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

Percentage Accurate: 99.1% → 98.4%

Time: 11.9s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = 1.0d0 + ((x / (y - t)) / (z - y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(1.0 + N[(N[(x / N[(y - t), $MachinePrecision]), $MachinePrecision] / N[(z - y), $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. Taylor expanded in x around 0 98.8%

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

   1. associate-/r* 98.1%

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

5. Simplified 98.1%

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

6. Final simplification 98.1%

   \[\leadsto 1 + \frac{\frac{x}{y - t}}{z - y} \]
7. Add Preprocessing

Alternative 2: 89.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{-49} \lor \neg \left(y \leq 8 \cdot 10^{-40}\right):\\ \;\;\;\;1 + \frac{x}{y \cdot \left(t - y\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{t \cdot \left(z - y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -7.5e-49) (not (<= y 8e-40)))
   (+ 1.0 (/ x (* y (- t y))))
   (- 1.0 (/ x (* t (- z y))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -7.5e-49) || !(y <= 8e-40)) {
		tmp = 1.0 + (x / (y * (t - y)));
	} else {
		tmp = 1.0 - (x / (t * (z - y)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((y <= (-7.5d-49)) .or. (.not. (y <= 8d-40))) then
        tmp = 1.0d0 + (x / (y * (t - y)))
    else
        tmp = 1.0d0 - (x / (t * (z - y)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -7.5e-49) || !(y <= 8e-40)) {
		tmp = 1.0 + (x / (y * (t - y)));
	} else {
		tmp = 1.0 - (x / (t * (z - y)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if (y <= -7.5e-49) or not (y <= 8e-40):
		tmp = 1.0 + (x / (y * (t - y)))
	else:
		tmp = 1.0 - (x / (t * (z - y)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -7.5e-49) || !(y <= 8e-40))
		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

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y <= -7.5e-49) || ~((y <= 8e-40)))
		tmp = 1.0 + (x / (y * (t - y)));
	else
		tmp = 1.0 - (x / (t * (z - y)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[Or[LessEqual[y, -7.5e-49], N[Not[LessEqual[y, 8e-40]], $MachinePrecision]], 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 2 regimes

2. if y < -7.4999999999999998e-49 or 7.9999999999999994e-40 < 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 92.3%

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

   if -7.4999999999999998e-49 < y < 7.9999999999999994e-40

   1. Initial program 97.0%

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

      1. *-commutative 97.0%

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

      2. sub-neg 97.0%

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

      3. distribute-lft-in 97.0%

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

   4. Applied egg-rr 97.0%

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

   5. Taylor expanded in t around inf 84.7%

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

      1. neg-mul-1 84.7%

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

      2. unsub-neg 84.7%

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

   7. Simplified 84.7%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{-49} \lor \neg \left(y \leq 8 \cdot 10^{-40}\right):\\ \;\;\;\;1 + \frac{x}{y \cdot \left(t - y\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{t \cdot \left(z - y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -8.6 \cdot 10^{-138} \lor \neg \left(t \leq 2.65 \cdot 10^{-83}\right):\\ \;\;\;\;1 - \frac{x}{t \cdot \left(z - y\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y \cdot \left(z - y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -8.6e-138) (not (<= t 2.65e-83)))
   (- 1.0 (/ x (* t (- z y))))
   (+ 1.0 (/ x (* y (- z y))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -8.6e-138) || !(t <= 2.65e-83)) {
		tmp = 1.0 - (x / (t * (z - y)));
	} else {
		tmp = 1.0 + (x / (y * (z - y)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-8.6d-138)) .or. (.not. (t <= 2.65d-83))) then
        tmp = 1.0d0 - (x / (t * (z - y)))
    else
        tmp = 1.0d0 + (x / (y * (z - y)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -8.6e-138) || !(t <= 2.65e-83)) {
		tmp = 1.0 - (x / (t * (z - y)));
	} else {
		tmp = 1.0 + (x / (y * (z - y)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if (t <= -8.6e-138) or not (t <= 2.65e-83):
		tmp = 1.0 - (x / (t * (z - y)))
	else:
		tmp = 1.0 + (x / (y * (z - y)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -8.6e-138) || !(t <= 2.65e-83))
		tmp = Float64(1.0 - Float64(x / Float64(t * Float64(z - y))));
	else
		tmp = Float64(1.0 + Float64(x / Float64(y * Float64(z - y))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -8.6e-138) || ~((t <= 2.65e-83)))
		tmp = 1.0 - (x / (t * (z - y)));
	else
		tmp = 1.0 + (x / (y * (z - y)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[Or[LessEqual[t, -8.6e-138], N[Not[LessEqual[t, 2.65e-83]], $MachinePrecision]], N[(1.0 - N[(x / N[(t * N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(x / N[(y * N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -8.6000000000000001e-138 or 2.65e-83 < t

   1. Initial program 99.4%

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

      1. *-commutative 99.4%

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

      2. sub-neg 99.4%

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

      3. distribute-lft-in 89.5%

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

   4. Applied egg-rr 89.5%

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

   5. Taylor expanded in t around inf 92.8%

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

      1. neg-mul-1 92.8%

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

      2. unsub-neg 92.8%

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

   7. Simplified 92.8%

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

   if -8.6000000000000001e-138 < t < 2.65e-83

   1. Initial program 97.2%

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

   3. Taylor expanded in t around 0 81.5%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8.6 \cdot 10^{-138} \lor \neg \left(t \leq 2.65 \cdot 10^{-83}\right):\\ \;\;\;\;1 - \frac{x}{t \cdot \left(z - y\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y \cdot \left(z - y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{-103}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{-18}:\\ \;\;\;\;1 + \frac{x}{t} \cdot \frac{-1}{z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y -1.3e-103)
   1.0
   (if (<= y 1.12e-18) (+ 1.0 (* (/ x t) (/ -1.0 z))) 1.0)))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.3e-103) {
		tmp = 1.0;
	} else if (y <= 1.12e-18) {
		tmp = 1.0 + ((x / t) * (-1.0 / z));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-1.3d-103)) then
        tmp = 1.0d0
    else if (y <= 1.12d-18) then
        tmp = 1.0d0 + ((x / t) * ((-1.0d0) / z))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.3e-103) {
		tmp = 1.0;
	} else if (y <= 1.12e-18) {
		tmp = 1.0 + ((x / t) * (-1.0 / z));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= -1.3e-103:
		tmp = 1.0
	elif y <= 1.12e-18:
		tmp = 1.0 + ((x / t) * (-1.0 / z))
	else:
		tmp = 1.0
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= -1.3e-103)
		tmp = 1.0;
	elseif (y <= 1.12e-18)
		tmp = Float64(1.0 + Float64(Float64(x / t) * Float64(-1.0 / z)));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -1.3e-103)
		tmp = 1.0;
	elseif (y <= 1.12e-18)
		tmp = 1.0 + ((x / t) * (-1.0 / z));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, -1.3e-103], 1.0, If[LessEqual[y, 1.12e-18], N[(1.0 + N[(N[(x / t), $MachinePrecision] * N[(-1.0 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -1.29999999999999998e-103 or 1.12000000000000001e-18 < 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 90.5%

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

      1. *-commutative 90.5%

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

      2. associate-/r* 90.5%

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

   5. Simplified 90.5%

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

   6. Taylor expanded in y around 0 72.3%

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

      1. mul-1-neg 72.3%

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

      2. distribute-neg-frac2 72.3%

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

   8. Simplified 72.3%

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

      1. add0 72.3%

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

      2. associate-/l/ 72.3%

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

      3. add-sqr-sqrt 40.5%

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

      4. sqrt-unprod 64.5%

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

      5. sqr-neg 64.5%

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

      6. sqrt-unprod 30.9%

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

      7. add-sqr-sqrt 69.5%

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

   10. Applied egg-rr 69.5%

       \[\leadsto 1 - \color{blue}{\left(\frac{x}{y \cdot t} + 0\right)} \]
   11. Step-by-step derivation

       1. associate-/r* 69.5%

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

       2. add0 69.5%

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

   12. Simplified 69.5%

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

   13. Taylor expanded in x around 0 87.6%

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

   if -1.29999999999999998e-103 < y < 1.12000000000000001e-18

   1. Initial program 96.6%

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

   3. Taylor expanded in y around 0 76.0%

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

      1. associate-/r* 74.8%

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

      2. div-inv 74.8%

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

   5. Applied egg-rr 74.8%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{-103}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{-18}:\\ \;\;\;\;1 + \frac{x}{t} \cdot \frac{-1}{z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+16}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 8.4 \cdot 10^{+56}:\\ \;\;\;\;1 - \frac{x}{t \cdot \left(z - y\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y -3.5e+16)
   1.0
   (if (<= y 8.4e+56) (- 1.0 (/ x (* t (- z y)))) (- 1.0 (/ (/ x y) y)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -3.5e+16) {
		tmp = 1.0;
	} else if (y <= 8.4e+56) {
		tmp = 1.0 - (x / (t * (z - y)));
	} else {
		tmp = 1.0 - ((x / y) / y);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-3.5d+16)) then
        tmp = 1.0d0
    else if (y <= 8.4d+56) then
        tmp = 1.0d0 - (x / (t * (z - y)))
    else
        tmp = 1.0d0 - ((x / y) / y)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -3.5e+16) {
		tmp = 1.0;
	} else if (y <= 8.4e+56) {
		tmp = 1.0 - (x / (t * (z - y)));
	} else {
		tmp = 1.0 - ((x / y) / y);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= -3.5e+16:
		tmp = 1.0
	elif y <= 8.4e+56:
		tmp = 1.0 - (x / (t * (z - y)))
	else:
		tmp = 1.0 - ((x / y) / y)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= -3.5e+16)
		tmp = 1.0;
	elseif (y <= 8.4e+56)
		tmp = Float64(1.0 - Float64(x / Float64(t * Float64(z - y))));
	else
		tmp = Float64(1.0 - Float64(Float64(x / y) / y));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -3.5e+16)
		tmp = 1.0;
	elseif (y <= 8.4e+56)
		tmp = 1.0 - (x / (t * (z - y)));
	else
		tmp = 1.0 - ((x / y) / y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, -3.5e+16], 1.0, If[LessEqual[y, 8.4e+56], N[(1.0 - N[(x / N[(t * N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.5e16

   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 95.3%

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

      1. *-commutative 95.3%

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

      2. associate-/r* 95.3%

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

   5. Simplified 95.3%

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

   6. Taylor expanded in y around 0 76.1%

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

      1. mul-1-neg 76.1%

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

      2. distribute-neg-frac2 76.1%

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

   8. Simplified 76.1%

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

      1. add0 76.1%

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

      2. associate-/l/ 76.2%

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

      3. add-sqr-sqrt 38.1%

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

      4. sqrt-unprod 64.3%

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

      5. sqr-neg 64.3%

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

      6. sqrt-unprod 38.2%

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

      7. add-sqr-sqrt 75.7%

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

   10. Applied egg-rr 75.7%

       \[\leadsto 1 - \color{blue}{\left(\frac{x}{y \cdot t} + 0\right)} \]
   11. Step-by-step derivation

       1. associate-/r* 75.7%

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

       2. add0 75.7%

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

   12. Simplified 75.7%

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

   13. Taylor expanded in x around 0 96.4%

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

   if -3.5e16 < y < 8.40000000000000068e56

   1. Initial program 97.7%

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

      1. *-commutative 97.7%

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

      2. sub-neg 97.7%

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

      3. distribute-lft-in 97.7%

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

   4. Applied egg-rr 97.7%

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

   5. Taylor expanded in t around inf 80.3%

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

      1. neg-mul-1 80.3%

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

      2. unsub-neg 80.3%

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

   7. Simplified 80.3%

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

   if 8.40000000000000068e56 < 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 0 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/r* 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around inf 100.0%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+16}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 8.4 \cdot 10^{+56}:\\ \;\;\;\;1 - \frac{x}{t \cdot \left(z - y\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{x}{y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -8.6 \cdot 10^{-138}:\\ \;\;\;\;1 - \frac{\frac{x}{t}}{y + z}\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{-83}:\\ \;\;\;\;1 + \frac{x}{y \cdot \left(z - y\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{t \cdot \left(z - y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= t -8.6e-138)
   (- 1.0 (/ (/ x t) (+ y z)))
   (if (<= t 2.1e-83)
     (+ 1.0 (/ x (* y (- z y))))
     (- 1.0 (/ x (* t (- z y)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -8.6e-138) {
		tmp = 1.0 - ((x / t) / (y + z));
	} else if (t <= 2.1e-83) {
		tmp = 1.0 + (x / (y * (z - y)));
	} else {
		tmp = 1.0 - (x / (t * (z - y)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-8.6d-138)) then
        tmp = 1.0d0 - ((x / t) / (y + z))
    else if (t <= 2.1d-83) then
        tmp = 1.0d0 + (x / (y * (z - y)))
    else
        tmp = 1.0d0 - (x / (t * (z - y)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -8.6e-138) {
		tmp = 1.0 - ((x / t) / (y + z));
	} else if (t <= 2.1e-83) {
		tmp = 1.0 + (x / (y * (z - y)));
	} else {
		tmp = 1.0 - (x / (t * (z - y)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if t <= -8.6e-138:
		tmp = 1.0 - ((x / t) / (y + z))
	elif t <= 2.1e-83:
		tmp = 1.0 + (x / (y * (z - y)))
	else:
		tmp = 1.0 - (x / (t * (z - y)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (t <= -8.6e-138)
		tmp = Float64(1.0 - Float64(Float64(x / t) / Float64(y + z)));
	elseif (t <= 2.1e-83)
		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

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -8.6e-138)
		tmp = 1.0 - ((x / t) / (y + z));
	elseif (t <= 2.1e-83)
		tmp = 1.0 + (x / (y * (z - y)));
	else
		tmp = 1.0 - (x / (t * (z - y)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[t, -8.6e-138], N[(1.0 - N[(N[(x / t), $MachinePrecision] / N[(y + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.1e-83], 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 3 regimes

2. if t < -8.6000000000000001e-138

   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 91.9%

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

      1. associate-*r/ 91.9%

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

      2. neg-mul-1 91.9%

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

   5. Simplified 91.9%

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

      1. add0 91.9%

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

      2. associate-/r* 92.8%

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

      3. add-sqr-sqrt 55.2%

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

      4. sqrt-unprod 69.9%

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

      5. sqr-neg 69.9%

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

      6. sqrt-unprod 32.5%

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

      7. add-sqr-sqrt 77.6%

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

      8. sub-neg 77.6%

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

      9. add-sqr-sqrt 43.1%

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

      10. sqrt-unprod 81.4%

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

      11. sqr-neg 81.4%

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

      12. sqrt-unprod 38.6%

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

      13. add-sqr-sqrt 87.2%

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

   7. Applied egg-rr 87.2%

      \[\leadsto 1 - \color{blue}{\left(\frac{\frac{x}{t}}{y + z} + 0\right)} \]
   8. Step-by-step derivation

      1. add0 87.2%

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

      2. +-commutative 87.2%

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

   9. Simplified 87.2%

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

   if -8.6000000000000001e-138 < t < 2.0999999999999999e-83

   1. Initial program 97.2%

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

   3. Taylor expanded in t around 0 81.5%

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

   if 2.0999999999999999e-83 < t

   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. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. distribute-lft-in 83.9%

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

   4. Applied egg-rr 83.9%

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

   5. Taylor expanded in t around inf 94.1%

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

      1. neg-mul-1 94.1%

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

      2. unsub-neg 94.1%

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

   7. Simplified 94.1%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8.6 \cdot 10^{-138}:\\ \;\;\;\;1 - \frac{\frac{x}{t}}{y + z}\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{-83}:\\ \;\;\;\;1 + \frac{x}{y \cdot \left(z - y\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{t \cdot \left(z - y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{-145}:\\ \;\;\;\;1 + \frac{x}{\left(y - t\right) \cdot z}\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{-206}:\\ \;\;\;\;1 + \frac{\frac{x}{t - y}}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{t \cdot \left(z - y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.7e-145)
   (+ 1.0 (/ x (* (- y t) z)))
   (if (<= z 3.5e-206)
     (+ 1.0 (/ (/ x (- t y)) y))
     (- 1.0 (/ x (* t (- z y)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.7e-145) {
		tmp = 1.0 + (x / ((y - t) * z));
	} else if (z <= 3.5e-206) {
		tmp = 1.0 + ((x / (t - y)) / y);
	} else {
		tmp = 1.0 - (x / (t * (z - y)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-1.7d-145)) then
        tmp = 1.0d0 + (x / ((y - t) * z))
    else if (z <= 3.5d-206) then
        tmp = 1.0d0 + ((x / (t - y)) / y)
    else
        tmp = 1.0d0 - (x / (t * (z - y)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.7e-145) {
		tmp = 1.0 + (x / ((y - t) * z));
	} else if (z <= 3.5e-206) {
		tmp = 1.0 + ((x / (t - y)) / y);
	} else {
		tmp = 1.0 - (x / (t * (z - y)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= -1.7e-145:
		tmp = 1.0 + (x / ((y - t) * z))
	elif z <= 3.5e-206:
		tmp = 1.0 + ((x / (t - y)) / y)
	else:
		tmp = 1.0 - (x / (t * (z - y)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.7e-145)
		tmp = Float64(1.0 + Float64(x / Float64(Float64(y - t) * z)));
	elseif (z <= 3.5e-206)
		tmp = Float64(1.0 + Float64(Float64(x / Float64(t - y)) / y));
	else
		tmp = Float64(1.0 - Float64(x / Float64(t * Float64(z - y))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -1.7e-145)
		tmp = 1.0 + (x / ((y - t) * z));
	elseif (z <= 3.5e-206)
		tmp = 1.0 + ((x / (t - y)) / y);
	else
		tmp = 1.0 - (x / (t * (z - y)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[z, -1.7e-145], N[(1.0 + N[(x / N[(N[(y - t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.5e-206], N[(1.0 + N[(N[(x / N[(t - y), $MachinePrecision]), $MachinePrecision] / y), $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 < -1.6999999999999999e-145

   1. Initial program 99.8%

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

   3. Taylor expanded in z around inf 94.5%

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

      1. associate-*r/ 94.5%

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

      2. neg-mul-1 94.5%

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

      3. *-commutative 94.5%

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

   5. Simplified 94.5%

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

   if -1.6999999999999999e-145 < z < 3.49999999999999989e-206

   1. Initial program 95.3%

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

   3. Taylor expanded in z around 0 95.3%

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

      1. *-commutative 95.3%

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

      2. associate-/r* 96.2%

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

   5. Simplified 96.2%

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

   if 3.49999999999999989e-206 < z

   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. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. distribute-lft-in 92.2%

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

   4. Applied egg-rr 92.2%

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

   5. Taylor expanded in t around inf 74.5%

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

      1. neg-mul-1 74.5%

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

      2. unsub-neg 74.5%

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

   7. Simplified 74.5%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{-145}:\\ \;\;\;\;1 + \frac{x}{\left(y - t\right) \cdot z}\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{-206}:\\ \;\;\;\;1 + \frac{\frac{x}{t - y}}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{t \cdot \left(z - y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 92.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{-145}:\\ \;\;\;\;1 - \frac{-1}{y - t} \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{-208}:\\ \;\;\;\;1 + \frac{\frac{x}{t - y}}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{t \cdot \left(z - y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.7e-145)
   (- 1.0 (* (/ -1.0 (- y t)) (/ x z)))
   (if (<= z 5.8e-208)
     (+ 1.0 (/ (/ x (- t y)) y))
     (- 1.0 (/ x (* t (- z y)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.7e-145) {
		tmp = 1.0 - ((-1.0 / (y - t)) * (x / z));
	} else if (z <= 5.8e-208) {
		tmp = 1.0 + ((x / (t - y)) / y);
	} else {
		tmp = 1.0 - (x / (t * (z - y)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-1.7d-145)) then
        tmp = 1.0d0 - (((-1.0d0) / (y - t)) * (x / z))
    else if (z <= 5.8d-208) then
        tmp = 1.0d0 + ((x / (t - y)) / y)
    else
        tmp = 1.0d0 - (x / (t * (z - y)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.7e-145) {
		tmp = 1.0 - ((-1.0 / (y - t)) * (x / z));
	} else if (z <= 5.8e-208) {
		tmp = 1.0 + ((x / (t - y)) / y);
	} else {
		tmp = 1.0 - (x / (t * (z - y)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= -1.7e-145:
		tmp = 1.0 - ((-1.0 / (y - t)) * (x / z))
	elif z <= 5.8e-208:
		tmp = 1.0 + ((x / (t - y)) / y)
	else:
		tmp = 1.0 - (x / (t * (z - y)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.7e-145)
		tmp = Float64(1.0 - Float64(Float64(-1.0 / Float64(y - t)) * Float64(x / z)));
	elseif (z <= 5.8e-208)
		tmp = Float64(1.0 + Float64(Float64(x / Float64(t - y)) / y));
	else
		tmp = Float64(1.0 - Float64(x / Float64(t * Float64(z - y))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -1.7e-145)
		tmp = 1.0 - ((-1.0 / (y - t)) * (x / z));
	elseif (z <= 5.8e-208)
		tmp = 1.0 + ((x / (t - y)) / y);
	else
		tmp = 1.0 - (x / (t * (z - y)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[z, -1.7e-145], N[(1.0 - N[(N[(-1.0 / N[(y - t), $MachinePrecision]), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 5.8e-208], N[(1.0 + N[(N[(x / N[(t - y), $MachinePrecision]), $MachinePrecision] / y), $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 < -1.6999999999999999e-145

   1. Initial program 99.8%

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

   3. Taylor expanded in z around inf 94.5%

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

      1. associate-*r/ 94.5%

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

      2. neg-mul-1 94.5%

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

      3. *-commutative 94.5%

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

   5. Simplified 94.5%

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

      1. neg-mul-1 94.5%

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

      2. times-frac 94.5%

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

   7. Applied egg-rr 94.5%

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

   if -1.6999999999999999e-145 < z < 5.7999999999999999e-208

   1. Initial program 95.3%

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

   3. Taylor expanded in z around 0 95.3%

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

      1. *-commutative 95.3%

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

      2. associate-/r* 96.2%

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

   5. Simplified 96.2%

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

   if 5.7999999999999999e-208 < z

   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. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. distribute-lft-in 92.2%

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

   4. Applied egg-rr 92.2%

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

   5. Taylor expanded in t around inf 74.5%

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

      1. neg-mul-1 74.5%

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

      2. unsub-neg 74.5%

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

   7. Simplified 74.5%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{-145}:\\ \;\;\;\;1 - \frac{-1}{y - t} \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{-208}:\\ \;\;\;\;1 + \frac{\frac{x}{t - y}}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{t \cdot \left(z - y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 83.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{-110}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{-18}:\\ \;\;\;\;1 - \frac{x}{t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y -5.2e-110) 1.0 (if (<= y 1.12e-18) (- 1.0 (/ x (* t z))) 1.0)))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -5.2e-110) {
		tmp = 1.0;
	} else if (y <= 1.12e-18) {
		tmp = 1.0 - (x / (t * z));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-5.2d-110)) then
        tmp = 1.0d0
    else if (y <= 1.12d-18) then
        tmp = 1.0d0 - (x / (t * z))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -5.2e-110) {
		tmp = 1.0;
	} else if (y <= 1.12e-18) {
		tmp = 1.0 - (x / (t * z));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= -5.2e-110:
		tmp = 1.0
	elif y <= 1.12e-18:
		tmp = 1.0 - (x / (t * z))
	else:
		tmp = 1.0
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= -5.2e-110)
		tmp = 1.0;
	elseif (y <= 1.12e-18)
		tmp = Float64(1.0 - Float64(x / Float64(t * z)));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -5.2e-110)
		tmp = 1.0;
	elseif (y <= 1.12e-18)
		tmp = 1.0 - (x / (t * z));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, -5.2e-110], 1.0, If[LessEqual[y, 1.12e-18], N[(1.0 - N[(x / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -5.19999999999999979e-110 or 1.12000000000000001e-18 < 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 90.5%

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

      1. *-commutative 90.5%

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

      2. associate-/r* 90.5%

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

   5. Simplified 90.5%

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

   6. Taylor expanded in y around 0 72.3%

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

      1. mul-1-neg 72.3%

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

      2. distribute-neg-frac2 72.3%

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

   8. Simplified 72.3%

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

      1. add0 72.3%

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

      2. associate-/l/ 72.3%

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

      3. add-sqr-sqrt 40.5%

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

      4. sqrt-unprod 64.5%

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

      5. sqr-neg 64.5%

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

      6. sqrt-unprod 30.9%

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

      7. add-sqr-sqrt 69.5%

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

   10. Applied egg-rr 69.5%

       \[\leadsto 1 - \color{blue}{\left(\frac{x}{y \cdot t} + 0\right)} \]
   11. Step-by-step derivation

       1. associate-/r* 69.5%

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

       2. add0 69.5%

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

   12. Simplified 69.5%

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

   13. Taylor expanded in x around 0 87.6%

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

   if -5.19999999999999979e-110 < y < 1.12000000000000001e-18

   1. Initial program 96.6%

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

   3. Taylor expanded in y around 0 76.0%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{-110}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{-18}:\\ \;\;\;\;1 - \frac{x}{t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 75.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{-147}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y \cdot t}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.7e-147) 1.0 (+ 1.0 (/ x (* y t)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.7e-147) {
		tmp = 1.0;
	} else {
		tmp = 1.0 + (x / (y * t));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-1.7d-147)) then
        tmp = 1.0d0
    else
        tmp = 1.0d0 + (x / (y * t))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.7e-147) {
		tmp = 1.0;
	} else {
		tmp = 1.0 + (x / (y * t));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= -1.7e-147:
		tmp = 1.0
	else:
		tmp = 1.0 + (x / (y * t))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.7e-147)
		tmp = 1.0;
	else
		tmp = Float64(1.0 + Float64(x / Float64(y * t)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -1.7e-147)
		tmp = 1.0;
	else
		tmp = 1.0 + (x / (y * t));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[z, -1.7e-147], 1.0, N[(1.0 + N[(x / N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.69999999999999998e-147

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0 64.8%

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

      1. *-commutative 64.8%

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

      2. associate-/r* 64.8%

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

   5. Simplified 64.8%

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

   6. Taylor expanded in y around 0 57.3%

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

      1. mul-1-neg 57.3%

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

      2. distribute-neg-frac2 57.3%

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

   8. Simplified 57.3%

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

      1. add0 57.3%

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

      2. associate-/l/ 57.3%

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

      3. add-sqr-sqrt 31.3%

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

      4. sqrt-unprod 58.1%

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

      5. sqr-neg 58.1%

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

      6. sqrt-unprod 26.0%

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

      7. add-sqr-sqrt 56.4%

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

   10. Applied egg-rr 56.4%

       \[\leadsto 1 - \color{blue}{\left(\frac{x}{y \cdot t} + 0\right)} \]
   11. Step-by-step derivation

       1. associate-/r* 56.4%

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

       2. add0 56.4%

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

   12. Simplified 56.4%

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

   13. Taylor expanded in x around 0 77.8%

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

   if -1.69999999999999998e-147 < z

   1. Initial program 98.2%

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

   3. Taylor expanded in z around 0 74.9%

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

      1. *-commutative 74.9%

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

      2. associate-/r* 75.3%

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

   5. Simplified 75.3%

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

   6. Taylor expanded in y around 0 58.7%

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

      1. mul-1-neg 58.7%

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

      2. distribute-neg-frac2 58.7%

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

   8. Simplified 58.7%

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

      1. add0 58.7%

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

      2. associate-/l/ 58.7%

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

      3. add-sqr-sqrt 32.6%

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

      4. sqrt-unprod 48.6%

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

      5. sqr-neg 48.6%

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

      6. sqrt-unprod 21.7%

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

      7. add-sqr-sqrt 50.0%

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

   10. Applied egg-rr 50.0%

       \[\leadsto 1 - \color{blue}{\left(\frac{x}{y \cdot t} + 0\right)} \]
   11. Step-by-step derivation

       1. associate-/r* 50.1%

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

       2. add0 50.1%

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

   12. Simplified 50.1%

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

       1. sub-neg 50.1%

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

       2. associate-/l/ 50.0%

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

       3. distribute-neg-frac 50.0%

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

       4. add-sqr-sqrt 29.8%

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

       5. sqrt-unprod 49.8%

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

       6. sqr-neg 49.8%

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

       7. sqrt-unprod 25.4%

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

       8. add-sqr-sqrt 58.7%

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

   14. Applied egg-rr 58.7%

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

       1. +-commutative 58.7%

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

   16. Simplified 58.7%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{-147}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = 1.0d0 + (x / ((y - t) * (z - y)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(1.0 + N[(x / N[(N[(y - t), $MachinePrecision] * N[(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 12: 75.5% accurate, 11.0× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
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 z around 0 71.3%

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

   1. *-commutative 71.3%

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

   2. associate-/r* 71.6%

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

5. Simplified 71.6%

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

6. Taylor expanded in y around 0 58.2%

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

   1. mul-1-neg 58.2%

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

   2. distribute-neg-frac2 58.2%

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

8. Simplified 58.2%

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

   1. add0 58.2%

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

   2. associate-/l/ 58.2%

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

   3. add-sqr-sqrt 32.1%

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

   4. sqrt-unprod 52.0%

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

   5. sqr-neg 52.0%

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

   6. sqrt-unprod 23.2%

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

   7. add-sqr-sqrt 52.3%

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

10. Applied egg-rr 52.3%

    \[\leadsto 1 - \color{blue}{\left(\frac{x}{y \cdot t} + 0\right)} \]
11. Step-by-step derivation

    1. associate-/r* 52.3%

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

    2. add0 52.3%

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

12. Simplified 52.3%

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

13. Taylor expanded in x around 0 73.4%

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

14. Final simplification 73.4%

    \[\leadsto 1 \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024046 
(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)))))