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

Percentage Accurate: 100.0% → 100.0%

Time: 13.9s

Alternatives: 5

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, 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. div-inv N/A

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

   2. *-commutative N/A

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

   3. distribute-lft-out N/A

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

   4. *-commutative N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. +-lowering-+.f64 N/A

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

   8. metadata-eval 100.0

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

4. Applied egg-rr 100.0%

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

Alternative 2: 61.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot x + \frac{x}{2}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+104}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;t\_0 \leq 10^{+95}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (* y x) (/ x 2.0))))
   (if (<= t_0 -1e+104) (* y x) (if (<= t_0 1e+95) z (* y x)))))

C

double code(double x, double y, double z) {
	double t_0 = (y * x) + (x / 2.0);
	double tmp;
	if (t_0 <= -1e+104) {
		tmp = y * x;
	} else if (t_0 <= 1e+95) {
		tmp = z;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = (y * x) + (x / 2.0d0)
    if (t_0 <= (-1d+104)) then
        tmp = y * x
    else if (t_0 <= 1d+95) then
        tmp = z
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (y * x) + (x / 2.0);
	double tmp;
	if (t_0 <= -1e+104) {
		tmp = y * x;
	} else if (t_0 <= 1e+95) {
		tmp = z;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y * x) + (x / 2.0)
	tmp = 0
	if t_0 <= -1e+104:
		tmp = y * x
	elif t_0 <= 1e+95:
		tmp = z
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y * x) + Float64(x / 2.0))
	tmp = 0.0
	if (t_0 <= -1e+104)
		tmp = Float64(y * x);
	elseif (t_0 <= 1e+95)
		tmp = z;
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y * x) + (x / 2.0);
	tmp = 0.0;
	if (t_0 <= -1e+104)
		tmp = y * x;
	elseif (t_0 <= 1e+95)
		tmp = z;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y * x), $MachinePrecision] + N[(x / 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+104], N[(y * x), $MachinePrecision], If[LessEqual[t$95$0, 1e+95], z, N[(y * x), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (/.f64 x #s(literal 2 binary64)) (*.f64 y x)) < -1e104 or 1.00000000000000002e95 < (+.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 y around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{x \cdot y + 0} \]

      2. accelerator-lowering-fma.f64 62.2

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, 0\right)} \]

   5. Simplified 62.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 62.2

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

   7. Applied egg-rr 62.2%

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

   if -1e104 < (+.f64 (/.f64 x #s(literal 2 binary64)) (*.f64 y x)) < 1.00000000000000002e95

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 62.9%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x + \frac{x}{2} \leq -1 \cdot 10^{+104}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \cdot x + \frac{x}{2} \leq 10^{+95}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -0.5], N[(y * x + z), $MachinePrecision], If[LessEqual[y, 0.5], N[(x * 0.5 + z), $MachinePrecision], N[(y * x + z), $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. Step-by-step derivation

      1. div-inv N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. +-lowering-+.f64 N/A

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

      8. metadata-eval 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{y}, x, z\right) \]
   6. Step-by-step derivation

   7. Simplified 98.2%

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

   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. accelerator-lowering-fma.f64 99.5

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

   5. Simplified 99.5%

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

Alternative 4: 85.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{+16}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 58000000:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5, z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.5e+16) (* y x) (if (<= y 58000000.0) (fma x 0.5 z) (* y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.5e+16) {
		tmp = y * x;
	} else if (y <= 58000000.0) {
		tmp = fma(x, 0.5, z);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.5e+16)
		tmp = Float64(y * x);
	elseif (y <= 58000000.0)
		tmp = fma(x, 0.5, z);
	else
		tmp = Float64(y * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.5e+16], N[(y * x), $MachinePrecision], If[LessEqual[y, 58000000.0], N[(x * 0.5 + z), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.5e16 or 5.8e7 < 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. +-rgt-identity N/A

         \[\leadsto \color{blue}{x \cdot y + 0} \]

      2. accelerator-lowering-fma.f64 76.4

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, 0\right)} \]

   5. Simplified 76.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 76.4

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

   7. Applied egg-rr 76.4%

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

   if -2.5e16 < y < 5.8e7

   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. accelerator-lowering-fma.f64 97.4

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

   5. Simplified 97.4%

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

Alternative 5: 39.5% accurate, 23.0× speedup

Math

\[\begin{array}{l} \\ z \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := z

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

5. Simplified 36.9%

   \[\leadsto \color{blue}{z} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024196 
(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))