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

Percentage Accurate: 99.2% → 99.2%

Time: 7.7s

Alternatives: 10

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: 99.2% 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: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

Alternative 2: 88.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{\left(z - y\right) \cdot \left(t - y\right)}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+18}:\\ \;\;\;\;\frac{x}{t \cdot \left(y - z\right)}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-107}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{\left(t - y\right) \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ x (* (- z y) (- t y)))))
   (if (<= t_1 -1e+18)
     (/ x (* t (- y z)))
     (if (<= t_1 5e-107) 1.0 (- 1.0 (/ x (* (- t y) z)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x / ((z - y) * (t - y));
	double tmp;
	if (t_1 <= -1e+18) {
		tmp = x / (t * (y - z));
	} else if (t_1 <= 5e-107) {
		tmp = 1.0;
	} else {
		tmp = 1.0 - (x / ((t - y) * z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x / ((z - y) * (t - y))
    if (t_1 <= (-1d+18)) then
        tmp = x / (t * (y - z))
    else if (t_1 <= 5d-107) then
        tmp = 1.0d0
    else
        tmp = 1.0d0 - (x / ((t - y) * z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x / ((z - y) * (t - y));
	double tmp;
	if (t_1 <= -1e+18) {
		tmp = x / (t * (y - z));
	} else if (t_1 <= 5e-107) {
		tmp = 1.0;
	} else {
		tmp = 1.0 - (x / ((t - y) * z));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x / ((z - y) * (t - y))
	tmp = 0
	if t_1 <= -1e+18:
		tmp = x / (t * (y - z))
	elif t_1 <= 5e-107:
		tmp = 1.0
	else:
		tmp = 1.0 - (x / ((t - y) * z))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x / Float64(Float64(z - y) * Float64(t - y)))
	tmp = 0.0
	if (t_1 <= -1e+18)
		tmp = Float64(x / Float64(t * Float64(y - z)));
	elseif (t_1 <= 5e-107)
		tmp = 1.0;
	else
		tmp = Float64(1.0 - Float64(x / Float64(Float64(t - y) * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x / ((z - y) * (t - y));
	tmp = 0.0;
	if (t_1 <= -1e+18)
		tmp = x / (t * (y - z));
	elseif (t_1 <= 5e-107)
		tmp = 1.0;
	else
		tmp = 1.0 - (x / ((t - y) * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x / N[(N[(z - y), $MachinePrecision] * N[(t - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+18], N[(x / N[(t * N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-107], 1.0, N[(1.0 - N[(x / N[(N[(t - y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))) < -1e18

   1. Initial program 93.1%

      \[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
   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. associate-/r* N/A

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

      2. associate-*r/ N/A

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

      3. lower-/.f64 N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-frac2 N/A

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

      6. mul-1-neg N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. mul-1-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. unsub-neg N/A

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

      14. mul-1-neg N/A

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

      15. remove-double-neg N/A

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

      16. lower--.f64 N/A

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

      17. lower--.f64 90.6

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

   5. Applied rewrites 90.6%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 53.0%

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

   if -1e18 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))) < 4.99999999999999971e-107

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 98.4%

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

   if 4.99999999999999971e-107 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t)))

   1. Initial program 99.8%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. unsub-neg N/A

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

      10. mul-1-neg N/A

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

      11. remove-double-neg N/A

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

      12. lower--.f64 81.9

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

   5. Applied rewrites 81.9%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\left(z - y\right) \cdot \left(t - y\right)} \leq -1 \cdot 10^{+18}:\\ \;\;\;\;\frac{x}{t \cdot \left(y - z\right)}\\ \mathbf{elif}\;\frac{x}{\left(z - y\right) \cdot \left(t - y\right)} \leq 5 \cdot 10^{-107}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{\left(t - y\right) \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{z \cdot y}\\ t_2 := 1 - \frac{x}{\left(z - y\right) \cdot \left(t - y\right)}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+34}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ x (* z y))) (t_2 (- 1.0 (/ x (* (- z y) (- t y))))))
   (if (<= t_2 -5e+34) t_1 (if (<= t_2 2.0) 1.0 t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x / (z * y);
	double t_2 = 1.0 - (x / ((z - y) * (t - y)));
	double tmp;
	if (t_2 <= -5e+34) {
		tmp = t_1;
	} else if (t_2 <= 2.0) {
		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 = x / (z * y)
    t_2 = 1.0d0 - (x / ((z - y) * (t - y)))
    if (t_2 <= (-5d+34)) then
        tmp = t_1
    else if (t_2 <= 2.0d0) 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 = x / (z * y);
	double t_2 = 1.0 - (x / ((z - y) * (t - y)));
	double tmp;
	if (t_2 <= -5e+34) {
		tmp = t_1;
	} else if (t_2 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x / (z * y)
	t_2 = 1.0 - (x / ((z - y) * (t - y)))
	tmp = 0
	if t_2 <= -5e+34:
		tmp = t_1
	elif t_2 <= 2.0:
		tmp = 1.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x / Float64(z * y))
	t_2 = Float64(1.0 - Float64(x / Float64(Float64(z - y) * Float64(t - y))))
	tmp = 0.0
	if (t_2 <= -5e+34)
		tmp = t_1;
	elseif (t_2 <= 2.0)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x / (z * y);
	t_2 = 1.0 - (x / ((z - y) * (t - y)));
	tmp = 0.0;
	if (t_2 <= -5e+34)
		tmp = t_1;
	elseif (t_2 <= 2.0)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x / N[(z * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[(x / N[(N[(z - y), $MachinePrecision] * N[(t - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+34], t$95$1, If[LessEqual[t$95$2, 2.0], 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)))) < -4.9999999999999998e34 or 2 < (-.f64 #s(literal 1 binary64) (/.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. 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. associate-/r* N/A

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

      2. associate-*r/ N/A

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

      3. lower-/.f64 N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-frac2 N/A

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

      6. mul-1-neg N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. mul-1-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. unsub-neg N/A

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

      14. mul-1-neg N/A

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

      15. remove-double-neg N/A

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

      16. lower--.f64 N/A

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

      17. lower--.f64 90.0

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

   5. Applied rewrites 90.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 69.4%

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

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 30.2%

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

   if -4.9999999999999998e34 < (-.f64 #s(literal 1 binary64) (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t)))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 97.3%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - \frac{x}{\left(z - y\right) \cdot \left(t - y\right)} \leq -5 \cdot 10^{+34}:\\ \;\;\;\;\frac{x}{z \cdot y}\\ \mathbf{elif}\;1 - \frac{x}{\left(z - y\right) \cdot \left(t - y\right)} \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{\left(z - y\right) \cdot \left(t - y\right)}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+18}:\\ \;\;\;\;\frac{x}{t \cdot \left(y - z\right)}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-11}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ x (* (- z y) (- t y)))))
   (if (<= t_1 -1e+18)
     (/ x (* t (- y z)))
     (if (<= t_1 5e-11) 1.0 (/ x (* (- y t) z))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x / ((z - y) * (t - y));
	double tmp;
	if (t_1 <= -1e+18) {
		tmp = x / (t * (y - z));
	} else if (t_1 <= 5e-11) {
		tmp = 1.0;
	} else {
		tmp = x / ((y - t) * z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x / ((z - y) * (t - y))
    if (t_1 <= (-1d+18)) then
        tmp = x / (t * (y - z))
    else if (t_1 <= 5d-11) then
        tmp = 1.0d0
    else
        tmp = x / ((y - t) * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x / ((z - y) * (t - y));
	double tmp;
	if (t_1 <= -1e+18) {
		tmp = x / (t * (y - z));
	} else if (t_1 <= 5e-11) {
		tmp = 1.0;
	} else {
		tmp = x / ((y - t) * z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x / ((z - y) * (t - y))
	tmp = 0
	if t_1 <= -1e+18:
		tmp = x / (t * (y - z))
	elif t_1 <= 5e-11:
		tmp = 1.0
	else:
		tmp = x / ((y - t) * z)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x / Float64(Float64(z - y) * Float64(t - y)))
	tmp = 0.0
	if (t_1 <= -1e+18)
		tmp = Float64(x / Float64(t * Float64(y - z)));
	elseif (t_1 <= 5e-11)
		tmp = 1.0;
	else
		tmp = Float64(x / Float64(Float64(y - t) * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x / ((z - y) * (t - y));
	tmp = 0.0;
	if (t_1 <= -1e+18)
		tmp = x / (t * (y - z));
	elseif (t_1 <= 5e-11)
		tmp = 1.0;
	else
		tmp = x / ((y - t) * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x / N[(N[(z - y), $MachinePrecision] * N[(t - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+18], N[(x / N[(t * N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-11], 1.0, N[(x / N[(N[(y - t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))) < -1e18

   1. Initial program 93.1%

      \[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
   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. associate-/r* N/A

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

      2. associate-*r/ N/A

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

      3. lower-/.f64 N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-frac2 N/A

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

      6. mul-1-neg N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. mul-1-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. unsub-neg N/A

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

      14. mul-1-neg N/A

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

      15. remove-double-neg N/A

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

      16. lower--.f64 N/A

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

      17. lower--.f64 90.6

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

   5. Applied rewrites 90.6%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 53.0%

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

   if -1e18 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))) < 5.00000000000000018e-11

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 98.3%

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

   if 5.00000000000000018e-11 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t)))

   1. Initial program 99.7%

      \[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
   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. associate-/r* N/A

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

      2. associate-*r/ N/A

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

      3. lower-/.f64 N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-frac2 N/A

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

      6. mul-1-neg N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. mul-1-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. unsub-neg N/A

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

      14. mul-1-neg N/A

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

      15. remove-double-neg N/A

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

      16. lower--.f64 N/A

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

      17. lower--.f64 89.1

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

   5. Applied rewrites 89.1%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 75.3%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\left(z - y\right) \cdot \left(t - y\right)} \leq -1 \cdot 10^{+18}:\\ \;\;\;\;\frac{x}{t \cdot \left(y - z\right)}\\ \mathbf{elif}\;\frac{x}{\left(z - y\right) \cdot \left(t - y\right)} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{\left(z - y\right) \cdot \left(t - y\right)}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+18}:\\ \;\;\;\;\frac{x}{t \cdot \left(y - z\right)}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-27}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{t \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ x (* (- z y) (- t y)))))
   (if (<= t_1 -1e+18)
     (/ x (* t (- y z)))
     (if (<= t_1 5e-27) 1.0 (- 1.0 (/ x (* t z)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x / ((z - y) * (t - y));
	double tmp;
	if (t_1 <= -1e+18) {
		tmp = x / (t * (y - z));
	} else if (t_1 <= 5e-27) {
		tmp = 1.0;
	} else {
		tmp = 1.0 - (x / (t * z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x / ((z - y) * (t - y))
    if (t_1 <= (-1d+18)) then
        tmp = x / (t * (y - z))
    else if (t_1 <= 5d-27) then
        tmp = 1.0d0
    else
        tmp = 1.0d0 - (x / (t * z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x / ((z - y) * (t - y));
	double tmp;
	if (t_1 <= -1e+18) {
		tmp = x / (t * (y - z));
	} else if (t_1 <= 5e-27) {
		tmp = 1.0;
	} else {
		tmp = 1.0 - (x / (t * z));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x / ((z - y) * (t - y))
	tmp = 0
	if t_1 <= -1e+18:
		tmp = x / (t * (y - z))
	elif t_1 <= 5e-27:
		tmp = 1.0
	else:
		tmp = 1.0 - (x / (t * z))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x / Float64(Float64(z - y) * Float64(t - y)))
	tmp = 0.0
	if (t_1 <= -1e+18)
		tmp = Float64(x / Float64(t * Float64(y - z)));
	elseif (t_1 <= 5e-27)
		tmp = 1.0;
	else
		tmp = Float64(1.0 - Float64(x / Float64(t * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x / ((z - y) * (t - y));
	tmp = 0.0;
	if (t_1 <= -1e+18)
		tmp = x / (t * (y - z));
	elseif (t_1 <= 5e-27)
		tmp = 1.0;
	else
		tmp = 1.0 - (x / (t * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x / N[(N[(z - y), $MachinePrecision] * N[(t - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+18], N[(x / N[(t * N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-27], 1.0, N[(1.0 - N[(x / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))) < -1e18

   1. Initial program 93.1%

      \[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
   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. associate-/r* N/A

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

      2. associate-*r/ N/A

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

      3. lower-/.f64 N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-frac2 N/A

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

      6. mul-1-neg N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. mul-1-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. unsub-neg N/A

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

      14. mul-1-neg N/A

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

      15. remove-double-neg N/A

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

      16. lower--.f64 N/A

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

      17. lower--.f64 90.6

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

   5. Applied rewrites 90.6%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 53.0%

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

   if -1e18 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))) < 5.0000000000000002e-27

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 98.5%

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

   if 5.0000000000000002e-27 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t)))

   1. Initial program 99.7%

      \[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
   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 64.8

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

   5. Applied rewrites 64.8%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\left(z - y\right) \cdot \left(t - y\right)} \leq -1 \cdot 10^{+18}:\\ \;\;\;\;\frac{x}{t \cdot \left(y - z\right)}\\ \mathbf{elif}\;\frac{x}{\left(z - y\right) \cdot \left(t - y\right)} \leq 5 \cdot 10^{-27}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{t \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{\left(z - y\right) \cdot \left(t - y\right)}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+18}:\\ \;\;\;\;\frac{x}{t \cdot \left(y - z\right)}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-11}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{t \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ x (* (- z y) (- t y)))))
   (if (<= t_1 -1e+18)
     (/ x (* t (- y z)))
     (if (<= t_1 5e-11) 1.0 (/ (- x) (* t z))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x / ((z - y) * (t - y));
	double tmp;
	if (t_1 <= -1e+18) {
		tmp = x / (t * (y - z));
	} else if (t_1 <= 5e-11) {
		tmp = 1.0;
	} else {
		tmp = -x / (t * z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x / ((z - y) * (t - y))
    if (t_1 <= (-1d+18)) then
        tmp = x / (t * (y - z))
    else if (t_1 <= 5d-11) then
        tmp = 1.0d0
    else
        tmp = -x / (t * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x / ((z - y) * (t - y));
	double tmp;
	if (t_1 <= -1e+18) {
		tmp = x / (t * (y - z));
	} else if (t_1 <= 5e-11) {
		tmp = 1.0;
	} else {
		tmp = -x / (t * z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x / ((z - y) * (t - y))
	tmp = 0
	if t_1 <= -1e+18:
		tmp = x / (t * (y - z))
	elif t_1 <= 5e-11:
		tmp = 1.0
	else:
		tmp = -x / (t * z)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x / Float64(Float64(z - y) * Float64(t - y)))
	tmp = 0.0
	if (t_1 <= -1e+18)
		tmp = Float64(x / Float64(t * Float64(y - z)));
	elseif (t_1 <= 5e-11)
		tmp = 1.0;
	else
		tmp = Float64(Float64(-x) / Float64(t * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x / ((z - y) * (t - y));
	tmp = 0.0;
	if (t_1 <= -1e+18)
		tmp = x / (t * (y - z));
	elseif (t_1 <= 5e-11)
		tmp = 1.0;
	else
		tmp = -x / (t * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x / N[(N[(z - y), $MachinePrecision] * N[(t - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+18], N[(x / N[(t * N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-11], 1.0, N[((-x) / N[(t * z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))) < -1e18

   1. Initial program 93.1%

      \[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
   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. associate-/r* N/A

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

      2. associate-*r/ N/A

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

      3. lower-/.f64 N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-frac2 N/A

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

      6. mul-1-neg N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. mul-1-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. unsub-neg N/A

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

      14. mul-1-neg N/A

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

      15. remove-double-neg N/A

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

      16. lower--.f64 N/A

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

      17. lower--.f64 90.6

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

   5. Applied rewrites 90.6%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 53.0%

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

   if -1e18 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))) < 5.00000000000000018e-11

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 98.3%

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

   if 5.00000000000000018e-11 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t)))

   1. Initial program 99.7%

      \[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
   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. associate-/r* N/A

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

      2. associate-*r/ N/A

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

      3. lower-/.f64 N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-frac2 N/A

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

      6. mul-1-neg N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. mul-1-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. unsub-neg N/A

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

      14. mul-1-neg N/A

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

      15. remove-double-neg N/A

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

      16. lower--.f64 N/A

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

      17. lower--.f64 89.1

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

   5. Applied rewrites 89.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 63.8%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\left(z - y\right) \cdot \left(t - y\right)} \leq -1 \cdot 10^{+18}:\\ \;\;\;\;\frac{x}{t \cdot \left(y - z\right)}\\ \mathbf{elif}\;\frac{x}{\left(z - y\right) \cdot \left(t - y\right)} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{t \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{\left(z - y\right) \cdot \left(t - y\right)}\\ t_2 := \frac{-x}{t \cdot z}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+18}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-11}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ x (* (- z y) (- t y)))) (t_2 (/ (- x) (* t z))))
   (if (<= t_1 -1e+18) t_2 (if (<= t_1 5e-11) 1.0 t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x / ((z - y) * (t - y));
	double t_2 = -x / (t * z);
	double tmp;
	if (t_1 <= -1e+18) {
		tmp = t_2;
	} else if (t_1 <= 5e-11) {
		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 / ((z - y) * (t - y))
    t_2 = -x / (t * z)
    if (t_1 <= (-1d+18)) then
        tmp = t_2
    else if (t_1 <= 5d-11) 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 / ((z - y) * (t - y));
	double t_2 = -x / (t * z);
	double tmp;
	if (t_1 <= -1e+18) {
		tmp = t_2;
	} else if (t_1 <= 5e-11) {
		tmp = 1.0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x / ((z - y) * (t - y))
	t_2 = -x / (t * z)
	tmp = 0
	if t_1 <= -1e+18:
		tmp = t_2
	elif t_1 <= 5e-11:
		tmp = 1.0
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x / Float64(Float64(z - y) * Float64(t - y)))
	t_2 = Float64(Float64(-x) / Float64(t * z))
	tmp = 0.0
	if (t_1 <= -1e+18)
		tmp = t_2;
	elseif (t_1 <= 5e-11)
		tmp = 1.0;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x / ((z - y) * (t - y));
	t_2 = -x / (t * z);
	tmp = 0.0;
	if (t_1 <= -1e+18)
		tmp = t_2;
	elseif (t_1 <= 5e-11)
		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[(z - y), $MachinePrecision] * N[(t - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[((-x) / N[(t * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+18], t$95$2, If[LessEqual[t$95$1, 5e-11], 1.0, t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))) < -1e18 or 5.00000000000000018e-11 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t)))

   1. Initial program 96.7%

      \[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
   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. associate-/r* N/A

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

      2. associate-*r/ N/A

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

      3. lower-/.f64 N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-frac2 N/A

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

      6. mul-1-neg N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. mul-1-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. unsub-neg N/A

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

      14. mul-1-neg N/A

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

      15. remove-double-neg N/A

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

      16. lower--.f64 N/A

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

      17. lower--.f64 89.8

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

   5. Applied rewrites 89.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 53.3%

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

   if -1e18 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 y t))) < 5.00000000000000018e-11

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 98.3%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\left(z - y\right) \cdot \left(t - y\right)} \leq -1 \cdot 10^{+18}:\\ \;\;\;\;\frac{-x}{t \cdot z}\\ \mathbf{elif}\;\frac{x}{\left(z - y\right) \cdot \left(t - y\right)} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{t \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 87.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{-84}:\\ \;\;\;\;1 - \frac{x}{\left(t - y\right) \cdot z}\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-43}:\\ \;\;\;\;1 - \frac{x}{\left(y - t\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{\left(z - y\right) \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -2.6e-84)
   (- 1.0 (/ x (* (- t y) z)))
   (if (<= z 8.5e-43)
     (- 1.0 (/ x (* (- y t) y)))
     (- 1.0 (/ x (* (- z y) t))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -2.6e-84) {
		tmp = 1.0 - (x / ((t - y) * z));
	} else if (z <= 8.5e-43) {
		tmp = 1.0 - (x / ((y - t) * y));
	} else {
		tmp = 1.0 - (x / ((z - y) * t));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-2.6d-84)) then
        tmp = 1.0d0 - (x / ((t - y) * z))
    else if (z <= 8.5d-43) then
        tmp = 1.0d0 - (x / ((y - t) * y))
    else
        tmp = 1.0d0 - (x / ((z - y) * t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -2.6e-84) {
		tmp = 1.0 - (x / ((t - y) * z));
	} else if (z <= 8.5e-43) {
		tmp = 1.0 - (x / ((y - t) * y));
	} else {
		tmp = 1.0 - (x / ((z - y) * t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -2.6e-84:
		tmp = 1.0 - (x / ((t - y) * z))
	elif z <= 8.5e-43:
		tmp = 1.0 - (x / ((y - t) * y))
	else:
		tmp = 1.0 - (x / ((z - y) * t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -2.6e-84)
		tmp = Float64(1.0 - Float64(x / Float64(Float64(t - y) * z)));
	elseif (z <= 8.5e-43)
		tmp = Float64(1.0 - Float64(x / Float64(Float64(y - t) * y)));
	else
		tmp = Float64(1.0 - Float64(x / Float64(Float64(z - y) * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -2.6e-84)
		tmp = 1.0 - (x / ((t - y) * z));
	elseif (z <= 8.5e-43)
		tmp = 1.0 - (x / ((y - t) * y));
	else
		tmp = 1.0 - (x / ((z - y) * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -2.6e-84], N[(1.0 - N[(x / N[(N[(t - y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 8.5e-43], N[(1.0 - N[(x / N[(N[(y - t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(x / N[(N[(z - y), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.6e-84

   1. Initial program 99.9%

      \[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
   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. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. unsub-neg N/A

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

      10. mul-1-neg N/A

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

      11. remove-double-neg N/A

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

      12. lower--.f64 96.7

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

   5. Applied rewrites 96.7%

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

   if -2.6e-84 < z < 8.50000000000000056e-43

   1. Initial program 97.7%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 89.9

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

   5. Applied rewrites 89.9%

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

   if 8.50000000000000056e-43 < z

   1. Initial program 100.0%

      \[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
   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. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. unsub-neg N/A

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

      10. mul-1-neg N/A

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

      11. remove-double-neg N/A

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

      12. lower--.f64 77.0

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

   5. Applied rewrites 77.0%

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

Alternative 9: 91.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 - \frac{x}{\left(t - y\right) \cdot z}\\ \mathbf{if}\;z \leq -2.6 \cdot 10^{-84}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-46}:\\ \;\;\;\;1 - \frac{x}{\left(y - t\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- 1.0 (/ x (* (- t y) z)))))
   (if (<= z -2.6e-84)
     t_1
     (if (<= z 8.5e-46) (- 1.0 (/ x (* (- y t) y))) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = 1.0 - (x / ((t - y) * z));
	double tmp;
	if (z <= -2.6e-84) {
		tmp = t_1;
	} else if (z <= 8.5e-46) {
		tmp = 1.0 - (x / ((y - t) * y));
	} 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) :: tmp
    t_1 = 1.0d0 - (x / ((t - y) * z))
    if (z <= (-2.6d-84)) then
        tmp = t_1
    else if (z <= 8.5d-46) then
        tmp = 1.0d0 - (x / ((y - t) * y))
    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 - y) * z));
	double tmp;
	if (z <= -2.6e-84) {
		tmp = t_1;
	} else if (z <= 8.5e-46) {
		tmp = 1.0 - (x / ((y - t) * y));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = 1.0 - (x / ((t - y) * z))
	tmp = 0
	if z <= -2.6e-84:
		tmp = t_1
	elif z <= 8.5e-46:
		tmp = 1.0 - (x / ((y - t) * y))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(1.0 - Float64(x / Float64(Float64(t - y) * z)))
	tmp = 0.0
	if (z <= -2.6e-84)
		tmp = t_1;
	elseif (z <= 8.5e-46)
		tmp = Float64(1.0 - Float64(x / Float64(Float64(y - t) * y)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = 1.0 - (x / ((t - y) * z));
	tmp = 0.0;
	if (z <= -2.6e-84)
		tmp = t_1;
	elseif (z <= 8.5e-46)
		tmp = 1.0 - (x / ((y - t) * y));
	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[(N[(t - y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.6e-84], t$95$1, If[LessEqual[z, 8.5e-46], N[(1.0 - N[(x / N[(N[(y - t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.6e-84 or 8.5000000000000001e-46 < z

   1. Initial program 99.9%

      \[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)} \]
   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. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. unsub-neg N/A

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

      10. mul-1-neg N/A

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

      11. remove-double-neg N/A

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

      12. lower--.f64 96.0

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

   5. Applied rewrites 96.0%

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

   if -2.6e-84 < z < 8.5000000000000001e-46

   1. Initial program 97.7%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 89.9

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

   5. Applied rewrites 89.9%

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

Alternative 10: 75.4% 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 99.2%

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

3. Taylor expanded in t around inf

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

5. Applied rewrites 74.1%

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

Reproduce

herbie shell --seed 2024254 
(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)))))