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

Percentage Accurate: 99.0% → 99.0%

Time: 9.3s

Alternatives: 6

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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 99.4%

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

3. Final simplification 99.4%

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

Alternative 2: 88.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 + \frac{x}{\left(y - z\right) \cdot \left(t - y\right)}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+20}:\\ \;\;\;\;1 - \frac{x}{z \cdot \left(t - y\right)}\\ \mathbf{elif}\;t\_1 \leq 1.001:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{t \cdot \left(z - y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ 1.0 (/ x (* (- y z) (- t y))))))
   (if (<= t_1 -4e+20)
     (- 1.0 (/ x (* z (- t y))))
     (if (<= t_1 1.001) 1.0 (- 1.0 (/ x (* t (- z y))))))))

C

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 <= -4e+20) {
		tmp = 1.0 - (x / (z * (t - y)));
	} else if (t_1 <= 1.001) {
		tmp = 1.0;
	} else {
		tmp = 1.0 - (x / (t * (z - y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 + (x / ((y - z) * (t - y)))
    if (t_1 <= (-4d+20)) then
        tmp = 1.0d0 - (x / (z * (t - y)))
    else if (t_1 <= 1.001d0) then
        tmp = 1.0d0
    else
        tmp = 1.0d0 - (x / (t * (z - y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = 1.0 + (x / ((y - z) * (t - y)));
	double tmp;
	if (t_1 <= -4e+20) {
		tmp = 1.0 - (x / (z * (t - y)));
	} else if (t_1 <= 1.001) {
		tmp = 1.0;
	} else {
		tmp = 1.0 - (x / (t * (z - y)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = 1.0 + (x / ((y - z) * (t - y)))
	tmp = 0
	if t_1 <= -4e+20:
		tmp = 1.0 - (x / (z * (t - y)))
	elif t_1 <= 1.001:
		tmp = 1.0
	else:
		tmp = 1.0 - (x / (t * (z - y)))
	return tmp

Julia

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 <= -4e+20)
		tmp = Float64(1.0 - Float64(x / Float64(z * Float64(t - y))));
	elseif (t_1 <= 1.001)
		tmp = 1.0;
	else
		tmp = Float64(1.0 - Float64(x / Float64(t * Float64(z - y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = 1.0 + (x / ((y - z) * (t - y)));
	tmp = 0.0;
	if (t_1 <= -4e+20)
		tmp = 1.0 - (x / (z * (t - y)));
	elseif (t_1 <= 1.001)
		tmp = 1.0;
	else
		tmp = 1.0 - (x / (t * (z - y)));
	end
	tmp_2 = tmp;
end

Wolfram

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, -4e+20], N[(1.0 - N[(x / N[(z * N[(t - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.001], 1.0, N[(1.0 - N[(x / N[(t * N[(z - y), $MachinePrecision]), $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)))) < -4e20

   1. Initial program 94.5%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. +-commutative N/A

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

      9. distribute-neg-in N/A

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

      10. unsub-neg N/A

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

      11. mul-1-neg N/A

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

      12. remove-double-neg N/A

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

      13. lower--.f64 74.2

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

   5. Applied rewrites 74.2%

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

   if -4e20 < (-.f64 #s(literal 1 binary64) (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t)))) < 1.0009999999999999

   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. Applied rewrites 99.6%

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

   if 1.0009999999999999 < (-.f64 #s(literal 1 binary64) (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))))

   1. Initial program 99.9%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. +-commutative N/A

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

      9. distribute-neg-in N/A

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

      10. unsub-neg N/A

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

      11. mul-1-neg N/A

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

      12. remove-double-neg N/A

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

      13. lower--.f64 68.3

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

   5. Applied rewrites 68.3%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 + \frac{x}{\left(y - z\right) \cdot \left(t - y\right)} \leq -4 \cdot 10^{+20}:\\ \;\;\;\;1 - \frac{x}{z \cdot \left(t - y\right)}\\ \mathbf{elif}\;1 + \frac{x}{\left(y - z\right) \cdot \left(t - y\right)} \leq 1.001:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{t \cdot \left(z - y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 - \frac{x}{t \cdot \left(z - y\right)}\\ t_2 := 1 + \frac{x}{\left(y - z\right) \cdot \left(t - y\right)}\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{+20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 1.001:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- 1.0 (/ x (* t (- z y)))))
        (t_2 (+ 1.0 (/ x (* (- y z) (- t y))))))
   (if (<= t_2 -4e+20) t_1 (if (<= t_2 1.001) 1.0 t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = 1.0 - (x / (t * (z - y)));
	double t_2 = 1.0 + (x / ((y - z) * (t - y)));
	double tmp;
	if (t_2 <= -4e+20) {
		tmp = t_1;
	} else if (t_2 <= 1.001) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = 1.0d0 - (x / (t * (z - y)))
    t_2 = 1.0d0 + (x / ((y - z) * (t - y)))
    if (t_2 <= (-4d+20)) then
        tmp = t_1
    else if (t_2 <= 1.001d0) then
        tmp = 1.0d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = 1.0 - (x / (t * (z - y)));
	double t_2 = 1.0 + (x / ((y - z) * (t - y)));
	double tmp;
	if (t_2 <= -4e+20) {
		tmp = t_1;
	} else if (t_2 <= 1.001) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = 1.0 - (x / (t * (z - y)))
	t_2 = 1.0 + (x / ((y - z) * (t - y)))
	tmp = 0
	if t_2 <= -4e+20:
		tmp = t_1
	elif t_2 <= 1.001:
		tmp = 1.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(1.0 - Float64(x / Float64(t * Float64(z - y))))
	t_2 = Float64(1.0 + Float64(x / Float64(Float64(y - z) * Float64(t - y))))
	tmp = 0.0
	if (t_2 <= -4e+20)
		tmp = t_1;
	elseif (t_2 <= 1.001)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = 1.0 - (x / (t * (z - y)));
	t_2 = 1.0 + (x / ((y - z) * (t - y)));
	tmp = 0.0;
	if (t_2 <= -4e+20)
		tmp = t_1;
	elseif (t_2 <= 1.001)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(1.0 - N[(x / N[(t * N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + N[(x / N[(N[(y - z), $MachinePrecision] * N[(t - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -4e+20], t$95$1, If[LessEqual[t$95$2, 1.001], 1.0, t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t)))) < -4e20 or 1.0009999999999999 < (-.f64 #s(literal 1 binary64) (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))))

   1. Initial program 97.4%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. +-commutative N/A

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

      9. distribute-neg-in N/A

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

      10. unsub-neg N/A

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

      11. mul-1-neg N/A

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

      12. remove-double-neg N/A

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

      13. lower--.f64 63.8

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

   5. Applied rewrites 63.8%

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

   if -4e20 < (-.f64 #s(literal 1 binary64) (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t)))) < 1.0009999999999999

   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. Applied rewrites 99.6%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 + \frac{x}{\left(y - z\right) \cdot \left(t - y\right)} \leq -4 \cdot 10^{+20}:\\ \;\;\;\;1 - \frac{x}{t \cdot \left(z - y\right)}\\ \mathbf{elif}\;1 + \frac{x}{\left(y - z\right) \cdot \left(t - y\right)} \leq 1.001:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{t \cdot \left(z - y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{\left(y - z\right) \cdot \left(y - t\right)}\\ \mathbf{if}\;t\_1 \leq -0.001:\\ \;\;\;\;1 - \frac{x}{-y \cdot t}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-26}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{z \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ x (* (- y z) (- y t)))))
   (if (<= t_1 -0.001)
     (- 1.0 (/ x (- (* y t))))
     (if (<= t_1 5e-26) 1.0 (- 1.0 (/ x (* z t)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x / ((y - z) * (y - t));
	double tmp;
	if (t_1 <= -0.001) {
		tmp = 1.0 - (x / -(y * t));
	} else if (t_1 <= 5e-26) {
		tmp = 1.0;
	} 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) :: t_1
    real(8) :: tmp
    t_1 = x / ((y - z) * (y - t))
    if (t_1 <= (-0.001d0)) then
        tmp = 1.0d0 - (x / -(y * t))
    else if (t_1 <= 5d-26) then
        tmp = 1.0d0
    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 t_1 = x / ((y - z) * (y - t));
	double tmp;
	if (t_1 <= -0.001) {
		tmp = 1.0 - (x / -(y * t));
	} else if (t_1 <= 5e-26) {
		tmp = 1.0;
	} else {
		tmp = 1.0 - (x / (z * t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x / ((y - z) * (y - t))
	tmp = 0
	if t_1 <= -0.001:
		tmp = 1.0 - (x / -(y * t))
	elif t_1 <= 5e-26:
		tmp = 1.0
	else:
		tmp = 1.0 - (x / (z * t))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x / Float64(Float64(y - z) * Float64(y - t)))
	tmp = 0.0
	if (t_1 <= -0.001)
		tmp = Float64(1.0 - Float64(x / Float64(-Float64(y * t))));
	elseif (t_1 <= 5e-26)
		tmp = 1.0;
	else
		tmp = Float64(1.0 - Float64(x / Float64(z * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x / ((y - z) * (y - t));
	tmp = 0.0;
	if (t_1 <= -0.001)
		tmp = 1.0 - (x / -(y * t));
	elseif (t_1 <= 5e-26)
		tmp = 1.0;
	else
		tmp = 1.0 - (x / (z * t));
	end
	tmp_2 = tmp;
end

Wolfram

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, -0.001], N[(1.0 - N[(x / (-N[(y * t), $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-26], 1.0, N[(1.0 - N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))) < -1e-3

   1. Initial program 99.9%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. +-commutative N/A

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

      9. distribute-neg-in N/A

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

      10. unsub-neg N/A

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

      11. mul-1-neg N/A

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

      12. remove-double-neg N/A

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

      13. lower--.f64 68.3

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

   5. Applied rewrites 68.3%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 40.0%

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

   if -1e-3 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))) < 5.00000000000000019e-26

   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. Applied rewrites 99.6%

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

   if 5.00000000000000019e-26 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t)))

   1. Initial program 94.5%

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

   3. Taylor expanded in y around 0

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

      1. lower-*.f64 48.8

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

   5. Applied rewrites 48.8%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \leq -0.001:\\ \;\;\;\;1 - \frac{x}{-y \cdot t}\\ \mathbf{elif}\;\frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \leq 5 \cdot 10^{-26}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{z \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{\left(y - z\right) \cdot \left(y - t\right)}\\ t_2 := 1 - \frac{x}{z \cdot t}\\ \mathbf{if}\;t\_1 \leq -100000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-26}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(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 -100000000.0) t_2 (if (<= t_1 5e-26) 1.0 t_2))))

C

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 <= -100000000.0) {
		tmp = t_2;
	} else if (t_1 <= 5e-26) {
		tmp = 1.0;
	} else {
		tmp = t_2;
	}
	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) :: 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 <= (-100000000.0d0)) then
        tmp = t_2
    else if (t_1 <= 5d-26) then
        tmp = 1.0d0
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

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 <= -100000000.0) {
		tmp = t_2;
	} else if (t_1 <= 5e-26) {
		tmp = 1.0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

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 <= -100000000.0:
		tmp = t_2
	elif t_1 <= 5e-26:
		tmp = 1.0
	else:
		tmp = t_2
	return tmp

Julia

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 <= -100000000.0)
		tmp = t_2;
	elseif (t_1 <= 5e-26)
		tmp = 1.0;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

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 <= -100000000.0)
		tmp = t_2;
	elseif (t_1 <= 5e-26)
		tmp = 1.0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

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, -100000000.0], t$95$2, If[LessEqual[t$95$1, 5e-26], 1.0, t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))) < -1e8 or 5.00000000000000019e-26 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t)))

   1. Initial program 97.3%

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

   3. Taylor expanded in y around 0

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

      1. lower-*.f64 45.5

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

   5. Applied rewrites 45.5%

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

   if -1e8 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))) < 5.00000000000000019e-26

   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. Applied rewrites 98.9%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \leq -100000000:\\ \;\;\;\;1 - \frac{x}{z \cdot t}\\ \mathbf{elif}\;\frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \leq 5 \cdot 10^{-26}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{z \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.1% accurate, 26.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 77.9%

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

Reproduce

herbie shell --seed 2024221 
(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)))))