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

Percentage Accurate: 99.2% → 99.2%

Time: 6.7s

Alternatives: 7

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: 99.2% 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: 99.2% 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]

Derivation

1. Initial program 99.2%

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

2. Final simplification 99.2%

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

Alternative 2: 86.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.1 \cdot 10^{-31}:\\ \;\;\;\;1 - \frac{x}{z \cdot t}\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{-48}:\\ \;\;\;\;1 + \frac{-1}{y \cdot \frac{y - z}{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{x}{t}}{y - z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -3.1e-31)
   (- 1.0 (/ x (* z t)))
   (if (<= t 3.1e-48)
     (+ 1.0 (/ -1.0 (* y (/ (- y z) x))))
     (+ 1.0 (/ (/ x t) (- y z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -3.1e-31) {
		tmp = 1.0 - (x / (z * t));
	} else if (t <= 3.1e-48) {
		tmp = 1.0 + (-1.0 / (y * ((y - z) / x)));
	} else {
		tmp = 1.0 + ((x / t) / (y - z));
	}
	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
    real(8) :: tmp
    if (t <= (-3.1d-31)) then
        tmp = 1.0d0 - (x / (z * t))
    else if (t <= 3.1d-48) then
        tmp = 1.0d0 + ((-1.0d0) / (y * ((y - z) / x)))
    else
        tmp = 1.0d0 + ((x / t) / (y - z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -3.1e-31) {
		tmp = 1.0 - (x / (z * t));
	} else if (t <= 3.1e-48) {
		tmp = 1.0 + (-1.0 / (y * ((y - z) / x)));
	} else {
		tmp = 1.0 + ((x / t) / (y - z));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -3.1e-31:
		tmp = 1.0 - (x / (z * t))
	elif t <= 3.1e-48:
		tmp = 1.0 + (-1.0 / (y * ((y - z) / x)))
	else:
		tmp = 1.0 + ((x / t) / (y - z))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -3.1e-31)
		tmp = Float64(1.0 - Float64(x / Float64(z * t)));
	elseif (t <= 3.1e-48)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(y * Float64(Float64(y - z) / x))));
	else
		tmp = Float64(1.0 + Float64(Float64(x / t) / Float64(y - z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -3.1e-31)
		tmp = 1.0 - (x / (z * t));
	elseif (t <= 3.1e-48)
		tmp = 1.0 + (-1.0 / (y * ((y - z) / x)));
	else
		tmp = 1.0 + ((x / t) / (y - z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -3.1e-31], N[(1.0 - N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.1e-48], N[(1.0 + N[(-1.0 / N[(y * N[(N[(y - z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(x / t), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -3.1e-31

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 79.6%

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

   if -3.1e-31 < t < 3.10000000000000016e-48

   1. Initial program 98.0%

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

   2. Taylor expanded in t around 0 90.7%

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

      1. clear-num 90.6%

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

      2. inv-pow 90.6%

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

      3. *-commutative 90.6%

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

      4. associate-/l* 90.6%

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

   4. Applied egg-rr 90.6%

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

      1. unpow-1 90.6%

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

      2. associate-/r/ 91.6%

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

   6. Simplified 91.6%

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

   if 3.10000000000000016e-48 < t

   1. Initial program 99.9%

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

   2. Taylor expanded in t around inf 98.2%

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

      1. mul-1-neg 98.2%

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

      2. distribute-frac-neg 98.2%

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

      3. associate-/r* 98.2%

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

   4. Simplified 98.2%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.1 \cdot 10^{-31}:\\ \;\;\;\;1 - \frac{x}{z \cdot t}\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{-48}:\\ \;\;\;\;1 + \frac{-1}{y \cdot \frac{y - z}{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{x}{t}}{y - z}\\ \end{array} \]

Alternative 3: 83.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-25}:\\ \;\;\;\;1 - \frac{x}{y \cdot \left(y - z\right)}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-37}:\\ \;\;\;\;1 + \frac{\frac{-1}{t}}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -2e-25)
   (- 1.0 (/ x (* y (- y z))))
   (if (<= y 2.2e-37) (+ 1.0 (/ (/ -1.0 t) (/ z x))) 1.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -2e-25) {
		tmp = 1.0 - (x / (y * (y - z)));
	} else if (y <= 2.2e-37) {
		tmp = 1.0 + ((-1.0 / t) / (z / x));
	} else {
		tmp = 1.0;
	}
	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
    real(8) :: tmp
    if (y <= (-2d-25)) then
        tmp = 1.0d0 - (x / (y * (y - z)))
    else if (y <= 2.2d-37) then
        tmp = 1.0d0 + (((-1.0d0) / t) / (z / x))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -2e-25) {
		tmp = 1.0 - (x / (y * (y - z)));
	} else if (y <= 2.2e-37) {
		tmp = 1.0 + ((-1.0 / t) / (z / x));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -2e-25:
		tmp = 1.0 - (x / (y * (y - z)))
	elif y <= 2.2e-37:
		tmp = 1.0 + ((-1.0 / t) / (z / x))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -2e-25)
		tmp = Float64(1.0 - Float64(x / Float64(y * Float64(y - z))));
	elseif (y <= 2.2e-37)
		tmp = Float64(1.0 + Float64(Float64(-1.0 / t) / Float64(z / x)));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -2e-25)
		tmp = 1.0 - (x / (y * (y - z)));
	elseif (y <= 2.2e-37)
		tmp = 1.0 + ((-1.0 / t) / (z / x));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -2e-25], N[(1.0 - N[(x / N[(y * N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.2e-37], N[(1.0 + N[(N[(-1.0 / t), $MachinePrecision] / N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 3 regimes

2. if y < -2.00000000000000008e-25

   1. Initial program 99.9%

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

   2. Taylor expanded in t around 0 93.3%

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

   if -2.00000000000000008e-25 < y < 2.20000000000000002e-37

   1. Initial program 98.2%

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

   2. Taylor expanded in y around 0 78.5%

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

      1. *-un-lft-identity 78.5%

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

      2. times-frac 76.9%

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

   4. Applied egg-rr 76.9%

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

      1. clear-num 76.9%

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

      2. un-div-inv 76.9%

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

   6. Applied egg-rr 76.9%

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

   if 2.20000000000000002e-37 < 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 90.2%

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

      1. associate-/l/ 90.2%

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

   4. Simplified 90.2%

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

   5. Taylor expanded in y around 0 69.6%

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

      1. associate-*r/ 69.6%

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

      2. neg-mul-1 69.6%

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

      3. *-commutative 69.6%

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

   7. Simplified 69.6%

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

      1. expm1-log1p-u 67.1%

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

      2. expm1-udef 67.1%

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

      3. add-sqr-sqrt 35.1%

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

      4. sqrt-unprod 60.3%

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

      5. sqr-neg 60.3%

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

      6. sqrt-unprod 32.1%

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

      7. add-sqr-sqrt 67.3%

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

   9. Applied egg-rr 67.3%

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

       1. expm1-def 67.3%

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

       2. expm1-log1p 67.8%

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

   11. Simplified 67.8%

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

   12. Taylor expanded in x around 0 93.8%

       \[\leadsto \color{blue}{1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-25}:\\ \;\;\;\;1 - \frac{x}{y \cdot \left(y - z\right)}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-37}:\\ \;\;\;\;1 + \frac{\frac{-1}{t}}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4: 86.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.9 \cdot 10^{-31}:\\ \;\;\;\;1 - \frac{x}{z \cdot t}\\ \mathbf{elif}\;t \leq 2.25 \cdot 10^{-48}:\\ \;\;\;\;1 - \frac{x}{y \cdot \left(y - z\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{x}{t}}{y - z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -2.9e-31)
   (- 1.0 (/ x (* z t)))
   (if (<= t 2.25e-48)
     (- 1.0 (/ x (* y (- y z))))
     (+ 1.0 (/ (/ x t) (- y z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -2.9e-31) {
		tmp = 1.0 - (x / (z * t));
	} else if (t <= 2.25e-48) {
		tmp = 1.0 - (x / (y * (y - z)));
	} else {
		tmp = 1.0 + ((x / t) / (y - z));
	}
	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
    real(8) :: tmp
    if (t <= (-2.9d-31)) then
        tmp = 1.0d0 - (x / (z * t))
    else if (t <= 2.25d-48) then
        tmp = 1.0d0 - (x / (y * (y - z)))
    else
        tmp = 1.0d0 + ((x / t) / (y - z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -2.9e-31) {
		tmp = 1.0 - (x / (z * t));
	} else if (t <= 2.25e-48) {
		tmp = 1.0 - (x / (y * (y - z)));
	} else {
		tmp = 1.0 + ((x / t) / (y - z));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -2.9e-31:
		tmp = 1.0 - (x / (z * t))
	elif t <= 2.25e-48:
		tmp = 1.0 - (x / (y * (y - z)))
	else:
		tmp = 1.0 + ((x / t) / (y - z))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -2.9e-31)
		tmp = Float64(1.0 - Float64(x / Float64(z * t)));
	elseif (t <= 2.25e-48)
		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

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -2.9e-31)
		tmp = 1.0 - (x / (z * t));
	elseif (t <= 2.25e-48)
		tmp = 1.0 - (x / (y * (y - z)));
	else
		tmp = 1.0 + ((x / t) / (y - z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -2.9e-31], N[(1.0 - N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.25e-48], 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 3 regimes

2. if t < -2.9000000000000001e-31

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 79.6%

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

   if -2.9000000000000001e-31 < t < 2.24999999999999994e-48

   1. Initial program 98.0%

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

   2. Taylor expanded in t around 0 90.7%

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

   if 2.24999999999999994e-48 < t

   1. Initial program 99.9%

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

   2. Taylor expanded in t around inf 98.2%

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

      1. mul-1-neg 98.2%

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

      2. distribute-frac-neg 98.2%

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

      3. associate-/r* 98.2%

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

   4. Simplified 98.2%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.9 \cdot 10^{-31}:\\ \;\;\;\;1 - \frac{x}{z \cdot t}\\ \mathbf{elif}\;t \leq 2.25 \cdot 10^{-48}:\\ \;\;\;\;1 - \frac{x}{y \cdot \left(y - z\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{x}{t}}{y - z}\\ \end{array} \]

Alternative 5: 83.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{-25}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{-36}:\\ \;\;\;\;1 - \frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -2.3e-25) 1.0 (if (<= y 1.95e-36) (- 1.0 (/ x (* z t))) 1.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -2.3e-25) {
		tmp = 1.0;
	} else if (y <= 1.95e-36) {
		tmp = 1.0 - (x / (z * t));
	} else {
		tmp = 1.0;
	}
	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
    real(8) :: tmp
    if (y <= (-2.3d-25)) then
        tmp = 1.0d0
    else if (y <= 1.95d-36) then
        tmp = 1.0d0 - (x / (z * t))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -2.3e-25) {
		tmp = 1.0;
	} else if (y <= 1.95e-36) {
		tmp = 1.0 - (x / (z * t));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -2.3e-25:
		tmp = 1.0
	elif y <= 1.95e-36:
		tmp = 1.0 - (x / (z * t))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -2.3e-25)
		tmp = 1.0;
	elseif (y <= 1.95e-36)
		tmp = Float64(1.0 - Float64(x / Float64(z * t)));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -2.3e-25)
		tmp = 1.0;
	elseif (y <= 1.95e-36)
		tmp = 1.0 - (x / (z * t));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -2.3e-25], 1.0, If[LessEqual[y, 1.95e-36], N[(1.0 - N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -2.2999999999999999e-25 or 1.95e-36 < y

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0 91.9%

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

      1. associate-/l/ 91.9%

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

   4. Simplified 91.9%

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

   5. Taylor expanded in y around 0 72.7%

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

      1. associate-*r/ 72.7%

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

      2. neg-mul-1 72.7%

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

      3. *-commutative 72.7%

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

   7. Simplified 72.7%

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

      1. expm1-log1p-u 70.0%

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

      2. expm1-udef 70.0%

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

      3. add-sqr-sqrt 31.4%

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

      4. sqrt-unprod 59.3%

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

      5. sqr-neg 59.3%

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

      6. sqrt-unprod 38.7%

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

      7. add-sqr-sqrt 69.5%

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

   9. Applied egg-rr 69.5%

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

       1. expm1-def 69.5%

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

       2. expm1-log1p 69.9%

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

   11. Simplified 69.9%

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

   12. Taylor expanded in x around 0 94.3%

       \[\leadsto \color{blue}{1} \]

   if -2.2999999999999999e-25 < y < 1.95e-36

   1. Initial program 98.2%

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

   2. Taylor expanded in y around 0 78.5%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{-25}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{-36}:\\ \;\;\;\;1 - \frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6: 84.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-25}:\\ \;\;\;\;1 - \frac{x}{y \cdot \left(y - z\right)}\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-37}:\\ \;\;\;\;1 - \frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -2e-25)
   (- 1.0 (/ x (* y (- y z))))
   (if (<= y 3e-37) (- 1.0 (/ x (* z t))) 1.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -2e-25) {
		tmp = 1.0 - (x / (y * (y - z)));
	} else if (y <= 3e-37) {
		tmp = 1.0 - (x / (z * t));
	} else {
		tmp = 1.0;
	}
	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
    real(8) :: tmp
    if (y <= (-2d-25)) then
        tmp = 1.0d0 - (x / (y * (y - z)))
    else if (y <= 3d-37) then
        tmp = 1.0d0 - (x / (z * t))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -2e-25) {
		tmp = 1.0 - (x / (y * (y - z)));
	} else if (y <= 3e-37) {
		tmp = 1.0 - (x / (z * t));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -2e-25:
		tmp = 1.0 - (x / (y * (y - z)))
	elif y <= 3e-37:
		tmp = 1.0 - (x / (z * t))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -2e-25)
		tmp = Float64(1.0 - Float64(x / Float64(y * Float64(y - z))));
	elseif (y <= 3e-37)
		tmp = Float64(1.0 - Float64(x / Float64(z * t)));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -2e-25)
		tmp = 1.0 - (x / (y * (y - z)));
	elseif (y <= 3e-37)
		tmp = 1.0 - (x / (z * t));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -2e-25], N[(1.0 - N[(x / N[(y * N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3e-37], N[(1.0 - N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 3 regimes

2. if y < -2.00000000000000008e-25

   1. Initial program 99.9%

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

   2. Taylor expanded in t around 0 93.3%

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

   if -2.00000000000000008e-25 < y < 3e-37

   1. Initial program 98.2%

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

   2. Taylor expanded in y around 0 78.5%

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

   if 3e-37 < 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 90.2%

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

      1. associate-/l/ 90.2%

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

   4. Simplified 90.2%

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

   5. Taylor expanded in y around 0 69.6%

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

      1. associate-*r/ 69.6%

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

      2. neg-mul-1 69.6%

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

      3. *-commutative 69.6%

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

   7. Simplified 69.6%

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

      1. expm1-log1p-u 67.1%

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

      2. expm1-udef 67.1%

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

      3. add-sqr-sqrt 35.1%

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

      4. sqrt-unprod 60.3%

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

      5. sqr-neg 60.3%

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

      6. sqrt-unprod 32.1%

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

      7. add-sqr-sqrt 67.3%

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

   9. Applied egg-rr 67.3%

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

       1. expm1-def 67.3%

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

       2. expm1-log1p 67.8%

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

   11. Simplified 67.8%

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

   12. Taylor expanded in x around 0 93.8%

       \[\leadsto \color{blue}{1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-25}:\\ \;\;\;\;1 - \frac{x}{y \cdot \left(y - z\right)}\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-37}:\\ \;\;\;\;1 - \frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7: 75.8% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 1.0)

C

double code(double x, double y, double z, double t) {
	return 1.0;
}

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
end function

Java

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

Python

def code(x, y, z, t):
	return 1.0

Julia

function code(x, y, z, t)
	return 1.0
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = 1.0;
end

Wolfram

code[x_, y_, z_, t_] := 1.0

Derivation

1. Initial program 99.2%

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

2. Taylor expanded in z around 0 70.6%

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

   1. associate-/l/ 70.2%

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

4. Simplified 70.2%

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

5. Taylor expanded in y around 0 58.1%

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

   1. associate-*r/ 58.1%

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

   2. neg-mul-1 58.1%

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

   3. *-commutative 58.1%

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

7. Simplified 58.1%

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

   1. expm1-log1p-u 53.6%

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

   2. expm1-udef 53.6%

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

   3. add-sqr-sqrt 25.0%

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

   4. sqrt-unprod 47.8%

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

   5. sqr-neg 47.8%

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

   6. sqrt-unprod 29.0%

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

   7. add-sqr-sqrt 53.1%

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

9. Applied egg-rr 53.1%

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

    1. expm1-def 53.1%

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

    2. expm1-log1p 55.1%

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

11. Simplified 55.1%

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

12. Taylor expanded in x around 0 78.4%

    \[\leadsto \color{blue}{1} \]

13. Final simplification 78.4%

    \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023195 
(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)))))