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

Percentage Accurate: 99.0% → 99.0%

Time: 8.6s

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 99.3%

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

3. Final simplification 99.3%

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

Alternative 2: 84.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{-29} \lor \neg \left(y \leq 8.5 \cdot 10^{-125}\right):\\ \;\;\;\;1 + \frac{x}{y \cdot \left(t - y\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{z \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -9.5e-29) (not (<= y 8.5e-125)))
   (+ 1.0 (/ x (* y (- t y))))
   (- 1.0 (/ x (* z t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -9.5e-29) || !(y <= 8.5e-125)) {
		tmp = 1.0 + (x / (y * (t - 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 <= (-9.5d-29)) .or. (.not. (y <= 8.5d-125))) then
        tmp = 1.0d0 + (x / (y * (t - 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 <= -9.5e-29) || !(y <= 8.5e-125)) {
		tmp = 1.0 + (x / (y * (t - y)));
	} else {
		tmp = 1.0 - (x / (z * t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y <= -9.5e-29) or not (y <= 8.5e-125):
		tmp = 1.0 + (x / (y * (t - y)))
	else:
		tmp = 1.0 - (x / (z * t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -9.5e-29) || !(y <= 8.5e-125))
		tmp = Float64(1.0 + Float64(x / Float64(y * Float64(t - 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 <= -9.5e-29) || ~((y <= 8.5e-125)))
		tmp = 1.0 + (x / (y * (t - y)));
	else
		tmp = 1.0 - (x / (z * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -9.5e-29], N[Not[LessEqual[y, 8.5e-125]], $MachinePrecision]], N[(1.0 + N[(x / N[(y * N[(t - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -9.50000000000000023e-29 or 8.5000000000000002e-125 < 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 z around 0 91.0%

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

   if -9.50000000000000023e-29 < y < 8.5000000000000002e-125

   1. Initial program 98.0%

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

   3. Taylor expanded in y around 0 82.9%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{-29} \lor \neg \left(y \leq 8.5 \cdot 10^{-125}\right):\\ \;\;\;\;1 + \frac{x}{y \cdot \left(t - y\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{z \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{-144} \lor \neg \left(y \leq 2.4 \cdot 10^{-62}\right):\\ \;\;\;\;1 + \frac{x}{y \cdot \left(z - y\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{z \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -2.5e-144) (not (<= y 2.4e-62)))
   (+ 1.0 (/ x (* y (- z y))))
   (- 1.0 (/ x (* z t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -2.5e-144) || !(y <= 2.4e-62)) {
		tmp = 1.0 + (x / (y * (z - 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 <= (-2.5d-144)) .or. (.not. (y <= 2.4d-62))) then
        tmp = 1.0d0 + (x / (y * (z - 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 <= -2.5e-144) || !(y <= 2.4e-62)) {
		tmp = 1.0 + (x / (y * (z - y)));
	} else {
		tmp = 1.0 - (x / (z * t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y <= -2.5e-144) or not (y <= 2.4e-62):
		tmp = 1.0 + (x / (y * (z - y)))
	else:
		tmp = 1.0 - (x / (z * t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -2.5e-144) || !(y <= 2.4e-62))
		tmp = Float64(1.0 + Float64(x / Float64(y * Float64(z - 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 <= -2.5e-144) || ~((y <= 2.4e-62)))
		tmp = 1.0 + (x / (y * (z - y)));
	else
		tmp = 1.0 - (x / (z * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -2.5e-144], N[Not[LessEqual[y, 2.4e-62]], $MachinePrecision]], N[(1.0 + N[(x / N[(y * N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.4999999999999999e-144 or 2.39999999999999984e-62 < y

   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 92.9%

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

   if -2.4999999999999999e-144 < y < 2.39999999999999984e-62

   1. Initial program 97.9%

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

   3. Taylor expanded in y around 0 84.0%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{-144} \lor \neg \left(y \leq 2.4 \cdot 10^{-62}\right):\\ \;\;\;\;1 + \frac{x}{y \cdot \left(z - y\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{z \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{-102} \lor \neg \left(y \leq 8.5 \cdot 10^{-63}\right):\\ \;\;\;\;1 + \frac{x}{y \cdot \left(z - y\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{t \cdot \left(z - y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -1.4e-102) (not (<= y 8.5e-63)))
   (+ 1.0 (/ x (* y (- z y))))
   (- 1.0 (/ x (* t (- z y))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -1.4e-102) || !(y <= 8.5e-63)) {
		tmp = 1.0 + (x / (y * (z - 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 <= (-1.4d-102)) .or. (.not. (y <= 8.5d-63))) then
        tmp = 1.0d0 + (x / (y * (z - 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 <= -1.4e-102) || !(y <= 8.5e-63)) {
		tmp = 1.0 + (x / (y * (z - y)));
	} else {
		tmp = 1.0 - (x / (t * (z - y)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y <= -1.4e-102) or not (y <= 8.5e-63):
		tmp = 1.0 + (x / (y * (z - y)))
	else:
		tmp = 1.0 - (x / (t * (z - y)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -1.4e-102) || !(y <= 8.5e-63))
		tmp = Float64(1.0 + Float64(x / Float64(y * Float64(z - y))));
	else
		tmp = Float64(1.0 - Float64(x / Float64(t * Float64(z - y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y <= -1.4e-102) || ~((y <= 8.5e-63)))
		tmp = 1.0 + (x / (y * (z - y)));
	else
		tmp = 1.0 - (x / (t * (z - y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -1.4e-102], N[Not[LessEqual[y, 8.5e-63]], $MachinePrecision]], N[(1.0 + N[(x / N[(y * N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(x / N[(t * N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.40000000000000006e-102 or 8.49999999999999969e-63 < y

   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 93.4%

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

   if -1.40000000000000006e-102 < y < 8.49999999999999969e-63

   1. Initial program 98.0%

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

   3. Taylor expanded in t around inf 87.9%

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

      1. associate-*r/ 87.9%

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

      2. neg-mul-1 87.9%

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

   5. Simplified 87.9%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{-102} \lor \neg \left(y \leq 8.5 \cdot 10^{-63}\right):\\ \;\;\;\;1 + \frac{x}{y \cdot \left(z - y\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{t \cdot \left(z - y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.1 \cdot 10^{-25}:\\ \;\;\;\;1 + \frac{x}{z \cdot \left(y - t\right)}\\ \mathbf{elif}\;z \leq 9 \cdot 10^{-192}:\\ \;\;\;\;1 + \frac{x}{y \cdot \left(t - y\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{t \cdot \left(z - y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -5.1e-25)
   (+ 1.0 (/ x (* z (- y t))))
   (if (<= z 9e-192) (+ 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 <= -5.1e-25) {
		tmp = 1.0 + (x / (z * (y - t)));
	} else if (z <= 9e-192) {
		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 <= (-5.1d-25)) then
        tmp = 1.0d0 + (x / (z * (y - t)))
    else if (z <= 9d-192) 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 <= -5.1e-25) {
		tmp = 1.0 + (x / (z * (y - t)));
	} else if (z <= 9e-192) {
		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 <= -5.1e-25:
		tmp = 1.0 + (x / (z * (y - t)))
	elif z <= 9e-192:
		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 <= -5.1e-25)
		tmp = Float64(1.0 + Float64(x / Float64(z * Float64(y - t))));
	elseif (z <= 9e-192)
		tmp = Float64(1.0 + Float64(x / Float64(y * Float64(t - y))));
	else
		tmp = Float64(1.0 - Float64(x / Float64(t * Float64(z - 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 * (y - t)));
	elseif (z <= 9e-192)
		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, -5.1e-25], N[(1.0 + N[(x / N[(z * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 9e-192], N[(1.0 + N[(x / N[(y * N[(t - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(x / N[(t * N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 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 < 9.00000000000000048e-192

   1. Initial program 97.3%

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

   3. Taylor expanded in z around 0 90.4%

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

   if 9.00000000000000048e-192 < z

   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 inf 82.8%

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

      1. associate-*r/ 82.8%

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

      2. neg-mul-1 82.8%

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

   5. Simplified 82.8%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.1 \cdot 10^{-25}:\\ \;\;\;\;1 + \frac{x}{z \cdot \left(y - t\right)}\\ \mathbf{elif}\;z \leq 9 \cdot 10^{-192}:\\ \;\;\;\;1 + \frac{x}{y \cdot \left(t - y\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{t \cdot \left(z - y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.1 \cdot 10^{-144}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-61}:\\ \;\;\;\;1 - \frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -8.1e-144) 1.0 (if (<= y 1.1e-61) (- 1.0 (/ x (* z t))) 1.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -8.1e-144) {
		tmp = 1.0;
	} else if (y <= 1.1e-61) {
		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 <= (-8.1d-144)) then
        tmp = 1.0d0
    else if (y <= 1.1d-61) 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 <= -8.1e-144) {
		tmp = 1.0;
	} else if (y <= 1.1e-61) {
		tmp = 1.0 - (x / (z * t));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -8.1e-144:
		tmp = 1.0
	elif y <= 1.1e-61:
		tmp = 1.0 - (x / (z * t))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -8.1e-144)
		tmp = 1.0;
	elseif (y <= 1.1e-61)
		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 <= -8.1e-144)
		tmp = 1.0;
	elseif (y <= 1.1e-61)
		tmp = 1.0 - (x / (z * t));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -8.1e-144], 1.0, If[LessEqual[y, 1.1e-61], N[(1.0 - N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -8.0999999999999998e-144 or 1.10000000000000004e-61 < 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 inf 79.0%

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

      1. associate-*r/ 79.0%

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

      2. neg-mul-1 79.0%

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

   5. Simplified 79.0%

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

   6. Taylor expanded in y around inf 69.2%

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

      1. div-inv 69.2%

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

      2. add-sqr-sqrt 34.7%

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

      3. sqrt-unprod 60.2%

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

      4. sqr-neg 60.2%

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

      5. sqrt-unprod 34.0%

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

      6. add-sqr-sqrt 65.9%

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

      7. *-commutative 65.9%

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

   8. Applied egg-rr 65.9%

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

      1. associate-*r/ 65.9%

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

      2. *-rgt-identity 65.9%

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

      3. associate-/l/ 65.8%

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

   10. Simplified 65.8%

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

   11. Taylor expanded in x around 0 88.7%

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

   if -8.0999999999999998e-144 < y < 1.10000000000000004e-61

   1. Initial program 97.9%

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

   3. Taylor expanded in y around 0 84.0%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.1 \cdot 10^{-144}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-61}:\\ \;\;\;\;1 - \frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{-122}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{x}{y}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -2.8e-122) 1.0 (+ 1.0 (/ (/ x y) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -2.8e-122) {
		tmp = 1.0;
	} else {
		tmp = 1.0 + ((x / 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 (z <= (-2.8d-122)) then
        tmp = 1.0d0
    else
        tmp = 1.0d0 + ((x / y) / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -2.8e-122) {
		tmp = 1.0;
	} else {
		tmp = 1.0 + ((x / y) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -2.8e-122:
		tmp = 1.0
	else:
		tmp = 1.0 + ((x / y) / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -2.8e-122)
		tmp = 1.0;
	else
		tmp = Float64(1.0 + Float64(Float64(x / y) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -2.8e-122)
		tmp = 1.0;
	else
		tmp = 1.0 + ((x / y) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -2.8e-122], 1.0, N[(1.0 + N[(N[(x / y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.7999999999999999e-122

   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 inf 82.8%

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

      1. associate-*r/ 82.8%

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

      2. neg-mul-1 82.8%

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

   5. Simplified 82.8%

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

   6. Taylor expanded in y around inf 57.4%

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

      1. div-inv 57.4%

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

      2. add-sqr-sqrt 41.6%

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

      3. sqrt-unprod 54.2%

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

      4. sqr-neg 54.2%

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

      5. sqrt-unprod 16.0%

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

      6. add-sqr-sqrt 57.1%

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

      7. *-commutative 57.1%

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

   8. Applied egg-rr 57.1%

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

      1. associate-*r/ 57.1%

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

      2. *-rgt-identity 57.1%

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

      3. associate-/l/ 57.1%

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

   10. Simplified 57.1%

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

   11. Taylor expanded in x around 0 86.8%

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

   if -2.7999999999999999e-122 < z

   1. Initial program 98.9%

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

   3. Taylor expanded in t around inf 82.3%

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

      1. associate-*r/ 82.3%

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

      2. neg-mul-1 82.3%

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

   5. Simplified 82.3%

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

   6. Taylor expanded in y around inf 60.4%

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

      1. div-inv 60.4%

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

      2. add-sqr-sqrt 26.8%

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

      3. sqrt-unprod 51.5%

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

      4. sqr-neg 51.5%

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

      5. sqrt-unprod 31.3%

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

      6. add-sqr-sqrt 51.4%

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

      7. *-commutative 51.4%

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

   8. Applied egg-rr 51.4%

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

      1. associate-*r/ 51.4%

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

      2. *-rgt-identity 51.4%

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

      3. associate-/l/ 51.4%

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

   10. Simplified 51.4%

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

       1. sub-neg 51.4%

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

       2. distribute-neg-frac 51.4%

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

       3. distribute-neg-frac2 51.4%

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

       4. associate-/r* 51.4%

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

       5. associate-/l/ 51.4%

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

       6. add-sqr-sqrt 30.4%

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

       7. sqrt-unprod 53.5%

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

       8. sqr-neg 53.5%

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

       9. sqrt-unprod 27.1%

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

       10. add-sqr-sqrt 60.9%

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

       11. associate-/r* 60.4%

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

   12. Applied egg-rr 60.4%

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

       1. associate-/r* 60.9%

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

   14. Simplified 60.9%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{-122}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{x}{y}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{-125}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -7e-125) 1.0 (+ 1.0 (/ x (* y t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -7e-125) {
		tmp = 1.0;
	} else {
		tmp = 1.0 + (x / (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 (z <= (-7d-125)) then
        tmp = 1.0d0
    else
        tmp = 1.0d0 + (x / (y * t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -7e-125) {
		tmp = 1.0;
	} else {
		tmp = 1.0 + (x / (y * t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -7e-125:
		tmp = 1.0
	else:
		tmp = 1.0 + (x / (y * t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -7e-125)
		tmp = 1.0;
	else
		tmp = Float64(1.0 + Float64(x / Float64(y * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -7e-125)
		tmp = 1.0;
	else
		tmp = 1.0 + (x / (y * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -7e-125], 1.0, N[(1.0 + N[(x / N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.99999999999999995e-125

   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 inf 82.8%

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

      1. associate-*r/ 82.8%

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

      2. neg-mul-1 82.8%

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

   5. Simplified 82.8%

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

   6. Taylor expanded in y around inf 57.4%

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

      1. div-inv 57.4%

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

      2. add-sqr-sqrt 41.6%

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

      3. sqrt-unprod 54.2%

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

      4. sqr-neg 54.2%

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

      5. sqrt-unprod 16.0%

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

      6. add-sqr-sqrt 57.1%

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

      7. *-commutative 57.1%

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

   8. Applied egg-rr 57.1%

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

      1. associate-*r/ 57.1%

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

      2. *-rgt-identity 57.1%

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

      3. associate-/l/ 57.1%

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

   10. Simplified 57.1%

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

   11. Taylor expanded in x around 0 86.8%

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

   if -6.99999999999999995e-125 < z

   1. Initial program 98.9%

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

   3. Taylor expanded in t around inf 82.3%

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

      1. associate-*r/ 82.3%

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

      2. neg-mul-1 82.3%

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

   5. Simplified 82.3%

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

   6. Taylor expanded in y around inf 60.4%

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

      1. div-inv 60.4%

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

      2. add-sqr-sqrt 26.8%

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

      3. sqrt-unprod 51.5%

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

      4. sqr-neg 51.5%

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

      5. sqrt-unprod 31.3%

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

      6. add-sqr-sqrt 51.4%

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

      7. *-commutative 51.4%

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

   8. Applied egg-rr 51.4%

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

      1. associate-*r/ 51.4%

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

      2. *-rgt-identity 51.4%

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

      3. associate-/l/ 51.4%

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

   10. Simplified 51.4%

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

       1. sub-neg 51.4%

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

       2. distribute-neg-frac 51.4%

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

       3. distribute-neg-frac2 51.4%

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

       4. associate-/r* 51.4%

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

       5. associate-/l/ 51.4%

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

       6. +-commutative 51.4%

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

       7. add-sqr-sqrt 30.4%

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

       8. sqrt-unprod 53.5%

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

       9. sqr-neg 53.5%

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

       10. sqrt-unprod 27.1%

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

       11. add-sqr-sqrt 60.9%

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

       12. associate-/r* 60.4%

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

   12. Applied egg-rr 60.4%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{-125}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 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)))))