Main:bigenough2 from A

Percentage Accurate: 100.0% → 100.0%

Time: 5.4s

Alternatives: 9

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-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: 62.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+210}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq -4.1 \cdot 10^{+29}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq -3.1 \cdot 10^{-9}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq 4 \cdot 10^{-9}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 270000000000:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.3e+210)
   (* y z)
   (if (<= y -4.1e+29)
     (* y x)
     (if (<= y -3.1e-9)
       (* y z)
       (if (<= y 4e-9) x (if (<= y 270000000000.0) (* y z) (* y x)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.3e+210) {
		tmp = y * z;
	} else if (y <= -4.1e+29) {
		tmp = y * x;
	} else if (y <= -3.1e-9) {
		tmp = y * z;
	} else if (y <= 4e-9) {
		tmp = x;
	} else if (y <= 270000000000.0) {
		tmp = y * 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 <= (-2.3d+210)) then
        tmp = y * z
    else if (y <= (-4.1d+29)) then
        tmp = y * x
    else if (y <= (-3.1d-9)) then
        tmp = y * z
    else if (y <= 4d-9) then
        tmp = x
    else if (y <= 270000000000.0d0) then
        tmp = y * 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 <= -2.3e+210) {
		tmp = y * z;
	} else if (y <= -4.1e+29) {
		tmp = y * x;
	} else if (y <= -3.1e-9) {
		tmp = y * z;
	} else if (y <= 4e-9) {
		tmp = x;
	} else if (y <= 270000000000.0) {
		tmp = y * z;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2.3e+210:
		tmp = y * z
	elif y <= -4.1e+29:
		tmp = y * x
	elif y <= -3.1e-9:
		tmp = y * z
	elif y <= 4e-9:
		tmp = x
	elif y <= 270000000000.0:
		tmp = y * z
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.3e+210)
		tmp = Float64(y * z);
	elseif (y <= -4.1e+29)
		tmp = Float64(y * x);
	elseif (y <= -3.1e-9)
		tmp = Float64(y * z);
	elseif (y <= 4e-9)
		tmp = x;
	elseif (y <= 270000000000.0)
		tmp = Float64(y * z);
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2.3e+210)
		tmp = y * z;
	elseif (y <= -4.1e+29)
		tmp = y * x;
	elseif (y <= -3.1e-9)
		tmp = y * z;
	elseif (y <= 4e-9)
		tmp = x;
	elseif (y <= 270000000000.0)
		tmp = y * z;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.3e+210], N[(y * z), $MachinePrecision], If[LessEqual[y, -4.1e+29], N[(y * x), $MachinePrecision], If[LessEqual[y, -3.1e-9], N[(y * z), $MachinePrecision], If[LessEqual[y, 4e-9], x, If[LessEqual[y, 270000000000.0], N[(y * z), $MachinePrecision], N[(y * x), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.2999999999999999e210 or -4.1000000000000003e29 < y < -3.10000000000000005e-9 or 4.00000000000000025e-9 < y < 2.7e11

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 96.9%

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

      1. fma-define 99.9%

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

      2. +-commutative 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in x around 0 80.0%

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

   if -2.2999999999999999e210 < y < -4.1000000000000003e29 or 2.7e11 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 63.1%

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

      1. *-commutative 63.1%

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

   5. Simplified 63.1%

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

   6. Taylor expanded in y around inf 62.7%

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

      1. *-commutative 62.7%

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

   8. Simplified 62.7%

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

   if -3.10000000000000005e-9 < y < 4.00000000000000025e-9

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 75.2%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+210}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq -4.1 \cdot 10^{+29}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq -3.1 \cdot 10^{-9}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq 4 \cdot 10^{-9}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 270000000000:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{-107} \lor \neg \left(x \leq 1.25 \cdot 10^{-70}\right):\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.3e-107) (not (<= x 1.25e-70))) (* x (+ y 1.0)) (* y z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.3e-107) || !(x <= 1.25e-70)) {
		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 <= (-1.3d-107)) .or. (.not. (x <= 1.25d-70))) 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 <= -1.3e-107) || !(x <= 1.25e-70)) {
		tmp = x * (y + 1.0);
	} else {
		tmp = y * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.3e-107) or not (x <= 1.25e-70):
		tmp = x * (y + 1.0)
	else:
		tmp = y * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.3e-107) || !(x <= 1.25e-70))
		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 <= -1.3e-107) || ~((x <= 1.25e-70)))
		tmp = x * (y + 1.0);
	else
		tmp = y * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.3e-107], N[Not[LessEqual[x, 1.25e-70]], $MachinePrecision]], N[(x * N[(y + 1.0), $MachinePrecision]), $MachinePrecision], N[(y * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3e-107 or 1.25e-70 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 82.0%

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

      1. +-commutative 82.0%

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

   5. Simplified 82.0%

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

   if -1.3e-107 < x < 1.25e-70

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

      1. fma-define 100.0%

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

      2. +-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 75.9%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{-107} \lor \neg \left(x \leq 1.25 \cdot 10^{-70}\right):\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{-6} \lor \neg \left(y \leq 0.0021\right):\\ \;\;\;\;y \cdot \left(x + z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -7e-6) (not (<= y 0.0021))) (* y (+ x z)) (* x (+ y 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -7e-6) || !(y <= 0.0021)) {
		tmp = y * (x + z);
	} else {
		tmp = x * (y + 1.0);
	}
	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 <= (-7d-6)) .or. (.not. (y <= 0.0021d0))) then
        tmp = y * (x + z)
    else
        tmp = x * (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -7e-6) || !(y <= 0.0021)) {
		tmp = y * (x + z);
	} else {
		tmp = x * (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -7e-6) or not (y <= 0.0021):
		tmp = y * (x + z)
	else:
		tmp = x * (y + 1.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -7e-6) || !(y <= 0.0021))
		tmp = Float64(y * Float64(x + z));
	else
		tmp = Float64(x * Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -7e-6) || ~((y <= 0.0021)))
		tmp = y * (x + z);
	else
		tmp = x * (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -7e-6], N[Not[LessEqual[y, 0.0021]], $MachinePrecision]], N[(y * N[(x + z), $MachinePrecision]), $MachinePrecision], N[(x * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.99999999999999989e-6 or 0.00209999999999999987 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 96.7%

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

      1. fma-define 99.2%

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

      2. +-commutative 99.2%

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

   5. Simplified 99.2%

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

   6. Taylor expanded in y around inf 99.0%

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

      1. +-commutative 99.0%

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

   8. Simplified 99.0%

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

   if -6.99999999999999989e-6 < y < 0.00209999999999999987

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 75.2%

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

      1. +-commutative 75.2%

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

   5. Simplified 75.2%

      \[\leadsto \color{blue}{x \cdot \left(y + 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{-6} \lor \neg \left(y \leq 0.0021\right):\\ \;\;\;\;y \cdot \left(x + z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-6} \lor \neg \left(y \leq 0.0028\right):\\ \;\;\;\;y \cdot \left(x + z\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -4e-6) (not (<= y 0.0028))) (* y (+ x z)) (+ x (* y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -4e-6) || !(y <= 0.0028)) {
		tmp = y * (x + z);
	} else {
		tmp = x + (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 <= (-4d-6)) .or. (.not. (y <= 0.0028d0))) then
        tmp = y * (x + z)
    else
        tmp = x + (y * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -4e-6) || !(y <= 0.0028)) {
		tmp = y * (x + z);
	} else {
		tmp = x + (y * x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -4e-6) or not (y <= 0.0028):
		tmp = y * (x + z)
	else:
		tmp = x + (y * x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -4e-6) || !(y <= 0.0028))
		tmp = Float64(y * Float64(x + z));
	else
		tmp = Float64(x + Float64(y * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -4e-6) || ~((y <= 0.0028)))
		tmp = y * (x + z);
	else
		tmp = x + (y * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -4e-6], N[Not[LessEqual[y, 0.0028]], $MachinePrecision]], N[(y * N[(x + z), $MachinePrecision]), $MachinePrecision], N[(x + N[(y * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.99999999999999982e-6 or 0.00279999999999999997 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 96.7%

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

      1. fma-define 99.2%

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

      2. +-commutative 99.2%

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

   5. Simplified 99.2%

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

   6. Taylor expanded in y around inf 99.0%

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

      1. +-commutative 99.0%

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

   8. Simplified 99.0%

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

   if -3.99999999999999982e-6 < y < 0.00279999999999999997

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 75.2%

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

      1. *-commutative 75.2%

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

   5. Simplified 75.2%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-6} \lor \neg \left(y \leq 0.0028\right):\\ \;\;\;\;y \cdot \left(x + z\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.0)) {
		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 <= (-1.0d0)) .or. (.not. (y <= 1.0d0))) 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 <= -1.0) || !(y <= 1.0)) {
		tmp = y * (x + z);
	} else {
		tmp = x + (y * z);
	}
	return tmp;
}

Python

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

Julia

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

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 1.0]], $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 < -1 or 1 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 96.7%

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

      1. fma-define 99.2%

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

      2. +-commutative 99.2%

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

   5. Simplified 99.2%

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

   6. Taylor expanded in y around inf 99.0%

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

      1. +-commutative 99.0%

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

   8. Simplified 99.0%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 98.0%

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

4. Final simplification 98.5%

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

Alternative 7: 61.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 8.5\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 8.5))) (* y x) x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.0) || !(y <= 8.5)) {
		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 <= 8.5d0))) 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 <= 8.5)) {
		tmp = y * x;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.0) or not (y <= 8.5):
		tmp = y * x
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 8.5))
		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 <= 8.5)))
		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, 8.5]], $MachinePrecision]], N[(y * x), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 8.5 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0 53.9%

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

      1. *-commutative 53.9%

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

   5. Simplified 53.9%

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

   6. Taylor expanded in y around inf 52.9%

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

      1. *-commutative 52.9%

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

   8. Simplified 52.9%

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

   if -1 < y < 8.5

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 72.2%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 8.5\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 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 9: 36.8% 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 39.2%

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

4. Final simplification 39.2%

   \[\leadsto x \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024115 
(FPCore (x y z)
  :name "Main:bigenough2 from A"
  :precision binary64
  (+ x (* y (+ z x))))