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

Percentage Accurate: 99.2% → 99.2%

Time: 12.2s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.2% 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

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

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 99.9%

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

3. Final simplification 99.9%

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

Alternative 2: 86.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{-34}:\\ \;\;\;\;1 - \frac{\frac{x}{y}}{y}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-197}:\\ \;\;\;\;1 + \frac{x}{z \cdot \left(y - t\right)}\\ \mathbf{elif}\;y \leq 620000000000:\\ \;\;\;\;1 + \frac{\frac{x}{y - z}}{t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -1.25e-34)
   (- 1.0 (/ (/ x y) y))
   (if (<= y 1.1e-197)
     (+ 1.0 (/ x (* z (- y t))))
     (if (<= y 620000000000.0) (+ 1.0 (/ (/ x (- y z)) t)) 1.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.25e-34) {
		tmp = 1.0 - ((x / y) / y);
	} else if (y <= 1.1e-197) {
		tmp = 1.0 + (x / (z * (y - t)));
	} else if (y <= 620000000000.0) {
		tmp = 1.0 + ((x / (y - 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 <= (-1.25d-34)) then
        tmp = 1.0d0 - ((x / y) / y)
    else if (y <= 1.1d-197) then
        tmp = 1.0d0 + (x / (z * (y - t)))
    else if (y <= 620000000000.0d0) then
        tmp = 1.0d0 + ((x / (y - 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 <= -1.25e-34) {
		tmp = 1.0 - ((x / y) / y);
	} else if (y <= 1.1e-197) {
		tmp = 1.0 + (x / (z * (y - t)));
	} else if (y <= 620000000000.0) {
		tmp = 1.0 + ((x / (y - z)) / t);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -1.25e-34:
		tmp = 1.0 - ((x / y) / y)
	elif y <= 1.1e-197:
		tmp = 1.0 + (x / (z * (y - t)))
	elif y <= 620000000000.0:
		tmp = 1.0 + ((x / (y - z)) / t)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -1.25e-34)
		tmp = Float64(1.0 - Float64(Float64(x / y) / y));
	elseif (y <= 1.1e-197)
		tmp = Float64(1.0 + Float64(x / Float64(z * Float64(y - t))));
	elseif (y <= 620000000000.0)
		tmp = Float64(1.0 + Float64(Float64(x / Float64(y - z)) / t));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -1.25e-34)
		tmp = 1.0 - ((x / y) / y);
	elseif (y <= 1.1e-197)
		tmp = 1.0 + (x / (z * (y - t)));
	elseif (y <= 620000000000.0)
		tmp = 1.0 + ((x / (y - z)) / t);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -1.25e-34], N[(1.0 - N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.1e-197], N[(1.0 + N[(x / N[(z * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 620000000000.0], N[(1.0 + N[(N[(x / N[(y - z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.2500000000000001e-34

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

         \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{x}{{y}^{2}}\right)}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(x, \color{blue}{\left({y}^{2}\right)}\right)\right) \]

      5. unpow2 N/A

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

      6. *-lowering-*.f64 93.7%

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

   5. Simplified 93.7%

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

      1. associate-/r* N/A

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

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

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

      3. /-lowering-/.f64 93.8%

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

   7. Applied egg-rr 93.8%

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

   if -1.2500000000000001e-34 < y < 1.1e-197

   1. Initial program 99.9%

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

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

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

      2. associate-/r* N/A

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

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

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

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

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

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

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

      6. --lowering--.f64 94.1%

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

   4. Applied egg-rr 94.1%

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

   5. Taylor expanded in z around inf

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

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

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. --lowering--.f64 86.5%

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

   7. Simplified 86.5%

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

   if 1.1e-197 < y < 6.2e11

   1. Initial program 99.8%

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

   3. Taylor expanded in t around inf

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

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

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. associate-/l/ N/A

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

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

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

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

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

      9. --lowering--.f64 71.0%

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

   5. Simplified 71.0%

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

   if 6.2e11 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 98.0%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{-34}:\\ \;\;\;\;1 - \frac{\frac{x}{y}}{y}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-197}:\\ \;\;\;\;1 + \frac{x}{z \cdot \left(y - t\right)}\\ \mathbf{elif}\;y \leq 620000000000:\\ \;\;\;\;1 + \frac{\frac{x}{y - z}}{t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.25 \cdot 10^{-95}:\\ \;\;\;\;1 + \frac{x}{y \cdot \left(z - y\right)}\\ \mathbf{elif}\;y \leq 1.16 \cdot 10^{-46}:\\ \;\;\;\;1 + \frac{x}{\left(y - z\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y \cdot \left(t - y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -2.25e-95)
   (+ 1.0 (/ x (* y (- z y))))
   (if (<= y 1.16e-46)
     (+ 1.0 (/ x (* (- y z) t)))
     (+ 1.0 (/ x (* y (- t y)))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -2.25e-95) {
		tmp = 1.0 + (x / (y * (z - y)));
	} else if (y <= 1.16e-46) {
		tmp = 1.0 + (x / ((y - z) * t));
	} else {
		tmp = 1.0 + (x / (y * (t - y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-2.25d-95)) then
        tmp = 1.0d0 + (x / (y * (z - y)))
    else if (y <= 1.16d-46) then
        tmp = 1.0d0 + (x / ((y - z) * t))
    else
        tmp = 1.0d0 + (x / (y * (t - y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -2.25e-95) {
		tmp = 1.0 + (x / (y * (z - y)));
	} else if (y <= 1.16e-46) {
		tmp = 1.0 + (x / ((y - z) * t));
	} else {
		tmp = 1.0 + (x / (y * (t - y)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -2.25e-95:
		tmp = 1.0 + (x / (y * (z - y)))
	elif y <= 1.16e-46:
		tmp = 1.0 + (x / ((y - z) * t))
	else:
		tmp = 1.0 + (x / (y * (t - y)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -2.25e-95)
		tmp = Float64(1.0 + Float64(x / Float64(y * Float64(z - y))));
	elseif (y <= 1.16e-46)
		tmp = Float64(1.0 + Float64(x / Float64(Float64(y - z) * t)));
	else
		tmp = Float64(1.0 + Float64(x / Float64(y * Float64(t - y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -2.25e-95)
		tmp = 1.0 + (x / (y * (z - y)));
	elseif (y <= 1.16e-46)
		tmp = 1.0 + (x / ((y - z) * t));
	else
		tmp = 1.0 + (x / (y * (t - y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -2.25e-95], N[(1.0 + N[(x / N[(y * N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.16e-46], N[(1.0 + N[(x / N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(x / N[(y * N[(t - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.25e-95

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 93.0%

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

   if -2.25e-95 < y < 1.16e-46

   1. Initial program 99.9%

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

   3. Taylor expanded in t around inf

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

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

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. associate-/l/ N/A

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

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

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

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

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

      9. --lowering--.f64 80.6%

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

   5. Simplified 80.6%

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

      1. +-commutative N/A

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

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

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

      3. associate-/l/ N/A

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

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

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

      5. *-commutative N/A

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

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

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

      7. --lowering--.f64 86.1%

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

   7. Applied egg-rr 86.1%

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

   if 1.16e-46 < y

   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 0

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

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

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

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

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

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

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

      4. --lowering--.f64 96.3%

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

   5. Simplified 96.3%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.25 \cdot 10^{-95}:\\ \;\;\;\;1 + \frac{x}{y \cdot \left(z - y\right)}\\ \mathbf{elif}\;y \leq 1.16 \cdot 10^{-46}:\\ \;\;\;\;1 + \frac{x}{\left(y - z\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y \cdot \left(t - y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{-77}:\\ \;\;\;\;1 + \frac{x}{z \cdot \left(y - t\right)}\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-190}:\\ \;\;\;\;1 + \frac{x}{y \cdot \left(t - y\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{\left(y - z\right) \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -4.2e-77)
   (+ 1.0 (/ x (* z (- y t))))
   (if (<= z 1.7e-190)
     (+ 1.0 (/ x (* y (- t y))))
     (+ 1.0 (/ x (* (- y z) t))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -4.2e-77) {
		tmp = 1.0 + (x / (z * (y - t)));
	} else if (z <= 1.7e-190) {
		tmp = 1.0 + (x / (y * (t - y)));
	} else {
		tmp = 1.0 + (x / ((y - z) * t));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-4.2d-77)) then
        tmp = 1.0d0 + (x / (z * (y - t)))
    else if (z <= 1.7d-190) then
        tmp = 1.0d0 + (x / (y * (t - y)))
    else
        tmp = 1.0d0 + (x / ((y - z) * t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -4.2e-77) {
		tmp = 1.0 + (x / (z * (y - t)));
	} else if (z <= 1.7e-190) {
		tmp = 1.0 + (x / (y * (t - y)));
	} else {
		tmp = 1.0 + (x / ((y - z) * t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -4.2e-77:
		tmp = 1.0 + (x / (z * (y - t)))
	elif z <= 1.7e-190:
		tmp = 1.0 + (x / (y * (t - y)))
	else:
		tmp = 1.0 + (x / ((y - z) * t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -4.2e-77)
		tmp = Float64(1.0 + Float64(x / Float64(z * Float64(y - t))));
	elseif (z <= 1.7e-190)
		tmp = Float64(1.0 + Float64(x / Float64(y * Float64(t - y))));
	else
		tmp = Float64(1.0 + Float64(x / Float64(Float64(y - z) * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -4.2e-77)
		tmp = 1.0 + (x / (z * (y - t)));
	elseif (z <= 1.7e-190)
		tmp = 1.0 + (x / (y * (t - y)));
	else
		tmp = 1.0 + (x / ((y - z) * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -4.2e-77], N[(1.0 + N[(x / N[(z * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.7e-190], N[(1.0 + N[(x / N[(y * N[(t - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(x / N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.20000000000000031e-77

   1. Initial program 99.9%

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

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

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

      2. associate-/r* N/A

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

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

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

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

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

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

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

      6. --lowering--.f64 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in z around inf

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

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

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. --lowering--.f64 95.4%

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

   7. Simplified 95.4%

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

   if -4.20000000000000031e-77 < z < 1.69999999999999991e-190

   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 0

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

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

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

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

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

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

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

      4. --lowering--.f64 89.4%

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

   5. Simplified 89.4%

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

   if 1.69999999999999991e-190 < 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 t around inf

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

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

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. associate-/l/ N/A

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

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

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

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

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

      9. --lowering--.f64 78.0%

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

   5. Simplified 78.0%

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

      1. +-commutative N/A

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

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

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

      3. associate-/l/ N/A

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

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

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

      5. *-commutative N/A

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

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

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

      7. --lowering--.f64 79.9%

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

   7. Applied egg-rr 79.9%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{-77}:\\ \;\;\;\;1 + \frac{x}{z \cdot \left(y - t\right)}\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-190}:\\ \;\;\;\;1 + \frac{x}{y \cdot \left(t - y\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{\left(y - z\right) \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{-34}:\\ \;\;\;\;1 - \frac{\frac{x}{y}}{y}\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-53}:\\ \;\;\;\;1 + \frac{x}{z \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -1.25e-34)
   (- 1.0 (/ (/ x y) y))
   (if (<= y 4.5e-53) (+ 1.0 (/ x (* z (- y t)))) 1.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.25e-34) {
		tmp = 1.0 - ((x / y) / y);
	} else if (y <= 4.5e-53) {
		tmp = 1.0 + (x / (z * (y - 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 <= (-1.25d-34)) then
        tmp = 1.0d0 - ((x / y) / y)
    else if (y <= 4.5d-53) then
        tmp = 1.0d0 + (x / (z * (y - 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 <= -1.25e-34) {
		tmp = 1.0 - ((x / y) / y);
	} else if (y <= 4.5e-53) {
		tmp = 1.0 + (x / (z * (y - t)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -1.25e-34:
		tmp = 1.0 - ((x / y) / y)
	elif y <= 4.5e-53:
		tmp = 1.0 + (x / (z * (y - t)))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -1.25e-34)
		tmp = Float64(1.0 - Float64(Float64(x / y) / y));
	elseif (y <= 4.5e-53)
		tmp = Float64(1.0 + Float64(x / Float64(z * Float64(y - t))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -1.25e-34)
		tmp = 1.0 - ((x / y) / y);
	elseif (y <= 4.5e-53)
		tmp = 1.0 + (x / (z * (y - t)));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -1.25e-34], N[(1.0 - N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.5e-53], N[(1.0 + N[(x / N[(z * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 3 regimes

2. if y < -1.2500000000000001e-34

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

         \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{x}{{y}^{2}}\right)}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(x, \color{blue}{\left({y}^{2}\right)}\right)\right) \]

      5. unpow2 N/A

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

      6. *-lowering-*.f64 93.7%

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

   5. Simplified 93.7%

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

      1. associate-/r* N/A

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

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

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

      3. /-lowering-/.f64 93.8%

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

   7. Applied egg-rr 93.8%

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

   if -1.2500000000000001e-34 < y < 4.49999999999999985e-53

   1. Initial program 99.9%

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

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

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

      2. associate-/r* N/A

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

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

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

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

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

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

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

      6. --lowering--.f64 94.9%

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

   4. Applied egg-rr 94.9%

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

   5. Taylor expanded in z around inf

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

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

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. --lowering--.f64 87.7%

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

   7. Simplified 87.7%

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

   if 4.49999999999999985e-53 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 91.6%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{-34}:\\ \;\;\;\;1 - \frac{\frac{x}{y}}{y}\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-53}:\\ \;\;\;\;1 + \frac{x}{z \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{-120}:\\ \;\;\;\;1 - \frac{\frac{x}{y}}{y}\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{-99}:\\ \;\;\;\;1 - \frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -5.5e-120)
   (- 1.0 (/ (/ x y) y))
   (if (<= y 1.9e-99) (- 1.0 (/ x (* z t))) 1.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -5.5e-120) {
		tmp = 1.0 - ((x / y) / y);
	} else if (y <= 1.9e-99) {
		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 <= (-5.5d-120)) then
        tmp = 1.0d0 - ((x / y) / y)
    else if (y <= 1.9d-99) 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 <= -5.5e-120) {
		tmp = 1.0 - ((x / y) / y);
	} else if (y <= 1.9e-99) {
		tmp = 1.0 - (x / (z * t));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -5.5e-120:
		tmp = 1.0 - ((x / y) / y)
	elif y <= 1.9e-99:
		tmp = 1.0 - (x / (z * t))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -5.5e-120)
		tmp = Float64(1.0 - Float64(Float64(x / y) / y));
	elseif (y <= 1.9e-99)
		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 <= -5.5e-120)
		tmp = 1.0 - ((x / y) / y);
	elseif (y <= 1.9e-99)
		tmp = 1.0 - (x / (z * t));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -5.5e-120], N[(1.0 - N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.9e-99], N[(1.0 - N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 3 regimes

2. if y < -5.5000000000000001e-120

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

         \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{x}{{y}^{2}}\right)}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(x, \color{blue}{\left({y}^{2}\right)}\right)\right) \]

      5. unpow2 N/A

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

      6. *-lowering-*.f64 85.1%

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

   5. Simplified 85.1%

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

      1. associate-/r* N/A

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

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

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

      3. /-lowering-/.f64 85.1%

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

   7. Applied egg-rr 85.1%

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

   if -5.5000000000000001e-120 < y < 1.8999999999999998e-99

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

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

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

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

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

      3. *-lowering-*.f64 85.8%

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

   5. Simplified 85.8%

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

   if 1.8999999999999998e-99 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 89.1%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{-120}:\\ \;\;\;\;1 - \frac{\frac{x}{y}}{y}\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{-99}:\\ \;\;\;\;1 - \frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{-120}:\\ \;\;\;\;1 - \frac{x}{y \cdot y}\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-99}:\\ \;\;\;\;1 - \frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -5.5e-120)
   (- 1.0 (/ x (* y y)))
   (if (<= y 2.5e-99) (- 1.0 (/ x (* z t))) 1.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -5.5e-120) {
		tmp = 1.0 - (x / (y * y));
	} else if (y <= 2.5e-99) {
		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 <= (-5.5d-120)) then
        tmp = 1.0d0 - (x / (y * y))
    else if (y <= 2.5d-99) 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 <= -5.5e-120) {
		tmp = 1.0 - (x / (y * y));
	} else if (y <= 2.5e-99) {
		tmp = 1.0 - (x / (z * t));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -5.5e-120:
		tmp = 1.0 - (x / (y * y))
	elif y <= 2.5e-99:
		tmp = 1.0 - (x / (z * t))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -5.5e-120)
		tmp = Float64(1.0 - Float64(x / Float64(y * y)));
	elseif (y <= 2.5e-99)
		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 <= -5.5e-120)
		tmp = 1.0 - (x / (y * y));
	elseif (y <= 2.5e-99)
		tmp = 1.0 - (x / (z * t));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -5.5e-120], N[(1.0 - N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.5e-99], N[(1.0 - N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 3 regimes

2. if y < -5.5000000000000001e-120

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

         \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{x}{{y}^{2}}\right)}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(x, \color{blue}{\left({y}^{2}\right)}\right)\right) \]

      5. unpow2 N/A

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

      6. *-lowering-*.f64 85.1%

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

   5. Simplified 85.1%

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

   if -5.5000000000000001e-120 < y < 2.49999999999999985e-99

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

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

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

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

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

      3. *-lowering-*.f64 85.8%

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

   5. Simplified 85.8%

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

   if 2.49999999999999985e-99 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 89.1%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{-120}:\\ \;\;\;\;1 - \frac{x}{y \cdot y}\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-99}:\\ \;\;\;\;1 - \frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 83.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.85 \cdot 10^{-77}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{-99}:\\ \;\;\;\;1 - \frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -2.85e-77) 1.0 (if (<= y 6.2e-99) (- 1.0 (/ x (* z t))) 1.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -2.85e-77) {
		tmp = 1.0;
	} else if (y <= 6.2e-99) {
		tmp = 1.0 - (x / (z * t));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-2.85d-77)) then
        tmp = 1.0d0
    else if (y <= 6.2d-99) then
        tmp = 1.0d0 - (x / (z * t))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -2.85e-77) {
		tmp = 1.0;
	} else if (y <= 6.2e-99) {
		tmp = 1.0 - (x / (z * t));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -2.85e-77:
		tmp = 1.0
	elif y <= 6.2e-99:
		tmp = 1.0 - (x / (z * t))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -2.85e-77)
		tmp = 1.0;
	elseif (y <= 6.2e-99)
		tmp = Float64(1.0 - Float64(x / Float64(z * t)));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -2.85e-77)
		tmp = 1.0;
	elseif (y <= 6.2e-99)
		tmp = 1.0 - (x / (z * t));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -2.85e-77], 1.0, If[LessEqual[y, 6.2e-99], N[(1.0 - N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -2.84999999999999991e-77 or 6.1999999999999997e-99 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 87.9%

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

   if -2.84999999999999991e-77 < y < 6.1999999999999997e-99

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

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

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

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

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

      3. *-lowering-*.f64 83.1%

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

   5. Simplified 83.1%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.85 \cdot 10^{-77}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{-99}:\\ \;\;\;\;1 - \frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{-121}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{-86}:\\ \;\;\;\;1 + \frac{\frac{x}{y}}{t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -6.2e-121) 1.0 (if (<= z 1.9e-86) (+ 1.0 (/ (/ x y) t)) 1.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -6.2e-121) {
		tmp = 1.0;
	} else if (z <= 1.9e-86) {
		tmp = 1.0 + ((x / y) / 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 (z <= (-6.2d-121)) then
        tmp = 1.0d0
    else if (z <= 1.9d-86) then
        tmp = 1.0d0 + ((x / y) / 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 (z <= -6.2e-121) {
		tmp = 1.0;
	} else if (z <= 1.9e-86) {
		tmp = 1.0 + ((x / y) / t);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -6.2e-121:
		tmp = 1.0
	elif z <= 1.9e-86:
		tmp = 1.0 + ((x / y) / t)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -6.2e-121)
		tmp = 1.0;
	elseif (z <= 1.9e-86)
		tmp = Float64(1.0 + Float64(Float64(x / y) / t));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -6.2e-121)
		tmp = 1.0;
	elseif (z <= 1.9e-86)
		tmp = 1.0 + ((x / y) / t);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -6.2e-121], 1.0, If[LessEqual[z, 1.9e-86], N[(1.0 + N[(N[(x / y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if z < -6.1999999999999997e-121 or 1.9e-86 < 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 x around 0

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

   5. Simplified 84.3%

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

   if -6.1999999999999997e-121 < z < 1.9e-86

   1. Initial program 99.9%

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

   3. Taylor expanded in t around inf

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

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

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. associate-/l/ N/A

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

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

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

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

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

      9. --lowering--.f64 74.8%

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

   5. Simplified 74.8%

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

   6. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 69.4%

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

   8. Simplified 69.4%

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

Alternative 10: 83.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{-116}:\\ \;\;\;\;1 + \frac{x}{z \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{\left(y - z\right) \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.3e-116) (+ 1.0 (/ x (* z (- y t)))) (+ 1.0 (/ x (* (- y z) t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.3e-116) {
		tmp = 1.0 + (x / (z * (y - t)));
	} else {
		tmp = 1.0 + (x / ((y - z) * t));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-1.3d-116)) then
        tmp = 1.0d0 + (x / (z * (y - t)))
    else
        tmp = 1.0d0 + (x / ((y - z) * t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.3e-116) {
		tmp = 1.0 + (x / (z * (y - t)));
	} else {
		tmp = 1.0 + (x / ((y - z) * t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -1.3e-116:
		tmp = 1.0 + (x / (z * (y - t)))
	else:
		tmp = 1.0 + (x / ((y - z) * t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.3e-116)
		tmp = Float64(1.0 + Float64(x / Float64(z * Float64(y - t))));
	else
		tmp = Float64(1.0 + Float64(x / Float64(Float64(y - z) * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -1.3e-116)
		tmp = 1.0 + (x / (z * (y - t)));
	else
		tmp = 1.0 + (x / ((y - z) * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1.3e-116], N[(1.0 + N[(x / N[(z * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(x / N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.3e-116

   1. Initial program 99.9%

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

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

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

      2. associate-/r* N/A

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

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

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

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

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

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

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

      6. --lowering--.f64 99.0%

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

   4. Applied egg-rr 99.0%

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

   5. Taylor expanded in z around inf

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

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

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. --lowering--.f64 94.8%

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

   7. Simplified 94.8%

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

   if -1.3e-116 < 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 t around inf

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

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

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. associate-/l/ N/A

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

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

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

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

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

      9. --lowering--.f64 75.8%

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

   5. Simplified 75.8%

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

      1. +-commutative N/A

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

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

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

      3. associate-/l/ N/A

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

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

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

      5. *-commutative N/A

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

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

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

      7. --lowering--.f64 78.8%

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

   7. Applied egg-rr 78.8%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{-116}:\\ \;\;\;\;1 + \frac{x}{z \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{\left(y - z\right) \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 75.1% 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 99.9%

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

3. Taylor expanded in x around 0

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

5. Simplified 76.8%

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

Reproduce

herbie shell --seed 2024158 
(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)))))