Main:bigenough2 from A

Percentage Accurate: 100.0% → 100.0%

Time: 3.9s

Alternatives: 8

Speedup: 1.0×

• 22.2% 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: 61.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{+36}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq -1.7 \cdot 10^{-6}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq 520000000000:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{+251}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.9e+36)
   (* y x)
   (if (<= y -1.7e-6)
     (* y z)
     (if (<= y 520000000000.0) x (if (<= y 7.8e+251) (* y x) (* y z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.9e+36) {
		tmp = y * x;
	} else if (y <= -1.7e-6) {
		tmp = y * z;
	} else if (y <= 520000000000.0) {
		tmp = x;
	} else if (y <= 7.8e+251) {
		tmp = 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 (y <= (-1.9d+36)) then
        tmp = y * x
    else if (y <= (-1.7d-6)) then
        tmp = y * z
    else if (y <= 520000000000.0d0) then
        tmp = x
    else if (y <= 7.8d+251) then
        tmp = 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 (y <= -1.9e+36) {
		tmp = y * x;
	} else if (y <= -1.7e-6) {
		tmp = y * z;
	} else if (y <= 520000000000.0) {
		tmp = x;
	} else if (y <= 7.8e+251) {
		tmp = y * x;
	} else {
		tmp = y * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.9e+36:
		tmp = y * x
	elif y <= -1.7e-6:
		tmp = y * z
	elif y <= 520000000000.0:
		tmp = x
	elif y <= 7.8e+251:
		tmp = y * x
	else:
		tmp = y * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.9e+36)
		tmp = Float64(y * x);
	elseif (y <= -1.7e-6)
		tmp = Float64(y * z);
	elseif (y <= 520000000000.0)
		tmp = x;
	elseif (y <= 7.8e+251)
		tmp = Float64(y * x);
	else
		tmp = Float64(y * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.9e+36)
		tmp = y * x;
	elseif (y <= -1.7e-6)
		tmp = y * z;
	elseif (y <= 520000000000.0)
		tmp = x;
	elseif (y <= 7.8e+251)
		tmp = y * x;
	else
		tmp = y * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.9e+36], N[(y * x), $MachinePrecision], If[LessEqual[y, -1.7e-6], N[(y * z), $MachinePrecision], If[LessEqual[y, 520000000000.0], x, If[LessEqual[y, 7.8e+251], N[(y * x), $MachinePrecision], N[(y * z), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.90000000000000012e36 or 5.2e11 < y < 7.79999999999999951e251

   1. Initial program 100.0%

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

   2. Taylor expanded in x around -inf 99.1%

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

      1. +-commutative 99.1%

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

      2. mul-1-neg 99.1%

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

      3. unsub-neg 99.1%

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

      4. sub-neg 99.1%

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

      5. metadata-eval 99.1%

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

      6. +-commutative 99.1%

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

      7. mul-1-neg 99.1%

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

      8. unsub-neg 99.1%

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

   4. Simplified 99.1%

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

   5. Taylor expanded in y around inf 98.9%

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

      1. mul-1-neg 98.9%

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

      2. *-commutative 98.9%

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

      3. distribute-rgt-neg-in 98.9%

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

   7. Simplified 98.9%

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

   8. Taylor expanded in z around 0 61.9%

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

      1. *-commutative 61.9%

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

   10. Simplified 61.9%

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

   if -1.90000000000000012e36 < y < -1.70000000000000003e-6 or 7.79999999999999951e251 < y

   1. Initial program 99.9%

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

   2. Taylor expanded in x around -inf 96.2%

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

      1. +-commutative 96.2%

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

      2. mul-1-neg 96.2%

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

      3. unsub-neg 96.2%

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

      4. sub-neg 96.2%

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

      5. metadata-eval 96.2%

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

      6. +-commutative 96.2%

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

      7. mul-1-neg 96.2%

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

      8. unsub-neg 96.2%

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

   4. Simplified 96.2%

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

   5. Taylor expanded in y around inf 85.6%

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

      1. mul-1-neg 85.6%

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

      2. *-commutative 85.6%

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

      3. distribute-rgt-neg-in 85.6%

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

   7. Simplified 85.6%

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

   8. Taylor expanded in z around inf 68.2%

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

   if -1.70000000000000003e-6 < y < 5.2e11

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 66.6%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{+36}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq -1.7 \cdot 10^{-6}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq 520000000000:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{+251}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]

Alternative 3: 75.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{-18} \lor \neg \left(x \leq 5.2 \cdot 10^{-24}\right):\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -4.7e-18) (not (<= x 5.2e-24))) (* x (+ y 1.0)) (* y z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -4.7e-18) || !(x <= 5.2e-24)) {
		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 <= (-4.7d-18)) .or. (.not. (x <= 5.2d-24))) 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 <= -4.7e-18) || !(x <= 5.2e-24)) {
		tmp = x * (y + 1.0);
	} else {
		tmp = y * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -4.7e-18) or not (x <= 5.2e-24):
		tmp = x * (y + 1.0)
	else:
		tmp = y * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -4.7e-18) || !(x <= 5.2e-24))
		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 <= -4.7e-18) || ~((x <= 5.2e-24)))
		tmp = x * (y + 1.0);
	else
		tmp = y * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -4.7e-18], N[Not[LessEqual[x, 5.2e-24]], $MachinePrecision]], N[(x * N[(y + 1.0), $MachinePrecision]), $MachinePrecision], N[(y * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.6999999999999996e-18 or 5.2e-24 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 85.6%

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

      1. +-commutative 85.6%

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

   4. Simplified 85.6%

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

   if -4.6999999999999996e-18 < x < 5.2e-24

   1. Initial program 100.0%

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

   2. Taylor expanded in x around -inf 100.0%

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

      1. +-commutative 100.0%

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

      2. mul-1-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. metadata-eval 100.0%

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

      6. +-commutative 100.0%

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

      7. mul-1-neg 100.0%

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

      8. unsub-neg 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in y around inf 83.1%

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

      1. mul-1-neg 83.1%

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

      2. *-commutative 83.1%

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

      3. distribute-rgt-neg-in 83.1%

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

   7. Simplified 83.1%

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

   8. Taylor expanded in z around inf 69.6%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{-18} \lor \neg \left(x \leq 5.2 \cdot 10^{-24}\right):\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]

Alternative 4: 85.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.6 \lor \neg \left(y \leq 6 \cdot 10^{-61}\right):\\ \;\;\;\;y \cdot \left(x + z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -6.6) (not (<= y 6e-61))) (* y (+ x z)) (* x (+ y 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -6.6) || !(y <= 6e-61)) {
		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 <= (-6.6d0)) .or. (.not. (y <= 6d-61))) 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 <= -6.6) || !(y <= 6e-61)) {
		tmp = y * (x + z);
	} else {
		tmp = x * (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -6.6) or not (y <= 6e-61):
		tmp = y * (x + z)
	else:
		tmp = x * (y + 1.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -6.6) || !(y <= 6e-61))
		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 <= -6.6) || ~((y <= 6e-61)))
		tmp = y * (x + z);
	else
		tmp = x * (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -6.6], N[Not[LessEqual[y, 6e-61]], $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 < -6.5999999999999996 or 6.00000000000000024e-61 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in x around -inf 98.6%

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

      1. +-commutative 98.6%

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

      2. mul-1-neg 98.6%

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

      3. unsub-neg 98.6%

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

      4. sub-neg 98.6%

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

      5. metadata-eval 98.6%

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

      6. +-commutative 98.6%

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

      7. mul-1-neg 98.6%

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

      8. unsub-neg 98.6%

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

   4. Simplified 98.6%

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

   5. Taylor expanded in y around inf 95.3%

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

      1. mul-1-neg 95.3%

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

      2. *-commutative 95.3%

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

      3. distribute-rgt-neg-in 95.3%

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

   7. Simplified 95.3%

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

      1. distribute-lft-out-- 96.6%

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

      2. *-commutative 96.6%

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

      3. sub-neg 96.6%

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

      4. remove-double-neg 96.6%

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

   9. Applied egg-rr 96.6%

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

   if -6.5999999999999996 < y < 6.00000000000000024e-61

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 71.0%

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

      1. +-commutative 71.0%

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

   4. Simplified 71.0%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.6 \lor \neg \left(y \leq 6 \cdot 10^{-61}\right):\\ \;\;\;\;y \cdot \left(x + z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \end{array} \]

Alternative 5: 99.0% accurate, 0.8× 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 100.0%

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

   2. Taylor expanded in x around -inf 98.6%

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

      1. +-commutative 98.6%

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

      2. mul-1-neg 98.6%

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

      3. unsub-neg 98.6%

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

      4. sub-neg 98.6%

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

      5. metadata-eval 98.6%

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

      6. +-commutative 98.6%

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

      7. mul-1-neg 98.6%

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

      8. unsub-neg 98.6%

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

   4. Simplified 98.6%

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

   5. Taylor expanded in y around inf 97.1%

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

      1. mul-1-neg 97.1%

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

      2. *-commutative 97.1%

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

      3. distribute-rgt-neg-in 97.1%

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

   7. Simplified 97.1%

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

      1. distribute-lft-out-- 98.5%

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

      2. *-commutative 98.5%

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

      3. sub-neg 98.5%

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

      4. remove-double-neg 98.5%

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

   9. Applied egg-rr 98.5%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in z around inf 97.4%

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

4. Final simplification 98.0%

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

Alternative 6: 60.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 520000000000\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 520000000000.0))) (* y x) x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.0) || !(y <= 520000000000.0)) {
		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 <= 520000000000.0d0))) 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 <= 520000000000.0)) {
		tmp = y * x;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.0) or not (y <= 520000000000.0):
		tmp = y * x
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 520000000000.0))
		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 <= 520000000000.0)))
		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, 520000000000.0]], $MachinePrecision]], N[(y * x), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 5.2e11 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in x around -inf 98.5%

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

      1. +-commutative 98.5%

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

      2. mul-1-neg 98.5%

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

      3. unsub-neg 98.5%

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

      4. sub-neg 98.5%

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

      5. metadata-eval 98.5%

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

      6. +-commutative 98.5%

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

      7. mul-1-neg 98.5%

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

      8. unsub-neg 98.5%

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

   4. Simplified 98.5%

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

   5. Taylor expanded in y around inf 97.0%

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

      1. mul-1-neg 97.0%

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

      2. *-commutative 97.0%

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

      3. distribute-rgt-neg-in 97.0%

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

   7. Simplified 97.0%

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

   8. Taylor expanded in z around 0 56.1%

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

      1. *-commutative 56.1%

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

   10. Simplified 56.1%

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

   if -1 < y < 5.2e11

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 65.2%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 520000000000\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

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. Final simplification 100.0%

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

Alternative 8: 36.2% 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 31.3%

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

3. Final simplification 31.3%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023310 
(FPCore (x y z)
  :name "Main:bigenough2 from A"
  :precision binary64
  (+ x (* y (+ z x))))