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

Percentage Accurate: 100.0% → 100.0%

Time: 4.0s

Alternatives: 8

Speedup: 0.1×

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, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\left(\frac{x}{2} + y \cdot x\right) + z \]
2. Step-by-step derivation

   1. associate-+l+ 100.0%

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

   2. *-commutative 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in x around 0 100.0%

   \[\leadsto \color{blue}{z + \left(0.5 + y\right) \cdot x} \]
5. Step-by-step derivation

   1. +-commutative 100.0%

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

   2. *-commutative 100.0%

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

   3. fma-udef 100.0%

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

   4. +-commutative 100.0%

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

6. Simplified 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 60.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+161}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq -1 \cdot 10^{+147}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq -2.35 \cdot 10^{+24}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-298}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{-138}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{+22}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4.8e+161)
   (* x y)
   (if (<= y -1e+147)
     z
     (if (<= y -2.35e+24)
       (* x y)
       (if (<= y 7.5e-298)
         z
         (if (<= y 1.12e-138) (* x 0.5) (if (<= y 8.5e+22) z (* x y))))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4.8e+161) {
		tmp = x * y;
	} else if (y <= -1e+147) {
		tmp = z;
	} else if (y <= -2.35e+24) {
		tmp = x * y;
	} else if (y <= 7.5e-298) {
		tmp = z;
	} else if (y <= 1.12e-138) {
		tmp = x * 0.5;
	} else if (y <= 8.5e+22) {
		tmp = z;
	} 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) :: tmp
    if (y <= (-4.8d+161)) then
        tmp = x * y
    else if (y <= (-1d+147)) then
        tmp = z
    else if (y <= (-2.35d+24)) then
        tmp = x * y
    else if (y <= 7.5d-298) then
        tmp = z
    else if (y <= 1.12d-138) then
        tmp = x * 0.5d0
    else if (y <= 8.5d+22) then
        tmp = z
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -4.8e+161) {
		tmp = x * y;
	} else if (y <= -1e+147) {
		tmp = z;
	} else if (y <= -2.35e+24) {
		tmp = x * y;
	} else if (y <= 7.5e-298) {
		tmp = z;
	} else if (y <= 1.12e-138) {
		tmp = x * 0.5;
	} else if (y <= 8.5e+22) {
		tmp = z;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -4.8e+161:
		tmp = x * y
	elif y <= -1e+147:
		tmp = z
	elif y <= -2.35e+24:
		tmp = x * y
	elif y <= 7.5e-298:
		tmp = z
	elif y <= 1.12e-138:
		tmp = x * 0.5
	elif y <= 8.5e+22:
		tmp = z
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -4.8e+161)
		tmp = Float64(x * y);
	elseif (y <= -1e+147)
		tmp = z;
	elseif (y <= -2.35e+24)
		tmp = Float64(x * y);
	elseif (y <= 7.5e-298)
		tmp = z;
	elseif (y <= 1.12e-138)
		tmp = Float64(x * 0.5);
	elseif (y <= 8.5e+22)
		tmp = z;
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -4.8e+161)
		tmp = x * y;
	elseif (y <= -1e+147)
		tmp = z;
	elseif (y <= -2.35e+24)
		tmp = x * y;
	elseif (y <= 7.5e-298)
		tmp = z;
	elseif (y <= 1.12e-138)
		tmp = x * 0.5;
	elseif (y <= 8.5e+22)
		tmp = z;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -4.8e+161], N[(x * y), $MachinePrecision], If[LessEqual[y, -1e+147], z, If[LessEqual[y, -2.35e+24], N[(x * y), $MachinePrecision], If[LessEqual[y, 7.5e-298], z, If[LessEqual[y, 1.12e-138], N[(x * 0.5), $MachinePrecision], If[LessEqual[y, 8.5e+22], z, N[(x * y), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.7999999999999998e161 or -9.9999999999999998e146 < y < -2.35e24 or 8.49999999999999979e22 < y

   1. Initial program 100.0%

      \[\left(\frac{x}{2} + y \cdot x\right) + z \]
   2. Step-by-step derivation

      1. associate-+l+ 100.0%

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

      2. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{z + \left(0.5 + y\right) \cdot x} \]
   5. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. fma-udef 100.0%

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

      4. +-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in y around inf 73.8%

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

   if -4.7999999999999998e161 < y < -9.9999999999999998e146 or -2.35e24 < y < 7.49999999999999987e-298 or 1.1199999999999999e-138 < y < 8.49999999999999979e22

   1. Initial program 100.0%

      \[\left(\frac{x}{2} + y \cdot x\right) + z \]
   2. Step-by-step derivation

      1. associate-+l+ 100.0%

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

      2. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{z + \left(0.5 + y\right) \cdot x} \]
   5. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. fma-udef 100.0%

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

      4. +-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around 0 64.6%

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

   if 7.49999999999999987e-298 < y < 1.1199999999999999e-138

   1. Initial program 100.0%

      \[\left(\frac{x}{2} + y \cdot x\right) + z \]
   2. Step-by-step derivation

      1. associate-+l+ 100.0%

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

      2. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 60.1%

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

   5. Taylor expanded in y around 0 60.1%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+161}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq -1 \cdot 10^{+147}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq -2.35 \cdot 10^{+24}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-298}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{-138}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{+22}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 3: 71.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9.2 \cdot 10^{+57}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq -2.15 \cdot 10^{+26} \lor \neg \left(z \leq -2.35 \cdot 10^{-94}\right) \land z \leq 5.8 \cdot 10^{+103}:\\ \;\;\;\;x \cdot \left(y + 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -9.2e+57)
   z
   (if (or (<= z -2.15e+26) (and (not (<= z -2.35e-94)) (<= z 5.8e+103)))
     (* x (+ y 0.5))
     z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -9.2e+57) {
		tmp = z;
	} else if ((z <= -2.15e+26) || (!(z <= -2.35e-94) && (z <= 5.8e+103))) {
		tmp = x * (y + 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 <= (-9.2d+57)) then
        tmp = z
    else if ((z <= (-2.15d+26)) .or. (.not. (z <= (-2.35d-94))) .and. (z <= 5.8d+103)) then
        tmp = x * (y + 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 <= -9.2e+57) {
		tmp = z;
	} else if ((z <= -2.15e+26) || (!(z <= -2.35e-94) && (z <= 5.8e+103))) {
		tmp = x * (y + 0.5);
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -9.2e+57:
		tmp = z
	elif (z <= -2.15e+26) or (not (z <= -2.35e-94) and (z <= 5.8e+103)):
		tmp = x * (y + 0.5)
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -9.2e+57)
		tmp = z;
	elseif ((z <= -2.15e+26) || (!(z <= -2.35e-94) && (z <= 5.8e+103)))
		tmp = Float64(x * Float64(y + 0.5));
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -9.2e+57)
		tmp = z;
	elseif ((z <= -2.15e+26) || (~((z <= -2.35e-94)) && (z <= 5.8e+103)))
		tmp = x * (y + 0.5);
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -9.2e+57], z, If[Or[LessEqual[z, -2.15e+26], And[N[Not[LessEqual[z, -2.35e-94]], $MachinePrecision], LessEqual[z, 5.8e+103]]], N[(x * N[(y + 0.5), $MachinePrecision]), $MachinePrecision], z]]

Derivation

1. Split input into 2 regimes

2. if z < -9.1999999999999995e57 or -2.1499999999999999e26 < z < -2.35000000000000002e-94 or 5.7999999999999997e103 < z

   1. Initial program 100.0%

      \[\left(\frac{x}{2} + y \cdot x\right) + z \]
   2. Step-by-step derivation

      1. associate-+l+ 100.0%

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

      2. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{z + \left(0.5 + y\right) \cdot x} \]
   5. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. fma-udef 100.0%

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

      4. +-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around 0 76.8%

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

   if -9.1999999999999995e57 < z < -2.1499999999999999e26 or -2.35000000000000002e-94 < z < 5.7999999999999997e103

   1. Initial program 100.0%

      \[\left(\frac{x}{2} + y \cdot x\right) + z \]
   2. Step-by-step derivation

      1. associate-+l+ 100.0%

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

      2. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 82.2%

      \[\leadsto \color{blue}{\left(0.5 + y\right) \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.2 \cdot 10^{+57}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq -2.15 \cdot 10^{+26} \lor \neg \left(z \leq -2.35 \cdot 10^{-94}\right) \land z \leq 5.8 \cdot 10^{+103}:\\ \;\;\;\;x \cdot \left(y + 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 4: 85.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{-95} \lor \neg \left(z \leq 9.2 \cdot 10^{-79}\right):\\ \;\;\;\;z + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -3e-95) (not (<= z 9.2e-79))) (+ z (* x y)) (* x (+ y 0.5))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -3e-95) || !(z <= 9.2e-79)) {
		tmp = z + (x * y);
	} else {
		tmp = x * (y + 0.5);
	}
	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 <= (-3d-95)) .or. (.not. (z <= 9.2d-79))) then
        tmp = z + (x * y)
    else
        tmp = x * (y + 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -3e-95) || !(z <= 9.2e-79)) {
		tmp = z + (x * y);
	} else {
		tmp = x * (y + 0.5);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -3e-95) or not (z <= 9.2e-79):
		tmp = z + (x * y)
	else:
		tmp = x * (y + 0.5)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -3e-95) || !(z <= 9.2e-79))
		tmp = Float64(z + Float64(x * y));
	else
		tmp = Float64(x * Float64(y + 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -3e-95) || ~((z <= 9.2e-79)))
		tmp = z + (x * y);
	else
		tmp = x * (y + 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -3e-95], N[Not[LessEqual[z, 9.2e-79]], $MachinePrecision]], N[(z + N[(x * y), $MachinePrecision]), $MachinePrecision], N[(x * N[(y + 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3e-95 or 9.20000000000000047e-79 < z

   1. Initial program 100.0%

      \[\left(\frac{x}{2} + y \cdot x\right) + z \]
   2. Step-by-step derivation

      1. associate-+l+ 100.0%

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

      2. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{z + \left(0.5 + y\right) \cdot x} \]
   5. Step-by-step derivation

      1. flip-+ 81.4%

         \[\leadsto z + \color{blue}{\frac{0.5 \cdot 0.5 - y \cdot y}{0.5 - y}} \cdot x \]

      2. associate-*l/ 77.9%

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

      3. metadata-eval 77.9%

         \[\leadsto z + \frac{\left(\color{blue}{0.25} - y \cdot y\right) \cdot x}{0.5 - y} \]

   6. Applied egg-rr 77.9%

      \[\leadsto z + \color{blue}{\frac{\left(0.25 - y \cdot y\right) \cdot x}{0.5 - y}} \]

   7. Taylor expanded in y around inf 88.4%

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

   if -3e-95 < z < 9.20000000000000047e-79

   1. Initial program 100.0%

      \[\left(\frac{x}{2} + y \cdot x\right) + z \]
   2. Step-by-step derivation

      1. associate-+l+ 100.0%

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

      2. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 91.2%

      \[\leadsto \color{blue}{\left(0.5 + y\right) \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{-95} \lor \neg \left(z \leq 9.2 \cdot 10^{-79}\right):\\ \;\;\;\;z + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + 0.5\right)\\ \end{array} \]

Alternative 5: 98.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{+23} \lor \neg \left(y \leq 0.5\right):\\ \;\;\;\;z + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z + \frac{x}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -4.5e+23) (not (<= y 0.5))) (+ z (* x y)) (+ z (/ x 2.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -4.5e+23) || !(y <= 0.5)) {
		tmp = z + (x * y);
	} else {
		tmp = z + (x / 2.0);
	}
	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 <= (-4.5d+23)) .or. (.not. (y <= 0.5d0))) then
        tmp = z + (x * y)
    else
        tmp = z + (x / 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -4.5e+23) || !(y <= 0.5)) {
		tmp = z + (x * y);
	} else {
		tmp = z + (x / 2.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -4.5e+23) or not (y <= 0.5):
		tmp = z + (x * y)
	else:
		tmp = z + (x / 2.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -4.5e+23) || !(y <= 0.5))
		tmp = Float64(z + Float64(x * y));
	else
		tmp = Float64(z + Float64(x / 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -4.5e+23) || ~((y <= 0.5)))
		tmp = z + (x * y);
	else
		tmp = z + (x / 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -4.5e+23], N[Not[LessEqual[y, 0.5]], $MachinePrecision]], N[(z + N[(x * y), $MachinePrecision]), $MachinePrecision], N[(z + N[(x / 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.49999999999999979e23 or 0.5 < y

   1. Initial program 100.0%

      \[\left(\frac{x}{2} + y \cdot x\right) + z \]
   2. Step-by-step derivation

      1. associate-+l+ 100.0%

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

      2. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{z + \left(0.5 + y\right) \cdot x} \]
   5. Step-by-step derivation

      1. flip-+ 59.2%

         \[\leadsto z + \color{blue}{\frac{0.5 \cdot 0.5 - y \cdot y}{0.5 - y}} \cdot x \]

      2. associate-*l/ 53.3%

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

      3. metadata-eval 53.3%

         \[\leadsto z + \frac{\left(\color{blue}{0.25} - y \cdot y\right) \cdot x}{0.5 - y} \]

   6. Applied egg-rr 53.3%

      \[\leadsto z + \color{blue}{\frac{\left(0.25 - y \cdot y\right) \cdot x}{0.5 - y}} \]

   7. Taylor expanded in y around inf 99.5%

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

   if -4.49999999999999979e23 < y < 0.5

   1. Initial program 100.0%

      \[\left(\frac{x}{2} + y \cdot x\right) + z \]
   2. Step-by-step derivation

      1. associate-+l+ 100.0%

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

      2. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 98.9%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{+23} \lor \neg \left(y \leq 0.5\right):\\ \;\;\;\;z + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z + \frac{x}{2}\\ \end{array} \]

Alternative 6: 52.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{-104}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 5.4 \cdot 10^{-80}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5.2e-104) z (if (<= z 5.4e-80) (* x 0.5) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.2e-104) {
		tmp = z;
	} else if (z <= 5.4e-80) {
		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 <= (-5.2d-104)) then
        tmp = z
    else if (z <= 5.4d-80) 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 <= -5.2e-104) {
		tmp = z;
	} else if (z <= 5.4e-80) {
		tmp = x * 0.5;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -5.2e-104:
		tmp = z
	elif z <= 5.4e-80:
		tmp = x * 0.5
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5.2e-104)
		tmp = z;
	elseif (z <= 5.4e-80)
		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 <= -5.2e-104)
		tmp = z;
	elseif (z <= 5.4e-80)
		tmp = x * 0.5;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5.2e-104], z, If[LessEqual[z, 5.4e-80], N[(x * 0.5), $MachinePrecision], z]]

Derivation

1. Split input into 2 regimes

2. if z < -5.20000000000000005e-104 or 5.4000000000000004e-80 < z

   1. Initial program 100.0%

      \[\left(\frac{x}{2} + y \cdot x\right) + z \]
   2. Step-by-step derivation

      1. associate-+l+ 100.0%

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

      2. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{z + \left(0.5 + y\right) \cdot x} \]
   5. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. fma-udef 100.0%

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

      4. +-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around 0 64.6%

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

   if -5.20000000000000005e-104 < z < 5.4000000000000004e-80

   1. Initial program 100.0%

      \[\left(\frac{x}{2} + y \cdot x\right) + z \]
   2. Step-by-step derivation

      1. associate-+l+ 100.0%

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

      2. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 91.0%

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

   5. Taylor expanded in y around 0 41.1%

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

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{-104}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 5.4 \cdot 10^{-80}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 7: 100.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

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 + (x * (y + 0.5d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\left(\frac{x}{2} + y \cdot x\right) + z \]
2. Step-by-step derivation

   1. associate-+l+ 100.0%

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

   2. *-commutative 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in x around 0 100.0%

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

5. Final simplification 100.0%

   \[\leadsto z + x \cdot \left(y + 0.5\right) \]

Alternative 8: 40.9% accurate, 9.0× 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. Step-by-step derivation

   1. associate-+l+ 100.0%

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

   2. *-commutative 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in x around 0 100.0%

   \[\leadsto \color{blue}{z + \left(0.5 + y\right) \cdot x} \]
5. Step-by-step derivation

   1. +-commutative 100.0%

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

   2. *-commutative 100.0%

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

   3. fma-udef 100.0%

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

   4. +-commutative 100.0%

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

6. Simplified 100.0%

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

7. Taylor expanded in x around 0 44.8%

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

8. Final simplification 44.8%

   \[\leadsto z \]

Reproduce

herbie shell --seed 2023196 
(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))