Data.Number.Erf:$cinvnormcdf from erf-2.0.0.0, B

Percentage Accurate: 99.9% → 99.9%

Time: 4.5s

Alternatives: 7

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ x - \frac{y}{1 + \frac{x \cdot y}{2}} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- x (/ y (+ 1.0 (/ (* x y) 2.0)))))

C

double code(double x, double y) {
	return x - (y / (1.0 + ((x * y) / 2.0)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x - (y / (1.0d0 + ((x * y) / 2.0d0)))
end function

Java

public static double code(double x, double y) {
	return x - (y / (1.0 + ((x * y) / 2.0)));
}

Python

def code(x, y):
	return x - (y / (1.0 + ((x * y) / 2.0)))

Julia

function code(x, y)
	return Float64(x - Float64(y / Float64(1.0 + Float64(Float64(x * y) / 2.0))))
end

MATLAB

function tmp = code(x, y)
	tmp = x - (y / (1.0 + ((x * y) / 2.0)));
end

Wolfram

code[x_, y_] := N[(x - N[(y / N[(1.0 + N[(N[(x * y), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x - \frac{y}{1 + \frac{x \cdot y}{2}} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- x (/ y (+ 1.0 (/ (* x y) 2.0)))))

C

double code(double x, double y) {
	return x - (y / (1.0 + ((x * y) / 2.0)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x - (y / (1.0d0 + ((x * y) / 2.0d0)))
end function

Java

public static double code(double x, double y) {
	return x - (y / (1.0 + ((x * y) / 2.0)));
}

Python

def code(x, y):
	return x - (y / (1.0 + ((x * y) / 2.0)))

Julia

function code(x, y)
	return Float64(x - Float64(y / Float64(1.0 + Float64(Float64(x * y) / 2.0))))
end

MATLAB

function tmp = code(x, y)
	tmp = x - (y / (1.0 + ((x * y) / 2.0)));
end

Wolfram

code[x_, y_] := N[(x - N[(y / N[(1.0 + N[(N[(x * y), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x - \frac{y}{1 + \frac{x \cdot y}{2}} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- x (/ y (+ 1.0 (/ (* x y) 2.0)))))

C

double code(double x, double y) {
	return x - (y / (1.0 + ((x * y) / 2.0)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x - (y / (1.0d0 + ((x * y) / 2.0d0)))
end function

Java

public static double code(double x, double y) {
	return x - (y / (1.0 + ((x * y) / 2.0)));
}

Python

def code(x, y):
	return x - (y / (1.0 + ((x * y) / 2.0)))

Julia

function code(x, y)
	return Float64(x - Float64(y / Float64(1.0 + Float64(Float64(x * y) / 2.0))))
end

MATLAB

function tmp = code(x, y)
	tmp = x - (y / (1.0 + ((x * y) / 2.0)));
end

Wolfram

code[x_, y_] := N[(x - N[(y / N[(1.0 + N[(N[(x * y), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[x - \frac{y}{1 + \frac{x \cdot y}{2}} \]

2. Final simplification 99.9%

   \[\leadsto x - \frac{y}{1 + \frac{x \cdot y}{2}} \]

Alternative 2: 84.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.17:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.35 \cdot 10^{-51}:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{elif}\;x \leq -1.72 \cdot 10^{-93}:\\ \;\;\;\;x - y\\ \mathbf{elif}\;x \leq -1.5 \cdot 10^{-134}:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-59}:\\ \;\;\;\;x - y\\ \mathbf{elif}\;x \leq 0.0095:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.17)
   x
   (if (<= x -1.35e-51)
     (/ -2.0 x)
     (if (<= x -1.72e-93)
       (- x y)
       (if (<= x -1.5e-134)
         (/ -2.0 x)
         (if (<= x 1.7e-59) (- x y) (if (<= x 0.0095) (/ -2.0 x) x)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.17) {
		tmp = x;
	} else if (x <= -1.35e-51) {
		tmp = -2.0 / x;
	} else if (x <= -1.72e-93) {
		tmp = x - y;
	} else if (x <= -1.5e-134) {
		tmp = -2.0 / x;
	} else if (x <= 1.7e-59) {
		tmp = x - y;
	} else if (x <= 0.0095) {
		tmp = -2.0 / x;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-0.17d0)) then
        tmp = x
    else if (x <= (-1.35d-51)) then
        tmp = (-2.0d0) / x
    else if (x <= (-1.72d-93)) then
        tmp = x - y
    else if (x <= (-1.5d-134)) then
        tmp = (-2.0d0) / x
    else if (x <= 1.7d-59) then
        tmp = x - y
    else if (x <= 0.0095d0) then
        tmp = (-2.0d0) / x
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -0.17) {
		tmp = x;
	} else if (x <= -1.35e-51) {
		tmp = -2.0 / x;
	} else if (x <= -1.72e-93) {
		tmp = x - y;
	} else if (x <= -1.5e-134) {
		tmp = -2.0 / x;
	} else if (x <= 1.7e-59) {
		tmp = x - y;
	} else if (x <= 0.0095) {
		tmp = -2.0 / x;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.17:
		tmp = x
	elif x <= -1.35e-51:
		tmp = -2.0 / x
	elif x <= -1.72e-93:
		tmp = x - y
	elif x <= -1.5e-134:
		tmp = -2.0 / x
	elif x <= 1.7e-59:
		tmp = x - y
	elif x <= 0.0095:
		tmp = -2.0 / x
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.17)
		tmp = x;
	elseif (x <= -1.35e-51)
		tmp = Float64(-2.0 / x);
	elseif (x <= -1.72e-93)
		tmp = Float64(x - y);
	elseif (x <= -1.5e-134)
		tmp = Float64(-2.0 / x);
	elseif (x <= 1.7e-59)
		tmp = Float64(x - y);
	elseif (x <= 0.0095)
		tmp = Float64(-2.0 / x);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.17)
		tmp = x;
	elseif (x <= -1.35e-51)
		tmp = -2.0 / x;
	elseif (x <= -1.72e-93)
		tmp = x - y;
	elseif (x <= -1.5e-134)
		tmp = -2.0 / x;
	elseif (x <= 1.7e-59)
		tmp = x - y;
	elseif (x <= 0.0095)
		tmp = -2.0 / x;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.17], x, If[LessEqual[x, -1.35e-51], N[(-2.0 / x), $MachinePrecision], If[LessEqual[x, -1.72e-93], N[(x - y), $MachinePrecision], If[LessEqual[x, -1.5e-134], N[(-2.0 / x), $MachinePrecision], If[LessEqual[x, 1.7e-59], N[(x - y), $MachinePrecision], If[LessEqual[x, 0.0095], N[(-2.0 / x), $MachinePrecision], x]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.170000000000000012 or 0.00949999999999999976 < x

   1. Initial program 100.0%

      \[x - \frac{y}{1 + \frac{x \cdot y}{2}} \]
   2. Step-by-step derivation

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 98.9%

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

   if -0.170000000000000012 < x < -1.3499999999999999e-51 or -1.7199999999999999e-93 < x < -1.5e-134 or 1.70000000000000009e-59 < x < 0.00949999999999999976

   1. Initial program 99.6%

      \[x - \frac{y}{1 + \frac{x \cdot y}{2}} \]
   2. Step-by-step derivation

      1. associate-/l* 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around inf 73.4%

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

   5. Taylor expanded in x around 0 69.6%

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

   if -1.3499999999999999e-51 < x < -1.7199999999999999e-93 or -1.5e-134 < x < 1.70000000000000009e-59

   1. Initial program 100.0%

      \[x - \frac{y}{1 + \frac{x \cdot y}{2}} \]
   2. Step-by-step derivation

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 80.5%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.17:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.35 \cdot 10^{-51}:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{elif}\;x \leq -1.72 \cdot 10^{-93}:\\ \;\;\;\;x - y\\ \mathbf{elif}\;x \leq -1.5 \cdot 10^{-134}:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-59}:\\ \;\;\;\;x - y\\ \mathbf{elif}\;x \leq 0.0095:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x - \frac{y}{1 + \frac{x}{\frac{2}{y}}} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- x (/ y (+ 1.0 (/ x (/ 2.0 y))))))

C

double code(double x, double y) {
	return x - (y / (1.0 + (x / (2.0 / y))));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x - (y / (1.0d0 + (x / (2.0d0 / y))))
end function

Java

public static double code(double x, double y) {
	return x - (y / (1.0 + (x / (2.0 / y))));
}

Python

def code(x, y):
	return x - (y / (1.0 + (x / (2.0 / y))))

Julia

function code(x, y)
	return Float64(x - Float64(y / Float64(1.0 + Float64(x / Float64(2.0 / y)))))
end

MATLAB

function tmp = code(x, y)
	tmp = x - (y / (1.0 + (x / (2.0 / y))));
end

Wolfram

code[x_, y_] := N[(x - N[(y / N[(1.0 + N[(x / N[(2.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[x - \frac{y}{1 + \frac{x \cdot y}{2}} \]
2. Step-by-step derivation

   1. associate-/l* 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto x - \frac{y}{1 + \frac{x}{\frac{2}{y}}} \]

Alternative 4: 77.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-88}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-145}:\\ \;\;\;\;-y\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-101}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 6.9 \cdot 10^{-59}:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -9e-88)
   x
   (if (<= x 1.1e-145)
     (- y)
     (if (<= x 1.3e-101) x (if (<= x 6.9e-59) (- y) x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -9e-88) {
		tmp = x;
	} else if (x <= 1.1e-145) {
		tmp = -y;
	} else if (x <= 1.3e-101) {
		tmp = x;
	} else if (x <= 6.9e-59) {
		tmp = -y;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-9d-88)) then
        tmp = x
    else if (x <= 1.1d-145) then
        tmp = -y
    else if (x <= 1.3d-101) then
        tmp = x
    else if (x <= 6.9d-59) then
        tmp = -y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -9e-88) {
		tmp = x;
	} else if (x <= 1.1e-145) {
		tmp = -y;
	} else if (x <= 1.3e-101) {
		tmp = x;
	} else if (x <= 6.9e-59) {
		tmp = -y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -9e-88:
		tmp = x
	elif x <= 1.1e-145:
		tmp = -y
	elif x <= 1.3e-101:
		tmp = x
	elif x <= 6.9e-59:
		tmp = -y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -9e-88)
		tmp = x;
	elseif (x <= 1.1e-145)
		tmp = Float64(-y);
	elseif (x <= 1.3e-101)
		tmp = x;
	elseif (x <= 6.9e-59)
		tmp = Float64(-y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -9e-88)
		tmp = x;
	elseif (x <= 1.1e-145)
		tmp = -y;
	elseif (x <= 1.3e-101)
		tmp = x;
	elseif (x <= 6.9e-59)
		tmp = -y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -9e-88], x, If[LessEqual[x, 1.1e-145], (-y), If[LessEqual[x, 1.3e-101], x, If[LessEqual[x, 6.9e-59], (-y), x]]]]

Derivation

1. Split input into 2 regimes

2. if x < -8.99999999999999982e-88 or 1.1e-145 < x < 1.3000000000000001e-101 or 6.89999999999999982e-59 < x

   1. Initial program 99.9%

      \[x - \frac{y}{1 + \frac{x \cdot y}{2}} \]
   2. Step-by-step derivation

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 79.8%

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

   if -8.99999999999999982e-88 < x < 1.1e-145 or 1.3000000000000001e-101 < x < 6.89999999999999982e-59

   1. Initial program 99.9%

      \[x - \frac{y}{1 + \frac{x \cdot y}{2}} \]
   2. Step-by-step derivation

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 70.8%

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

      1. neg-mul-1 70.8%

         \[\leadsto \color{blue}{-y} \]

   6. Simplified 70.8%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-88}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-145}:\\ \;\;\;\;-y\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-101}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 6.9 \cdot 10^{-59}:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5: 90.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+72} \lor \neg \left(y \leq 1.7 \cdot 10^{+61}\right):\\ \;\;\;\;x - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -3.8e+72) (not (<= y 1.7e+61))) (- x (/ 2.0 x)) (- x y)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -3.8e+72) || !(y <= 1.7e+61)) {
		tmp = x - (2.0 / x);
	} else {
		tmp = x - y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-3.8d+72)) .or. (.not. (y <= 1.7d+61))) then
        tmp = x - (2.0d0 / x)
    else
        tmp = x - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -3.8e+72) || !(y <= 1.7e+61)) {
		tmp = x - (2.0 / x);
	} else {
		tmp = x - y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -3.8e+72) or not (y <= 1.7e+61):
		tmp = x - (2.0 / x)
	else:
		tmp = x - y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -3.8e+72) || !(y <= 1.7e+61))
		tmp = Float64(x - Float64(2.0 / x));
	else
		tmp = Float64(x - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -3.8e+72) || ~((y <= 1.7e+61)))
		tmp = x - (2.0 / x);
	else
		tmp = x - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -3.8e+72], N[Not[LessEqual[y, 1.7e+61]], $MachinePrecision]], N[(x - N[(2.0 / x), $MachinePrecision]), $MachinePrecision], N[(x - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.80000000000000006e72 or 1.70000000000000013e61 < y

   1. Initial program 99.9%

      \[x - \frac{y}{1 + \frac{x \cdot y}{2}} \]
   2. Step-by-step derivation

      1. associate-/l* 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around inf 83.8%

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

   if -3.80000000000000006e72 < y < 1.70000000000000013e61

   1. Initial program 100.0%

      \[x - \frac{y}{1 + \frac{x \cdot y}{2}} \]
   2. Step-by-step derivation

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 97.1%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+72} \lor \neg \left(y \leq 1.7 \cdot 10^{+61}\right):\\ \;\;\;\;x - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \]

Alternative 6: 86.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 0.0092:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.4) x (if (<= x 0.0092) (- x y) x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.4) {
		tmp = x;
	} else if (x <= 0.0092) {
		tmp = x - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.4d0)) then
        tmp = x
    else if (x <= 0.0092d0) then
        tmp = x - y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.4) {
		tmp = x;
	} else if (x <= 0.0092) {
		tmp = x - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.4:
		tmp = x
	elif x <= 0.0092:
		tmp = x - y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.4)
		tmp = x;
	elseif (x <= 0.0092)
		tmp = Float64(x - y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.4)
		tmp = x;
	elseif (x <= 0.0092)
		tmp = x - y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.4], x, If[LessEqual[x, 0.0092], N[(x - y), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3999999999999999 or 0.0091999999999999998 < x

   1. Initial program 100.0%

      \[x - \frac{y}{1 + \frac{x \cdot y}{2}} \]
   2. Step-by-step derivation

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 98.1%

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

   if -1.3999999999999999 < x < 0.0091999999999999998

   1. Initial program 99.9%

      \[x - \frac{y}{1 + \frac{x \cdot y}{2}} \]
   2. Step-by-step derivation

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 65.1%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 0.0092:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7: 61.7% accurate, 11.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 x)

C

double code(double x, double y) {
	return x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x
end function

Java

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

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

function tmp = code(x, y)
	tmp = x;
end

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 99.9%

   \[x - \frac{y}{1 + \frac{x \cdot y}{2}} \]
2. Step-by-step derivation

   1. associate-/l* 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in x around inf 57.9%

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

5. Final simplification 57.9%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023240 
(FPCore (x y)
  :name "Data.Number.Erf:$cinvnormcdf from erf-2.0.0.0, B"
  :precision binary64
  (- x (/ y (+ 1.0 (/ (* x y) 2.0)))))