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

Percentage Accurate: 99.1% → 99.1%

Time: 10.4s

Alternatives: 7

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 1.0× speedup

Math

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

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 99.2%

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

3. Final simplification 99.2%

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

Alternative 2: 83.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{-53}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{-104}:\\ \;\;\;\;1 - \frac{x}{z \cdot t}\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-88}:\\ \;\;\;\;\frac{x}{y \cdot \left(t - y\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -1.05e-53)
   1.0
   (if (<= y 3.3e-104)
     (- 1.0 (/ x (* z t)))
     (if (<= y 4.2e-88) (/ x (* y (- t y))) 1.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.05e-53) {
		tmp = 1.0;
	} else if (y <= 3.3e-104) {
		tmp = 1.0 - (x / (z * t));
	} else if (y <= 4.2e-88) {
		tmp = x / (y * (t - y));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-1.05d-53)) then
        tmp = 1.0d0
    else if (y <= 3.3d-104) then
        tmp = 1.0d0 - (x / (z * t))
    else if (y <= 4.2d-88) then
        tmp = x / (y * (t - y))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.05e-53) {
		tmp = 1.0;
	} else if (y <= 3.3e-104) {
		tmp = 1.0 - (x / (z * t));
	} else if (y <= 4.2e-88) {
		tmp = x / (y * (t - y));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -1.05e-53:
		tmp = 1.0
	elif y <= 3.3e-104:
		tmp = 1.0 - (x / (z * t))
	elif y <= 4.2e-88:
		tmp = x / (y * (t - y))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -1.05e-53)
		tmp = 1.0;
	elseif (y <= 3.3e-104)
		tmp = Float64(1.0 - Float64(x / Float64(z * t)));
	elseif (y <= 4.2e-88)
		tmp = Float64(x / Float64(y * Float64(t - y)));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -1.05e-53)
		tmp = 1.0;
	elseif (y <= 3.3e-104)
		tmp = 1.0 - (x / (z * t));
	elseif (y <= 4.2e-88)
		tmp = x / (y * (t - y));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -1.05e-53], 1.0, If[LessEqual[y, 3.3e-104], N[(1.0 - N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.2e-88], N[(x / N[(y * N[(t - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.04999999999999989e-53 or 4.1999999999999999e-88 < y

   1. Initial program 99.9%

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

      1. sub-neg N/A

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

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

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\frac{x}{\left(y - z\right) \cdot \left(y - t\right)}\right)\right)}\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{x}{y - z}}{y - t}\right)\right)\right) \]

      4. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\frac{x}{y - z}}{\color{blue}{\mathsf{neg}\left(\left(y - t\right)\right)}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{x}{y - z}\right), \color{blue}{\left(\mathsf{neg}\left(\left(y - t\right)\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(y - z\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(y - t\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \left(\mathsf{neg}\left(\left(y - \color{blue}{t}\right)\right)\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \left(\mathsf{neg}\left(\left(y + \left(\mathsf{neg}\left(t\right)\right)\right)\right)\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(t\right)\right) + y\right)\right)\right)\right)\right) \]

      10. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(t\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right)\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(t\right)\right)\right)\right) - \color{blue}{y}\right)\right)\right) \]

      12. remove-double-neg N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \left(t - y\right)\right)\right) \]

      13. --lowering--.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \color{blue}{y}\right)\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 89.6%

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

   if -1.04999999999999989e-53 < y < 3.30000000000000002e-104

   1. Initial program 98.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(x, \color{blue}{\left(t \cdot z\right)}\right)\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 77.4%

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(t, \color{blue}{z}\right)\right)\right) \]

   5. Simplified 77.4%

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

   if 3.30000000000000002e-104 < y < 4.1999999999999999e-88

   1. Initial program 99.7%

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

      1. sub-neg N/A

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

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

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\frac{x}{\left(y - z\right) \cdot \left(y - t\right)}\right)\right)}\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{x}{y - z}}{y - t}\right)\right)\right) \]

      4. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\frac{x}{y - z}}{\color{blue}{\mathsf{neg}\left(\left(y - t\right)\right)}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{x}{y - z}\right), \color{blue}{\left(\mathsf{neg}\left(\left(y - t\right)\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(y - z\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(y - t\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \left(\mathsf{neg}\left(\left(y - \color{blue}{t}\right)\right)\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \left(\mathsf{neg}\left(\left(y + \left(\mathsf{neg}\left(t\right)\right)\right)\right)\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(t\right)\right) + y\right)\right)\right)\right)\right) \]

      10. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(t\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right)\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(t\right)\right)\right)\right) - \color{blue}{y}\right)\right)\right) \]

      12. remove-double-neg N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \left(t - y\right)\right)\right) \]

      13. --lowering--.f64 99.7%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \color{blue}{y}\right)\right)\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in x around inf

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

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

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(\left(t - y\right) \cdot \left(y - z\right)\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\left(t - y\right), \color{blue}{\left(y - z\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(t, y\right), \left(\color{blue}{y} - z\right)\right)\right) \]

      4. --lowering--.f64 99.7%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(t, y\right), \mathsf{\_.f64}\left(y, \color{blue}{z}\right)\right)\right) \]

   7. Simplified 99.7%

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

   8. Taylor expanded in y around inf

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(t, y\right), \color{blue}{y}\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 99.7%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{-53}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{-104}:\\ \;\;\;\;1 - \frac{x}{z \cdot t}\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-88}:\\ \;\;\;\;\frac{x}{y \cdot \left(t - y\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 + \frac{\frac{x}{y}}{t - y}\\ \mathbf{if}\;y \leq -1.2 \cdot 10^{-53}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 5.7 \cdot 10^{-108}:\\ \;\;\;\;1 + \frac{\frac{x}{y - z}}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ 1.0 (/ (/ x y) (- t y)))))
   (if (<= y -1.2e-53)
     t_1
     (if (<= y 5.7e-108) (+ 1.0 (/ (/ x (- y z)) t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = 1.0 + ((x / y) / (t - y));
	double tmp;
	if (y <= -1.2e-53) {
		tmp = t_1;
	} else if (y <= 5.7e-108) {
		tmp = 1.0 + ((x / (y - z)) / t);
	} 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 / y) / (t - y))
    if (y <= (-1.2d-53)) then
        tmp = t_1
    else if (y <= 5.7d-108) then
        tmp = 1.0d0 + ((x / (y - z)) / t)
    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 / y) / (t - y));
	double tmp;
	if (y <= -1.2e-53) {
		tmp = t_1;
	} else if (y <= 5.7e-108) {
		tmp = 1.0 + ((x / (y - z)) / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = 1.0 + ((x / y) / (t - y))
	tmp = 0
	if y <= -1.2e-53:
		tmp = t_1
	elif y <= 5.7e-108:
		tmp = 1.0 + ((x / (y - z)) / t)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(1.0 + Float64(Float64(x / y) / Float64(t - y)))
	tmp = 0.0
	if (y <= -1.2e-53)
		tmp = t_1;
	elseif (y <= 5.7e-108)
		tmp = Float64(1.0 + Float64(Float64(x / Float64(y - z)) / t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = 1.0 + ((x / y) / (t - y));
	tmp = 0.0;
	if (y <= -1.2e-53)
		tmp = t_1;
	elseif (y <= 5.7e-108)
		tmp = 1.0 + ((x / (y - z)) / t);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(1.0 + N[(N[(x / y), $MachinePrecision] / N[(t - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.2e-53], t$95$1, If[LessEqual[y, 5.7e-108], N[(1.0 + N[(N[(x / N[(y - z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.20000000000000004e-53 or 5.7e-108 < y

   1. Initial program 99.9%

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

      1. sub-neg N/A

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

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

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\frac{x}{\left(y - z\right) \cdot \left(y - t\right)}\right)\right)}\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{x}{y - z}}{y - t}\right)\right)\right) \]

      4. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\frac{x}{y - z}}{\color{blue}{\mathsf{neg}\left(\left(y - t\right)\right)}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{x}{y - z}\right), \color{blue}{\left(\mathsf{neg}\left(\left(y - t\right)\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(y - z\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(y - t\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \left(\mathsf{neg}\left(\left(y - \color{blue}{t}\right)\right)\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \left(\mathsf{neg}\left(\left(y + \left(\mathsf{neg}\left(t\right)\right)\right)\right)\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(t\right)\right) + y\right)\right)\right)\right)\right) \]

      10. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(t\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right)\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(t\right)\right)\right)\right) - \color{blue}{y}\right)\right)\right) \]

      12. remove-double-neg N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \left(t - y\right)\right)\right) \]

      13. --lowering--.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \color{blue}{y}\right)\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in y around inf

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\color{blue}{\left(\frac{x}{y}\right)}, \mathsf{\_.f64}\left(t, y\right)\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 92.5%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), \mathsf{\_.f64}\left(\color{blue}{t}, y\right)\right)\right) \]

   7. Simplified 92.5%

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

   if -1.20000000000000004e-53 < y < 5.7e-108

   1. Initial program 98.0%

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

      1. sub-neg N/A

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

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

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\frac{x}{\left(y - z\right) \cdot \left(y - t\right)}\right)\right)}\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{x}{y - z}}{y - t}\right)\right)\right) \]

      4. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\frac{x}{y - z}}{\color{blue}{\mathsf{neg}\left(\left(y - t\right)\right)}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{x}{y - z}\right), \color{blue}{\left(\mathsf{neg}\left(\left(y - t\right)\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(y - z\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(y - t\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \left(\mathsf{neg}\left(\left(y - \color{blue}{t}\right)\right)\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \left(\mathsf{neg}\left(\left(y + \left(\mathsf{neg}\left(t\right)\right)\right)\right)\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(t\right)\right) + y\right)\right)\right)\right)\right) \]

      10. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(t\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right)\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(t\right)\right)\right)\right) - \color{blue}{y}\right)\right)\right) \]

      12. remove-double-neg N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \left(t - y\right)\right)\right) \]

      13. --lowering--.f64 95.1%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \color{blue}{y}\right)\right)\right) \]

   3. Simplified 95.1%

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

   5. Taylor expanded in t around inf

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \color{blue}{t}\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 86.2%

      \[\leadsto 1 + \frac{\frac{x}{y - z}}{\color{blue}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 85.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 + \frac{\frac{x}{y}}{t - y}\\ \mathbf{if}\;y \leq -1.1 \cdot 10^{-53}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.45 \cdot 10^{-108}:\\ \;\;\;\;1 - \frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ 1.0 (/ (/ x y) (- t y)))))
   (if (<= y -1.1e-53) t_1 (if (<= y 2.45e-108) (- 1.0 (/ x (* z t))) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = 1.0 + ((x / y) / (t - y));
	double tmp;
	if (y <= -1.1e-53) {
		tmp = t_1;
	} else if (y <= 2.45e-108) {
		tmp = 1.0 - (x / (z * t));
	} 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 / y) / (t - y))
    if (y <= (-1.1d-53)) then
        tmp = t_1
    else if (y <= 2.45d-108) then
        tmp = 1.0d0 - (x / (z * t))
    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 / y) / (t - y));
	double tmp;
	if (y <= -1.1e-53) {
		tmp = t_1;
	} else if (y <= 2.45e-108) {
		tmp = 1.0 - (x / (z * t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = 1.0 + ((x / y) / (t - y))
	tmp = 0
	if y <= -1.1e-53:
		tmp = t_1
	elif y <= 2.45e-108:
		tmp = 1.0 - (x / (z * t))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(1.0 + Float64(Float64(x / y) / Float64(t - y)))
	tmp = 0.0
	if (y <= -1.1e-53)
		tmp = t_1;
	elseif (y <= 2.45e-108)
		tmp = Float64(1.0 - Float64(x / Float64(z * t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = 1.0 + ((x / y) / (t - y));
	tmp = 0.0;
	if (y <= -1.1e-53)
		tmp = t_1;
	elseif (y <= 2.45e-108)
		tmp = 1.0 - (x / (z * t));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(1.0 + N[(N[(x / y), $MachinePrecision] / N[(t - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.1e-53], t$95$1, If[LessEqual[y, 2.45e-108], N[(1.0 - N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.10000000000000009e-53 or 2.4499999999999999e-108 < y

   1. Initial program 99.9%

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

      1. sub-neg N/A

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

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

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\frac{x}{\left(y - z\right) \cdot \left(y - t\right)}\right)\right)}\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{x}{y - z}}{y - t}\right)\right)\right) \]

      4. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\frac{x}{y - z}}{\color{blue}{\mathsf{neg}\left(\left(y - t\right)\right)}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{x}{y - z}\right), \color{blue}{\left(\mathsf{neg}\left(\left(y - t\right)\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(y - z\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(y - t\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \left(\mathsf{neg}\left(\left(y - \color{blue}{t}\right)\right)\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \left(\mathsf{neg}\left(\left(y + \left(\mathsf{neg}\left(t\right)\right)\right)\right)\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(t\right)\right) + y\right)\right)\right)\right)\right) \]

      10. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(t\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right)\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(t\right)\right)\right)\right) - \color{blue}{y}\right)\right)\right) \]

      12. remove-double-neg N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \left(t - y\right)\right)\right) \]

      13. --lowering--.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \color{blue}{y}\right)\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in y around inf

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\color{blue}{\left(\frac{x}{y}\right)}, \mathsf{\_.f64}\left(t, y\right)\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 92.5%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), \mathsf{\_.f64}\left(\color{blue}{t}, y\right)\right)\right) \]

   7. Simplified 92.5%

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

   if -1.10000000000000009e-53 < y < 2.4499999999999999e-108

   1. Initial program 98.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(x, \color{blue}{\left(t \cdot z\right)}\right)\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 77.2%

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(t, \color{blue}{z}\right)\right)\right) \]

   5. Simplified 77.2%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{-53}:\\ \;\;\;\;1 + \frac{\frac{x}{y}}{t - y}\\ \mathbf{elif}\;y \leq 2.45 \cdot 10^{-108}:\\ \;\;\;\;1 - \frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{x}{y}}{t - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-53}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 2.55 \cdot 10^{-145}:\\ \;\;\;\;1 - \frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -1e-53) 1.0 (if (<= y 2.55e-145) (- 1.0 (/ x (* z t))) 1.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1e-53) {
		tmp = 1.0;
	} else if (y <= 2.55e-145) {
		tmp = 1.0 - (x / (z * t));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-1d-53)) then
        tmp = 1.0d0
    else if (y <= 2.55d-145) then
        tmp = 1.0d0 - (x / (z * t))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1e-53) {
		tmp = 1.0;
	} else if (y <= 2.55e-145) {
		tmp = 1.0 - (x / (z * t));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -1e-53:
		tmp = 1.0
	elif y <= 2.55e-145:
		tmp = 1.0 - (x / (z * t))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -1e-53)
		tmp = 1.0;
	elseif (y <= 2.55e-145)
		tmp = Float64(1.0 - Float64(x / Float64(z * t)));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -1e-53)
		tmp = 1.0;
	elseif (y <= 2.55e-145)
		tmp = 1.0 - (x / (z * t));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -1e-53], 1.0, If[LessEqual[y, 2.55e-145], N[(1.0 - N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -1.00000000000000003e-53 or 2.54999999999999992e-145 < y

   1. Initial program 99.9%

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

      1. sub-neg N/A

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

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

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\frac{x}{\left(y - z\right) \cdot \left(y - t\right)}\right)\right)}\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{x}{y - z}}{y - t}\right)\right)\right) \]

      4. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\frac{x}{y - z}}{\color{blue}{\mathsf{neg}\left(\left(y - t\right)\right)}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{x}{y - z}\right), \color{blue}{\left(\mathsf{neg}\left(\left(y - t\right)\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(y - z\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(y - t\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \left(\mathsf{neg}\left(\left(y - \color{blue}{t}\right)\right)\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \left(\mathsf{neg}\left(\left(y + \left(\mathsf{neg}\left(t\right)\right)\right)\right)\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(t\right)\right) + y\right)\right)\right)\right)\right) \]

      10. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(t\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right)\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(t\right)\right)\right)\right) - \color{blue}{y}\right)\right)\right) \]

      12. remove-double-neg N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \left(t - y\right)\right)\right) \]

      13. --lowering--.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \color{blue}{y}\right)\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 84.6%

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

   if -1.00000000000000003e-53 < y < 2.54999999999999992e-145

   1. Initial program 97.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(x, \color{blue}{\left(t \cdot z\right)}\right)\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 80.6%

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(t, \color{blue}{z}\right)\right)\right) \]

   5. Simplified 80.6%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-53}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 2.55 \cdot 10^{-145}:\\ \;\;\;\;1 - \frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{\frac{x}{y - z}}{t - y} \end{array} \]

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(Float64(x / 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[(N[(x / N[(y - z), $MachinePrecision]), $MachinePrecision] / N[(t - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

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

   1. sub-neg N/A

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

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

      \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\frac{x}{\left(y - z\right) \cdot \left(y - t\right)}\right)\right)}\right) \]

   3. associate-/r* N/A

      \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{x}{y - z}}{y - t}\right)\right)\right) \]

   4. distribute-neg-frac2 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\frac{x}{y - z}}{\color{blue}{\mathsf{neg}\left(\left(y - t\right)\right)}}\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{x}{y - z}\right), \color{blue}{\left(\mathsf{neg}\left(\left(y - t\right)\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(y - z\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(y - t\right)}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \left(\mathsf{neg}\left(\left(y - \color{blue}{t}\right)\right)\right)\right)\right) \]

   8. sub-neg N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \left(\mathsf{neg}\left(\left(y + \left(\mathsf{neg}\left(t\right)\right)\right)\right)\right)\right)\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(t\right)\right) + y\right)\right)\right)\right)\right) \]

   10. distribute-neg-in N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(t\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right)\right) \]

   11. unsub-neg N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(t\right)\right)\right)\right) - \color{blue}{y}\right)\right)\right) \]

   12. remove-double-neg N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \left(t - y\right)\right)\right) \]

   13. --lowering--.f64 98.1%

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \color{blue}{y}\right)\right)\right) \]

3. Simplified 98.1%

   \[\leadsto \color{blue}{1 + \frac{\frac{x}{y - z}}{t - y}} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 7: 75.4% accurate, 11.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

   1. sub-neg N/A

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

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

      \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\frac{x}{\left(y - z\right) \cdot \left(y - t\right)}\right)\right)}\right) \]

   3. associate-/r* N/A

      \[\leadsto \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{x}{y - z}}{y - t}\right)\right)\right) \]

   4. distribute-neg-frac2 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\frac{x}{y - z}}{\color{blue}{\mathsf{neg}\left(\left(y - t\right)\right)}}\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{x}{y - z}\right), \color{blue}{\left(\mathsf{neg}\left(\left(y - t\right)\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(y - z\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(y - t\right)}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \left(\mathsf{neg}\left(\left(y - \color{blue}{t}\right)\right)\right)\right)\right) \]

   8. sub-neg N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \left(\mathsf{neg}\left(\left(y + \left(\mathsf{neg}\left(t\right)\right)\right)\right)\right)\right)\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(t\right)\right) + y\right)\right)\right)\right)\right) \]

   10. distribute-neg-in N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(t\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right)\right) \]

   11. unsub-neg N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(t\right)\right)\right)\right) - \color{blue}{y}\right)\right)\right) \]

   12. remove-double-neg N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \left(t - y\right)\right)\right) \]

   13. --lowering--.f64 98.1%

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \color{blue}{y}\right)\right)\right) \]

3. Simplified 98.1%

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

5. Taylor expanded in x around 0

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

7. Simplified 74.0%

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

Reproduce

herbie shell --seed 2024191 
(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)))))