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

Percentage Accurate: 100.0% → 100.0%

Time: 4.8s

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

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(y * x + N[(0.5 * x + 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. clear-num N/A

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

   9. associate-/r/ N/A

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

   10. lower-fma.f64 N/A

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

   11. metadata-eval 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 85.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+48}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x, z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y - -0.5\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.4e+48) (* y x) (if (<= y 1e+15) (fma 0.5 x z) (* (- y -0.5) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.4e+48) {
		tmp = y * x;
	} else if (y <= 1e+15) {
		tmp = fma(0.5, x, z);
	} else {
		tmp = (y - -0.5) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.4e+48)
		tmp = Float64(y * x);
	elseif (y <= 1e+15)
		tmp = fma(0.5, x, z);
	else
		tmp = Float64(Float64(y - -0.5) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.4e+48], N[(y * x), $MachinePrecision], If[LessEqual[y, 1e+15], N[(0.5 * x + z), $MachinePrecision], N[(N[(y - -0.5), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.40000000000000006e48

   1. Initial program 99.9%

      \[\left(\frac{x}{2} + y \cdot x\right) + z \]
   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. lower-fma.f64 20.0

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

   5. Applied rewrites 20.0%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 79.6

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

   8. Applied rewrites 79.6%

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

   if -1.40000000000000006e48 < y < 1e15

   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. lower-fma.f64 95.9

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

   5. Applied rewrites 95.9%

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

   if 1e15 < y

   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. lower-fma.f64 35.0

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

   5. Applied rewrites 35.0%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

      5. sub-neg N/A

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

      6. lower--.f64 69.6

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

   8. Applied rewrites 69.6%

      \[\leadsto \color{blue}{\left(y - -0.5\right) \cdot x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 85.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+48} \lor \neg \left(y \leq 10^{+15}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.4e+48) (not (<= y 1e+15))) (* y x) (fma 0.5 x z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.4e+48) || !(y <= 1e+15)) {
		tmp = y * x;
	} else {
		tmp = fma(0.5, x, z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.4e+48) || !(y <= 1e+15))
		tmp = Float64(y * x);
	else
		tmp = fma(0.5, x, z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.4e+48], N[Not[LessEqual[y, 1e+15]], $MachinePrecision]], N[(y * x), $MachinePrecision], N[(0.5 * x + z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.40000000000000006e48 or 1e15 < y

   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. lower-fma.f64 28.5

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

   5. Applied rewrites 28.5%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 74.0

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

   8. Applied rewrites 74.0%

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

   if -1.40000000000000006e48 < y < 1e15

   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. lower-fma.f64 95.9

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

   5. Applied rewrites 95.9%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+48} \lor \neg \left(y \leq 10^{+15}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x, z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 59.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -380 \lor \neg \left(y \leq 1.1 \cdot 10^{-7}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -380.0) (not (<= y 1.1e-7))) (* y x) (* 0.5 x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -380.0) || !(y <= 1.1e-7)) {
		tmp = y * x;
	} else {
		tmp = 0.5 * 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 <= (-380.0d0)) .or. (.not. (y <= 1.1d-7))) then
        tmp = y * x
    else
        tmp = 0.5d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -380.0) || !(y <= 1.1e-7)) {
		tmp = y * x;
	} else {
		tmp = 0.5 * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -380.0) or not (y <= 1.1e-7):
		tmp = y * x
	else:
		tmp = 0.5 * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -380.0) || !(y <= 1.1e-7))
		tmp = Float64(y * x);
	else
		tmp = Float64(0.5 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -380.0) || ~((y <= 1.1e-7)))
		tmp = y * x;
	else
		tmp = 0.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -380.0], N[Not[LessEqual[y, 1.1e-7]], $MachinePrecision]], N[(y * x), $MachinePrecision], N[(0.5 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -380 or 1.1000000000000001e-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 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. lower-fma.f64 33.0

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

   5. Applied rewrites 33.0%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 68.4

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

   8. Applied rewrites 68.4%

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

   if -380 < y < 1.1000000000000001e-7

   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. lower-fma.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 44.1%

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

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -380 \lor \neg \left(y \leq 1.1 \cdot 10^{-7}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 100.0% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. lift-+.f64 N/A

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

   3. 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. lower-+.f64 N/A

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

   11. metadata-eval 100.0

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

4. Applied rewrites 100.0%

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

Alternative 6: 26.5% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(0.5 * 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 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. lower-fma.f64 66.1

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

5. Applied rewrites 66.1%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 23.6%

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

Reproduce

herbie shell --seed 2024324 
(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))