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

Percentage Accurate: 100.0% → 100.0%

Time: 4.9s

Alternatives: 6

Speedup: 2.3×

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, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. +-commutative N/A

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

   4. associate-+l+ N/A

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

   5. lift-*.f64 N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. div-inv N/A

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

   9. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, z\right)}\right) \]

   10. metadata-eval 100.0

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

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, \mathsf{fma}\left(x, 0.5, z\right)\right)} \]
5. Add Preprocessing

Alternative 2: 98.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.2 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(y, x, z\right)\\ \mathbf{elif}\;y \leq 0.0105:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5, z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -9.2e+16) (fma y x z) (if (<= y 0.0105) (fma x 0.5 z) (fma y x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -9.2e+16) {
		tmp = fma(y, x, z);
	} else if (y <= 0.0105) {
		tmp = fma(x, 0.5, z);
	} else {
		tmp = fma(y, x, z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -9.2e+16)
		tmp = fma(y, x, z);
	elseif (y <= 0.0105)
		tmp = fma(x, 0.5, z);
	else
		tmp = fma(y, x, z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -9.2e+16], N[(y * x + z), $MachinePrecision], If[LessEqual[y, 0.0105], N[(x * 0.5 + z), $MachinePrecision], N[(y * x + z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -9.2e16 or 0.0105000000000000007 < 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 100.0

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

   5. Applied rewrites 100.0%

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

      1. *-commutative N/A

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

      2. lower-fma.f64 100.0

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

   7. Applied rewrites 100.0%

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

   if -9.2e16 < y < 0.0105000000000000007

   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 3: 84.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+17}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{+81}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5, z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4.8e+17) (* y x) (if (<= y 5.8e+81) (fma x 0.5 z) (* y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4.8e+17) {
		tmp = y * x;
	} else if (y <= 5.8e+81) {
		tmp = fma(x, 0.5, z);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -4.8e+17)
		tmp = Float64(y * x);
	elseif (y <= 5.8e+81)
		tmp = fma(x, 0.5, z);
	else
		tmp = Float64(y * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -4.8e+17], N[(y * x), $MachinePrecision], If[LessEqual[y, 5.8e+81], N[(x * 0.5 + z), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.8e17 or 5.7999999999999999e81 < 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 79.0

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

   5. Applied rewrites 79.0%

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

   if -4.8e17 < y < 5.7999999999999999e81

   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 96.0

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

   5. Applied rewrites 96.0%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+17}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{+81}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5, z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 59.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -0.5) {
		tmp = y * x;
	} else if (y <= 0.5) {
		tmp = x * 0.5;
	} else {
		tmp = y * x;
	}
	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 = y * x
    else if (y <= 0.5d0) then
        tmp = x * 0.5d0
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -0.5 or 0.5 < 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 72.7

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

   5. Applied rewrites 72.7%

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

   if -0.5 < y < 0.5

   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 49.3

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

   5. Applied rewrites 49.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 47.9%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.5:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 0.5:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 100.0% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. div-inv N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. distribute-lft-out N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2} + y, x, z\right)} \]

   10. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{y + \frac{1}{2}}, x, z\right) \]

   11. lower-+.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{y + \frac{1}{2}}, x, z\right) \]

   12. metadata-eval 100.0

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

4. Applied rewrites 100.0%

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

Alternative 6: 26.3% accurate, 3.8× 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. 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 61.0

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

5. Applied rewrites 61.0%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 25.6%

   \[\leadsto x \cdot \color{blue}{0.5} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024216 
(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))