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

Percentage Accurate: 99.1% → 98.5%

Time: 10.2s

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: 98.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

   1. associate-/r* N/A

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

   2. /-lowering-/.f64 N/A

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

   3. /-lowering-/.f64 N/A

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

   4. --lowering--.f64 N/A

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

   5. --lowering--.f64 98.1

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

4. Applied egg-rr 98.1%

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

5. Final simplification 98.1%

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

Alternative 2: 85.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := 1 + \frac{x}{\left(y - z\right) \cdot \left(t - y\right)}\\ \mathbf{if}\;t\_1 \leq 0.99999999:\\ \;\;\;\;1 - \frac{x}{z \cdot t}\\ \mathbf{elif}\;t\_1 \leq 10^{+20}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(z - y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ 1.0 (/ x (* (- y z) (- t y))))))
   (if (<= t_1 0.99999999)
     (- 1.0 (/ x (* z t)))
     (if (<= t_1 1e+20) 1.0 (/ x (* y (- z y)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = 1.0 + (x / ((y - z) * (t - y)));
	double tmp;
	if (t_1 <= 0.99999999) {
		tmp = 1.0 - (x / (z * t));
	} else if (t_1 <= 1e+20) {
		tmp = 1.0;
	} else {
		tmp = x / (y * (z - y));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 + (x / ((y - z) * (t - y)))
    if (t_1 <= 0.99999999d0) then
        tmp = 1.0d0 - (x / (z * t))
    else if (t_1 <= 1d+20) then
        tmp = 1.0d0
    else
        tmp = x / (y * (z - y))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = 1.0 + (x / ((y - z) * (t - y)));
	double tmp;
	if (t_1 <= 0.99999999) {
		tmp = 1.0 - (x / (z * t));
	} else if (t_1 <= 1e+20) {
		tmp = 1.0;
	} else {
		tmp = x / (y * (z - y));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = 1.0 + (x / ((y - z) * (t - y)))
	tmp = 0
	if t_1 <= 0.99999999:
		tmp = 1.0 - (x / (z * t))
	elif t_1 <= 1e+20:
		tmp = 1.0
	else:
		tmp = x / (y * (z - y))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(1.0 + Float64(x / Float64(Float64(y - z) * Float64(t - y))))
	tmp = 0.0
	if (t_1 <= 0.99999999)
		tmp = Float64(1.0 - Float64(x / Float64(z * t)));
	elseif (t_1 <= 1e+20)
		tmp = 1.0;
	else
		tmp = Float64(x / Float64(y * Float64(z - y)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = 1.0 + (x / ((y - z) * (t - y)));
	tmp = 0.0;
	if (t_1 <= 0.99999999)
		tmp = 1.0 - (x / (z * t));
	elseif (t_1 <= 1e+20)
		tmp = 1.0;
	else
		tmp = x / (y * (z - y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(1.0 + N[(x / N[(N[(y - z), $MachinePrecision] * N[(t - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.99999999], N[(1.0 - N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+20], 1.0, N[(x / N[(y * N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 #s(literal 1 binary64) (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t)))) < 0.99999998999999995

   1. Initial program 96.8%

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

   3. Taylor expanded in y around 0

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

      1. --lowering--.f64 N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-lowering-*.f64 46.1

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

   5. Simplified 46.1%

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

   if 0.99999998999999995 < (-.f64 #s(literal 1 binary64) (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t)))) < 1e20

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 99.0%

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

   if 1e20 < (-.f64 #s(literal 1 binary64) (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))))

   1. Initial program 96.9%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. *-lowering-*.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. mul-1-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. unsub-neg N/A

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

      14. mul-1-neg N/A

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

      15. remove-double-neg N/A

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

      16. --lowering--.f64 96.9

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

   5. Simplified 96.9%

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

   6. Taylor expanded in y around inf

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

   8. Simplified 58.4%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 + \frac{x}{\left(y - z\right) \cdot \left(t - y\right)} \leq 0.99999999:\\ \;\;\;\;1 - \frac{x}{z \cdot t}\\ \mathbf{elif}\;1 + \frac{x}{\left(y - z\right) \cdot \left(t - y\right)} \leq 10^{+20}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(z - y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{x}{\left(y - z\right) \cdot \left(y - t\right)}\\ t_2 := \frac{x}{\left(y - z\right) \cdot \left(t - y\right)}\\ \mathbf{if}\;t\_1 \leq -1000000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-8}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ x (* (- y z) (- y t)))) (t_2 (/ x (* (- y z) (- t y)))))
   (if (<= t_1 -1000000000000.0) t_2 (if (<= t_1 2e-8) 1.0 t_2))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = x / ((y - z) * (y - t));
	double t_2 = x / ((y - z) * (t - y));
	double tmp;
	if (t_1 <= -1000000000000.0) {
		tmp = t_2;
	} else if (t_1 <= 2e-8) {
		tmp = 1.0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x / ((y - z) * (y - t))
    t_2 = x / ((y - z) * (t - y))
    if (t_1 <= (-1000000000000.0d0)) then
        tmp = t_2
    else if (t_1 <= 2d-8) then
        tmp = 1.0d0
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = x / ((y - z) * (y - t));
	double t_2 = x / ((y - z) * (t - y));
	double tmp;
	if (t_1 <= -1000000000000.0) {
		tmp = t_2;
	} else if (t_1 <= 2e-8) {
		tmp = 1.0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = x / ((y - z) * (y - t))
	t_2 = x / ((y - z) * (t - y))
	tmp = 0
	if t_1 <= -1000000000000.0:
		tmp = t_2
	elif t_1 <= 2e-8:
		tmp = 1.0
	else:
		tmp = t_2
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(x / Float64(Float64(y - z) * Float64(y - t)))
	t_2 = Float64(x / Float64(Float64(y - z) * Float64(t - y)))
	tmp = 0.0
	if (t_1 <= -1000000000000.0)
		tmp = t_2;
	elseif (t_1 <= 2e-8)
		tmp = 1.0;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = x / ((y - z) * (y - t));
	t_2 = x / ((y - z) * (t - y));
	tmp = 0.0;
	if (t_1 <= -1000000000000.0)
		tmp = t_2;
	elseif (t_1 <= 2e-8)
		tmp = 1.0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x / N[(N[(y - z), $MachinePrecision] * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x / N[(N[(y - z), $MachinePrecision] * N[(t - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1000000000000.0], t$95$2, If[LessEqual[t$95$1, 2e-8], 1.0, t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))) < -1e12 or 2e-8 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t)))

   1. Initial program 96.9%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. *-lowering-*.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. mul-1-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. unsub-neg N/A

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

      14. mul-1-neg N/A

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

      15. remove-double-neg N/A

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

      16. --lowering--.f64 95.6

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

   5. Simplified 95.6%

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

   if -1e12 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))) < 2e-8

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 99.7%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \leq -1000000000000:\\ \;\;\;\;\frac{x}{\left(y - z\right) \cdot \left(t - y\right)}\\ \mathbf{elif}\;\frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - z\right) \cdot \left(t - y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := 1 + \frac{x}{\left(y - z\right) \cdot \left(t - y\right)}\\ \mathbf{if}\;t\_1 \leq -200:\\ \;\;\;\;\frac{x}{y \cdot \left(-y\right)}\\ \mathbf{elif}\;t\_1 \leq 10^{+20}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ 1.0 (/ x (* (- y z) (- t y))))))
   (if (<= t_1 -200.0)
     (/ x (* y (- y)))
     (if (<= t_1 1e+20) 1.0 (/ x (* y z))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = 1.0 + (x / ((y - z) * (t - y)));
	double tmp;
	if (t_1 <= -200.0) {
		tmp = x / (y * -y);
	} else if (t_1 <= 1e+20) {
		tmp = 1.0;
	} else {
		tmp = x / (y * z);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 + (x / ((y - z) * (t - y)))
    if (t_1 <= (-200.0d0)) then
        tmp = x / (y * -y)
    else if (t_1 <= 1d+20) then
        tmp = 1.0d0
    else
        tmp = x / (y * z)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = 1.0 + (x / ((y - z) * (t - y)));
	double tmp;
	if (t_1 <= -200.0) {
		tmp = x / (y * -y);
	} else if (t_1 <= 1e+20) {
		tmp = 1.0;
	} else {
		tmp = x / (y * z);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = 1.0 + (x / ((y - z) * (t - y)))
	tmp = 0
	if t_1 <= -200.0:
		tmp = x / (y * -y)
	elif t_1 <= 1e+20:
		tmp = 1.0
	else:
		tmp = x / (y * z)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(1.0 + Float64(x / Float64(Float64(y - z) * Float64(t - y))))
	tmp = 0.0
	if (t_1 <= -200.0)
		tmp = Float64(x / Float64(y * Float64(-y)));
	elseif (t_1 <= 1e+20)
		tmp = 1.0;
	else
		tmp = Float64(x / Float64(y * z));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = 1.0 + (x / ((y - z) * (t - y)));
	tmp = 0.0;
	if (t_1 <= -200.0)
		tmp = x / (y * -y);
	elseif (t_1 <= 1e+20)
		tmp = 1.0;
	else
		tmp = x / (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(1.0 + N[(x / N[(N[(y - z), $MachinePrecision] * N[(t - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -200.0], N[(x / N[(y * (-y)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+20], 1.0, N[(x / N[(y * z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 #s(literal 1 binary64) (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t)))) < -200

   1. Initial program 96.8%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

         \[\leadsto 1 + \color{blue}{\left(\mathsf{neg}\left(\frac{x}{{y}^{2}}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{1 - \frac{x}{{y}^{2}}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{1 - \frac{x}{{y}^{2}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto 1 - \color{blue}{\frac{x}{{y}^{2}}} \]

      5. unpow2 N/A

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

      6. *-lowering-*.f64 26.2

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

   5. Simplified 26.2%

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

   6. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{x}{{y}^{2}}\right)} \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{x}{\mathsf{neg}\left({y}^{2}\right)}} \]

      3. neg-mul-1 N/A

         \[\leadsto \frac{x}{\color{blue}{-1 \cdot {y}^{2}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{-1 \cdot {y}^{2}}} \]

      5. neg-mul-1 N/A

         \[\leadsto \frac{x}{\color{blue}{\mathsf{neg}\left({y}^{2}\right)}} \]

      6. unpow2 N/A

         \[\leadsto \frac{x}{\mathsf{neg}\left(\color{blue}{y \cdot y}\right)} \]

      7. distribute-rgt-neg-in N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot \left(\mathsf{neg}\left(y\right)\right)}} \]

      8. mul-1-neg N/A

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

      9. *-lowering-*.f64 N/A

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

      10. mul-1-neg N/A

          \[\leadsto \frac{x}{y \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}} \]

      11. neg-lowering-neg.f64 24.2

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

   8. Simplified 24.2%

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

   if -200 < (-.f64 #s(literal 1 binary64) (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t)))) < 1e20

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 98.8%

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

   if 1e20 < (-.f64 #s(literal 1 binary64) (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))))

   1. Initial program 96.9%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. *-lowering-*.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. mul-1-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. unsub-neg N/A

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

      14. mul-1-neg N/A

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

      15. remove-double-neg N/A

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

      16. --lowering--.f64 96.9

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

   5. Simplified 96.9%

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

   6. Taylor expanded in y around inf

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

   8. Simplified 58.4%

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

   9. Taylor expanded in y around 0

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

       1. /-lowering-/.f64 N/A

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 42.7

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

   11. Simplified 42.7%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 + \frac{x}{\left(y - z\right) \cdot \left(t - y\right)} \leq -200:\\ \;\;\;\;\frac{x}{y \cdot \left(-y\right)}\\ \mathbf{elif}\;1 + \frac{x}{\left(y - z\right) \cdot \left(t - y\right)} \leq 10^{+20}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{x}{\left(y - z\right) \cdot \left(y - t\right)}\\ t_2 := \frac{x}{z \cdot \left(y - t\right)}\\ \mathbf{if}\;t\_1 \leq -1000000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-8}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ x (* (- y z) (- y t)))) (t_2 (/ x (* z (- y t)))))
   (if (<= t_1 -1000000000000.0) t_2 (if (<= t_1 2e-8) 1.0 t_2))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = x / ((y - z) * (y - t));
	double t_2 = x / (z * (y - t));
	double tmp;
	if (t_1 <= -1000000000000.0) {
		tmp = t_2;
	} else if (t_1 <= 2e-8) {
		tmp = 1.0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x / ((y - z) * (y - t))
    t_2 = x / (z * (y - t))
    if (t_1 <= (-1000000000000.0d0)) then
        tmp = t_2
    else if (t_1 <= 2d-8) then
        tmp = 1.0d0
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = x / ((y - z) * (y - t));
	double t_2 = x / (z * (y - t));
	double tmp;
	if (t_1 <= -1000000000000.0) {
		tmp = t_2;
	} else if (t_1 <= 2e-8) {
		tmp = 1.0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = x / ((y - z) * (y - t))
	t_2 = x / (z * (y - t))
	tmp = 0
	if t_1 <= -1000000000000.0:
		tmp = t_2
	elif t_1 <= 2e-8:
		tmp = 1.0
	else:
		tmp = t_2
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(x / Float64(Float64(y - z) * Float64(y - t)))
	t_2 = Float64(x / Float64(z * Float64(y - t)))
	tmp = 0.0
	if (t_1 <= -1000000000000.0)
		tmp = t_2;
	elseif (t_1 <= 2e-8)
		tmp = 1.0;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = x / ((y - z) * (y - t));
	t_2 = x / (z * (y - t));
	tmp = 0.0;
	if (t_1 <= -1000000000000.0)
		tmp = t_2;
	elseif (t_1 <= 2e-8)
		tmp = 1.0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x / N[(N[(y - z), $MachinePrecision] * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x / N[(z * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1000000000000.0], t$95$2, If[LessEqual[t$95$1, 2e-8], 1.0, t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))) < -1e12 or 2e-8 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t)))

   1. Initial program 96.9%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. *-lowering-*.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. mul-1-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. unsub-neg N/A

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

      14. mul-1-neg N/A

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

      15. remove-double-neg N/A

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

      16. --lowering--.f64 95.6

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

   5. Simplified 95.6%

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

   6. Taylor expanded in z around inf

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

   8. Simplified 62.3%

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

   if -1e12 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))) < 2e-8

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 99.7%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \leq -1000000000000:\\ \;\;\;\;\frac{x}{z \cdot \left(y - t\right)}\\ \mathbf{elif}\;\frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot \left(y - t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{x}{\left(y - z\right) \cdot \left(y - t\right)}\\ t_2 := 1 - \frac{x}{z \cdot t}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-15}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-14}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ x (* (- y z) (- y t)))) (t_2 (- 1.0 (/ x (* z t)))))
   (if (<= t_1 -2e-15) t_2 (if (<= t_1 2e-14) 1.0 t_2))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = x / ((y - z) * (y - t));
	double t_2 = 1.0 - (x / (z * t));
	double tmp;
	if (t_1 <= -2e-15) {
		tmp = t_2;
	} else if (t_1 <= 2e-14) {
		tmp = 1.0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x / ((y - z) * (y - t))
    t_2 = 1.0d0 - (x / (z * t))
    if (t_1 <= (-2d-15)) then
        tmp = t_2
    else if (t_1 <= 2d-14) then
        tmp = 1.0d0
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = x / ((y - z) * (y - t));
	double t_2 = 1.0 - (x / (z * t));
	double tmp;
	if (t_1 <= -2e-15) {
		tmp = t_2;
	} else if (t_1 <= 2e-14) {
		tmp = 1.0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = x / ((y - z) * (y - t))
	t_2 = 1.0 - (x / (z * t))
	tmp = 0
	if t_1 <= -2e-15:
		tmp = t_2
	elif t_1 <= 2e-14:
		tmp = 1.0
	else:
		tmp = t_2
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(x / Float64(Float64(y - z) * Float64(y - t)))
	t_2 = Float64(1.0 - Float64(x / Float64(z * t)))
	tmp = 0.0
	if (t_1 <= -2e-15)
		tmp = t_2;
	elseif (t_1 <= 2e-14)
		tmp = 1.0;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = x / ((y - z) * (y - t));
	t_2 = 1.0 - (x / (z * t));
	tmp = 0.0;
	if (t_1 <= -2e-15)
		tmp = t_2;
	elseif (t_1 <= 2e-14)
		tmp = 1.0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x / N[(N[(y - z), $MachinePrecision] * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-15], t$95$2, If[LessEqual[t$95$1, 2e-14], 1.0, t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))) < -2.0000000000000002e-15 or 2e-14 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t)))

   1. Initial program 97.0%

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

   3. Taylor expanded in y around 0

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

      1. --lowering--.f64 N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-lowering-*.f64 49.7

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

   5. Simplified 49.7%

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

   if -2.0000000000000002e-15 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))) < 2e-14

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 99.9%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \leq -2 \cdot 10^{-15}:\\ \;\;\;\;1 - \frac{x}{z \cdot t}\\ \mathbf{elif}\;\frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \leq 2 \cdot 10^{-14}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{z \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{x}{\left(y - z\right) \cdot \left(y - t\right)}\\ t_2 := \frac{x}{z \cdot \left(-t\right)}\\ \mathbf{if}\;t\_1 \leq -1000000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 500:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ x (* (- y z) (- y t)))) (t_2 (/ x (* z (- t)))))
   (if (<= t_1 -1000000000000.0) t_2 (if (<= t_1 500.0) 1.0 t_2))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = x / ((y - z) * (y - t));
	double t_2 = x / (z * -t);
	double tmp;
	if (t_1 <= -1000000000000.0) {
		tmp = t_2;
	} else if (t_1 <= 500.0) {
		tmp = 1.0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x / ((y - z) * (y - t))
    t_2 = x / (z * -t)
    if (t_1 <= (-1000000000000.0d0)) then
        tmp = t_2
    else if (t_1 <= 500.0d0) then
        tmp = 1.0d0
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = x / ((y - z) * (y - t));
	double t_2 = x / (z * -t);
	double tmp;
	if (t_1 <= -1000000000000.0) {
		tmp = t_2;
	} else if (t_1 <= 500.0) {
		tmp = 1.0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = x / ((y - z) * (y - t))
	t_2 = x / (z * -t)
	tmp = 0
	if t_1 <= -1000000000000.0:
		tmp = t_2
	elif t_1 <= 500.0:
		tmp = 1.0
	else:
		tmp = t_2
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(x / Float64(Float64(y - z) * Float64(y - t)))
	t_2 = Float64(x / Float64(z * Float64(-t)))
	tmp = 0.0
	if (t_1 <= -1000000000000.0)
		tmp = t_2;
	elseif (t_1 <= 500.0)
		tmp = 1.0;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = x / ((y - z) * (y - t));
	t_2 = x / (z * -t);
	tmp = 0.0;
	if (t_1 <= -1000000000000.0)
		tmp = t_2;
	elseif (t_1 <= 500.0)
		tmp = 1.0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x / N[(N[(y - z), $MachinePrecision] * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x / N[(z * (-t)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1000000000000.0], t$95$2, If[LessEqual[t$95$1, 500.0], 1.0, t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))) < -1e12 or 500 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t)))

   1. Initial program 96.9%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. *-lowering-*.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. mul-1-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. unsub-neg N/A

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

      14. mul-1-neg N/A

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

      15. remove-double-neg N/A

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

      16. --lowering--.f64 96.6

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

   5. Simplified 96.6%

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

   6. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{x}{t \cdot z}\right)} \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{x}{\mathsf{neg}\left(t \cdot z\right)}} \]

      3. mul-1-neg N/A

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

      4. /-lowering-/.f64 N/A

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

      5. mul-1-neg N/A

         \[\leadsto \frac{x}{\color{blue}{\mathsf{neg}\left(t \cdot z\right)}} \]

      6. *-commutative N/A

         \[\leadsto \frac{x}{\mathsf{neg}\left(\color{blue}{z \cdot t}\right)} \]

      7. distribute-rgt-neg-in N/A

         \[\leadsto \frac{x}{\color{blue}{z \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{z \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      9. neg-lowering-neg.f64 48.6

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

   8. Simplified 48.6%

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

   if -1e12 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))) < 500

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 99.2%

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

Alternative 8: 80.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{x}{\left(y - z\right) \cdot \left(y - t\right)}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+25}:\\ \;\;\;\;\frac{x}{y \cdot z}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+61}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot t}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ x (* (- y z) (- y t)))))
   (if (<= t_1 -2e+25) (/ x (* y z)) (if (<= t_1 5e+61) 1.0 (/ x (* y t))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = x / ((y - z) * (y - t));
	double tmp;
	if (t_1 <= -2e+25) {
		tmp = x / (y * z);
	} else if (t_1 <= 5e+61) {
		tmp = 1.0;
	} else {
		tmp = x / (y * t);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x / ((y - z) * (y - t))
    if (t_1 <= (-2d+25)) then
        tmp = x / (y * z)
    else if (t_1 <= 5d+61) then
        tmp = 1.0d0
    else
        tmp = x / (y * t)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = x / ((y - z) * (y - t));
	double tmp;
	if (t_1 <= -2e+25) {
		tmp = x / (y * z);
	} else if (t_1 <= 5e+61) {
		tmp = 1.0;
	} else {
		tmp = x / (y * t);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = x / ((y - z) * (y - t))
	tmp = 0
	if t_1 <= -2e+25:
		tmp = x / (y * z)
	elif t_1 <= 5e+61:
		tmp = 1.0
	else:
		tmp = x / (y * t)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(x / Float64(Float64(y - z) * Float64(y - t)))
	tmp = 0.0
	if (t_1 <= -2e+25)
		tmp = Float64(x / Float64(y * z));
	elseif (t_1 <= 5e+61)
		tmp = 1.0;
	else
		tmp = Float64(x / Float64(y * t));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = x / ((y - z) * (y - t));
	tmp = 0.0;
	if (t_1 <= -2e+25)
		tmp = x / (y * z);
	elseif (t_1 <= 5e+61)
		tmp = 1.0;
	else
		tmp = x / (y * t);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x / N[(N[(y - z), $MachinePrecision] * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+25], N[(x / N[(y * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+61], 1.0, N[(x / N[(y * t), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))) < -2.00000000000000018e25

   1. Initial program 96.9%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. *-lowering-*.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. mul-1-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. unsub-neg N/A

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

      14. mul-1-neg N/A

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

      15. remove-double-neg N/A

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

      16. --lowering--.f64 96.9

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

   5. Simplified 96.9%

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

   6. Taylor expanded in y around inf

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

   8. Simplified 58.4%

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

   9. Taylor expanded in y around 0

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

       1. /-lowering-/.f64 N/A

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 42.7

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

   11. Simplified 42.7%

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

   if -2.00000000000000018e25 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))) < 5.00000000000000018e61

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 97.7%

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

   if 5.00000000000000018e61 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t)))

   1. Initial program 96.6%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. --lowering--.f64 N/A

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

      5. --lowering--.f64 90.6

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

   4. Applied egg-rr 90.6%

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

   5. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 35.3

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

   7. Simplified 35.3%

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

   8. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 24.8

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

   10. Simplified 24.8%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \leq -2 \cdot 10^{+25}:\\ \;\;\;\;\frac{x}{y \cdot z}\\ \mathbf{elif}\;\frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \leq 5 \cdot 10^{+61}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{x}{\left(y - z\right) \cdot \left(y - t\right)}\\ t_2 := \frac{x}{y \cdot t}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+42}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+61}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ x (* (- y z) (- y t)))) (t_2 (/ x (* y t))))
   (if (<= t_1 -2e+42) t_2 (if (<= t_1 5e+61) 1.0 t_2))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = x / ((y - z) * (y - t));
	double t_2 = x / (y * t);
	double tmp;
	if (t_1 <= -2e+42) {
		tmp = t_2;
	} else if (t_1 <= 5e+61) {
		tmp = 1.0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x / ((y - z) * (y - t))
    t_2 = x / (y * t)
    if (t_1 <= (-2d+42)) then
        tmp = t_2
    else if (t_1 <= 5d+61) then
        tmp = 1.0d0
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = x / ((y - z) * (y - t));
	double t_2 = x / (y * t);
	double tmp;
	if (t_1 <= -2e+42) {
		tmp = t_2;
	} else if (t_1 <= 5e+61) {
		tmp = 1.0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = x / ((y - z) * (y - t))
	t_2 = x / (y * t)
	tmp = 0
	if t_1 <= -2e+42:
		tmp = t_2
	elif t_1 <= 5e+61:
		tmp = 1.0
	else:
		tmp = t_2
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(x / Float64(Float64(y - z) * Float64(y - t)))
	t_2 = Float64(x / Float64(y * t))
	tmp = 0.0
	if (t_1 <= -2e+42)
		tmp = t_2;
	elseif (t_1 <= 5e+61)
		tmp = 1.0;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = x / ((y - z) * (y - t));
	t_2 = x / (y * t);
	tmp = 0.0;
	if (t_1 <= -2e+42)
		tmp = t_2;
	elseif (t_1 <= 5e+61)
		tmp = 1.0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x / N[(N[(y - z), $MachinePrecision] * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x / N[(y * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+42], t$95$2, If[LessEqual[t$95$1, 5e+61], 1.0, t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))) < -2.00000000000000009e42 or 5.00000000000000018e61 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t)))

   1. Initial program 96.6%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. --lowering--.f64 N/A

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

      5. --lowering--.f64 93.6

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

   4. Applied egg-rr 93.6%

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

   5. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 46.2

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

   7. Simplified 46.2%

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

   8. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 29.1

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

   10. Simplified 29.1%

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

   if -2.00000000000000009e42 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))) < 5.00000000000000018e61

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 96.8%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \leq -2 \cdot 10^{+42}:\\ \;\;\;\;\frac{x}{y \cdot t}\\ \mathbf{elif}\;\frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \leq 5 \cdot 10^{+61}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

3. Final simplification 99.2%

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

Alternative 11: 75.3% accurate, 26.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := 1.0

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \]
4. Step-by-step derivation

5. Simplified 74.4%

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

Reproduce

herbie shell --seed 2024198 
(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)))))