Main:bigenough2 from A

Percentage Accurate: 100.0% → 100.0%

Time: 2.6s

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, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. distribute-rgt-in 98.4%

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

   2. fma-def 99.6%

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

   3. *-commutative 99.6%

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

3. Applied egg-rr 99.6%

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

   1. fma-udef 98.4%

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

   2. flip-+ 52.4%

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

   3. pow1 52.4%

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

   4. pow1 52.4%

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

   5. pow-prod-up 52.4%

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

   6. *-commutative 52.4%

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

   7. metadata-eval 52.4%

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

   8. pow2 52.4%

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

   9. *-commutative 52.4%

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

   10. *-commutative 52.4%

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

   11. *-commutative 52.4%

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

5. Applied egg-rr 52.4%

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

   1. unpow2 52.4%

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

   2. unpow2 52.4%

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

   3. difference-of-squares 52.5%

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

   4. *-commutative 52.5%

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

   5. distribute-rgt-in 52.5%

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

   6. *-commutative 52.5%

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

   7. distribute-lft-out-- 52.5%

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

   8. times-frac 98.0%

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

   9. +-commutative 98.0%

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

   10. *-commutative 98.0%

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

   11. distribute-lft-out-- 99.9%

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

7. Simplified 99.9%

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

8. Taylor expanded in y around 0 100.0%

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

9. Final simplification 100.0%

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

Alternative 2: 87.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{-72} \lor \neg \left(z \leq 5.2 \cdot 10^{-86}\right):\\ \;\;\;\;x + z \cdot y\\ \mathbf{else}:\\ \;\;\;\;x + x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2.9e-72) (not (<= z 5.2e-86))) (+ x (* z y)) (+ x (* x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.9e-72) || !(z <= 5.2e-86)) {
		tmp = x + (z * y);
	} else {
		tmp = x + (x * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-2.9d-72)) .or. (.not. (z <= 5.2d-86))) then
        tmp = x + (z * y)
    else
        tmp = x + (x * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.9e-72) || !(z <= 5.2e-86)) {
		tmp = x + (z * y);
	} else {
		tmp = x + (x * y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -2.9e-72) or not (z <= 5.2e-86):
		tmp = x + (z * y)
	else:
		tmp = x + (x * y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -2.9e-72) || !(z <= 5.2e-86))
		tmp = Float64(x + Float64(z * y));
	else
		tmp = Float64(x + Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -2.9e-72) || ~((z <= 5.2e-86)))
		tmp = x + (z * y);
	else
		tmp = x + (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -2.9e-72], N[Not[LessEqual[z, 5.2e-86]], $MachinePrecision]], N[(x + N[(z * y), $MachinePrecision]), $MachinePrecision], N[(x + N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.89999999999999998e-72 or 5.2000000000000002e-86 < z

   1. Initial program 100.0%

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

   2. Taylor expanded in z around inf 89.8%

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

   if -2.89999999999999998e-72 < z < 5.2000000000000002e-86

   1. Initial program 100.0%

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

   2. Taylor expanded in z around 0 90.9%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{-72} \lor \neg \left(z \leq 5.2 \cdot 10^{-86}\right):\\ \;\;\;\;x + z \cdot y\\ \mathbf{else}:\\ \;\;\;\;x + x \cdot y\\ \end{array} \]

Alternative 3: 61.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.0) (* x y) (if (<= y 1.0) x (* x y))))

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-1.0d0)) then
        tmp = x * y
    else if (y <= 1.0d0) then
        tmp = x
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

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

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.0:
		tmp = x * y
	elif y <= 1.0:
		tmp = x
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.0)
		tmp = Float64(x * y);
	elseif (y <= 1.0)
		tmp = x;
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

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

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.0], N[(x * y), $MachinePrecision], If[LessEqual[y, 1.0], x, N[(x * y), $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 z around 0 51.1%

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

   3. Taylor expanded in y around inf 50.6%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in z around 0 68.4%

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

   3. Taylor expanded in y around 0 67.3%

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

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 4: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 + ((x + z) * y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Final simplification 100.0%

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

Alternative 5: 62.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ x + x \cdot y \end{array} \]

FPCore

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

C

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

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 + (x * y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in z around 0 59.7%

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

3. Final simplification 59.7%

   \[\leadsto x + x \cdot y \]

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 z around 0 59.7%

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

3. Taylor expanded in y around 0 34.9%

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

4. Final simplification 34.9%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023196 
(FPCore (x y z)
  :name "Main:bigenough2 from A"
  :precision binary64
  (+ x (* y (+ z x))))