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

Percentage Accurate: 100.0% → 100.0%

Time: 2.3s

Alternatives: 6

Speedup: 1.0×

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

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(N[(x / 2.0), $MachinePrecision] + N[(x * y + z), $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. fma-def 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \frac{x}{2} + \mathsf{fma}\left(x, y, z\right) \]

Alternative 2: 82.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{2} + x \cdot y\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{+248} \lor \neg \left(t_0 \leq 2 \cdot 10^{+174}\right):\\ \;\;\;\;x \cdot \left(y + 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{2} + z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (/ x 2.0) (* x y))))
   (if (or (<= t_0 -1e+248) (not (<= t_0 2e+174)))
     (* x (+ y 0.5))
     (+ (/ x 2.0) z))))

C

double code(double x, double y, double z) {
	double t_0 = (x / 2.0) + (x * y);
	double tmp;
	if ((t_0 <= -1e+248) || !(t_0 <= 2e+174)) {
		tmp = x * (y + 0.5);
	} else {
		tmp = (x / 2.0) + 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) :: t_0
    real(8) :: tmp
    t_0 = (x / 2.0d0) + (x * y)
    if ((t_0 <= (-1d+248)) .or. (.not. (t_0 <= 2d+174))) then
        tmp = x * (y + 0.5d0)
    else
        tmp = (x / 2.0d0) + z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x / 2.0) + (x * y);
	double tmp;
	if ((t_0 <= -1e+248) || !(t_0 <= 2e+174)) {
		tmp = x * (y + 0.5);
	} else {
		tmp = (x / 2.0) + z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x / 2.0) + (x * y)
	tmp = 0
	if (t_0 <= -1e+248) or not (t_0 <= 2e+174):
		tmp = x * (y + 0.5)
	else:
		tmp = (x / 2.0) + z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x / 2.0) + Float64(x * y))
	tmp = 0.0
	if ((t_0 <= -1e+248) || !(t_0 <= 2e+174))
		tmp = Float64(x * Float64(y + 0.5));
	else
		tmp = Float64(Float64(x / 2.0) + z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x / 2.0) + (x * y);
	tmp = 0.0;
	if ((t_0 <= -1e+248) || ~((t_0 <= 2e+174)))
		tmp = x * (y + 0.5);
	else
		tmp = (x / 2.0) + z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x / 2.0), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -1e+248], N[Not[LessEqual[t$95$0, 2e+174]], $MachinePrecision]], N[(x * N[(y + 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(x / 2.0), $MachinePrecision] + z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (/.f64 x 2) (*.f64 y x)) < -1.00000000000000005e248 or 2.00000000000000014e174 < (+.f64 (/.f64 x 2) (*.f64 y 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 94.8%

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

   5. Taylor expanded in x around 0 94.8%

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

   if -1.00000000000000005e248 < (+.f64 (/.f64 x 2) (*.f64 y x)) < 2.00000000000000014e174

   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 83.8%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{2} + x \cdot y \leq -1 \cdot 10^{+248} \lor \neg \left(\frac{x}{2} + x \cdot y \leq 2 \cdot 10^{+174}\right):\\ \;\;\;\;x \cdot \left(y + 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{2} + z\\ \end{array} \]

Alternative 3: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 / 2.0d0) + (x * y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Final simplification 100.0%

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

Alternative 4: 59.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{-29}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 0.5:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.4e-29) (* x y) (if (<= y 0.5) (* x 0.5) (* x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.4e-29) {
		tmp = x * y;
	} else if (y <= 0.5) {
		tmp = x * 0.5;
	} 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 <= (-2.4d-29)) then
        tmp = x * y
    else if (y <= 0.5d0) then
        tmp = x * 0.5d0
    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 <= -2.4e-29) {
		tmp = x * y;
	} else if (y <= 0.5) {
		tmp = x * 0.5;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2.4e-29:
		tmp = x * y
	elif y <= 0.5:
		tmp = x * 0.5
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.4e-29)
		tmp = Float64(x * y);
	elseif (y <= 0.5)
		tmp = Float64(x * 0.5);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2.4e-29)
		tmp = x * y;
	elseif (y <= 0.5)
		tmp = x * 0.5;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.4e-29], N[(x * y), $MachinePrecision], If[LessEqual[y, 0.5], N[(x * 0.5), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.39999999999999992e-29 or 0.5 < 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 inf 64.5%

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

   5. Taylor expanded in y around inf 64.0%

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

   if -2.39999999999999992e-29 < 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 inf 55.0%

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

   5. Taylor expanded in y around 0 54.1%

      \[\leadsto \color{blue}{0.5 \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{-29}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 0.5:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 5: 60.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return 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 = x * (y + 0.5d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(x * N[(y + 0.5), $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 inf 59.5%

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

5. Taylor expanded in x around 0 59.5%

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

6. Final simplification 59.5%

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

Alternative 6: 26.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ x \cdot 0.5 \end{array} \]

FPCore

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

C

double code(double x, double y, double z) {
	return x * 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 = x * 0.5d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(x * 0.5), $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 inf 59.5%

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

5. Taylor expanded in y around 0 29.6%

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

6. Final simplification 29.6%

   \[\leadsto x \cdot 0.5 \]

Reproduce

herbie shell --seed 2023221 
(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))