Data.Histogram.Bin.BinF:$cfromIndex from histogram-fill-0.8.4.1

Percentage Accurate: 100.0% → 100.0%

Time: 4.6s

Alternatives: 8

Speedup: 1.3×

Specification

Math

\[\begin{array}{l} \\ \left(\frac{x}{2} + y \cdot x\right) + z \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\frac{x}{2} + y \cdot x\right) + z \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, y + 0.5, z\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma x (+ y 0.5) z))

C

double code(double x, double y, double z) {
	return fma(x, (y + 0.5), z);
}

Julia

function code(x, y, z)
	return fma(x, Float64(y + 0.5), z)
end

Wolfram

code[x_, y_, z_] := N[(x * N[(y + 0.5), $MachinePrecision] + z), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. associate-+l+ 100.0%

      \[\leadsto \color{blue}{\frac{x}{2} + \left(y \cdot x + z\right)} \]

   2. *-commutative 100.0%

      \[\leadsto \frac{x}{2} + \left(\color{blue}{x \cdot y} + z\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{x}{2} + \left(x \cdot y + z\right)} \]

4. Taylor expanded in x around 0 100.0%

   \[\leadsto \color{blue}{z + \left(0.5 + y\right) \cdot x} \]
5. Step-by-step derivation

   1. +-commutative 100.0%

      \[\leadsto \color{blue}{\left(0.5 + y\right) \cdot x + z} \]

   2. *-commutative 100.0%

      \[\leadsto \color{blue}{x \cdot \left(0.5 + y\right)} + z \]

   3. fma-udef 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.5 + y, z\right)} \]

   4. +-commutative 100.0%

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{y + 0.5}, z\right) \]

6. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, y + 0.5, z\right)} \]

7. Final simplification 100.0%

   \[\leadsto \mathsf{fma}\left(x, y + 0.5, z\right) \]

Alternative 2: 59.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+97}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq -3.9 \cdot 10^{-53}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq -4.6 \cdot 10^{-220}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-116}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq 0.00365:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+122}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -8e+97)
   (* x y)
   (if (<= y -3.9e-53)
     z
     (if (<= y -4.6e-220)
       (* x 0.5)
       (if (<= y 4.5e-116)
         z
         (if (<= y 0.00365) (* x 0.5) (if (<= y 6.5e+122) z (* x y))))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -8e+97) {
		tmp = x * y;
	} else if (y <= -3.9e-53) {
		tmp = z;
	} else if (y <= -4.6e-220) {
		tmp = x * 0.5;
	} else if (y <= 4.5e-116) {
		tmp = z;
	} else if (y <= 0.00365) {
		tmp = x * 0.5;
	} else if (y <= 6.5e+122) {
		tmp = z;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-8d+97)) then
        tmp = x * y
    else if (y <= (-3.9d-53)) then
        tmp = z
    else if (y <= (-4.6d-220)) then
        tmp = x * 0.5d0
    else if (y <= 4.5d-116) then
        tmp = z
    else if (y <= 0.00365d0) then
        tmp = x * 0.5d0
    else if (y <= 6.5d+122) then
        tmp = z
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -8e+97) {
		tmp = x * y;
	} else if (y <= -3.9e-53) {
		tmp = z;
	} else if (y <= -4.6e-220) {
		tmp = x * 0.5;
	} else if (y <= 4.5e-116) {
		tmp = z;
	} else if (y <= 0.00365) {
		tmp = x * 0.5;
	} else if (y <= 6.5e+122) {
		tmp = z;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -8e+97:
		tmp = x * y
	elif y <= -3.9e-53:
		tmp = z
	elif y <= -4.6e-220:
		tmp = x * 0.5
	elif y <= 4.5e-116:
		tmp = z
	elif y <= 0.00365:
		tmp = x * 0.5
	elif y <= 6.5e+122:
		tmp = z
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -8e+97)
		tmp = Float64(x * y);
	elseif (y <= -3.9e-53)
		tmp = z;
	elseif (y <= -4.6e-220)
		tmp = Float64(x * 0.5);
	elseif (y <= 4.5e-116)
		tmp = z;
	elseif (y <= 0.00365)
		tmp = Float64(x * 0.5);
	elseif (y <= 6.5e+122)
		tmp = z;
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -8e+97)
		tmp = x * y;
	elseif (y <= -3.9e-53)
		tmp = z;
	elseif (y <= -4.6e-220)
		tmp = x * 0.5;
	elseif (y <= 4.5e-116)
		tmp = z;
	elseif (y <= 0.00365)
		tmp = x * 0.5;
	elseif (y <= 6.5e+122)
		tmp = z;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -8e+97], N[(x * y), $MachinePrecision], If[LessEqual[y, -3.9e-53], z, If[LessEqual[y, -4.6e-220], N[(x * 0.5), $MachinePrecision], If[LessEqual[y, 4.5e-116], z, If[LessEqual[y, 0.00365], N[(x * 0.5), $MachinePrecision], If[LessEqual[y, 6.5e+122], z, N[(x * y), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.0000000000000006e97 or 6.49999999999999963e122 < y

   1. Initial program 100.0%

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

      1. associate-+l+ 100.0%

         \[\leadsto \color{blue}{\frac{x}{2} + \left(y \cdot x + z\right)} \]

      2. *-commutative 100.0%

         \[\leadsto \frac{x}{2} + \left(\color{blue}{x \cdot y} + z\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{x}{2} + \left(x \cdot y + z\right)} \]

   4. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{z + \left(0.5 + y\right) \cdot x} \]
   5. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{\left(0.5 + y\right) \cdot x + z} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{x \cdot \left(0.5 + y\right)} + z \]

      3. fma-udef 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.5 + y, z\right)} \]

      4. +-commutative 100.0%

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{y + 0.5}, z\right) \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y + 0.5, z\right)} \]

   7. Taylor expanded in y around inf 84.3%

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

   if -8.0000000000000006e97 < y < -3.9000000000000002e-53 or -4.59999999999999961e-220 < y < 4.50000000000000012e-116 or 0.00365000000000000003 < y < 6.49999999999999963e122

   1. Initial program 99.9%

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

      1. associate-+l+ 99.9%

         \[\leadsto \color{blue}{\frac{x}{2} + \left(y \cdot x + z\right)} \]

      2. *-commutative 99.9%

         \[\leadsto \frac{x}{2} + \left(\color{blue}{x \cdot y} + z\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{x}{2} + \left(x \cdot y + z\right)} \]

   4. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{z + \left(0.5 + y\right) \cdot x} \]
   5. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{\left(0.5 + y\right) \cdot x + z} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{x \cdot \left(0.5 + y\right)} + z \]

      3. fma-udef 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.5 + y, z\right)} \]

      4. +-commutative 100.0%

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{y + 0.5}, z\right) \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y + 0.5, z\right)} \]

   7. Taylor expanded in x around 0 59.3%

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

   if -3.9000000000000002e-53 < y < -4.59999999999999961e-220 or 4.50000000000000012e-116 < y < 0.00365000000000000003

   1. Initial program 100.0%

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

      1. associate-+l+ 100.0%

         \[\leadsto \color{blue}{\frac{x}{2} + \left(y \cdot x + z\right)} \]

      2. *-commutative 100.0%

         \[\leadsto \frac{x}{2} + \left(\color{blue}{x \cdot y} + z\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{x}{2} + \left(x \cdot y + z\right)} \]

   4. Taylor expanded in x around inf 67.1%

      \[\leadsto \color{blue}{\left(0.5 + y\right) \cdot x} \]

   5. Taylor expanded in y around 0 63.8%

      \[\leadsto \color{blue}{0.5 \cdot x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+97}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq -3.9 \cdot 10^{-53}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq -4.6 \cdot 10^{-220}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-116}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq 0.00365:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+122}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 3: 74.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -15 \lor \neg \left(x \leq 2.45 \cdot 10^{-76}\right):\\ \;\;\;\;x \cdot \left(y + 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -15.0) (not (<= x 2.45e-76))) (* x (+ y 0.5)) z))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -15.0) || !(x <= 2.45e-76)) {
		tmp = x * (y + 0.5);
	} else {
		tmp = z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-15.0d0)) .or. (.not. (x <= 2.45d-76))) then
        tmp = x * (y + 0.5d0)
    else
        tmp = z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -15.0) || !(x <= 2.45e-76)) {
		tmp = x * (y + 0.5);
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -15.0) or not (x <= 2.45e-76):
		tmp = x * (y + 0.5)
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -15.0) || !(x <= 2.45e-76))
		tmp = Float64(x * Float64(y + 0.5));
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -15.0) || ~((x <= 2.45e-76)))
		tmp = x * (y + 0.5);
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -15.0], N[Not[LessEqual[x, 2.45e-76]], $MachinePrecision]], N[(x * N[(y + 0.5), $MachinePrecision]), $MachinePrecision], z]

Derivation

1. Split input into 2 regimes

2. if x < -15 or 2.44999999999999986e-76 < x

   1. Initial program 100.0%

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

      1. associate-+l+ 100.0%

         \[\leadsto \color{blue}{\frac{x}{2} + \left(y \cdot x + z\right)} \]

      2. *-commutative 100.0%

         \[\leadsto \frac{x}{2} + \left(\color{blue}{x \cdot y} + z\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{x}{2} + \left(x \cdot y + z\right)} \]

   4. Taylor expanded in x around inf 83.8%

      \[\leadsto \color{blue}{\left(0.5 + y\right) \cdot x} \]

   if -15 < x < 2.44999999999999986e-76

   1. Initial program 100.0%

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

      1. associate-+l+ 100.0%

         \[\leadsto \color{blue}{\frac{x}{2} + \left(y \cdot x + z\right)} \]

      2. *-commutative 100.0%

         \[\leadsto \frac{x}{2} + \left(\color{blue}{x \cdot y} + z\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{x}{2} + \left(x \cdot y + z\right)} \]

   4. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{z + \left(0.5 + y\right) \cdot x} \]
   5. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{\left(0.5 + y\right) \cdot x + z} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{x \cdot \left(0.5 + y\right)} + z \]

      3. fma-udef 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.5 + y, z\right)} \]

      4. +-commutative 100.0%

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{y + 0.5}, z\right) \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y + 0.5, z\right)} \]

   7. Taylor expanded in x around 0 69.1%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -15 \lor \neg \left(x \leq 2.45 \cdot 10^{-76}\right):\\ \;\;\;\;x \cdot \left(y + 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 4: 85.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -34000 \lor \neg \left(x \leq 1.65 \cdot 10^{+109}\right):\\ \;\;\;\;x \cdot \left(y + 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;z + x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -34000.0) (not (<= x 1.65e+109)))
   (* x (+ y 0.5))
   (+ z (* x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -34000.0) || !(x <= 1.65e+109)) {
		tmp = x * (y + 0.5);
	} else {
		tmp = z + (x * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-34000.0d0)) .or. (.not. (x <= 1.65d+109))) then
        tmp = x * (y + 0.5d0)
    else
        tmp = z + (x * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -34000.0) || !(x <= 1.65e+109)) {
		tmp = x * (y + 0.5);
	} else {
		tmp = z + (x * y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -34000.0) or not (x <= 1.65e+109):
		tmp = x * (y + 0.5)
	else:
		tmp = z + (x * y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -34000.0) || !(x <= 1.65e+109))
		tmp = Float64(x * Float64(y + 0.5));
	else
		tmp = Float64(z + Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -34000.0) || ~((x <= 1.65e+109)))
		tmp = x * (y + 0.5);
	else
		tmp = z + (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -34000.0], N[Not[LessEqual[x, 1.65e+109]], $MachinePrecision]], N[(x * N[(y + 0.5), $MachinePrecision]), $MachinePrecision], N[(z + N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -34000 or 1.6499999999999999e109 < x

   1. Initial program 100.0%

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

      1. associate-+l+ 100.0%

         \[\leadsto \color{blue}{\frac{x}{2} + \left(y \cdot x + z\right)} \]

      2. *-commutative 100.0%

         \[\leadsto \frac{x}{2} + \left(\color{blue}{x \cdot y} + z\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{x}{2} + \left(x \cdot y + z\right)} \]

   4. Taylor expanded in x around inf 92.6%

      \[\leadsto \color{blue}{\left(0.5 + y\right) \cdot x} \]

   if -34000 < x < 1.6499999999999999e109

   1. Initial program 100.0%

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

      1. associate-+l+ 100.0%

         \[\leadsto \color{blue}{\frac{x}{2} + \left(y \cdot x + z\right)} \]

      2. *-commutative 100.0%

         \[\leadsto \frac{x}{2} + \left(\color{blue}{x \cdot y} + z\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{x}{2} + \left(x \cdot y + z\right)} \]

   4. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{z + \left(0.5 + y\right) \cdot x} \]
   5. Step-by-step derivation

      1. flip-+ 81.1%

         \[\leadsto z + \color{blue}{\frac{0.5 \cdot 0.5 - y \cdot y}{0.5 - y}} \cdot x \]

      2. associate-*l/ 79.9%

         \[\leadsto z + \color{blue}{\frac{\left(0.5 \cdot 0.5 - y \cdot y\right) \cdot x}{0.5 - y}} \]

      3. metadata-eval 79.9%

         \[\leadsto z + \frac{\left(\color{blue}{0.25} - y \cdot y\right) \cdot x}{0.5 - y} \]

   6. Applied egg-rr 79.9%

      \[\leadsto z + \color{blue}{\frac{\left(0.25 - y \cdot y\right) \cdot x}{0.5 - y}} \]

   7. Taylor expanded in y around inf 83.0%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -34000 \lor \neg \left(x \leq 1.65 \cdot 10^{+109}\right):\\ \;\;\;\;x \cdot \left(y + 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;z + x \cdot y\\ \end{array} \]

Alternative 5: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6600 \lor \neg \left(y \leq 0.5\right):\\ \;\;\;\;z + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z + \frac{x}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -6600.0) (not (<= y 0.5))) (+ z (* x y)) (+ z (/ x 2.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -6600.0) || !(y <= 0.5)) {
		tmp = z + (x * y);
	} else {
		tmp = z + (x / 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-6600.0d0)) .or. (.not. (y <= 0.5d0))) then
        tmp = z + (x * y)
    else
        tmp = z + (x / 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -6600.0) || !(y <= 0.5)) {
		tmp = z + (x * y);
	} else {
		tmp = z + (x / 2.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -6600.0) or not (y <= 0.5):
		tmp = z + (x * y)
	else:
		tmp = z + (x / 2.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -6600.0) || !(y <= 0.5))
		tmp = Float64(z + Float64(x * y));
	else
		tmp = Float64(z + Float64(x / 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -6600.0) || ~((y <= 0.5)))
		tmp = z + (x * y);
	else
		tmp = z + (x / 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -6600.0], N[Not[LessEqual[y, 0.5]], $MachinePrecision]], N[(z + N[(x * y), $MachinePrecision]), $MachinePrecision], N[(z + N[(x / 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6600 or 0.5 < y

   1. Initial program 99.9%

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

      1. associate-+l+ 99.9%

         \[\leadsto \color{blue}{\frac{x}{2} + \left(y \cdot x + z\right)} \]

      2. *-commutative 99.9%

         \[\leadsto \frac{x}{2} + \left(\color{blue}{x \cdot y} + z\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{x}{2} + \left(x \cdot y + z\right)} \]

   4. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{z + \left(0.5 + y\right) \cdot x} \]
   5. Step-by-step derivation

      1. flip-+ 70.0%

         \[\leadsto z + \color{blue}{\frac{0.5 \cdot 0.5 - y \cdot y}{0.5 - y}} \cdot x \]

      2. associate-*l/ 62.4%

         \[\leadsto z + \color{blue}{\frac{\left(0.5 \cdot 0.5 - y \cdot y\right) \cdot x}{0.5 - y}} \]

      3. metadata-eval 62.4%

         \[\leadsto z + \frac{\left(\color{blue}{0.25} - y \cdot y\right) \cdot x}{0.5 - y} \]

   6. Applied egg-rr 62.4%

      \[\leadsto z + \color{blue}{\frac{\left(0.25 - y \cdot y\right) \cdot x}{0.5 - y}} \]

   7. Taylor expanded in y around inf 99.8%

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

   if -6600 < y < 0.5

   1. Initial program 100.0%

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

      1. associate-+l+ 100.0%

         \[\leadsto \color{blue}{\frac{x}{2} + \left(y \cdot x + z\right)} \]

      2. *-commutative 100.0%

         \[\leadsto \frac{x}{2} + \left(\color{blue}{x \cdot y} + z\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{x}{2} + \left(x \cdot y + z\right)} \]

   4. Taylor expanded in x around 0 97.8%

      \[\leadsto \frac{x}{2} + \color{blue}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6600 \lor \neg \left(y \leq 0.5\right):\\ \;\;\;\;z + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z + \frac{x}{2}\\ \end{array} \]

Alternative 6: 50.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -42000:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+65}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -42000.0) (* x 0.5) (if (<= x 1.4e+65) z (* x 0.5))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -42000.0) {
		tmp = x * 0.5;
	} else if (x <= 1.4e+65) {
		tmp = z;
	} else {
		tmp = x * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-42000.0d0)) then
        tmp = x * 0.5d0
    else if (x <= 1.4d+65) then
        tmp = z
    else
        tmp = x * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -42000.0) {
		tmp = x * 0.5;
	} else if (x <= 1.4e+65) {
		tmp = z;
	} else {
		tmp = x * 0.5;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -42000.0:
		tmp = x * 0.5
	elif x <= 1.4e+65:
		tmp = z
	else:
		tmp = x * 0.5
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -42000.0)
		tmp = Float64(x * 0.5);
	elseif (x <= 1.4e+65)
		tmp = z;
	else
		tmp = Float64(x * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -42000.0)
		tmp = x * 0.5;
	elseif (x <= 1.4e+65)
		tmp = z;
	else
		tmp = x * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -42000.0], N[(x * 0.5), $MachinePrecision], If[LessEqual[x, 1.4e+65], z, N[(x * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -42000 or 1.3999999999999999e65 < x

   1. Initial program 100.0%

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

      1. associate-+l+ 100.0%

         \[\leadsto \color{blue}{\frac{x}{2} + \left(y \cdot x + z\right)} \]

      2. *-commutative 100.0%

         \[\leadsto \frac{x}{2} + \left(\color{blue}{x \cdot y} + z\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{x}{2} + \left(x \cdot y + z\right)} \]

   4. Taylor expanded in x around inf 91.3%

      \[\leadsto \color{blue}{\left(0.5 + y\right) \cdot x} \]

   5. Taylor expanded in y around 0 52.3%

      \[\leadsto \color{blue}{0.5 \cdot x} \]

   if -42000 < x < 1.3999999999999999e65

   1. Initial program 100.0%

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

      1. associate-+l+ 100.0%

         \[\leadsto \color{blue}{\frac{x}{2} + \left(y \cdot x + z\right)} \]

      2. *-commutative 100.0%

         \[\leadsto \frac{x}{2} + \left(\color{blue}{x \cdot y} + z\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{x}{2} + \left(x \cdot y + z\right)} \]

   4. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{z + \left(0.5 + y\right) \cdot x} \]
   5. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{\left(0.5 + y\right) \cdot x + z} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{x \cdot \left(0.5 + y\right)} + z \]

      3. fma-udef 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.5 + y, z\right)} \]

      4. +-commutative 100.0%

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{y + 0.5}, z\right) \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y + 0.5, z\right)} \]

   7. Taylor expanded in x around 0 62.9%

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

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -42000:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+65}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5\\ \end{array} \]

Alternative 7: 100.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ z + x \cdot \left(y + 0.5\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ z (* x (+ y 0.5))))

C

double code(double x, double y, double z) {
	return z + (x * (y + 0.5));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = z + (x * (y + 0.5d0))
end function

Java

public static double code(double x, double y, double z) {
	return z + (x * (y + 0.5));
}

Python

def code(x, y, z):
	return z + (x * (y + 0.5))

Julia

function code(x, y, z)
	return Float64(z + Float64(x * Float64(y + 0.5)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = z + (x * (y + 0.5));
end

Wolfram

code[x_, y_, z_] := N[(z + N[(x * N[(y + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. associate-+l+ 100.0%

      \[\leadsto \color{blue}{\frac{x}{2} + \left(y \cdot x + z\right)} \]

   2. *-commutative 100.0%

      \[\leadsto \frac{x}{2} + \left(\color{blue}{x \cdot y} + z\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{x}{2} + \left(x \cdot y + z\right)} \]

4. Taylor expanded in x around 0 100.0%

   \[\leadsto \color{blue}{z + \left(0.5 + y\right) \cdot x} \]

5. Final simplification 100.0%

   \[\leadsto z + x \cdot \left(y + 0.5\right) \]

Alternative 8: 40.0% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := z

Derivation

1. Initial program 100.0%

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

   1. associate-+l+ 100.0%

      \[\leadsto \color{blue}{\frac{x}{2} + \left(y \cdot x + z\right)} \]

   2. *-commutative 100.0%

      \[\leadsto \frac{x}{2} + \left(\color{blue}{x \cdot y} + z\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{x}{2} + \left(x \cdot y + z\right)} \]

4. Taylor expanded in x around 0 100.0%

   \[\leadsto \color{blue}{z + \left(0.5 + y\right) \cdot x} \]
5. Step-by-step derivation

   1. +-commutative 100.0%

      \[\leadsto \color{blue}{\left(0.5 + y\right) \cdot x + z} \]

   2. *-commutative 100.0%

      \[\leadsto \color{blue}{x \cdot \left(0.5 + y\right)} + z \]

   3. fma-udef 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.5 + y, z\right)} \]

   4. +-commutative 100.0%

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{y + 0.5}, z\right) \]

6. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, y + 0.5, z\right)} \]

7. Taylor expanded in x around 0 40.5%

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

8. Final simplification 40.5%

   \[\leadsto z \]

Reproduce

herbie shell --seed 2023171 
(FPCore (x y z)
  :name "Data.Histogram.Bin.BinF:$cfromIndex from histogram-fill-0.8.4.1"
  :precision binary64
  (+ (+ (/ x 2.0) (* y x)) z))