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

Percentage Accurate: 99.0% → 99.0%

Time: 9.4s

Alternatives: 8

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: 99.0% 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: 99.0% accurate, 1.0× speedup

Math

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

FPCore

(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) * (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) * (t - y)))
end function

Java

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) * (t - y)))

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Final simplification 99.3%

   \[\leadsto 1 + \frac{x}{\left(y - z\right) \cdot \left(t - y\right)} \]
4. Add Preprocessing

Alternative 2: 81.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.1 \cdot 10^{-25}:\\ \;\;\;\;1 - \frac{x}{z \cdot \left(t - y\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{1}{\frac{y}{x} \cdot \left(t - y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -5.1e-25)
   (- 1.0 (/ x (* z (- t y))))
   (+ 1.0 (/ 1.0 (* (/ y x) (- t y))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -5.1e-25) {
		tmp = 1.0 - (x / (z * (t - y)));
	} else {
		tmp = 1.0 + (1.0 / ((y / x) * (t - 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
    real(8) :: tmp
    if (z <= (-5.1d-25)) then
        tmp = 1.0d0 - (x / (z * (t - y)))
    else
        tmp = 1.0d0 + (1.0d0 / ((y / x) * (t - y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -5.1e-25) {
		tmp = 1.0 - (x / (z * (t - y)));
	} else {
		tmp = 1.0 + (1.0 / ((y / x) * (t - y)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -5.1e-25:
		tmp = 1.0 - (x / (z * (t - y)))
	else:
		tmp = 1.0 + (1.0 / ((y / x) * (t - y)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -5.1e-25)
		tmp = Float64(1.0 - Float64(x / Float64(z * Float64(t - y))));
	else
		tmp = Float64(1.0 + Float64(1.0 / Float64(Float64(y / x) * Float64(t - y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -5.1e-25)
		tmp = 1.0 - (x / (z * (t - y)));
	else
		tmp = 1.0 + (1.0 / ((y / x) * (t - y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -5.1e-25], N[(1.0 - N[(x / N[(z * N[(t - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(1.0 / N[(N[(y / x), $MachinePrecision] * N[(t - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.1000000000000003e-25

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 95.4%

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

      1. associate-*r/ 95.4%

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

      2. neg-mul-1 95.4%

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

      3. *-commutative 95.4%

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

   5. Simplified 95.4%

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

   if -5.1000000000000003e-25 < z

   1. Initial program 99.0%

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

   3. Taylor expanded in z around 0 77.3%

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

      1. associate-/r* 77.8%

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

   5. Simplified 77.8%

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

      1. clear-num 77.8%

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

      2. inv-pow 77.8%

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

      3. div-inv 77.8%

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

      4. clear-num 77.8%

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

   7. Applied egg-rr 77.8%

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

      1. unpow-1 77.8%

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

   9. Simplified 77.8%

      \[\leadsto 1 - \color{blue}{\frac{1}{\left(y - t\right) \cdot \frac{y}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.1 \cdot 10^{-25}:\\ \;\;\;\;1 - \frac{x}{z \cdot \left(t - y\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{1}{\frac{y}{x} \cdot \left(t - y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{-102}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{x}{y}}{z - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 1.8e-102) 1.0 (+ 1.0 (/ (/ x y) (- z y)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.8e-102) {
		tmp = 1.0;
	} else {
		tmp = 1.0 + ((x / y) / (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
    real(8) :: tmp
    if (y <= 1.8d-102) then
        tmp = 1.0d0
    else
        tmp = 1.0d0 + ((x / y) / (z - y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.8e-102) {
		tmp = 1.0;
	} else {
		tmp = 1.0 + ((x / y) / (z - y));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 1.8e-102:
		tmp = 1.0
	else:
		tmp = 1.0 + ((x / y) / (z - y))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 1.8e-102)
		tmp = 1.0;
	else
		tmp = Float64(1.0 + Float64(Float64(x / y) / Float64(z - y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 1.8e-102)
		tmp = 1.0;
	else
		tmp = 1.0 + ((x / y) / (z - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 1.8e-102], 1.0, N[(1.0 + N[(N[(x / y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.8e-102

   1. Initial program 99.0%

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

   3. Taylor expanded in t around inf 84.7%

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

      1. associate-*r/ 84.7%

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

      2. neg-mul-1 84.7%

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

   5. Simplified 84.7%

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

   6. Taylor expanded in y around inf 56.9%

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

      1. div-inv 56.9%

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

      2. add-sqr-sqrt 31.4%

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

      3. sqrt-unprod 50.6%

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

      4. sqr-neg 50.6%

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

      5. sqrt-unprod 23.5%

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

      6. add-sqr-sqrt 49.2%

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

      7. *-commutative 49.2%

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

   8. Applied egg-rr 49.2%

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

      1. associate-*r/ 49.2%

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

      2. *-rgt-identity 49.2%

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

      3. associate-/l/ 49.2%

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

   10. Simplified 49.2%

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

   11. Taylor expanded in x around 0 74.8%

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

   if 1.8e-102 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0 90.9%

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

      1. associate-/r* 90.9%

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

   5. Simplified 90.9%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{-102}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{x}{y}}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 3100000000000:\\ \;\;\;\;1 + \frac{\frac{x}{y}}{z - y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{1}{t \cdot \frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 3100000000000.0)
   (+ 1.0 (/ (/ x y) (- z y)))
   (+ 1.0 (/ 1.0 (* t (/ y x))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 3100000000000.0) {
		tmp = 1.0 + ((x / y) / (z - y));
	} else {
		tmp = 1.0 + (1.0 / (t * (y / x)));
	}
	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 <= 3100000000000.0d0) then
        tmp = 1.0d0 + ((x / y) / (z - y))
    else
        tmp = 1.0d0 + (1.0d0 / (t * (y / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 3100000000000.0) {
		tmp = 1.0 + ((x / y) / (z - y));
	} else {
		tmp = 1.0 + (1.0 / (t * (y / x)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= 3100000000000.0:
		tmp = 1.0 + ((x / y) / (z - y))
	else:
		tmp = 1.0 + (1.0 / (t * (y / x)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 3100000000000.0)
		tmp = Float64(1.0 + Float64(Float64(x / y) / Float64(z - y)));
	else
		tmp = Float64(1.0 + Float64(1.0 / Float64(t * Float64(y / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 3100000000000.0)
		tmp = 1.0 + ((x / y) / (z - y));
	else
		tmp = 1.0 + (1.0 / (t * (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 3100000000000.0], N[(1.0 + N[(N[(x / y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(1.0 / N[(t * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 3.1e12

   1. Initial program 99.0%

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

   3. Taylor expanded in t around 0 77.2%

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

      1. associate-/r* 76.7%

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

   5. Simplified 76.7%

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

   if 3.1e12 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 82.1%

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

      1. associate-/r* 80.4%

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

   5. Simplified 80.4%

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

      1. clear-num 80.4%

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

      2. inv-pow 80.4%

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

      3. div-inv 80.4%

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

      4. clear-num 80.4%

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

   7. Applied egg-rr 80.4%

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

      1. unpow-1 80.4%

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

   9. Simplified 80.4%

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

   10. Taylor expanded in y around 0 81.4%

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

       1. mul-1-neg 81.4%

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

       2. associate-/l* 79.7%

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

       3. distribute-rgt-neg-in 79.7%

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

       4. distribute-neg-frac2 79.7%

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

   12. Simplified 79.7%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3100000000000:\\ \;\;\;\;1 + \frac{\frac{x}{y}}{z - y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{1}{t \cdot \frac{y}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{-102}:\\ \;\;\;\;1 + \frac{x}{y \cdot \left(z - y\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{\left(y - z\right) \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -1.4e-102) (+ 1.0 (/ x (* y (- z y)))) (+ 1.0 (/ x (* (- y z) t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.4e-102) {
		tmp = 1.0 + (x / (y * (z - y)));
	} else {
		tmp = 1.0 + (x / ((y - z) * t));
	}
	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 <= (-1.4d-102)) then
        tmp = 1.0d0 + (x / (y * (z - y)))
    else
        tmp = 1.0d0 + (x / ((y - z) * t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.4e-102) {
		tmp = 1.0 + (x / (y * (z - y)));
	} else {
		tmp = 1.0 + (x / ((y - z) * t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -1.4e-102:
		tmp = 1.0 + (x / (y * (z - y)))
	else:
		tmp = 1.0 + (x / ((y - z) * t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -1.4e-102)
		tmp = Float64(1.0 + Float64(x / Float64(y * Float64(z - y))));
	else
		tmp = Float64(1.0 + Float64(x / Float64(Float64(y - z) * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -1.4e-102)
		tmp = 1.0 + (x / (y * (z - y)));
	else
		tmp = 1.0 + (x / ((y - z) * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -1.4e-102], N[(1.0 + N[(x / N[(y * N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(x / N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.40000000000000006e-102

   1. Initial program 100.0%

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

      1. clear-num 100.0%

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

      2. associate-/r/ 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in t around 0 93.0%

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

   if -1.40000000000000006e-102 < y

   1. Initial program 98.8%

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

   3. Taylor expanded in t around inf 83.8%

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

      1. associate-*r/ 83.8%

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

      2. neg-mul-1 83.8%

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

   5. Simplified 83.8%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{-102}:\\ \;\;\;\;1 + \frac{x}{y \cdot \left(z - y\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{\left(y - z\right) \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.1% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -9e-29) (+ 1.0 (/ x (* y (- t y)))) (- 1.0 (/ x (* z (- t y))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -9e-29) {
		tmp = 1.0 + (x / (y * (t - y)));
	} else {
		tmp = 1.0 - (x / (z * (t - 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
    real(8) :: tmp
    if (y <= (-9d-29)) then
        tmp = 1.0d0 + (x / (y * (t - y)))
    else
        tmp = 1.0d0 - (x / (z * (t - y)))
    end if
    code = tmp
end function

Java

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

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -9e-29:
		tmp = 1.0 + (x / (y * (t - y)))
	else:
		tmp = 1.0 - (x / (z * (t - y)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -9e-29)
		tmp = Float64(1.0 + Float64(x / Float64(y * Float64(t - y))));
	else
		tmp = Float64(1.0 - Float64(x / Float64(z * Float64(t - y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -9e-29)
		tmp = 1.0 + (x / (y * (t - y)));
	else
		tmp = 1.0 - (x / (z * (t - y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -9e-29], N[(1.0 + N[(x / N[(y * N[(t - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(x / N[(z * N[(t - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -8.9999999999999996e-29

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 94.1%

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

   if -8.9999999999999996e-29 < y

   1. Initial program 98.9%

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

   3. Taylor expanded in z around inf 83.6%

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

      1. associate-*r/ 83.6%

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

      2. neg-mul-1 83.6%

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

      3. *-commutative 83.6%

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

   5. Simplified 83.6%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{-29}:\\ \;\;\;\;1 + \frac{x}{y \cdot \left(t - y\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{z \cdot \left(t - y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{-28}:\\ \;\;\;\;1 + \frac{x}{y \cdot \left(t - y\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{x}{z}}{y - t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -1.25e-28) (+ 1.0 (/ x (* y (- t y)))) (+ 1.0 (/ (/ x z) (- y t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.25e-28) {
		tmp = 1.0 + (x / (y * (t - y)));
	} else {
		tmp = 1.0 + ((x / z) / (y - t));
	}
	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 <= (-1.25d-28)) then
        tmp = 1.0d0 + (x / (y * (t - y)))
    else
        tmp = 1.0d0 + ((x / z) / (y - t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.25e-28) {
		tmp = 1.0 + (x / (y * (t - y)));
	} else {
		tmp = 1.0 + ((x / z) / (y - t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -1.25e-28:
		tmp = 1.0 + (x / (y * (t - y)))
	else:
		tmp = 1.0 + ((x / z) / (y - t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -1.25e-28)
		tmp = Float64(1.0 + Float64(x / Float64(y * Float64(t - y))));
	else
		tmp = Float64(1.0 + Float64(Float64(x / z) / Float64(y - t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -1.25e-28)
		tmp = 1.0 + (x / (y * (t - y)));
	else
		tmp = 1.0 + ((x / z) / (y - t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -1.25e-28], N[(1.0 + N[(x / N[(y * N[(t - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(x / z), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.25e-28

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 94.1%

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

   if -1.25e-28 < y

   1. Initial program 98.9%

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

   3. Taylor expanded in z around inf 83.6%

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

      1. mul-1-neg 83.6%

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

      2. associate-/r* 83.0%

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

      3. distribute-neg-frac 83.0%

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

   5. Simplified 83.0%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{-28}:\\ \;\;\;\;1 + \frac{x}{y \cdot \left(t - y\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{x}{z}}{y - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.2% 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 99.3%

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

3. Taylor expanded in t around inf 82.5%

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

   1. associate-*r/ 82.5%

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

   2. neg-mul-1 82.5%

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

5. Simplified 82.5%

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

6. Taylor expanded in y around inf 59.5%

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

   1. div-inv 59.5%

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

   2. add-sqr-sqrt 31.4%

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

   3. sqrt-unprod 52.3%

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

   4. sqr-neg 52.3%

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

   5. sqrt-unprod 26.5%

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

   6. add-sqr-sqrt 53.2%

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

   7. *-commutative 53.2%

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

8. Applied egg-rr 53.2%

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

   1. associate-*r/ 53.2%

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

   2. *-rgt-identity 53.2%

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

   3. associate-/l/ 53.1%

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

10. Simplified 53.1%

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

11. Taylor expanded in x around 0 77.5%

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

12. Final simplification 77.5%

    \[\leadsto 1 \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024066 
(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)))))