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

Percentage Accurate: 99.1% → 98.9%

Time: 7.3s

Alternatives: 11

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: 98.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: 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 - (((1.0d0 / (y - t)) / (y - z)) * x)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. clear-num 99.3%

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

   2. associate-/r/ 99.3%

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

   3. *-commutative 99.3%

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

   4. associate-/r* 99.6%

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

3. Applied egg-rr 99.6%

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

4. Final simplification 99.6%

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

Alternative 2: 91.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -2.5 \cdot 10^{-146}:\\ \;\;\;\;1 + x \cdot \frac{\frac{-1}{t}}{z}\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{-98}:\\ \;\;\;\;1 + x \cdot \frac{\frac{-1}{y}}{y - z}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{x}{t}}{y - z}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= t -2.5e-146)
   (+ 1.0 (* x (/ (/ -1.0 t) z)))
   (if (<= t 2.7e-98)
     (+ 1.0 (* x (/ (/ -1.0 y) (- y z))))
     (+ 1.0 (/ (/ x t) (- y z))))))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -2.5e-146) {
		tmp = 1.0 + (x * ((-1.0 / t) / z));
	} else if (t <= 2.7e-98) {
		tmp = 1.0 + (x * ((-1.0 / y) / (y - z)));
	} else {
		tmp = 1.0 + ((x / t) / (y - z));
	}
	return tmp;
}

Fortran

NOTE: 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 <= (-2.5d-146)) then
        tmp = 1.0d0 + (x * (((-1.0d0) / t) / z))
    else if (t <= 2.7d-98) then
        tmp = 1.0d0 + (x * (((-1.0d0) / y) / (y - z)))
    else
        tmp = 1.0d0 + ((x / t) / (y - z))
    end if
    code = tmp
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -2.5e-146) {
		tmp = 1.0 + (x * ((-1.0 / t) / z));
	} else if (t <= 2.7e-98) {
		tmp = 1.0 + (x * ((-1.0 / y) / (y - z)));
	} else {
		tmp = 1.0 + ((x / t) / (y - z));
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	tmp = 0
	if t <= -2.5e-146:
		tmp = 1.0 + (x * ((-1.0 / t) / z))
	elif t <= 2.7e-98:
		tmp = 1.0 + (x * ((-1.0 / y) / (y - z)))
	else:
		tmp = 1.0 + ((x / t) / (y - z))
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (t <= -2.5e-146)
		tmp = Float64(1.0 + Float64(x * Float64(Float64(-1.0 / t) / z)));
	elseif (t <= 2.7e-98)
		tmp = Float64(1.0 + Float64(x * Float64(Float64(-1.0 / y) / Float64(y - z))));
	else
		tmp = Float64(1.0 + Float64(Float64(x / t) / Float64(y - z)));
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -2.5e-146)
		tmp = 1.0 + (x * ((-1.0 / t) / z));
	elseif (t <= 2.7e-98)
		tmp = 1.0 + (x * ((-1.0 / y) / (y - z)));
	else
		tmp = 1.0 + ((x / t) / (y - z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[t, -2.5e-146], N[(1.0 + N[(x * N[(N[(-1.0 / t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.7e-98], N[(1.0 + N[(x * N[(N[(-1.0 / y), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(x / t), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.49999999999999979e-146

   1. Initial program 99.3%

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

      1. clear-num 99.3%

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

      2. associate-/r/ 99.3%

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

      3. *-commutative 99.3%

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

      4. associate-/r* 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in y around 0 67.1%

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

      1. associate-/r* 67.1%

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

   6. Simplified 67.1%

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

   if -2.49999999999999979e-146 < t < 2.6999999999999999e-98

   1. Initial program 98.7%

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

      1. clear-num 98.6%

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

      2. associate-/r/ 98.7%

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

      3. *-commutative 98.7%

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

      4. associate-/r* 98.7%

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

   3. Applied egg-rr 98.7%

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

   4. Taylor expanded in t around 0 93.4%

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

      1. associate-/r* 93.4%

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

   6. Simplified 93.4%

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

   if 2.6999999999999999e-98 < t

   1. Initial program 99.9%

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

   2. Taylor expanded in t around inf 95.8%

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

      1. mul-1-neg 95.8%

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

      2. distribute-frac-neg 95.8%

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

      3. associate-/r* 94.7%

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

   4. Simplified 94.7%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.5 \cdot 10^{-146}:\\ \;\;\;\;1 + x \cdot \frac{\frac{-1}{t}}{z}\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{-98}:\\ \;\;\;\;1 + x \cdot \frac{\frac{-1}{y}}{y - z}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{x}{t}}{y - z}\\ \end{array} \]

Alternative 3: 86.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{-111} \lor \neg \left(y \leq 1.48 \cdot 10^{-120}\right):\\ \;\;\;\;1 - \frac{x}{y \cdot \left(y - z\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{t \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -1.8e-111) (not (<= y 1.48e-120)))
   (- 1.0 (/ x (* y (- y z))))
   (- 1.0 (/ x (* t z)))))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -1.8e-111) || !(y <= 1.48e-120)) {
		tmp = 1.0 - (x / (y * (y - z)));
	} else {
		tmp = 1.0 - (x / (t * z));
	}
	return tmp;
}

Fortran

NOTE: 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 <= (-1.8d-111)) .or. (.not. (y <= 1.48d-120))) then
        tmp = 1.0d0 - (x / (y * (y - z)))
    else
        tmp = 1.0d0 - (x / (t * z))
    end if
    code = tmp
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -1.8e-111) || !(y <= 1.48e-120)) {
		tmp = 1.0 - (x / (y * (y - z)));
	} else {
		tmp = 1.0 - (x / (t * z));
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	tmp = 0
	if (y <= -1.8e-111) or not (y <= 1.48e-120):
		tmp = 1.0 - (x / (y * (y - z)))
	else:
		tmp = 1.0 - (x / (t * z))
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -1.8e-111) || !(y <= 1.48e-120))
		tmp = Float64(1.0 - Float64(x / Float64(y * Float64(y - z))));
	else
		tmp = Float64(1.0 - Float64(x / Float64(t * z)));
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y <= -1.8e-111) || ~((y <= 1.48e-120)))
		tmp = 1.0 - (x / (y * (y - z)));
	else
		tmp = 1.0 - (x / (t * z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[Or[LessEqual[y, -1.8e-111], N[Not[LessEqual[y, 1.48e-120]], $MachinePrecision]], N[(1.0 - N[(x / N[(y * N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(x / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.80000000000000005e-111 or 1.4800000000000001e-120 < y

   1. Initial program 99.6%

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

   2. Taylor expanded in t around 0 91.5%

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

   if -1.80000000000000005e-111 < y < 1.4800000000000001e-120

   1. Initial program 98.9%

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

   2. Taylor expanded in y around 0 83.0%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{-111} \lor \neg \left(y \leq 1.48 \cdot 10^{-120}\right):\\ \;\;\;\;1 - \frac{x}{y \cdot \left(y - z\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{t \cdot z}\\ \end{array} \]

Alternative 4: 86.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{-110}:\\ \;\;\;\;1 - \frac{x}{y \cdot \left(y - z\right)}\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{-137}:\\ \;\;\;\;1 - \frac{x}{t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y \cdot \left(y - t\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y -2.7e-110)
   (- 1.0 (/ x (* y (- y z))))
   (if (<= y 3.9e-137) (- 1.0 (/ x (* t z))) (- 1.0 (/ x (* y (- y t)))))))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -2.7e-110) {
		tmp = 1.0 - (x / (y * (y - z)));
	} else if (y <= 3.9e-137) {
		tmp = 1.0 - (x / (t * z));
	} else {
		tmp = 1.0 - (x / (y * (y - t)));
	}
	return tmp;
}

Fortran

NOTE: 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 <= (-2.7d-110)) then
        tmp = 1.0d0 - (x / (y * (y - z)))
    else if (y <= 3.9d-137) then
        tmp = 1.0d0 - (x / (t * z))
    else
        tmp = 1.0d0 - (x / (y * (y - t)))
    end if
    code = tmp
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -2.7e-110) {
		tmp = 1.0 - (x / (y * (y - z)));
	} else if (y <= 3.9e-137) {
		tmp = 1.0 - (x / (t * z));
	} else {
		tmp = 1.0 - (x / (y * (y - t)));
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= -2.7e-110:
		tmp = 1.0 - (x / (y * (y - z)))
	elif y <= 3.9e-137:
		tmp = 1.0 - (x / (t * z))
	else:
		tmp = 1.0 - (x / (y * (y - t)))
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= -2.7e-110)
		tmp = Float64(1.0 - Float64(x / Float64(y * Float64(y - z))));
	elseif (y <= 3.9e-137)
		tmp = Float64(1.0 - Float64(x / Float64(t * z)));
	else
		tmp = Float64(1.0 - Float64(x / Float64(y * Float64(y - t))));
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -2.7e-110)
		tmp = 1.0 - (x / (y * (y - z)));
	elseif (y <= 3.9e-137)
		tmp = 1.0 - (x / (t * z));
	else
		tmp = 1.0 - (x / (y * (y - t)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, -2.7e-110], N[(1.0 - N[(x / N[(y * N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.9e-137], N[(1.0 - N[(x / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(x / N[(y * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.6999999999999998e-110

   1. Initial program 99.9%

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

   2. Taylor expanded in t around 0 92.9%

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

   if -2.6999999999999998e-110 < y < 3.8999999999999999e-137

   1. Initial program 98.8%

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

   2. Taylor expanded in y around 0 84.3%

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

   if 3.8999999999999999e-137 < y

   1. Initial program 99.3%

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

   2. Taylor expanded in z around 0 85.6%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{-110}:\\ \;\;\;\;1 - \frac{x}{y \cdot \left(y - z\right)}\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{-137}:\\ \;\;\;\;1 - \frac{x}{t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y \cdot \left(y - t\right)}\\ \end{array} \]

Alternative 5: 86.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{-109}:\\ \;\;\;\;1 - \frac{x}{y \cdot \left(y - z\right)}\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-137}:\\ \;\;\;\;1 - \frac{x}{t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{x}{y}}{y - t}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y -1.05e-109)
   (- 1.0 (/ x (* y (- y z))))
   (if (<= y 9.5e-137) (- 1.0 (/ x (* t z))) (- 1.0 (/ (/ x y) (- y t))))))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.05e-109) {
		tmp = 1.0 - (x / (y * (y - z)));
	} else if (y <= 9.5e-137) {
		tmp = 1.0 - (x / (t * z));
	} else {
		tmp = 1.0 - ((x / y) / (y - t));
	}
	return tmp;
}

Fortran

NOTE: 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 <= (-1.05d-109)) then
        tmp = 1.0d0 - (x / (y * (y - z)))
    else if (y <= 9.5d-137) then
        tmp = 1.0d0 - (x / (t * z))
    else
        tmp = 1.0d0 - ((x / y) / (y - t))
    end if
    code = tmp
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.05e-109) {
		tmp = 1.0 - (x / (y * (y - z)));
	} else if (y <= 9.5e-137) {
		tmp = 1.0 - (x / (t * z));
	} else {
		tmp = 1.0 - ((x / y) / (y - t));
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= -1.05e-109:
		tmp = 1.0 - (x / (y * (y - z)))
	elif y <= 9.5e-137:
		tmp = 1.0 - (x / (t * z))
	else:
		tmp = 1.0 - ((x / y) / (y - t))
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= -1.05e-109)
		tmp = Float64(1.0 - Float64(x / Float64(y * Float64(y - z))));
	elseif (y <= 9.5e-137)
		tmp = Float64(1.0 - Float64(x / Float64(t * z)));
	else
		tmp = Float64(1.0 - Float64(Float64(x / y) / Float64(y - t)));
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -1.05e-109)
		tmp = 1.0 - (x / (y * (y - z)));
	elseif (y <= 9.5e-137)
		tmp = 1.0 - (x / (t * z));
	else
		tmp = 1.0 - ((x / y) / (y - t));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, -1.05e-109], N[(1.0 - N[(x / N[(y * N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9.5e-137], N[(1.0 - N[(x / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(x / y), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.04999999999999998e-109

   1. Initial program 99.9%

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

   2. Taylor expanded in t around 0 92.9%

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

   if -1.04999999999999998e-109 < y < 9.5000000000000007e-137

   1. Initial program 98.8%

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

   2. Taylor expanded in y around 0 84.3%

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

   if 9.5000000000000007e-137 < y

   1. Initial program 99.3%

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

   2. Taylor expanded in z around 0 85.6%

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

      1. associate-/l/ 85.7%

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

   4. Simplified 85.7%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{-109}:\\ \;\;\;\;1 - \frac{x}{y \cdot \left(y - z\right)}\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-137}:\\ \;\;\;\;1 - \frac{x}{t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{x}{y}}{y - t}\\ \end{array} \]

Alternative 6: 91.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -1.56 \cdot 10^{-145}:\\ \;\;\;\;1 + x \cdot \frac{\frac{-1}{t}}{z}\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{-97}:\\ \;\;\;\;1 - \frac{x}{y \cdot \left(y - z\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{x}{t}}{y - z}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= t -1.56e-145)
   (+ 1.0 (* x (/ (/ -1.0 t) z)))
   (if (<= t 1.7e-97)
     (- 1.0 (/ x (* y (- y z))))
     (+ 1.0 (/ (/ x t) (- y z))))))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.56e-145) {
		tmp = 1.0 + (x * ((-1.0 / t) / z));
	} else if (t <= 1.7e-97) {
		tmp = 1.0 - (x / (y * (y - z)));
	} else {
		tmp = 1.0 + ((x / t) / (y - z));
	}
	return tmp;
}

Fortran

NOTE: 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 <= (-1.56d-145)) then
        tmp = 1.0d0 + (x * (((-1.0d0) / t) / z))
    else if (t <= 1.7d-97) then
        tmp = 1.0d0 - (x / (y * (y - z)))
    else
        tmp = 1.0d0 + ((x / t) / (y - z))
    end if
    code = tmp
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.56e-145) {
		tmp = 1.0 + (x * ((-1.0 / t) / z));
	} else if (t <= 1.7e-97) {
		tmp = 1.0 - (x / (y * (y - z)));
	} else {
		tmp = 1.0 + ((x / t) / (y - z));
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	tmp = 0
	if t <= -1.56e-145:
		tmp = 1.0 + (x * ((-1.0 / t) / z))
	elif t <= 1.7e-97:
		tmp = 1.0 - (x / (y * (y - z)))
	else:
		tmp = 1.0 + ((x / t) / (y - z))
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (t <= -1.56e-145)
		tmp = Float64(1.0 + Float64(x * Float64(Float64(-1.0 / t) / z)));
	elseif (t <= 1.7e-97)
		tmp = Float64(1.0 - Float64(x / Float64(y * Float64(y - z))));
	else
		tmp = Float64(1.0 + Float64(Float64(x / t) / Float64(y - z)));
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -1.56e-145)
		tmp = 1.0 + (x * ((-1.0 / t) / z));
	elseif (t <= 1.7e-97)
		tmp = 1.0 - (x / (y * (y - z)));
	else
		tmp = 1.0 + ((x / t) / (y - z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[t, -1.56e-145], N[(1.0 + N[(x * N[(N[(-1.0 / t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.7e-97], N[(1.0 - N[(x / N[(y * N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(x / t), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.56000000000000001e-145

   1. Initial program 99.3%

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

      1. clear-num 99.3%

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

      2. associate-/r/ 99.3%

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

      3. *-commutative 99.3%

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

      4. associate-/r* 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in y around 0 67.1%

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

      1. associate-/r* 67.1%

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

   6. Simplified 67.1%

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

   if -1.56000000000000001e-145 < t < 1.6999999999999999e-97

   1. Initial program 98.7%

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

   2. Taylor expanded in t around 0 93.4%

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

   if 1.6999999999999999e-97 < t

   1. Initial program 99.9%

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

   2. Taylor expanded in t around inf 95.8%

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

      1. mul-1-neg 95.8%

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

      2. distribute-frac-neg 95.8%

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

      3. associate-/r* 94.7%

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

   4. Simplified 94.7%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.56 \cdot 10^{-145}:\\ \;\;\;\;1 + x \cdot \frac{\frac{-1}{t}}{z}\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{-97}:\\ \;\;\;\;1 - \frac{x}{y \cdot \left(y - z\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{x}{t}}{y - z}\\ \end{array} \]

Alternative 7: 71.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{+19} \lor \neg \left(y \leq 1.85 \cdot 10^{-17}\right):\\ \;\;\;\;1 - \frac{x}{y \cdot t}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{t \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -1.65e+19) (not (<= y 1.85e-17)))
   (- 1.0 (/ x (* y t)))
   (- 1.0 (/ x (* t z)))))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -1.65e+19) || !(y <= 1.85e-17)) {
		tmp = 1.0 - (x / (y * t));
	} else {
		tmp = 1.0 - (x / (t * z));
	}
	return tmp;
}

Fortran

NOTE: 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 <= (-1.65d+19)) .or. (.not. (y <= 1.85d-17))) then
        tmp = 1.0d0 - (x / (y * t))
    else
        tmp = 1.0d0 - (x / (t * z))
    end if
    code = tmp
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -1.65e+19) || !(y <= 1.85e-17)) {
		tmp = 1.0 - (x / (y * t));
	} else {
		tmp = 1.0 - (x / (t * z));
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	tmp = 0
	if (y <= -1.65e+19) or not (y <= 1.85e-17):
		tmp = 1.0 - (x / (y * t))
	else:
		tmp = 1.0 - (x / (t * z))
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -1.65e+19) || !(y <= 1.85e-17))
		tmp = Float64(1.0 - Float64(x / Float64(y * t)));
	else
		tmp = Float64(1.0 - Float64(x / Float64(t * z)));
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y <= -1.65e+19) || ~((y <= 1.85e-17)))
		tmp = 1.0 - (x / (y * t));
	else
		tmp = 1.0 - (x / (t * z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[Or[LessEqual[y, -1.65e+19], N[Not[LessEqual[y, 1.85e-17]], $MachinePrecision]], N[(1.0 - N[(x / N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(x / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.65e19 or 1.8499999999999999e-17 < y

   1. Initial program 99.6%

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

   2. Taylor expanded in z around 0 95.2%

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

      1. associate-/l/ 95.2%

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

   4. Simplified 95.2%

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

   5. Taylor expanded in y around 0 70.9%

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

      1. associate-*r/ 70.9%

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

      2. neg-mul-1 70.9%

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

      3. *-commutative 70.9%

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

   7. Simplified 70.9%

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

      1. expm1-log1p-u 69.5%

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

      2. expm1-udef 69.5%

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

      3. add-sqr-sqrt 44.3%

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

      4. sqrt-unprod 63.1%

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

      5. sqr-neg 63.1%

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

      6. sqrt-unprod 25.1%

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

      7. add-sqr-sqrt 69.3%

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

   9. Applied egg-rr 69.3%

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

       1. expm1-def 69.3%

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

       2. expm1-log1p 70.1%

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

   11. Simplified 70.1%

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

   if -1.65e19 < y < 1.8499999999999999e-17

   1. Initial program 99.2%

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

   2. Taylor expanded in y around 0 74.6%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{+19} \lor \neg \left(y \leq 1.85 \cdot 10^{-17}\right):\\ \;\;\;\;1 - \frac{x}{y \cdot t}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{t \cdot z}\\ \end{array} \]

Alternative 8: 81.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{-74} \lor \neg \left(y \leq 1.95 \cdot 10^{-44}\right):\\ \;\;\;\;1 - \frac{x}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{t \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -1.3e-74) (not (<= y 1.95e-44)))
   (- 1.0 (/ x (* y y)))
   (- 1.0 (/ x (* t z)))))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -1.3e-74) || !(y <= 1.95e-44)) {
		tmp = 1.0 - (x / (y * y));
	} else {
		tmp = 1.0 - (x / (t * z));
	}
	return tmp;
}

Fortran

NOTE: 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-74)) .or. (.not. (y <= 1.95d-44))) then
        tmp = 1.0d0 - (x / (y * y))
    else
        tmp = 1.0d0 - (x / (t * z))
    end if
    code = tmp
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -1.3e-74) || !(y <= 1.95e-44)) {
		tmp = 1.0 - (x / (y * y));
	} else {
		tmp = 1.0 - (x / (t * z));
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	tmp = 0
	if (y <= -1.3e-74) or not (y <= 1.95e-44):
		tmp = 1.0 - (x / (y * y))
	else:
		tmp = 1.0 - (x / (t * z))
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -1.3e-74) || !(y <= 1.95e-44))
		tmp = Float64(1.0 - Float64(x / Float64(y * y)));
	else
		tmp = Float64(1.0 - Float64(x / Float64(t * z)));
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y <= -1.3e-74) || ~((y <= 1.95e-44)))
		tmp = 1.0 - (x / (y * y));
	else
		tmp = 1.0 - (x / (t * z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[Or[LessEqual[y, -1.3e-74], N[Not[LessEqual[y, 1.95e-44]], $MachinePrecision]], N[(1.0 - N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(x / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.3e-74 or 1.9500000000000001e-44 < y

   1. Initial program 99.6%

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

   2. Taylor expanded in y around inf 88.7%

      \[\leadsto 1 - \color{blue}{\frac{x}{{y}^{2}}} \]
   3. Step-by-step derivation

      1. unpow2 88.7%

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

   4. Simplified 88.7%

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

   if -1.3e-74 < y < 1.9500000000000001e-44

   1. Initial program 99.1%

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

   2. Taylor expanded in y around 0 78.8%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{-74} \lor \neg \left(y \leq 1.95 \cdot 10^{-44}\right):\\ \;\;\;\;1 - \frac{x}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{t \cdot z}\\ \end{array} \]

Alternative 9: 81.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -6.4 \cdot 10^{-75} \lor \neg \left(y \leq 4.6 \cdot 10^{-45}\right):\\ \;\;\;\;1 - \frac{\frac{x}{y}}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{t \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -6.4e-75) (not (<= y 4.6e-45)))
   (- 1.0 (/ (/ x y) y))
   (- 1.0 (/ x (* t z)))))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -6.4e-75) || !(y <= 4.6e-45)) {
		tmp = 1.0 - ((x / y) / y);
	} else {
		tmp = 1.0 - (x / (t * z));
	}
	return tmp;
}

Fortran

NOTE: 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 <= (-6.4d-75)) .or. (.not. (y <= 4.6d-45))) then
        tmp = 1.0d0 - ((x / y) / y)
    else
        tmp = 1.0d0 - (x / (t * z))
    end if
    code = tmp
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -6.4e-75) || !(y <= 4.6e-45)) {
		tmp = 1.0 - ((x / y) / y);
	} else {
		tmp = 1.0 - (x / (t * z));
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	tmp = 0
	if (y <= -6.4e-75) or not (y <= 4.6e-45):
		tmp = 1.0 - ((x / y) / y)
	else:
		tmp = 1.0 - (x / (t * z))
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -6.4e-75) || !(y <= 4.6e-45))
		tmp = Float64(1.0 - Float64(Float64(x / y) / y));
	else
		tmp = Float64(1.0 - Float64(x / Float64(t * z)));
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y <= -6.4e-75) || ~((y <= 4.6e-45)))
		tmp = 1.0 - ((x / y) / y);
	else
		tmp = 1.0 - (x / (t * z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[Or[LessEqual[y, -6.4e-75], N[Not[LessEqual[y, 4.6e-45]], $MachinePrecision]], N[(1.0 - N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(x / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.39999999999999953e-75 or 4.59999999999999983e-45 < y

   1. Initial program 99.6%

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

   2. Taylor expanded in y around inf 88.7%

      \[\leadsto 1 - \color{blue}{\frac{x}{{y}^{2}}} \]
   3. Step-by-step derivation

      1. unpow2 88.7%

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

      2. associate-/r* 88.7%

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

   4. Simplified 88.7%

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

   if -6.39999999999999953e-75 < y < 4.59999999999999983e-45

   1. Initial program 99.1%

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

   2. Taylor expanded in y around 0 78.8%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.4 \cdot 10^{-75} \lor \neg \left(y \leq 4.6 \cdot 10^{-45}\right):\\ \;\;\;\;1 - \frac{\frac{x}{y}}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{t \cdot z}\\ \end{array} \]

Alternative 10: 99.1% accurate, 1.0× speedup

Math

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

FPCore

NOTE: 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) (- y z)))))

C

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

Fortran

NOTE: 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) * (y - z)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: 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[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

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

2. Final simplification 99.4%

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

Alternative 11: 54.3% accurate, 1.6× speedup

Math

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

FPCore

NOTE: 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))))

C

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

Fortran

NOTE: 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))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in z around 0 72.4%

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

   1. associate-/l/ 72.0%

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

4. Simplified 72.0%

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

5. Taylor expanded in y around 0 57.0%

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

   1. associate-*r/ 57.0%

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

   2. neg-mul-1 57.0%

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

   3. *-commutative 57.0%

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

7. Simplified 57.0%

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

   1. expm1-log1p-u 50.8%

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

   2. expm1-udef 50.8%

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

   3. add-sqr-sqrt 30.0%

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

   4. sqrt-unprod 46.3%

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

   5. sqr-neg 46.3%

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

   6. sqrt-unprod 17.9%

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

   7. add-sqr-sqrt 47.1%

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

9. Applied egg-rr 47.1%

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

    1. expm1-def 47.1%

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

    2. expm1-log1p 49.2%

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

11. Simplified 49.2%

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

12. Final simplification 49.2%

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

Reproduce

herbie shell --seed 2023257 
(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)))))