmp = 0
	if (y <= -9.6e-40) or not (y <= 3.4e-15):
		tmp = 1.0 - (x / (y * y))
	else:
		tmp = 1.0 - ((x / t) / z)
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -9.6e-40) || !(y <= 3.4e-15))
		tmp = Float64(1.0 - Float64(x / Float64(y * y)));
	else
		tmp = Float64(1.0 - Float64(Float64(x / t) / z));
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y <= -9.6e-40) || ~((y <= 3.4e-15)))
		tmp = 1.0 - (x / (y * y));
	else
		tmp = 1.0 - ((x / t) / z);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[Or[LessEqual[y, -9.6e-40], N[Not[LessEqual[y, 3.4e-15]], $MachinePrecision]], N[(1.0 - N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(x / t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -9.59999999999999965e-40 or 3.4e-15 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 91.6%

      \[\leadsto 1 - \color{blue}{\frac{x}{{y}^{2}}} \]
   3. Step-by-step derivation

      1. unpow2 91.6%

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

   4. Simplified 91.6%

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

   if -9.59999999999999965e-40 < y < 3.4e-15

   1. Initial program 97.3%

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

   2. Taylor expanded in y around 0 75.2%

      \[\leadsto 1 - \color{blue}{\frac{x}{t \cdot z}} \]
   3. Step-by-step derivation

      1. associate-/r* 76.0%

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

   4. Simplified 76.0%

      \[\leadsto 1 - \color{blue}{\frac{\frac{x}{t}}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.6 \cdot 10^{-40} \lor \neg \left(y \leq 3.4 \cdot 10^{-15}\right):\\ \;\;\;\;1 - \frac{x}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{x}{t}}{z}\\ \end{array} \]

Alternative 6: 81.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{-40}:\\ \;\;\;\;1 - \frac{\frac{x}{y}}{y}\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-13}:\\ \;\;\;\;1 - \frac{\frac{x}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y -1.8e-40)
   (- 1.0 (/ (/ x y) y))
   (if (<= y 9.5e-13) (- 1.0 (/ (/ x t) z)) (- 1.0 (/ x (* y y))))))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.8e-40) {
		tmp = 1.0 - ((x / y) / y);
	} else if (y <= 9.5e-13) {
		tmp = 1.0 - ((x / t) / z);
	} else {
		tmp = 1.0 - (x / (y * y));
	}
	return tmp;
}

Fortran

NOTE: z and t should be sorted in increasing order before calling this 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) :: tmp
    if (y <= (-1.8d-40)) then
        tmp = 1.0d0 - ((x / y) / y)
    else if (y <= 9.5d-13) then
        tmp = 1.0d0 - ((x / t) / z)
    else
        tmp = 1.0d0 - (x / (y * y))
    end if
    code = tmp
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.8e-40) {
		tmp = 1.0 - ((x / y) / y);
	} else if (y <= 9.5e-13) {
		tmp = 1.0 - ((x / t) / z);
	} else {
		tmp = 1.0 - (x / (y * y));
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= -1.8e-40:
		tmp = 1.0 - ((x / y) / y)
	elif y <= 9.5e-13:
		tmp = 1.0 - ((x / t) / z)
	else:
		tmp = 1.0 - (x / (y * y))
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= -1.8e-40)
		tmp = Float64(1.0 - Float64(Float64(x / y) / y));
	elseif (y <= 9.5e-13)
		tmp = Float64(1.0 - Float64(Float64(x / t) / z));
	else
		tmp = Float64(1.0 - Float64(x / Float64(y * y)));
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -1.8e-40)
		tmp = 1.0 - ((x / y) / y);
	elseif (y <= 9.5e-13)
		tmp = 1.0 - ((x / t) / z);
	else
		tmp = 1.0 - (x / (y * y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, -1.8e-40], N[(1.0 - N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9.5e-13], N[(1.0 - N[(N[(x / t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.8e-40

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 92.4%

      \[\leadsto 1 - \color{blue}{\frac{x}{{y}^{2}}} \]
   3. Step-by-step derivation

      1. unpow2 92.4%

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

      2. associate-/r* 92.5%

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

   4. Simplified 92.5%

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

   if -1.8e-40 < y < 9.49999999999999991e-13

   1. Initial program 97.3%

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

   2. Taylor expanded in y around 0 75.2%

      \[\leadsto 1 - \color{blue}{\frac{x}{t \cdot z}} \]
   3. Step-by-step derivation

      1. associate-/r* 76.0%

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

   4. Simplified 76.0%

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

   if 9.49999999999999991e-13 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 90.8%

      \[\leadsto 1 - \color{blue}{\frac{x}{{y}^{2}}} \]
   3. Step-by-step derivation

      1. unpow2 90.8%

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

   4. Simplified 90.8%

      \[\leadsto 1 - \color{blue}{\frac{x}{y \cdot y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{-40}:\\ \;\;\;\;1 - \frac{\frac{x}{y}}{y}\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-13}:\\ \;\;\;\;1 - \frac{\frac{x}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y \cdot y}\\ \end{array} \]

Alternative 7: 90.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 2.3 \cdot 10^{-112}:\\ \;\;\;\;1 - \frac{x}{y \cdot \left(y - z\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{x}{t}}{y - z}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= t 2.3e-112) (- 1.0 (/ x (* y (- y z)))) (+ 1.0 (/ (/ x t) (- y z)))))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 2.3e-112) {
		tmp = 1.0 - (x / (y * (y - z)));
	} else {
		tmp = 1.0 + ((x / t) / (y - z));
	}
	return tmp;
}

Fortran

NOTE: z and t should be sorted in increasing order before calling this 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) :: tmp
    if (t <= 2.3d-112) then
        tmp = 1.0d0 - (x / (y * (y - z)))
    else
        tmp = 1.0d0 + ((x / t) / (y - z))
    end if
    code = tmp
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 2.3e-112) {
		tmp = 1.0 - (x / (y * (y - z)));
	} else {
		tmp = 1.0 + ((x / t) / (y - z));
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	tmp = 0
	if t <= 2.3e-112:
		tmp = 1.0 - (x / (y * (y - z)))
	else:
		tmp = 1.0 + ((x / t) / (y - z))
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (t <= 2.3e-112)
		tmp = Float64(1.0 - Float64(x / Float64(y * Float64(y - z))));
	else
		tmp = Float64(1.0 + Float64(Float64(x / t) / Float64(y - z)));
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 2.3e-112)
		tmp = 1.0 - (x / (y * (y - z)));
	else
		tmp = 1.0 + ((x / t) / (y - z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[t, 2.3e-112], N[(1.0 - N[(x / N[(y * N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(x / t), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.29999999999999991e-112

   1. Initial program 97.9%

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

   2. Taylor expanded in t around 0 71.8%

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

   if 2.29999999999999991e-112 < t

   1. Initial program 100.0%

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

   2. Taylor expanded in t around inf 98.3%

      \[\leadsto 1 - \color{blue}{-1 \cdot \frac{x}{t \cdot \left(y - z\right)}} \]
   3. Step-by-step derivation

      1. mul-1-neg 98.3%

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

      2. distribute-frac-neg 98.3%

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

      3. associate-/r* 98.3%

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

   4. Simplified 98.3%

      \[\leadsto 1 - \color{blue}{\frac{\frac{-x}{t}}{y - z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.3 \cdot 10^{-112}:\\ \;\;\;\;1 - \frac{x}{y \cdot \left(y - z\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{x}{t}}{y - z}\\ \end{array} \]

Alternative 8: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ 1 - \frac{x}{\left(y - t\right) \cdot \left(y - z\right)} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (- 1.0 (/ x (* (- y t) (- y z)))))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	return 1.0 - (x / ((y - t) * (y - z)));
}

Fortran

NOTE: z and t should be sorted in increasing order before calling this 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 - t) * (y - z)))
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t) {
	return 1.0 - (x / ((y - t) * (y - z)));
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	return 1.0 - (x / ((y - t) * (y - z)))

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	return Float64(1.0 - Float64(x / Float64(Float64(y - t) * Float64(y - z))))
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp = code(x, y, z, t)
	tmp = 1.0 - (x / ((y - t) * (y - z)));
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(1.0 - N[(x / N[(N[(y - t), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.6%

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

2. Final simplification 98.6%

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

Alternative 9: 52.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ 1 - \frac{x}{y \cdot t} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (- 1.0 (/ x (* y t))))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	return 1.0 - (x / (y * t));
}

Fortran

NOTE: z and t should be sorted in increasing order before calling this 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 * t))
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t) {
	return 1.0 - (x / (y * t));
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	return 1.0 - (x / (y * t))

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	return Float64(1.0 - Float64(x / Float64(y * t)))
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp = code(x, y, z, t)
	tmp = 1.0 - (x / (y * t));
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(1.0 - N[(x / N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.6%

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

2. Taylor expanded in z around 0 71.2%

   \[\leadsto 1 - \color{blue}{\frac{x}{\left(y - t\right) \cdot y}} \]
3. Step-by-step derivation

   1. associate-/l/ 72.2%

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

4. Simplified 72.2%

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

5. Taylor expanded in y around 0 58.5%

   \[\leadsto 1 - \color{blue}{-1 \cdot \frac{x}{y \cdot t}} \]
6. Step-by-step derivation

   1. associate-*r/ 58.5%

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

   2. neg-mul-1 58.5%

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

7. Simplified 58.5%

   \[\leadsto 1 - \color{blue}{\frac{-x}{y \cdot t}} \]
8. Step-by-step derivation

   1. expm1-log1p-u 54.2%

      \[\leadsto 1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-x}{y \cdot t}\right)\right)} \]

   2. expm1-udef 54.2%

      \[\leadsto 1 - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{-x}{y \cdot t}\right)} - 1\right)} \]

   3. add-sqr-sqrt 27.7%

      \[\leadsto 1 - \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}}{y \cdot t}\right)} - 1\right) \]

   4. sqrt-unprod 53.8%

      \[\leadsto 1 - \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}}{y \cdot t}\right)} - 1\right) \]

   5. sqr-neg 53.8%

      \[\leadsto 1 - \left(e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{x \cdot x}}}{y \cdot t}\right)} - 1\right) \]

   6. sqrt-unprod 28.3%

      \[\leadsto 1 - \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{y \cdot t}\right)} - 1\right) \]

   7. add-sqr-sqrt 54.7%

      \[\leadsto 1 - \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{x}}{y \cdot t}\right)} - 1\right) \]

   8. *-commutative 54.7%

      \[\leadsto 1 - \left(e^{\mathsf{log1p}\left(\frac{x}{\color{blue}{t \cdot y}}\right)} - 1\right) \]

9. Applied egg-rr 54.7%

   \[\leadsto 1 - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x}{t \cdot y}\right)} - 1\right)} \]
10. Step-by-step derivation

    1. expm1-def 54.7%

       \[\leadsto 1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{t \cdot y}\right)\right)} \]

    2. expm1-log1p 56.6%

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

11. Simplified 56.6%

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

12. Final simplification 56.6%

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

Reproduce

herbie shell --seed 2023228 
(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)))))