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

Percentage Accurate: 99.0% → 99.0%

Time: 11.4s

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: 99.0% 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

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

FPCore

(FPCore (x y z t) :precision binary64 (+ 1.0 (/ x (* (- y t) (- z y)))))

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = 1.0d0 + (x / ((y - t) * (z - y)))
end function

Java

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

Python

def code(x, y, z, t):
	return 1.0 + (x / ((y - t) * (z - y)))

Julia

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

MATLAB

function tmp = code(x, y, z, t)
	tmp = 1.0 + (x / ((y - t) * (z - y)));
end

Wolfram

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

Derivation

1. Initial program 98.8%

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

3. Final simplification 98.8%

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

Alternative 2: 78.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.95 \cdot 10^{-51}:\\ \;\;\;\;1 - \frac{x}{z \cdot t}\\ \mathbf{elif}\;t \leq 9 \cdot 10^{-288}:\\ \;\;\;\;1 + \frac{x}{y \cdot z}\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{-165}:\\ \;\;\;\;1 - \frac{\frac{x}{y}}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{x}{t}}{z - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -1.95e-51)
   (- 1.0 (/ x (* z t)))
   (if (<= t 9e-288)
     (+ 1.0 (/ x (* y z)))
     (if (<= t 1.9e-165) (- 1.0 (/ (/ x y) y)) (- 1.0 (/ (/ x t) (- z y)))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.95e-51) {
		tmp = 1.0 - (x / (z * t));
	} else if (t <= 9e-288) {
		tmp = 1.0 + (x / (y * z));
	} else if (t <= 1.9e-165) {
		tmp = 1.0 - ((x / y) / y);
	} else {
		tmp = 1.0 - ((x / t) / (z - y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-1.95d-51)) then
        tmp = 1.0d0 - (x / (z * t))
    else if (t <= 9d-288) then
        tmp = 1.0d0 + (x / (y * z))
    else if (t <= 1.9d-165) then
        tmp = 1.0d0 - ((x / y) / y)
    else
        tmp = 1.0d0 - ((x / t) / (z - y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.95e-51) {
		tmp = 1.0 - (x / (z * t));
	} else if (t <= 9e-288) {
		tmp = 1.0 + (x / (y * z));
	} else if (t <= 1.9e-165) {
		tmp = 1.0 - ((x / y) / y);
	} else {
		tmp = 1.0 - ((x / t) / (z - y));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -1.95e-51:
		tmp = 1.0 - (x / (z * t))
	elif t <= 9e-288:
		tmp = 1.0 + (x / (y * z))
	elif t <= 1.9e-165:
		tmp = 1.0 - ((x / y) / y)
	else:
		tmp = 1.0 - ((x / t) / (z - y))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -1.95e-51)
		tmp = Float64(1.0 - Float64(x / Float64(z * t)));
	elseif (t <= 9e-288)
		tmp = Float64(1.0 + Float64(x / Float64(y * z)));
	elseif (t <= 1.9e-165)
		tmp = Float64(1.0 - Float64(Float64(x / y) / y));
	else
		tmp = Float64(1.0 - Float64(Float64(x / t) / Float64(z - y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -1.95e-51)
		tmp = 1.0 - (x / (z * t));
	elseif (t <= 9e-288)
		tmp = 1.0 + (x / (y * z));
	elseif (t <= 1.9e-165)
		tmp = 1.0 - ((x / y) / y);
	else
		tmp = 1.0 - ((x / t) / (z - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -1.95e-51], N[(1.0 - N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 9e-288], N[(1.0 + N[(x / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.9e-165], N[(1.0 - N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(x / t), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < -1.9499999999999999e-51

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 73.5%

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

   if -1.9499999999999999e-51 < t < 9.0000000000000003e-288

   1. Initial program 98.3%

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

   3. Taylor expanded in z around inf 81.6%

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

      1. associate-/r* 83.1%

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

   5. Simplified 83.1%

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

   6. Taylor expanded in y around inf 76.0%

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

      1. *-commutative 76.0%

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

   8. Simplified 76.0%

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

   if 9.0000000000000003e-288 < t < 1.90000000000000009e-165

   1. Initial program 95.3%

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

      1. *-un-lft-identity 95.3%

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

      2. times-frac 90.6%

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

   4. Applied egg-rr 90.6%

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

   5. Taylor expanded in t around 0 77.2%

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

      1. associate-/r* 73.7%

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

   7. Simplified 73.7%

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

   8. Taylor expanded in y around inf 59.2%

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

   if 1.90000000000000009e-165 < t

   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 88.9%

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

      1. associate-/r* 88.9%

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

   5. Simplified 88.9%

      \[\leadsto \color{blue}{1 + \frac{\frac{x}{t}}{y - z}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.95 \cdot 10^{-51}:\\ \;\;\;\;1 - \frac{x}{z \cdot t}\\ \mathbf{elif}\;t \leq 9 \cdot 10^{-288}:\\ \;\;\;\;1 + \frac{x}{y \cdot z}\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{-165}:\\ \;\;\;\;1 - \frac{\frac{x}{y}}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{x}{t}}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{-109} \lor \neg \left(y \leq 2.1 \cdot 10^{-67}\right):\\ \;\;\;\;1 + \frac{\frac{x}{y}}{t - y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{x}{t}}{z - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -1.3e-109) (not (<= y 2.1e-67)))
   (+ 1.0 (/ (/ x y) (- t y)))
   (- 1.0 (/ (/ x t) (- z y)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -1.3e-109) || !(y <= 2.1e-67)) {
		tmp = 1.0 + ((x / y) / (t - y));
	} else {
		tmp = 1.0 - ((x / t) / (z - y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((y <= (-1.3d-109)) .or. (.not. (y <= 2.1d-67))) then
        tmp = 1.0d0 + ((x / y) / (t - y))
    else
        tmp = 1.0d0 - ((x / t) / (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 <= -1.3e-109) || !(y <= 2.1e-67)) {
		tmp = 1.0 + ((x / y) / (t - y));
	} else {
		tmp = 1.0 - ((x / t) / (z - y));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y <= -1.3e-109) or not (y <= 2.1e-67):
		tmp = 1.0 + ((x / y) / (t - y))
	else:
		tmp = 1.0 - ((x / t) / (z - y))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -1.3e-109) || !(y <= 2.1e-67))
		tmp = Float64(1.0 + Float64(Float64(x / y) / Float64(t - y)));
	else
		tmp = Float64(1.0 - Float64(Float64(x / t) / Float64(z - y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y <= -1.3e-109) || ~((y <= 2.1e-67)))
		tmp = 1.0 + ((x / y) / (t - y));
	else
		tmp = 1.0 - ((x / t) / (z - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -1.3e-109], N[Not[LessEqual[y, 2.1e-67]], $MachinePrecision]], N[(1.0 + N[(N[(x / y), $MachinePrecision] / N[(t - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(x / t), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.2999999999999999e-109 or 2.1000000000000002e-67 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0 90.5%

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

      1. sub-neg 90.5%

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

      2. associate-/r* 90.6%

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

      3. distribute-neg-frac2 90.6%

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

      4. neg-sub0 90.6%

         \[\leadsto 1 + \frac{\frac{x}{y}}{\color{blue}{0 - \left(y - t\right)}} \]

      5. sub-neg 90.6%

         \[\leadsto 1 + \frac{\frac{x}{y}}{0 - \color{blue}{\left(y + \left(-t\right)\right)}} \]

      6. +-commutative 90.6%

         \[\leadsto 1 + \frac{\frac{x}{y}}{0 - \color{blue}{\left(\left(-t\right) + y\right)}} \]

      7. associate--r+ 90.6%

         \[\leadsto 1 + \frac{\frac{x}{y}}{\color{blue}{\left(0 - \left(-t\right)\right) - y}} \]

      8. neg-sub0 90.6%

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

      9. remove-double-neg 90.6%

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

   5. Simplified 90.6%

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

   if -1.2999999999999999e-109 < y < 2.1000000000000002e-67

   1. Initial program 96.8%

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

   3. Taylor expanded in t around inf 87.4%

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

      1. associate-/r* 86.6%

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

   5. Simplified 86.6%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{-109} \lor \neg \left(y \leq 2.1 \cdot 10^{-67}\right):\\ \;\;\;\;1 + \frac{\frac{x}{y}}{t - y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{x}{t}}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{-86}:\\ \;\;\;\;1 + \frac{\frac{x}{z}}{y - t}\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-251}:\\ \;\;\;\;1 + \frac{\frac{x}{y}}{t - y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{x}{t}}{z - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.55e-86)
   (+ 1.0 (/ (/ x z) (- y t)))
   (if (<= z 1.1e-251)
     (+ 1.0 (/ (/ x y) (- t y)))
     (- 1.0 (/ (/ x t) (- z y))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.55e-86) {
		tmp = 1.0 + ((x / z) / (y - t));
	} else if (z <= 1.1e-251) {
		tmp = 1.0 + ((x / y) / (t - y));
	} else {
		tmp = 1.0 - ((x / t) / (z - y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-1.55d-86)) then
        tmp = 1.0d0 + ((x / z) / (y - t))
    else if (z <= 1.1d-251) then
        tmp = 1.0d0 + ((x / y) / (t - y))
    else
        tmp = 1.0d0 - ((x / t) / (z - y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.55e-86) {
		tmp = 1.0 + ((x / z) / (y - t));
	} else if (z <= 1.1e-251) {
		tmp = 1.0 + ((x / y) / (t - y));
	} else {
		tmp = 1.0 - ((x / t) / (z - y));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -1.55e-86:
		tmp = 1.0 + ((x / z) / (y - t))
	elif z <= 1.1e-251:
		tmp = 1.0 + ((x / y) / (t - y))
	else:
		tmp = 1.0 - ((x / t) / (z - y))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.55e-86)
		tmp = Float64(1.0 + Float64(Float64(x / z) / Float64(y - t)));
	elseif (z <= 1.1e-251)
		tmp = Float64(1.0 + Float64(Float64(x / y) / Float64(t - y)));
	else
		tmp = Float64(1.0 - Float64(Float64(x / t) / Float64(z - y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -1.55e-86)
		tmp = 1.0 + ((x / z) / (y - t));
	elseif (z <= 1.1e-251)
		tmp = 1.0 + ((x / y) / (t - y));
	else
		tmp = 1.0 - ((x / t) / (z - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1.55e-86], N[(1.0 + N[(N[(x / z), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.1e-251], N[(1.0 + N[(N[(x / y), $MachinePrecision] / N[(t - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(x / t), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.54999999999999994e-86

   1. Initial program 98.7%

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

   3. Taylor expanded in z around inf 93.7%

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

      1. associate-/r* 94.9%

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

   5. Simplified 94.9%

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

   if -1.54999999999999994e-86 < z < 1.1e-251

   1. Initial program 96.7%

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

   3. Taylor expanded in z around 0 86.0%

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

      1. sub-neg 86.0%

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

      2. associate-/r* 86.1%

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

      3. distribute-neg-frac2 86.1%

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

      4. neg-sub0 86.1%

         \[\leadsto 1 + \frac{\frac{x}{y}}{\color{blue}{0 - \left(y - t\right)}} \]

      5. sub-neg 86.1%

         \[\leadsto 1 + \frac{\frac{x}{y}}{0 - \color{blue}{\left(y + \left(-t\right)\right)}} \]

      6. +-commutative 86.1%

         \[\leadsto 1 + \frac{\frac{x}{y}}{0 - \color{blue}{\left(\left(-t\right) + y\right)}} \]

      7. associate--r+ 86.1%

         \[\leadsto 1 + \frac{\frac{x}{y}}{\color{blue}{\left(0 - \left(-t\right)\right) - y}} \]

      8. neg-sub0 86.1%

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

      9. remove-double-neg 86.1%

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

   5. Simplified 86.1%

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

   if 1.1e-251 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in t around inf 79.2%

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

      1. associate-/r* 77.6%

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

   5. Simplified 77.6%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{-86}:\\ \;\;\;\;1 + \frac{\frac{x}{z}}{y - t}\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-251}:\\ \;\;\;\;1 + \frac{\frac{x}{y}}{t - y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{x}{t}}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{-11} \lor \neg \left(y \leq 1.15 \cdot 10^{-59}\right):\\ \;\;\;\;1 - \frac{\frac{x}{y}}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{z \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -1.9e-11) (not (<= y 1.15e-59)))
   (- 1.0 (/ (/ x y) y))
   (- 1.0 (/ x (* z t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -1.9e-11) || !(y <= 1.15e-59)) {
		tmp = 1.0 - ((x / y) / y);
	} else {
		tmp = 1.0 - (x / (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 <= (-1.9d-11)) .or. (.not. (y <= 1.15d-59))) then
        tmp = 1.0d0 - ((x / y) / y)
    else
        tmp = 1.0d0 - (x / (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 <= -1.9e-11) || !(y <= 1.15e-59)) {
		tmp = 1.0 - ((x / y) / y);
	} else {
		tmp = 1.0 - (x / (z * t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y <= -1.9e-11) or not (y <= 1.15e-59):
		tmp = 1.0 - ((x / y) / y)
	else:
		tmp = 1.0 - (x / (z * t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -1.9e-11) || !(y <= 1.15e-59))
		tmp = Float64(1.0 - Float64(Float64(x / y) / y));
	else
		tmp = Float64(1.0 - Float64(x / Float64(z * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y <= -1.9e-11) || ~((y <= 1.15e-59)))
		tmp = 1.0 - ((x / y) / y);
	else
		tmp = 1.0 - (x / (z * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -1.9e-11], N[Not[LessEqual[y, 1.15e-59]], $MachinePrecision]], N[(1.0 - N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.8999999999999999e-11 or 1.1499999999999999e-59 < y

   1. Initial program 99.9%

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

      1. *-un-lft-identity 99.9%

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

      2. times-frac 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in t around 0 95.6%

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

      1. associate-/r* 95.7%

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

   7. Simplified 95.7%

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

   8. Taylor expanded in y around inf 88.8%

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

   if -1.8999999999999999e-11 < y < 1.1499999999999999e-59

   1. Initial program 97.5%

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

   3. Taylor expanded in y around 0 74.7%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{-11} \lor \neg \left(y \leq 1.15 \cdot 10^{-59}\right):\\ \;\;\;\;1 - \frac{\frac{x}{y}}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{z \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{-64}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{-88}:\\ \;\;\;\;1 - \frac{\frac{x}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -2.8e-64) 1.0 (if (<= y 3.6e-88) (- 1.0 (/ (/ x t) z)) 1.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -2.8e-64) {
		tmp = 1.0;
	} else if (y <= 3.6e-88) {
		tmp = 1.0 - ((x / t) / z);
	} 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 <= (-2.8d-64)) then
        tmp = 1.0d0
    else if (y <= 3.6d-88) then
        tmp = 1.0d0 - ((x / t) / z)
    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 <= -2.8e-64) {
		tmp = 1.0;
	} else if (y <= 3.6e-88) {
		tmp = 1.0 - ((x / t) / z);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -2.8e-64:
		tmp = 1.0
	elif y <= 3.6e-88:
		tmp = 1.0 - ((x / t) / z)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -2.8e-64)
		tmp = 1.0;
	elseif (y <= 3.6e-88)
		tmp = Float64(1.0 - Float64(Float64(x / t) / z));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -2.8e-64)
		tmp = 1.0;
	elseif (y <= 3.6e-88)
		tmp = 1.0 - ((x / t) / z);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -2.8e-64], 1.0, If[LessEqual[y, 3.6e-88], N[(1.0 - N[(N[(x / t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -2.80000000000000004e-64 or 3.5999999999999999e-88 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 84.8%

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

   if -2.80000000000000004e-64 < y < 3.5999999999999999e-88

   1. Initial program 97.1%

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

      1. *-un-lft-identity 97.1%

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

      2. times-frac 95.5%

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

   4. Applied egg-rr 95.5%

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

   5. Taylor expanded in y around 0 76.5%

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

      1. associate-/r* 74.8%

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

   7. Simplified 74.8%

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

Alternative 7: 83.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{-64}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.16 \cdot 10^{-86}:\\ \;\;\;\;1 - \frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -2.7e-64) 1.0 (if (<= y 1.16e-86) (- 1.0 (/ x (* z t))) 1.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -2.7e-64) {
		tmp = 1.0;
	} else if (y <= 1.16e-86) {
		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 <= (-2.7d-64)) then
        tmp = 1.0d0
    else if (y <= 1.16d-86) 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 <= -2.7e-64) {
		tmp = 1.0;
	} else if (y <= 1.16e-86) {
		tmp = 1.0 - (x / (z * t));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -2.7e-64:
		tmp = 1.0
	elif y <= 1.16e-86:
		tmp = 1.0 - (x / (z * t))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -2.7e-64)
		tmp = 1.0;
	elseif (y <= 1.16e-86)
		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 <= -2.7e-64)
		tmp = 1.0;
	elseif (y <= 1.16e-86)
		tmp = 1.0 - (x / (z * t));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -2.7e-64], 1.0, If[LessEqual[y, 1.16e-86], N[(1.0 - N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -2.69999999999999986e-64 or 1.16e-86 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 84.8%

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

   if -2.69999999999999986e-64 < y < 1.16e-86

   1. Initial program 97.1%

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

   3. Taylor expanded in y around 0 76.5%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{-64}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.16 \cdot 10^{-86}:\\ \;\;\;\;1 - \frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.8 \cdot 10^{-52}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 4.4 \cdot 10^{-65}:\\ \;\;\;\;1 + \frac{x}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -4.8e-52) 1.0 (if (<= t 4.4e-65) (+ 1.0 (/ x (* y z))) 1.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -4.8e-52) {
		tmp = 1.0;
	} else if (t <= 4.4e-65) {
		tmp = 1.0 + (x / (y * z));
	} 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 (t <= (-4.8d-52)) then
        tmp = 1.0d0
    else if (t <= 4.4d-65) then
        tmp = 1.0d0 + (x / (y * z))
    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 (t <= -4.8e-52) {
		tmp = 1.0;
	} else if (t <= 4.4e-65) {
		tmp = 1.0 + (x / (y * z));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -4.8e-52:
		tmp = 1.0
	elif t <= 4.4e-65:
		tmp = 1.0 + (x / (y * z))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -4.8e-52)
		tmp = 1.0;
	elseif (t <= 4.4e-65)
		tmp = Float64(1.0 + Float64(x / Float64(y * z)));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -4.8e-52)
		tmp = 1.0;
	elseif (t <= 4.4e-65)
		tmp = 1.0 + (x / (y * z));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -4.8e-52], 1.0, If[LessEqual[t, 4.4e-65], N[(1.0 + N[(x / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if t < -4.8000000000000003e-52 or 4.40000000000000042e-65 < t

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 83.9%

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

   if -4.8000000000000003e-52 < t < 4.40000000000000042e-65

   1. Initial program 97.1%

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

   3. Taylor expanded in z around inf 80.4%

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

      1. associate-/r* 81.3%

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

   5. Simplified 81.3%

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

   6. Taylor expanded in y around inf 70.8%

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

      1. *-commutative 70.8%

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

   8. Simplified 70.8%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.8 \cdot 10^{-52}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 4.4 \cdot 10^{-65}:\\ \;\;\;\;1 + \frac{x}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.5% 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 98.8%

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

3. Taylor expanded in x around 0 71.0%

   \[\leadsto \color{blue}{1} \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2024170 
(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)))))