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

Percentage Accurate: 100.0% → 100.0%

Time: 4.9s

Alternatives: 6

Speedup: 1.8×

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

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

   2. lift-+.f64 N/A

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

   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: 83.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{2} + y \cdot x\\ t_1 := x \cdot \left(y + 0.5\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+128}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{+190}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5, z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (/ x 2.0) (* y x))) (t_1 (* x (+ y 0.5))))
   (if (<= t_0 -2e+128) t_1 (if (<= t_0 1e+190) (fma x 0.5 z) t_1))))

C

double code(double x, double y, double z) {
	double t_0 = (x / 2.0) + (y * x);
	double t_1 = x * (y + 0.5);
	double tmp;
	if (t_0 <= -2e+128) {
		tmp = t_1;
	} else if (t_0 <= 1e+190) {
		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(y * x))
	t_1 = Float64(x * Float64(y + 0.5))
	tmp = 0.0
	if (t_0 <= -2e+128)
		tmp = t_1;
	elseif (t_0 <= 1e+190)
		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[(y * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(y + 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+128], t$95$1, If[LessEqual[t$95$0, 1e+190], 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)) < -2.0000000000000002e128 or 1.0000000000000001e190 < (+.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 93.7

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

   5. Applied rewrites 93.7%

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

   if -2.0000000000000002e128 < (+.f64 (/.f64 x #s(literal 2 binary64)) (*.f64 y x)) < 1.0000000000000001e190

   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 86.0

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

   5. Applied rewrites 86.0%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{2} + y \cdot x \leq -2 \cdot 10^{+128}:\\ \;\;\;\;x \cdot \left(y + 0.5\right)\\ \mathbf{elif}\;\frac{x}{2} + y \cdot x \leq 10^{+190}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5, z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, y, z)
	t_0 = Float64(z + Float64(y * x))
	tmp = 0.0
	if (y <= -0.5)
		tmp = t_0;
	elseif (y <= 0.5)
		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[(y * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.5], t$95$0, If[LessEqual[y, 0.5], N[(x * 0.5 + z), $MachinePrecision], t$95$0]]]

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

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

      1. lower-*.f64 98.8

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

   5. Applied rewrites 98.8%

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

   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 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 98.5

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

   5. Applied rewrites 98.5%

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

4. Final simplification 98.7%

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

Alternative 4: 83.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+124}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 8.8 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5, z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4e+124) (* y x) (if (<= y 8.8e+17) (fma x 0.5 z) (* y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4e+124) {
		tmp = y * x;
	} else if (y <= 8.8e+17) {
		tmp = fma(x, 0.5, z);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -4e+124)
		tmp = Float64(y * x);
	elseif (y <= 8.8e+17)
		tmp = fma(x, 0.5, z);
	else
		tmp = Float64(y * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -4e+124], N[(y * x), $MachinePrecision], If[LessEqual[y, 8.8e+17], N[(x * 0.5 + z), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.99999999999999979e124 or 8.8e17 < 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 78.9

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

   5. Applied rewrites 78.9%

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

   if -3.99999999999999979e124 < y < 8.8e17

   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 92.5

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

   5. Applied rewrites 92.5%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+124}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 8.8 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5, z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 58.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.5:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{+15}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -0.5) (* y x) (if (<= y 1.95e+15) (* 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 <= 1.95e+15) {
		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 <= 1.95d+15) 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 <= 1.95e+15) {
		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 <= 1.95e+15:
		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 <= 1.95e+15)
		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 <= 1.95e+15)
		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, 1.95e+15], N[(x * 0.5), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.5 or 1.95e15 < 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 70.9

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

   5. Applied rewrites 70.9%

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

   if -0.5 < y < 1.95e15

   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 48.7

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

   5. Applied rewrites 48.7%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 46.7%

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

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.5:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{+15}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 36.6% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(y * x), $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 36.4

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

5. Applied rewrites 36.4%

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

6. Final simplification 36.4%

   \[\leadsto y \cdot x \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024221 
(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))