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

Percentage Accurate: 100.0% → 100.0%

Time: 4.6s

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, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 2: 85.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{2} + x \cdot y\\ t_1 := x \cdot \left(y + 0.5\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+135}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+50}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5, z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (/ x 2.0) (* x y))) (t_1 (* x (+ y 0.5))))
   (if (<= t_0 -1e+135) t_1 (if (<= t_0 2e+50) (fma x 0.5 z) t_1))))

C

double code(double x, double y, double z) {
	double t_0 = (x / 2.0) + (x * y);
	double t_1 = x * (y + 0.5);
	double tmp;
	if (t_0 <= -1e+135) {
		tmp = t_1;
	} else if (t_0 <= 2e+50) {
		tmp = fma(x, 0.5, z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x / 2.0) + Float64(x * y))
	t_1 = Float64(x * Float64(y + 0.5))
	tmp = 0.0
	if (t_0 <= -1e+135)
		tmp = t_1;
	elseif (t_0 <= 2e+50)
		tmp = fma(x, 0.5, z);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x / 2.0), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(y + 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+135], t$95$1, If[LessEqual[t$95$0, 2e+50], N[(x * 0.5 + z), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (/.f64 x #s(literal 2 binary64)) (*.f64 y x)) < -9.99999999999999962e134 or 2.0000000000000002e50 < (+.f64 (/.f64 x #s(literal 2 binary64)) (*.f64 y x))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} + y\right)} \]

      2. lower-+.f64 88.9

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

   5. Applied rewrites 88.9%

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

   if -9.99999999999999962e134 < (+.f64 (/.f64 x #s(literal 2 binary64)) (*.f64 y x)) < 2.0000000000000002e50

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot x + z} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \frac{1}{2}} + z \]

      3. lower-fma.f64 86.2

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

   5. Applied rewrites 86.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.5, z\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{2} + x \cdot y \leq -1 \cdot 10^{+135}:\\ \;\;\;\;x \cdot \left(y + 0.5\right)\\ \mathbf{elif}\;\frac{x}{2} + x \cdot y \leq 2 \cdot 10^{+50}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5, z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z + x \cdot y\\ \mathbf{if}\;y \leq -0.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{-25}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5, z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ z (* x y))))
   (if (<= y -0.5) t_0 (if (<= y 4.4e-25) (fma x 0.5 z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = z + (x * y);
	double tmp;
	if (y <= -0.5) {
		tmp = t_0;
	} else if (y <= 4.4e-25) {
		tmp = fma(x, 0.5, z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(z + Float64(x * y))
	tmp = 0.0
	if (y <= -0.5)
		tmp = t_0;
	elseif (y <= 4.4e-25)
		tmp = fma(x, 0.5, z);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z + N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.5], t$95$0, If[LessEqual[y, 4.4e-25], N[(x * 0.5 + z), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.5 or 4.4000000000000004e-25 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{x \cdot y} + z \]
   4. Step-by-step derivation

      1. lower-*.f64 98.9

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

   5. Applied rewrites 98.9%

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

   if -0.5 < y < 4.4000000000000004e-25

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot x + z} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \frac{1}{2}} + z \]

      3. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.5, z\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.5:\\ \;\;\;\;z + x \cdot y\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{-25}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5, z\right)\\ \mathbf{else}:\\ \;\;\;\;z + x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+15}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5, z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3.5e+15) (* x y) (if (<= y 7.8e+19) (fma x 0.5 z) (* x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.5e+15) {
		tmp = x * y;
	} else if (y <= 7.8e+19) {
		tmp = fma(x, 0.5, z);
	} else {
		tmp = x * y;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -3.5e+15)
		tmp = Float64(x * y);
	elseif (y <= 7.8e+19)
		tmp = fma(x, 0.5, z);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3.5e+15], N[(x * y), $MachinePrecision], If[LessEqual[y, 7.8e+19], N[(x * 0.5 + z), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.5e15 or 7.8e19 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 71.5

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

   5. Applied rewrites 71.5%

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

   if -3.5e15 < y < 7.8e19

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot x + z} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \frac{1}{2}} + z \]

      3. lower-fma.f64 98.6

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

   5. Applied rewrites 98.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.5, z\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 59.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.5:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-7}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -0.5) (* x y) (if (<= y 4.2e-7) (* x 0.5) (* x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -0.5) {
		tmp = x * y;
	} else if (y <= 4.2e-7) {
		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 <= (-0.5d0)) then
        tmp = x * y
    else if (y <= 4.2d-7) 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 <= -0.5) {
		tmp = x * y;
	} else if (y <= 4.2e-7) {
		tmp = x * 0.5;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -0.5:
		tmp = x * y
	elif y <= 4.2e-7:
		tmp = x * 0.5
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -0.5)
		tmp = Float64(x * y);
	elseif (y <= 4.2e-7)
		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 <= -0.5)
		tmp = x * y;
	elseif (y <= 4.2e-7)
		tmp = x * 0.5;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -0.5], N[(x * y), $MachinePrecision], If[LessEqual[y, 4.2e-7], N[(x * 0.5), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.5 or 4.2e-7 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 68.7

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

   5. Applied rewrites 68.7%

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

   if -0.5 < y < 4.2e-7

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} + y\right)} \]

      2. lower-+.f64 47.9

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

   5. Applied rewrites 47.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 47.9%

      \[\leadsto x \cdot 0.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 36.3% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around inf

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

   1. lower-*.f64 35.8

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

5. Applied rewrites 35.8%

   \[\leadsto \color{blue}{x \cdot y} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024219 
(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))