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

Percentage Accurate: 100.0% → 100.0%

Time: 7.8s

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

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

   2. *-commutative N/A

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

   3. distribute-lft-out N/A

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

   4. *-commutative N/A

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

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

      \[\leadsto \mathsf{fma.f64}\left(\left(\frac{1}{2} + y\right), \color{blue}{x}, z\right) \]

   6. +-commutative N/A

      \[\leadsto \mathsf{fma.f64}\left(\left(y + \frac{1}{2}\right), x, z\right) \]

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

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{+.f64}\left(y, \left(\frac{1}{2}\right)\right), x, z\right) \]

   8. metadata-eval 100.0%

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{+.f64}\left(y, \frac{1}{2}\right), x, z\right) \]

4. Applied egg-rr 100.0%

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

Alternative 2: 98.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.5:\\ \;\;\;\;\mathsf{fma}\left(y, x, z\right)\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5, z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -0.5) (fma y x z) (if (<= y 1.75e-9) (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 <= 1.75e-9) {
		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 <= 1.75e-9)
		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, 1.75e-9], N[(x * 0.5 + z), $MachinePrecision], N[(y * x + z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.5 or 1.75e-9 < 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

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

      2. *-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. *-commutative N/A

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

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

         \[\leadsto \mathsf{fma.f64}\left(\left(\frac{1}{2} + y\right), \color{blue}{x}, z\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(y + \frac{1}{2}\right), x, z\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{+.f64}\left(y, \left(\frac{1}{2}\right)\right), x, z\right) \]

      8. metadata-eval 100.0%

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{+.f64}\left(y, \frac{1}{2}\right), x, z\right) \]

   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

      \[\leadsto \mathsf{fma.f64}\left(\color{blue}{y}, x, z\right) \]
   6. Step-by-step derivation

   7. Simplified 99.1%

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

   if -0.5 < y < 1.75e-9

   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

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 99.0%

         \[\leadsto \mathsf{fma.f64}\left(x, \color{blue}{\frac{1}{2}}, z\right) \]

   5. Simplified 99.0%

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

Alternative 3: 84.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{+37}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5, z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.1e+37) (* y x) (if (<= y 4.2e+51) (fma x 0.5 z) (* y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.1e+37) {
		tmp = y * x;
	} else if (y <= 4.2e+51) {
		tmp = fma(x, 0.5, z);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.1e+37)
		tmp = Float64(y * x);
	elseif (y <= 4.2e+51)
		tmp = fma(x, 0.5, z);
	else
		tmp = Float64(y * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.1e+37], N[(y * x), $MachinePrecision], If[LessEqual[y, 4.2e+51], N[(x * 0.5 + z), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.1000000000000001e37 or 4.2000000000000002e51 < 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

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

      2. accelerator-lowering-fma.f64 75.6%

         \[\leadsto \mathsf{fma.f64}\left(x, \color{blue}{y}, 0\right) \]

   5. Simplified 75.6%

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

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 75.6%

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{x}\right) \]

   7. Applied egg-rr 75.6%

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

   if -2.1000000000000001e37 < y < 4.2000000000000002e51

   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

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 95.4%

         \[\leadsto \mathsf{fma.f64}\left(x, \color{blue}{\frac{1}{2}}, z\right) \]

   5. Simplified 95.4%

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

Alternative 4: 59.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.4 \cdot 10^{+43}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+51}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -8.4e+43) (* y x) (if (<= y 4.5e+51) z (* y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -8.4e+43) {
		tmp = y * x;
	} else if (y <= 4.5e+51) {
		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) :: tmp
    if (y <= (-8.4d+43)) then
        tmp = y * x
    else if (y <= 4.5d+51) 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 tmp;
	if (y <= -8.4e+43) {
		tmp = y * x;
	} else if (y <= 4.5e+51) {
		tmp = z;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -8.4e+43:
		tmp = y * x
	elif y <= 4.5e+51:
		tmp = z
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -8.4e+43)
		tmp = Float64(y * x);
	elseif (y <= 4.5e+51)
		tmp = z;
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -8.4e+43)
		tmp = y * x;
	elseif (y <= 4.5e+51)
		tmp = z;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -8.4e+43], N[(y * x), $MachinePrecision], If[LessEqual[y, 4.5e+51], z, N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -8.40000000000000007e43 or 4.5e51 < 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

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

      2. accelerator-lowering-fma.f64 75.6%

         \[\leadsto \mathsf{fma.f64}\left(x, \color{blue}{y}, 0\right) \]

   5. Simplified 75.6%

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

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 75.6%

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{x}\right) \]

   7. Applied egg-rr 75.6%

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

   if -8.40000000000000007e43 < y < 4.5e51

   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 52.3%

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

Alternative 5: 40.5% 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 40.2%

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

Reproduce

herbie shell --seed 2024193 
(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))