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

Percentage Accurate: 100.0% → 100.0%

Time: 5.6s

Alternatives: 6

Speedup: 1.0×

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, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\frac{x}{2} + x \cdot y\right) + z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (+ (/ x 2.0) (* x y)) z))

C

double code(double x, double y, double z) {
	return ((x / 2.0) + (x * y)) + 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) + (x * y)) + z
end function

Java

public static double code(double x, double y, double z) {
	return ((x / 2.0) + (x * y)) + z;
}

Python

def code(x, y, z):
	return ((x / 2.0) + (x * y)) + z

Julia

function code(x, y, z)
	return Float64(Float64(Float64(x / 2.0) + Float64(x * y)) + z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((x / 2.0) + (x * y)) + z;
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(x / 2.0), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 2: 60.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{2} + x \cdot y\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+246}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;t\_0 \leq -4 \cdot 10^{+121}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;t\_0 \leq 10^{+132}:\\ \;\;\;\;z\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+247}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (/ x 2.0) (* x y))))
   (if (<= t_0 -2e+246)
     (* x y)
     (if (<= t_0 -4e+121)
       (* x 0.5)
       (if (<= t_0 1e+132) z (if (<= t_0 2e+247) (* x 0.5) (* x y)))))))

C

double code(double x, double y, double z) {
	double t_0 = (x / 2.0) + (x * y);
	double tmp;
	if (t_0 <= -2e+246) {
		tmp = x * y;
	} else if (t_0 <= -4e+121) {
		tmp = x * 0.5;
	} else if (t_0 <= 1e+132) {
		tmp = z;
	} else if (t_0 <= 2e+247) {
		tmp = x * 0.5;
	} else {
		tmp = x * y;
	}
	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 = (x / 2.0d0) + (x * y)
    if (t_0 <= (-2d+246)) then
        tmp = x * y
    else if (t_0 <= (-4d+121)) then
        tmp = x * 0.5d0
    else if (t_0 <= 1d+132) then
        tmp = z
    else if (t_0 <= 2d+247) then
        tmp = x * 0.5d0
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x / 2.0) + (x * y);
	double tmp;
	if (t_0 <= -2e+246) {
		tmp = x * y;
	} else if (t_0 <= -4e+121) {
		tmp = x * 0.5;
	} else if (t_0 <= 1e+132) {
		tmp = z;
	} else if (t_0 <= 2e+247) {
		tmp = x * 0.5;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x / 2.0) + (x * y)
	tmp = 0
	if t_0 <= -2e+246:
		tmp = x * y
	elif t_0 <= -4e+121:
		tmp = x * 0.5
	elif t_0 <= 1e+132:
		tmp = z
	elif t_0 <= 2e+247:
		tmp = x * 0.5
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x / 2.0) + Float64(x * y))
	tmp = 0.0
	if (t_0 <= -2e+246)
		tmp = Float64(x * y);
	elseif (t_0 <= -4e+121)
		tmp = Float64(x * 0.5);
	elseif (t_0 <= 1e+132)
		tmp = z;
	elseif (t_0 <= 2e+247)
		tmp = Float64(x * 0.5);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x / 2.0) + (x * y);
	tmp = 0.0;
	if (t_0 <= -2e+246)
		tmp = x * y;
	elseif (t_0 <= -4e+121)
		tmp = x * 0.5;
	elseif (t_0 <= 1e+132)
		tmp = z;
	elseif (t_0 <= 2e+247)
		tmp = x * 0.5;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x / 2.0), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+246], N[(x * y), $MachinePrecision], If[LessEqual[t$95$0, -4e+121], N[(x * 0.5), $MachinePrecision], If[LessEqual[t$95$0, 1e+132], z, If[LessEqual[t$95$0, 2e+247], N[(x * 0.5), $MachinePrecision], N[(x * y), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (/.f64 x #s(literal 2 binary64)) (*.f64 y x)) < -2.00000000000000014e246 or 1.9999999999999999e247 < (+.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. *-lowering-*.f64 88.6

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

   5. Simplified 88.6%

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

   if -2.00000000000000014e246 < (+.f64 (/.f64 x #s(literal 2 binary64)) (*.f64 y x)) < -4.00000000000000015e121 or 9.99999999999999991e131 < (+.f64 (/.f64 x #s(literal 2 binary64)) (*.f64 y x)) < 1.9999999999999999e247

   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 79.3

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

   5. Simplified 79.3%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 55.4

         \[\leadsto \color{blue}{x \cdot 0.5} \]

   8. Simplified 55.4%

      \[\leadsto \color{blue}{x \cdot 0.5} \]

   if -4.00000000000000015e121 < (+.f64 (/.f64 x #s(literal 2 binary64)) (*.f64 y x)) < 9.99999999999999991e131

   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 61.8%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{2} + x \cdot y \leq -2 \cdot 10^{+246}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;\frac{x}{2} + x \cdot y \leq -4 \cdot 10^{+121}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;\frac{x}{2} + x \cdot y \leq 10^{+132}:\\ \;\;\;\;z\\ \mathbf{elif}\;\frac{x}{2} + x \cdot y \leq 2 \cdot 10^{+247}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -52000000:\\ \;\;\;\;\mathsf{fma}\left(y, x, z\right)\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5, z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -52000000.0)
   (fma y x z)
   (if (<= y 1.3e-16) (fma x 0.5 z) (fma y x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -52000000.0) {
		tmp = fma(y, x, z);
	} else if (y <= 1.3e-16) {
		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 <= -52000000.0)
		tmp = fma(y, x, z);
	elseif (y <= 1.3e-16)
		tmp = fma(x, 0.5, z);
	else
		tmp = fma(y, x, z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -52000000.0], N[(y * x + z), $MachinePrecision], If[LessEqual[y, 1.3e-16], N[(x * 0.5 + z), $MachinePrecision], N[(y * x + z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.2e7 or 1.2999999999999999e-16 < 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. *-lowering-*.f64 99.4

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

   5. Simplified 99.4%

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

      1. *-commutative N/A

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

      2. accelerator-lowering-fma.f64 99.4

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

   7. Applied egg-rr 99.4%

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

   if -5.2e7 < y < 1.2999999999999999e-16

   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 100.0

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

   5. Simplified 100.0%

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

Alternative 4: 84.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -23000000000000:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+77}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5, z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -23000000000000.0) (* x y) (if (<= y 4e+77) (fma x 0.5 z) (* x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -23000000000000.0) {
		tmp = x * y;
	} else if (y <= 4e+77) {
		tmp = fma(x, 0.5, z);
	} else {
		tmp = x * y;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -23000000000000.0)
		tmp = Float64(x * y);
	elseif (y <= 4e+77)
		tmp = fma(x, 0.5, z);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -23000000000000.0], N[(x * y), $MachinePrecision], If[LessEqual[y, 4e+77], N[(x * 0.5 + z), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.3e13 or 3.99999999999999993e77 < 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. *-lowering-*.f64 73.7

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

   5. Simplified 73.7%

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

   if -2.3e13 < y < 3.99999999999999993e77

   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 95.5

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

   5. Simplified 95.5%

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

Alternative 5: 50.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-70}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{-93}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5e-70) z (if (<= z 1.35e-93) (* x 0.5) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5e-70) {
		tmp = z;
	} else if (z <= 1.35e-93) {
		tmp = x * 0.5;
	} else {
		tmp = z;
	}
	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 (z <= (-5d-70)) then
        tmp = z
    else if (z <= 1.35d-93) then
        tmp = x * 0.5d0
    else
        tmp = z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -5e-70) {
		tmp = z;
	} else if (z <= 1.35e-93) {
		tmp = x * 0.5;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -5e-70:
		tmp = z
	elif z <= 1.35e-93:
		tmp = x * 0.5
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5e-70)
		tmp = z;
	elseif (z <= 1.35e-93)
		tmp = Float64(x * 0.5);
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -5e-70)
		tmp = z;
	elseif (z <= 1.35e-93)
		tmp = x * 0.5;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5e-70], z, If[LessEqual[z, 1.35e-93], N[(x * 0.5), $MachinePrecision], z]]

Derivation

1. Split input into 2 regimes

2. if z < -4.9999999999999998e-70 or 1.3500000000000001e-93 < z

   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 58.3%

      \[\leadsto \color{blue}{z} \]

   if -4.9999999999999998e-70 < z < 1.3500000000000001e-93

   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 51.4

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

   5. Simplified 51.4%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 42.5

         \[\leadsto \color{blue}{x \cdot 0.5} \]

   8. Simplified 42.5%

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

Alternative 6: 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 41.1%

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

Reproduce

herbie shell --seed 2024198 
(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))