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

Percentage Accurate: 99.1% → 99.1%

Time: 7.8s

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: 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]

Derivation

1. Initial program 98.9%

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

2. Final simplification 98.9%

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

Alternative 2: 86.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.15 \cdot 10^{-137}:\\ \;\;\;\;1 + \frac{\frac{x}{z}}{y - t}\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{-145}:\\ \;\;\;\;1 + \frac{-1}{y \cdot \frac{y - z}{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{\left(y - z\right) \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -1.15e-137)
   (+ 1.0 (/ (/ x z) (- y t)))
   (if (<= t 3.2e-145)
     (+ 1.0 (/ -1.0 (* y (/ (- y z) x))))
     (+ 1.0 (/ x (* (- y z) t))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.15e-137) {
		tmp = 1.0 + ((x / z) / (y - t));
	} else if (t <= 3.2e-145) {
		tmp = 1.0 + (-1.0 / (y * ((y - z) / x)));
	} else {
		tmp = 1.0 + (x / ((y - z) * t));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-1.15d-137)) then
        tmp = 1.0d0 + ((x / z) / (y - t))
    else if (t <= 3.2d-145) then
        tmp = 1.0d0 + ((-1.0d0) / (y * ((y - z) / x)))
    else
        tmp = 1.0d0 + (x / ((y - z) * t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.15e-137) {
		tmp = 1.0 + ((x / z) / (y - t));
	} else if (t <= 3.2e-145) {
		tmp = 1.0 + (-1.0 / (y * ((y - z) / x)));
	} else {
		tmp = 1.0 + (x / ((y - z) * t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -1.15e-137:
		tmp = 1.0 + ((x / z) / (y - t))
	elif t <= 3.2e-145:
		tmp = 1.0 + (-1.0 / (y * ((y - z) / x)))
	else:
		tmp = 1.0 + (x / ((y - z) * t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -1.15e-137)
		tmp = Float64(1.0 + Float64(Float64(x / z) / Float64(y - t)));
	elseif (t <= 3.2e-145)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(y * Float64(Float64(y - z) / x))));
	else
		tmp = Float64(1.0 + Float64(x / Float64(Float64(y - z) * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -1.15e-137)
		tmp = 1.0 + ((x / z) / (y - t));
	elseif (t <= 3.2e-145)
		tmp = 1.0 + (-1.0 / (y * ((y - z) / x)));
	else
		tmp = 1.0 + (x / ((y - z) * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -1.15e-137], N[(1.0 + N[(N[(x / z), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.2e-145], N[(1.0 + N[(-1.0 / N[(y * N[(N[(y - z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(x / N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.15000000000000004e-137

   1. Initial program 98.2%

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

      1. associate-/r* 97.7%

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

      2. div-inv 97.7%

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

   3. Applied egg-rr 97.7%

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

   4. Taylor expanded in z around inf 81.0%

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

      1. associate-*r/ 81.0%

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

      2. times-frac 79.9%

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

      3. associate-*l/ 79.9%

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

      4. associate-*r/ 79.9%

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

      5. neg-mul-1 79.9%

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

   6. Simplified 79.9%

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

   if -1.15000000000000004e-137 < t < 3.20000000000000008e-145

   1. Initial program 98.5%

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

      1. associate-/r* 99.8%

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

      2. div-inv 99.8%

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

   3. Applied egg-rr 99.8%

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

      1. clear-num 99.7%

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

      2. frac-times 99.8%

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

      3. metadata-eval 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in t around 0 84.5%

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

      1. associate-*r/ 85.2%

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

   8. Simplified 85.2%

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

   if 3.20000000000000008e-145 < 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 96.5%

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

      1. associate-*r/ 96.5%

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

      2. neg-mul-1 96.5%

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

   4. Simplified 96.5%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.15 \cdot 10^{-137}:\\ \;\;\;\;1 + \frac{\frac{x}{z}}{y - t}\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{-145}:\\ \;\;\;\;1 + \frac{-1}{y \cdot \frac{y - z}{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{\left(y - z\right) \cdot t}\\ \end{array} \]

Alternative 3: 86.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{-77} \lor \neg \left(y \leq 3.15 \cdot 10^{-52}\right):\\ \;\;\;\;1 - \frac{x}{y \cdot \left(y - z\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{z \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -4.2e-77) (not (<= y 3.15e-52)))
   (- 1.0 (/ x (* y (- y z))))
   (- 1.0 (/ x (* z t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -4.2e-77) || !(y <= 3.15e-52)) {
		tmp = 1.0 - (x / (y * (y - z)));
	} else {
		tmp = 1.0 - (x / (z * t));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((y <= (-4.2d-77)) .or. (.not. (y <= 3.15d-52))) then
        tmp = 1.0d0 - (x / (y * (y - z)))
    else
        tmp = 1.0d0 - (x / (z * t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -4.2e-77) || !(y <= 3.15e-52)) {
		tmp = 1.0 - (x / (y * (y - z)));
	} else {
		tmp = 1.0 - (x / (z * t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y <= -4.2e-77) or not (y <= 3.15e-52):
		tmp = 1.0 - (x / (y * (y - z)))
	else:
		tmp = 1.0 - (x / (z * t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -4.2e-77) || !(y <= 3.15e-52))
		tmp = Float64(1.0 - Float64(x / Float64(y * Float64(y - z))));
	else
		tmp = Float64(1.0 - Float64(x / Float64(z * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y <= -4.2e-77) || ~((y <= 3.15e-52)))
		tmp = 1.0 - (x / (y * (y - z)));
	else
		tmp = 1.0 - (x / (z * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -4.2e-77], N[Not[LessEqual[y, 3.15e-52]], $MachinePrecision]], N[(1.0 - N[(x / N[(y * N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.20000000000000031e-77 or 3.1500000000000002e-52 < 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 93.4%

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

   if -4.20000000000000031e-77 < y < 3.1500000000000002e-52

   1. Initial program 97.8%

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

   2. Taylor expanded in y around 0 75.5%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{-77} \lor \neg \left(y \leq 3.15 \cdot 10^{-52}\right):\\ \;\;\;\;1 - \frac{x}{y \cdot \left(y - z\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{z \cdot t}\\ \end{array} \]

Alternative 4: 86.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{-75}:\\ \;\;\;\;1 - \frac{x}{y \cdot \left(y - t\right)}\\ \mathbf{elif}\;y \leq 7 \cdot 10^{-54}:\\ \;\;\;\;1 - \frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y \cdot \left(y - z\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -6.8e-75)
   (- 1.0 (/ x (* y (- y t))))
   (if (<= y 7e-54) (- 1.0 (/ x (* z t))) (- 1.0 (/ x (* y (- y z)))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -6.8e-75) {
		tmp = 1.0 - (x / (y * (y - t)));
	} else if (y <= 7e-54) {
		tmp = 1.0 - (x / (z * t));
	} else {
		tmp = 1.0 - (x / (y * (y - z)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-6.8d-75)) then
        tmp = 1.0d0 - (x / (y * (y - t)))
    else if (y <= 7d-54) then
        tmp = 1.0d0 - (x / (z * t))
    else
        tmp = 1.0d0 - (x / (y * (y - z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -6.8e-75) {
		tmp = 1.0 - (x / (y * (y - t)));
	} else if (y <= 7e-54) {
		tmp = 1.0 - (x / (z * t));
	} else {
		tmp = 1.0 - (x / (y * (y - z)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -6.8e-75:
		tmp = 1.0 - (x / (y * (y - t)))
	elif y <= 7e-54:
		tmp = 1.0 - (x / (z * t))
	else:
		tmp = 1.0 - (x / (y * (y - z)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -6.8e-75)
		tmp = Float64(1.0 - Float64(x / Float64(y * Float64(y - t))));
	elseif (y <= 7e-54)
		tmp = Float64(1.0 - Float64(x / Float64(z * t)));
	else
		tmp = Float64(1.0 - Float64(x / Float64(y * Float64(y - z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -6.8e-75)
		tmp = 1.0 - (x / (y * (y - t)));
	elseif (y <= 7e-54)
		tmp = 1.0 - (x / (z * t));
	else
		tmp = 1.0 - (x / (y * (y - z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -6.8e-75], N[(1.0 - N[(x / N[(y * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7e-54], N[(1.0 - N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(x / N[(y * N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.8000000000000003e-75

   1. Initial program 100.0%

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

   2. Taylor expanded in z around 0 90.7%

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

   if -6.8000000000000003e-75 < y < 6.99999999999999964e-54

   1. Initial program 97.8%

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

   2. Taylor expanded in y around 0 75.5%

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

   if 6.99999999999999964e-54 < y

   1. Initial program 99.3%

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

   2. Taylor expanded in t around 0 95.5%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{-75}:\\ \;\;\;\;1 - \frac{x}{y \cdot \left(y - t\right)}\\ \mathbf{elif}\;y \leq 7 \cdot 10^{-54}:\\ \;\;\;\;1 - \frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y \cdot \left(y - z\right)}\\ \end{array} \]

Alternative 5: 86.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -7.6 \cdot 10^{-153}:\\ \;\;\;\;1 + \frac{x}{z \cdot \left(y - t\right)}\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{-146}:\\ \;\;\;\;1 - \frac{x}{y \cdot \left(y - z\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{\left(y - z\right) \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -7.6e-153)
   (+ 1.0 (/ x (* z (- y t))))
   (if (<= t 7.2e-146)
     (- 1.0 (/ x (* y (- y z))))
     (+ 1.0 (/ x (* (- y z) t))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -7.6e-153) {
		tmp = 1.0 + (x / (z * (y - t)));
	} else if (t <= 7.2e-146) {
		tmp = 1.0 - (x / (y * (y - z)));
	} else {
		tmp = 1.0 + (x / ((y - z) * t));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-7.6d-153)) then
        tmp = 1.0d0 + (x / (z * (y - t)))
    else if (t <= 7.2d-146) then
        tmp = 1.0d0 - (x / (y * (y - z)))
    else
        tmp = 1.0d0 + (x / ((y - z) * t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -7.6e-153) {
		tmp = 1.0 + (x / (z * (y - t)));
	} else if (t <= 7.2e-146) {
		tmp = 1.0 - (x / (y * (y - z)));
	} else {
		tmp = 1.0 + (x / ((y - z) * t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -7.6e-153:
		tmp = 1.0 + (x / (z * (y - t)))
	elif t <= 7.2e-146:
		tmp = 1.0 - (x / (y * (y - z)))
	else:
		tmp = 1.0 + (x / ((y - z) * t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -7.6e-153)
		tmp = Float64(1.0 + Float64(x / Float64(z * Float64(y - t))));
	elseif (t <= 7.2e-146)
		tmp = Float64(1.0 - Float64(x / Float64(y * Float64(y - z))));
	else
		tmp = Float64(1.0 + Float64(x / Float64(Float64(y - z) * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -7.6e-153)
		tmp = 1.0 + (x / (z * (y - t)));
	elseif (t <= 7.2e-146)
		tmp = 1.0 - (x / (y * (y - z)));
	else
		tmp = 1.0 + (x / ((y - z) * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -7.6e-153], N[(1.0 + N[(x / N[(z * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 7.2e-146], N[(1.0 - N[(x / N[(y * N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(x / N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -7.60000000000000046e-153

   1. Initial program 98.2%

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

   2. Taylor expanded in z around inf 81.2%

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

      1. associate-*r/ 81.2%

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

      2. neg-mul-1 81.2%

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

   4. Simplified 81.2%

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

   if -7.60000000000000046e-153 < t < 7.19999999999999957e-146

   1. Initial program 98.4%

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

   2. Taylor expanded in t around 0 84.4%

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

   if 7.19999999999999957e-146 < 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 96.5%

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

      1. associate-*r/ 96.5%

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

      2. neg-mul-1 96.5%

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

   4. Simplified 96.5%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.6 \cdot 10^{-153}:\\ \;\;\;\;1 + \frac{x}{z \cdot \left(y - t\right)}\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{-146}:\\ \;\;\;\;1 - \frac{x}{y \cdot \left(y - z\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{\left(y - z\right) \cdot t}\\ \end{array} \]

Alternative 6: 86.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5.2 \cdot 10^{-143}:\\ \;\;\;\;1 + \frac{\frac{x}{z}}{y - t}\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{-144}:\\ \;\;\;\;1 - \frac{x}{y \cdot \left(y - z\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{\left(y - z\right) \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -5.2e-143)
   (+ 1.0 (/ (/ x z) (- y t)))
   (if (<= t 1.2e-144)
     (- 1.0 (/ x (* y (- y z))))
     (+ 1.0 (/ x (* (- y z) t))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -5.2e-143) {
		tmp = 1.0 + ((x / z) / (y - t));
	} else if (t <= 1.2e-144) {
		tmp = 1.0 - (x / (y * (y - z)));
	} else {
		tmp = 1.0 + (x / ((y - z) * t));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-5.2d-143)) then
        tmp = 1.0d0 + ((x / z) / (y - t))
    else if (t <= 1.2d-144) then
        tmp = 1.0d0 - (x / (y * (y - z)))
    else
        tmp = 1.0d0 + (x / ((y - z) * t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -5.2e-143) {
		tmp = 1.0 + ((x / z) / (y - t));
	} else if (t <= 1.2e-144) {
		tmp = 1.0 - (x / (y * (y - z)));
	} else {
		tmp = 1.0 + (x / ((y - z) * t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -5.2e-143:
		tmp = 1.0 + ((x / z) / (y - t))
	elif t <= 1.2e-144:
		tmp = 1.0 - (x / (y * (y - z)))
	else:
		tmp = 1.0 + (x / ((y - z) * t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -5.2e-143)
		tmp = Float64(1.0 + Float64(Float64(x / z) / Float64(y - t)));
	elseif (t <= 1.2e-144)
		tmp = Float64(1.0 - Float64(x / Float64(y * Float64(y - z))));
	else
		tmp = Float64(1.0 + Float64(x / Float64(Float64(y - z) * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -5.2e-143)
		tmp = 1.0 + ((x / z) / (y - t));
	elseif (t <= 1.2e-144)
		tmp = 1.0 - (x / (y * (y - z)));
	else
		tmp = 1.0 + (x / ((y - z) * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -5.2e-143], N[(1.0 + N[(N[(x / z), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.2e-144], N[(1.0 - N[(x / N[(y * N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(x / N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -5.19999999999999974e-143

   1. Initial program 98.2%

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

      1. associate-/r* 97.7%

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

      2. div-inv 97.7%

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

   3. Applied egg-rr 97.7%

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

   4. Taylor expanded in z around inf 81.0%

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

      1. associate-*r/ 81.0%

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

      2. times-frac 79.9%

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

      3. associate-*l/ 79.9%

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

      4. associate-*r/ 79.9%

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

      5. neg-mul-1 79.9%

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

   6. Simplified 79.9%

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

   if -5.19999999999999974e-143 < t < 1.19999999999999997e-144

   1. Initial program 98.5%

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

   2. Taylor expanded in t around 0 84.6%

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

   if 1.19999999999999997e-144 < 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 96.5%

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

      1. associate-*r/ 96.5%

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

      2. neg-mul-1 96.5%

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

   4. Simplified 96.5%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.2 \cdot 10^{-143}:\\ \;\;\;\;1 + \frac{\frac{x}{z}}{y - t}\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{-144}:\\ \;\;\;\;1 - \frac{x}{y \cdot \left(y - z\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{\left(y - z\right) \cdot t}\\ \end{array} \]

Alternative 7: 84.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{-59}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 7 \cdot 10^{-54}:\\ \;\;\;\;1 - \frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -2.1e-59) 1.0 (if (<= y 7e-54) (- 1.0 (/ x (* z t))) 1.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -2.1e-59) {
		tmp = 1.0;
	} else if (y <= 7e-54) {
		tmp = 1.0 - (x / (z * t));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-2.1d-59)) then
        tmp = 1.0d0
    else if (y <= 7d-54) then
        tmp = 1.0d0 - (x / (z * t))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -2.1e-59) {
		tmp = 1.0;
	} else if (y <= 7e-54) {
		tmp = 1.0 - (x / (z * t));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -2.1e-59:
		tmp = 1.0
	elif y <= 7e-54:
		tmp = 1.0 - (x / (z * t))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -2.1e-59)
		tmp = 1.0;
	elseif (y <= 7e-54)
		tmp = Float64(1.0 - Float64(x / Float64(z * t)));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -2.1e-59)
		tmp = 1.0;
	elseif (y <= 7e-54)
		tmp = 1.0 - (x / (z * t));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -2.1e-59], 1.0, If[LessEqual[y, 7e-54], N[(1.0 - N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -2.09999999999999997e-59 or 6.99999999999999964e-54 < 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 93.0%

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

      1. associate-/l/ 93.3%

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

   4. Simplified 93.3%

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

   5. Taylor expanded in y around 0 69.2%

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

      1. associate-*r/ 69.2%

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

      2. neg-mul-1 69.2%

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

   7. Simplified 69.2%

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

      1. expm1-log1p-u 67.7%

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

      2. expm1-udef 67.7%

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

      3. add-sqr-sqrt 31.6%

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

      4. sqrt-unprod 59.0%

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

      5. sqr-neg 59.0%

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

      6. sqrt-unprod 36.2%

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

      7. add-sqr-sqrt 67.0%

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

      8. associate-/r* 67.0%

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

   9. Applied egg-rr 67.0%

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

       1. expm1-def 67.0%

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

       2. expm1-log1p 68.5%

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

       3. associate-/l/ 68.5%

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

   11. Simplified 68.5%

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

   12. Taylor expanded in x around 0 87.7%

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

   if -2.09999999999999997e-59 < y < 6.99999999999999964e-54

   1. Initial program 97.9%

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

   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 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{-59}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 7 \cdot 10^{-54}:\\ \;\;\;\;1 - \frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8: 83.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.7 \cdot 10^{-54}:\\ \;\;\;\;1 - \frac{x}{y \cdot y}\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-52}:\\ \;\;\;\;1 - \frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -4.7e-54)
   (- 1.0 (/ x (* y y)))
   (if (<= y 4.2e-52) (- 1.0 (/ x (* z t))) 1.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -4.7e-54) {
		tmp = 1.0 - (x / (y * y));
	} else if (y <= 4.2e-52) {
		tmp = 1.0 - (x / (z * t));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-4.7d-54)) then
        tmp = 1.0d0 - (x / (y * y))
    else if (y <= 4.2d-52) then
        tmp = 1.0d0 - (x / (z * t))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -4.7e-54) {
		tmp = 1.0 - (x / (y * y));
	} else if (y <= 4.2e-52) {
		tmp = 1.0 - (x / (z * t));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -4.7e-54:
		tmp = 1.0 - (x / (y * y))
	elif y <= 4.2e-52:
		tmp = 1.0 - (x / (z * t))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -4.7e-54)
		tmp = Float64(1.0 - Float64(x / Float64(y * y)));
	elseif (y <= 4.2e-52)
		tmp = Float64(1.0 - Float64(x / Float64(z * t)));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -4.7e-54)
		tmp = 1.0 - (x / (y * y));
	elseif (y <= 4.2e-52)
		tmp = 1.0 - (x / (z * t));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -4.7e-54], N[(1.0 - N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.2e-52], N[(1.0 - N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 3 regimes

2. if y < -4.7e-54

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 84.7%

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

      1. unpow2 84.7%

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

   4. Simplified 84.7%

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

   if -4.7e-54 < y < 4.1999999999999997e-52

   1. Initial program 98.0%

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

   2. Taylor expanded in y around 0 74.2%

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

   if 4.1999999999999997e-52 < 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 95.5%

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

      1. associate-/l/ 96.1%

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

   4. Simplified 96.1%

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

   5. Taylor expanded in y around 0 71.4%

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

      1. associate-*r/ 71.4%

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

      2. neg-mul-1 71.4%

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

   7. Simplified 71.4%

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

      1. expm1-log1p-u 70.5%

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

      2. expm1-udef 70.5%

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

      3. add-sqr-sqrt 31.2%

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

      4. sqrt-unprod 62.3%

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

      5. sqr-neg 62.3%

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

      6. sqrt-unprod 39.3%

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

      7. add-sqr-sqrt 70.4%

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

      8. associate-/r* 70.4%

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

   9. Applied egg-rr 70.4%

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

       1. expm1-def 70.4%

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

       2. expm1-log1p 71.3%

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

       3. associate-/l/ 71.3%

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

   11. Simplified 71.3%

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

   12. Taylor expanded in x around 0 91.7%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.7 \cdot 10^{-54}:\\ \;\;\;\;1 - \frac{x}{y \cdot y}\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-52}:\\ \;\;\;\;1 - \frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9: 81.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 8.5 \cdot 10^{-145}:\\ \;\;\;\;1 - \frac{x}{y \cdot \left(y - z\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{\left(y - z\right) \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 8.5e-145) (- 1.0 (/ x (* y (- y z)))) (+ 1.0 (/ x (* (- y z) t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 8.5e-145) {
		tmp = 1.0 - (x / (y * (y - z)));
	} else {
		tmp = 1.0 + (x / ((y - z) * t));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= 8.5d-145) then
        tmp = 1.0d0 - (x / (y * (y - z)))
    else
        tmp = 1.0d0 + (x / ((y - z) * t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 8.5e-145) {
		tmp = 1.0 - (x / (y * (y - z)));
	} else {
		tmp = 1.0 + (x / ((y - z) * t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= 8.5e-145:
		tmp = 1.0 - (x / (y * (y - z)))
	else:
		tmp = 1.0 + (x / ((y - z) * t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 8.5e-145)
		tmp = Float64(1.0 - Float64(x / Float64(y * Float64(y - z))));
	else
		tmp = Float64(1.0 + Float64(x / Float64(Float64(y - z) * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 8.5e-145)
		tmp = 1.0 - (x / (y * (y - z)));
	else
		tmp = 1.0 + (x / ((y - z) * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 8.5e-145], N[(1.0 - N[(x / N[(y * N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(x / N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 8.50000000000000043e-145

   1. Initial program 98.3%

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

   2. Taylor expanded in t around 0 74.9%

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

   if 8.50000000000000043e-145 < 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 96.5%

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

      1. associate-*r/ 96.5%

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

      2. neg-mul-1 96.5%

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

   4. Simplified 96.5%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8.5 \cdot 10^{-145}:\\ \;\;\;\;1 - \frac{x}{y \cdot \left(y - z\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{\left(y - z\right) \cdot t}\\ \end{array} \]

Alternative 10: 66.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{-224}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.15e-224) 1.0 (+ 1.0 (/ x (* y t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.15e-224) {
		tmp = 1.0;
	} else {
		tmp = 1.0 + (x / (y * t));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-1.15d-224)) then
        tmp = 1.0d0
    else
        tmp = 1.0d0 + (x / (y * t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.15e-224) {
		tmp = 1.0;
	} else {
		tmp = 1.0 + (x / (y * t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -1.15e-224:
		tmp = 1.0
	else:
		tmp = 1.0 + (x / (y * t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.15e-224)
		tmp = 1.0;
	else
		tmp = Float64(1.0 + Float64(x / Float64(y * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -1.15e-224)
		tmp = 1.0;
	else
		tmp = 1.0 + (x / (y * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1.15e-224], 1.0, N[(1.0 + N[(x / N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.14999999999999994e-224

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0 70.8%

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

      1. associate-/l/ 70.8%

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

   4. Simplified 70.8%

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

   5. Taylor expanded in y around 0 54.2%

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

      1. associate-*r/ 54.2%

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

      2. neg-mul-1 54.2%

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

   7. Simplified 54.2%

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

      1. expm1-log1p-u 53.2%

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

      2. expm1-udef 53.2%

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

      3. add-sqr-sqrt 24.5%

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

      4. sqrt-unprod 49.2%

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

      5. sqr-neg 49.2%

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

      6. sqrt-unprod 28.1%

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

      7. add-sqr-sqrt 52.4%

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

      8. associate-/r* 52.4%

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

   9. Applied egg-rr 52.4%

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

       1. expm1-def 52.4%

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

       2. expm1-log1p 53.4%

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

       3. associate-/l/ 53.4%

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

   11. Simplified 53.4%

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

   12. Taylor expanded in x around 0 74.6%

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

   if -1.14999999999999994e-224 < z

   1. Initial program 98.0%

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

   2. Taylor expanded in z around 0 73.8%

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

      1. associate-/l/ 74.1%

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

   4. Simplified 74.1%

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

   5. Taylor expanded in y around 0 59.6%

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

      1. associate-*r/ 59.6%

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

      2. neg-mul-1 59.6%

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

   7. Simplified 59.6%

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

      1. expm1-log1p-u 53.3%

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

      2. expm1-udef 53.3%

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

      3. add-sqr-sqrt 21.9%

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

      4. sqrt-unprod 47.1%

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

      5. sqr-neg 47.1%

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

      6. sqrt-unprod 30.1%

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

      7. add-sqr-sqrt 51.4%

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

      8. associate-/r* 51.4%

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

   9. Applied egg-rr 51.4%

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

       1. expm1-def 51.4%

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

       2. expm1-log1p 53.5%

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

       3. associate-/l/ 53.5%

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

   11. Simplified 53.5%

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

       1. sub-neg 53.5%

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

       2. distribute-neg-frac 53.5%

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

       3. add-sqr-sqrt 23.1%

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

       4. sqrt-unprod 49.0%

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

       5. sqr-neg 49.0%

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

       6. sqrt-unprod 32.0%

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

       7. add-sqr-sqrt 59.6%

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

       8. *-commutative 59.6%

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

   13. Applied egg-rr 59.6%

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

       1. +-commutative 59.6%

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

   15. Simplified 59.6%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{-224}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y \cdot t}\\ \end{array} \]

Alternative 11: 75.4% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x, y, z, t)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := 1.0

Derivation

1. Initial program 98.9%

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

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/ 72.6%

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

4. Simplified 72.6%

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

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%

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

7. Simplified 57.0%

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

   1. expm1-log1p-u 53.2%

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

   2. expm1-udef 53.2%

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

   3. add-sqr-sqrt 23.1%

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

   4. sqrt-unprod 48.1%

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

   5. sqr-neg 48.1%

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

   6. sqrt-unprod 29.2%

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

   7. add-sqr-sqrt 51.9%

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

   8. associate-/r* 51.9%

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

9. Applied egg-rr 51.9%

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

    1. expm1-def 51.9%

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

    2. expm1-log1p 53.5%

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

    3. associate-/l/ 53.4%

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

11. Simplified 53.4%

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

12. Taylor expanded in x around 0 69.9%

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

13. Final simplification 69.9%

    \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023192 
(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)))))