Main:bigenough2 from A

Percentage Accurate: 100.0% → 100.0%

Time: 3.9s

Alternatives: 8

Speedup: 1.0×

• 20.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. +-commutative 100.0%

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

   2. fma-define 100.0%

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

   3. +-commutative 100.0%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 54.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{+15}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{+35}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+258}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.8e+15)
   x
   (if (<= x 2.15e+35) (* y z) (if (<= x 2.2e+258) x (* y x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.8e+15) {
		tmp = x;
	} else if (x <= 2.15e+35) {
		tmp = y * z;
	} else if (x <= 2.2e+258) {
		tmp = x;
	} 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 (x <= (-1.8d+15)) then
        tmp = x
    else if (x <= 2.15d+35) then
        tmp = y * z
    else if (x <= 2.2d+258) then
        tmp = x
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.8e+15) {
		tmp = x;
	} else if (x <= 2.15e+35) {
		tmp = y * z;
	} else if (x <= 2.2e+258) {
		tmp = x;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.8e+15:
		tmp = x
	elif x <= 2.15e+35:
		tmp = y * z
	elif x <= 2.2e+258:
		tmp = x
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.8e+15)
		tmp = x;
	elseif (x <= 2.15e+35)
		tmp = Float64(y * z);
	elseif (x <= 2.2e+258)
		tmp = x;
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.8e+15)
		tmp = x;
	elseif (x <= 2.15e+35)
		tmp = y * z;
	elseif (x <= 2.2e+258)
		tmp = x;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.8e+15], x, If[LessEqual[x, 2.15e+35], N[(y * z), $MachinePrecision], If[LessEqual[x, 2.2e+258], x, N[(y * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.8e15 or 2.1499999999999999e35 < x < 2.19999999999999983e258

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 58.0%

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

   if -1.8e15 < x < 2.1499999999999999e35

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 87.8%

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

   4. Taylor expanded in x around 0 69.9%

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

   if 2.19999999999999983e258 < x

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

      3. +-commutative 100.0%

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

   3. Simplified 100.0%

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

      1. fma-undefine 100.0%

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

      2. +-commutative 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-lft-in 81.8%

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

      5. associate-+r+ 81.8%

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

   6. Applied egg-rr 81.8%

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

   7. Taylor expanded in x around 0 55.2%

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

   8. Taylor expanded in z around 0 73.4%

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

      1. *-commutative 73.4%

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

   10. Simplified 73.4%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{+15}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{+35}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+258}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{-34} \lor \neg \left(x \leq 3.6 \cdot 10^{-19}\right):\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -7.5e-34) (not (<= x 3.6e-19))) (* x (+ y 1.0)) (* y z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -7.5e-34) || !(x <= 3.6e-19)) {
		tmp = x * (y + 1.0);
	} else {
		tmp = y * 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 ((x <= (-7.5d-34)) .or. (.not. (x <= 3.6d-19))) then
        tmp = x * (y + 1.0d0)
    else
        tmp = y * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -7.5e-34) || !(x <= 3.6e-19)) {
		tmp = x * (y + 1.0);
	} else {
		tmp = y * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -7.5e-34) or not (x <= 3.6e-19):
		tmp = x * (y + 1.0)
	else:
		tmp = y * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -7.5e-34) || !(x <= 3.6e-19))
		tmp = Float64(x * Float64(y + 1.0));
	else
		tmp = Float64(y * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -7.5e-34) || ~((x <= 3.6e-19)))
		tmp = x * (y + 1.0);
	else
		tmp = y * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -7.5e-34], N[Not[LessEqual[x, 3.6e-19]], $MachinePrecision]], N[(x * N[(y + 1.0), $MachinePrecision]), $MachinePrecision], N[(y * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.5000000000000004e-34 or 3.6000000000000001e-19 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 85.0%

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

      1. +-commutative 85.0%

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

   5. Simplified 85.0%

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

   if -7.5000000000000004e-34 < x < 3.6000000000000001e-19

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 92.0%

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

   4. Taylor expanded in x around 0 75.7%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{-34} \lor \neg \left(x \leq 3.6 \cdot 10^{-19}\right):\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -35000000000000 \lor \neg \left(x \leq 2.15 \cdot 10^{+35}\right):\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -35000000000000.0) (not (<= x 2.15e+35)))
   (* x (+ y 1.0))
   (* y (+ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -35000000000000.0) || !(x <= 2.15e+35)) {
		tmp = x * (y + 1.0);
	} else {
		tmp = y * (x + 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 ((x <= (-35000000000000.0d0)) .or. (.not. (x <= 2.15d+35))) then
        tmp = x * (y + 1.0d0)
    else
        tmp = y * (x + z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -35000000000000.0) || !(x <= 2.15e+35)) {
		tmp = x * (y + 1.0);
	} else {
		tmp = y * (x + z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -35000000000000.0) or not (x <= 2.15e+35):
		tmp = x * (y + 1.0)
	else:
		tmp = y * (x + z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -35000000000000.0) || !(x <= 2.15e+35))
		tmp = Float64(x * Float64(y + 1.0));
	else
		tmp = Float64(y * Float64(x + z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -35000000000000.0) || ~((x <= 2.15e+35)))
		tmp = x * (y + 1.0);
	else
		tmp = y * (x + z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -35000000000000.0], N[Not[LessEqual[x, 2.15e+35]], $MachinePrecision]], N[(x * N[(y + 1.0), $MachinePrecision]), $MachinePrecision], N[(y * N[(x + z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.5e13 or 2.1499999999999999e35 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 90.5%

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

      1. +-commutative 90.5%

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

   5. Simplified 90.5%

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

   if -3.5e13 < x < 2.1499999999999999e35

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

      3. +-commutative 100.0%

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

   3. Simplified 100.0%

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

      1. fma-undefine 100.0%

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

      2. +-commutative 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. associate-+r+ 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in y around inf 81.1%

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

      1. +-commutative 81.1%

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

   9. Simplified 81.1%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -35000000000000 \lor \neg \left(x \leq 2.15 \cdot 10^{+35}\right):\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.85 \cdot 10^{+16} \lor \neg \left(y \leq 1.2 \cdot 10^{-22}\right):\\ \;\;\;\;y \cdot \left(x + z\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2.85e+16) (not (<= y 1.2e-22))) (* y (+ x z)) (+ x (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.85e+16) || !(y <= 1.2e-22)) {
		tmp = y * (x + z);
	} else {
		tmp = x + (y * 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 ((y <= (-2.85d+16)) .or. (.not. (y <= 1.2d-22))) then
        tmp = y * (x + z)
    else
        tmp = x + (y * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.85e+16) || !(y <= 1.2e-22)) {
		tmp = y * (x + z);
	} else {
		tmp = x + (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -2.85e+16) or not (y <= 1.2e-22):
		tmp = y * (x + z)
	else:
		tmp = x + (y * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -2.85e+16) || !(y <= 1.2e-22))
		tmp = Float64(y * Float64(x + z));
	else
		tmp = Float64(x + Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -2.85e+16) || ~((y <= 1.2e-22)))
		tmp = y * (x + z);
	else
		tmp = x + (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -2.85e+16], N[Not[LessEqual[y, 1.2e-22]], $MachinePrecision]], N[(y * N[(x + z), $MachinePrecision]), $MachinePrecision], N[(x + N[(y * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.85e16 or 1.20000000000000001e-22 < y

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

      3. +-commutative 100.0%

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

   3. Simplified 100.0%

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

      1. fma-undefine 100.0%

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

      2. +-commutative 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-lft-in 98.3%

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

      5. associate-+r+ 98.3%

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

   6. Applied egg-rr 98.3%

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

   7. Taylor expanded in y around inf 99.9%

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

      1. +-commutative 99.9%

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

   9. Simplified 99.9%

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

   if -2.85e16 < y < 1.20000000000000001e-22

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 99.0%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.85 \cdot 10^{+16} \lor \neg \left(y \leq 1.2 \cdot 10^{-22}\right):\\ \;\;\;\;y \cdot \left(x + z\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1.22 \cdot 10^{-26}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 1.22e-26))) (* y x) x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.22e-26)) {
		tmp = y * x;
	} else {
		tmp = 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 <= (-1.0d0)) .or. (.not. (y <= 1.22d-26))) then
        tmp = y * x
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.22e-26)) {
		tmp = y * x;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.0) or not (y <= 1.22e-26):
		tmp = y * x
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 1.22e-26))
		tmp = Float64(y * x);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -1.0) || ~((y <= 1.22e-26)))
		tmp = y * x;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 1.22e-26]], $MachinePrecision]], N[(y * x), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1.22e-26 < y

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

      3. +-commutative 100.0%

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

   3. Simplified 100.0%

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

      1. fma-undefine 100.0%

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

      2. +-commutative 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-lft-in 98.4%

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

      5. associate-+r+ 98.4%

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

   6. Applied egg-rr 98.4%

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

   7. Taylor expanded in x around 0 97.8%

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

   8. Taylor expanded in z around 0 47.0%

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

      1. *-commutative 47.0%

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

   10. Simplified 47.0%

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

   if -1 < y < 1.22e-26

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 65.0%

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

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1.22 \cdot 10^{-26}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 8: 36.4% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around 0 34.8%

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

4. Final simplification 34.8%

   \[\leadsto x \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024067 
(FPCore (x y z)
  :name "Main:bigenough2 from A"
  :precision binary64
  (+ x (* y (+ z x))))