Main:bigenough2 from A

Percentage Accurate: 100.0% → 100.0%

Time: 2.8s

Alternatives: 6

Speedup: 1.0×

• 20.9% 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-def 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. Final simplification 100.0%

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

Alternative 2: 75.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{-176} \lor \neg \left(x \leq 10^{-15}\right):\\ \;\;\;\;x + y \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -6e-176) (not (<= x 1e-15))) (+ x (* y x)) (* y z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -6e-176) || !(x <= 1e-15)) {
		tmp = x + (y * x);
	} 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 <= (-6d-176)) .or. (.not. (x <= 1d-15))) then
        tmp = x + (y * x)
    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 <= -6e-176) || !(x <= 1e-15)) {
		tmp = x + (y * x);
	} else {
		tmp = y * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -6e-176) or not (x <= 1e-15):
		tmp = x + (y * x)
	else:
		tmp = y * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -6e-176) || !(x <= 1e-15))
		tmp = Float64(x + Float64(y * x));
	else
		tmp = Float64(y * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -6e-176) || ~((x <= 1e-15)))
		tmp = x + (y * x);
	else
		tmp = y * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -6e-176], N[Not[LessEqual[x, 1e-15]], $MachinePrecision]], N[(x + N[(y * x), $MachinePrecision]), $MachinePrecision], N[(y * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6e-176 or 1.0000000000000001e-15 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in z around 0 85.2%

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

      1. *-commutative 85.2%

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

   4. Simplified 85.2%

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

   if -6e-176 < x < 1.0000000000000001e-15

   1. Initial program 100.0%

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

   2. Taylor expanded in z around inf 94.4%

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

   3. Taylor expanded in x around 0 73.2%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{-176} \lor \neg \left(x \leq 10^{-15}\right):\\ \;\;\;\;x + y \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]

Alternative 3: 86.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{-41} \lor \neg \left(x \leq 4.4 \cdot 10^{-8}\right):\\ \;\;\;\;x + y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.2e-41) (not (<= x 4.4e-8))) (+ x (* y x)) (+ x (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.2e-41) || !(x <= 4.4e-8)) {
		tmp = x + (y * x);
	} 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 ((x <= (-1.2d-41)) .or. (.not. (x <= 4.4d-8))) then
        tmp = x + (y * x)
    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 ((x <= -1.2e-41) || !(x <= 4.4e-8)) {
		tmp = x + (y * x);
	} else {
		tmp = x + (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.2e-41) or not (x <= 4.4e-8):
		tmp = x + (y * x)
	else:
		tmp = x + (y * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.2e-41) || !(x <= 4.4e-8))
		tmp = Float64(x + Float64(y * x));
	else
		tmp = Float64(x + Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.2e-41) || ~((x <= 4.4e-8)))
		tmp = x + (y * x);
	else
		tmp = x + (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.2e-41], N[Not[LessEqual[x, 4.4e-8]], $MachinePrecision]], N[(x + N[(y * x), $MachinePrecision]), $MachinePrecision], N[(x + N[(y * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.20000000000000011e-41 or 4.3999999999999997e-8 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in z around 0 91.8%

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

      1. *-commutative 91.8%

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

   4. Simplified 91.8%

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

   if -1.20000000000000011e-41 < x < 4.3999999999999997e-8

   1. Initial program 100.0%

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

   2. Taylor expanded in z around inf 92.1%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{-41} \lor \neg \left(x \leq 4.4 \cdot 10^{-8}\right):\\ \;\;\;\;x + y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot z\\ \end{array} \]

Alternative 4: 61.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-46}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq 0.00052:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -5e-46) (* y z) (if (<= y 0.00052) x (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -5e-46) {
		tmp = y * z;
	} else if (y <= 0.00052) {
		tmp = x;
	} 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 (y <= (-5d-46)) then
        tmp = y * z
    else if (y <= 0.00052d0) then
        tmp = x
    else
        tmp = y * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -5e-46) {
		tmp = y * z;
	} else if (y <= 0.00052) {
		tmp = x;
	} else {
		tmp = y * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -5e-46:
		tmp = y * z
	elif y <= 0.00052:
		tmp = x
	else:
		tmp = y * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -5e-46)
		tmp = Float64(y * z);
	elseif (y <= 0.00052)
		tmp = x;
	else
		tmp = Float64(y * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -5e-46)
		tmp = y * z;
	elseif (y <= 0.00052)
		tmp = x;
	else
		tmp = y * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -5e-46], N[(y * z), $MachinePrecision], If[LessEqual[y, 0.00052], x, N[(y * z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.99999999999999992e-46 or 5.19999999999999954e-4 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in z around inf 55.2%

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

   3. Taylor expanded in x around 0 52.0%

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

   if -4.99999999999999992e-46 < y < 5.19999999999999954e-4

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 80.0%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-46}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq 0.00052:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]

Alternative 5: 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. Final simplification 100.0%

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

Alternative 6: 37.1% 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. Taylor expanded in y around 0 44.2%

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

3. Final simplification 44.2%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023293 
(FPCore (x y z)
  :name "Main:bigenough2 from A"
  :precision binary64
  (+ x (* y (+ z x))))