Main:bigenough2 from A

Percentage Accurate: 100.0% → 100.0%

Time: 5.1s

Alternatives: 7

Speedup: 1.0×

• 21.4% 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: 75.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-89} \lor \neg \left(x \leq 1.7 \cdot 10^{-23}\right):\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -4e-89) (not (<= x 1.7e-23))) (* x (+ y 1.0)) (* y z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -4e-89) || !(x <= 1.7e-23)) {
		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 <= (-4d-89)) .or. (.not. (x <= 1.7d-23))) 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 <= -4e-89) || !(x <= 1.7e-23)) {
		tmp = x * (y + 1.0);
	} else {
		tmp = y * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -4e-89) or not (x <= 1.7e-23):
		tmp = x * (y + 1.0)
	else:
		tmp = y * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -4e-89) || !(x <= 1.7e-23))
		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 <= -4e-89) || ~((x <= 1.7e-23)))
		tmp = x * (y + 1.0);
	else
		tmp = y * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -4e-89], N[Not[LessEqual[x, 1.7e-23]], $MachinePrecision]], N[(x * N[(y + 1.0), $MachinePrecision]), $MachinePrecision], N[(y * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.00000000000000015e-89 or 1.7e-23 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 83.0%

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

      1. +-commutative 83.0%

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

   4. Simplified 83.0%

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

   if -4.00000000000000015e-89 < x < 1.7e-23

   1. Initial program 100.0%

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

      1. distribute-lft-in 99.9%

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

      2. flip-+ 55.2%

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

   3. Applied egg-rr 55.2%

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

      1. distribute-lft-out-- 55.2%

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

      2. difference-of-squares 55.2%

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

      3. +-commutative 55.2%

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

      4. distribute-lft-in 55.2%

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

      5. distribute-lft-out-- 55.2%

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

   5. Simplified 55.2%

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

      1. *-un-lft-identity 55.2%

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

      2. *-commutative 55.2%

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

      3. metadata-eval 55.2%

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

      4. div-inv 55.2%

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

      5. *-commutative 55.2%

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

      6. associate-/l* 55.0%

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

   7. Applied egg-rr 55.0%

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

      1. associate-/l* 75.8%

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

      2. frac-2neg 75.8%

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

      3. *-inverses 99.7%

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

      4. associate-/l/ 99.7%

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

      5. *-un-lft-identity 99.7%

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

      6. distribute-frac-neg 99.7%

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

      7. +-commutative 99.7%

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

      8. distribute-neg-frac 99.7%

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

      9. metadata-eval 99.7%

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

   9. Applied egg-rr 99.7%

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

   10. Taylor expanded in x around 0 71.3%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-89} \lor \neg \left(x \leq 1.7 \cdot 10^{-23}\right):\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]

Alternative 3: 84.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{-75} \lor \neg \left(y \leq 1.1 \cdot 10^{-33}\right):\\ \;\;\;\;y \cdot \left(x + z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -3.4e-75) (not (<= y 1.1e-33))) (* y (+ x z)) (* x (+ y 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3.4e-75) || !(y <= 1.1e-33)) {
		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 <= (-3.4d-75)) .or. (.not. (y <= 1.1d-33))) 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 <= -3.4e-75) || !(y <= 1.1e-33)) {
		tmp = y * (x + z);
	} else {
		tmp = x * (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -3.4e-75) or not (y <= 1.1e-33):
		tmp = y * (x + z)
	else:
		tmp = x * (y + 1.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -3.4e-75) || !(y <= 1.1e-33))
		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 <= -3.4e-75) || ~((y <= 1.1e-33)))
		tmp = y * (x + z);
	else
		tmp = x * (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -3.4e-75], N[Not[LessEqual[y, 1.1e-33]], $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 < -3.40000000000000015e-75 or 1.10000000000000003e-33 < y

   1. Initial program 100.0%

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

      1. distribute-lft-in 93.2%

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

      2. flip-+ 36.4%

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

   3. Applied egg-rr 36.4%

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

      1. distribute-lft-out-- 36.4%

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

      2. difference-of-squares 36.8%

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

      3. +-commutative 36.8%

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

      4. distribute-lft-in 36.8%

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

      5. distribute-lft-out-- 36.8%

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

   5. Simplified 36.8%

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

      1. *-un-lft-identity 36.8%

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

      2. *-commutative 36.8%

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

      3. metadata-eval 36.8%

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

      4. div-inv 36.8%

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

      5. *-commutative 36.8%

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

      6. associate-/l* 36.7%

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

   7. Applied egg-rr 36.7%

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

      1. associate-/l* 66.8%

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

      2. frac-2neg 66.8%

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

      3. *-inverses 99.7%

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

      4. associate-/l/ 99.7%

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

      5. *-un-lft-identity 99.7%

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

      6. distribute-frac-neg 99.7%

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

      7. +-commutative 99.7%

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

      8. distribute-neg-frac 99.7%

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

      9. metadata-eval 99.7%

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

   9. Applied egg-rr 99.7%

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

   10. Taylor expanded in y around inf 92.7%

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

   if -3.40000000000000015e-75 < y < 1.10000000000000003e-33

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 82.6%

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

      1. +-commutative 82.6%

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

   4. Simplified 82.6%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{-75} \lor \neg \left(y \leq 1.1 \cdot 10^{-33}\right):\\ \;\;\;\;y \cdot \left(x + z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \end{array} \]

Alternative 4: 98.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -42000 \lor \neg \left(y \leq 3.35 \cdot 10^{-6}\right):\\ \;\;\;\;y \cdot \left(x + z\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -42000.0) (not (<= y 3.35e-6))) (* y (+ x z)) (+ x (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -42000.0) || !(y <= 3.35e-6)) {
		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 <= (-42000.0d0)) .or. (.not. (y <= 3.35d-6))) 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 <= -42000.0) || !(y <= 3.35e-6)) {
		tmp = y * (x + z);
	} else {
		tmp = x + (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -42000.0) or not (y <= 3.35e-6):
		tmp = y * (x + z)
	else:
		tmp = x + (y * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -42000.0) || !(y <= 3.35e-6))
		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 <= -42000.0) || ~((y <= 3.35e-6)))
		tmp = y * (x + z);
	else
		tmp = x + (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -42000.0], N[Not[LessEqual[y, 3.35e-6]], $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 < -42000 or 3.35e-6 < y

   1. Initial program 99.9%

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

      1. distribute-lft-in 91.9%

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

      2. flip-+ 30.0%

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

   3. Applied egg-rr 30.0%

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

      1. distribute-lft-out-- 30.0%

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

      2. difference-of-squares 30.4%

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

      3. +-commutative 30.4%

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

      4. distribute-lft-in 30.4%

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

      5. distribute-lft-out-- 30.4%

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

   5. Simplified 30.4%

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

      1. *-un-lft-identity 30.4%

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

      2. *-commutative 30.4%

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

      3. metadata-eval 30.4%

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

      4. div-inv 30.4%

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

      5. *-commutative 30.4%

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

      6. associate-/l* 30.3%

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

   7. Applied egg-rr 30.3%

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

      1. associate-/l* 60.2%

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

      2. frac-2neg 60.2%

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

      3. *-inverses 99.7%

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

      4. associate-/l/ 99.7%

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

      5. *-un-lft-identity 99.7%

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

      6. distribute-frac-neg 99.7%

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

      7. +-commutative 99.7%

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

      8. distribute-neg-frac 99.7%

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

      9. metadata-eval 99.7%

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

   9. Applied egg-rr 99.7%

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

   10. Taylor expanded in y around inf 98.9%

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

   if -42000 < y < 3.35e-6

   1. Initial program 100.0%

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

   2. Taylor expanded in z around inf 98.9%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -42000 \lor \neg \left(y \leq 3.35 \cdot 10^{-6}\right):\\ \;\;\;\;y \cdot \left(x + z\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot z\\ \end{array} \]

Alternative 5: 61.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{-76} \lor \neg \left(y \leq 2.5 \cdot 10^{-33}\right):\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -3.4e-76) (not (<= y 2.5e-33))) (* y z) x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3.4e-76) || !(y <= 2.5e-33)) {
		tmp = y * z;
	} 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 <= (-3.4d-76)) .or. (.not. (y <= 2.5d-33))) then
        tmp = y * z
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3.4e-76) || !(y <= 2.5e-33)) {
		tmp = y * z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -3.4e-76) or not (y <= 2.5e-33):
		tmp = y * z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -3.4e-76) || !(y <= 2.5e-33))
		tmp = Float64(y * z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -3.4e-76) || ~((y <= 2.5e-33)))
		tmp = y * z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -3.4e-76], N[Not[LessEqual[y, 2.5e-33]], $MachinePrecision]], N[(y * z), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -3.3999999999999999e-76 or 2.50000000000000014e-33 < y

   1. Initial program 100.0%

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

      1. distribute-lft-in 93.2%

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

      2. flip-+ 36.4%

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

   3. Applied egg-rr 36.4%

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

      1. distribute-lft-out-- 36.4%

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

      2. difference-of-squares 36.8%

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

      3. +-commutative 36.8%

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

      4. distribute-lft-in 36.8%

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

      5. distribute-lft-out-- 36.8%

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

   5. Simplified 36.8%

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

      1. *-un-lft-identity 36.8%

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

      2. *-commutative 36.8%

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

      3. metadata-eval 36.8%

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

      4. div-inv 36.8%

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

      5. *-commutative 36.8%

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

      6. associate-/l* 36.7%

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

   7. Applied egg-rr 36.7%

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

      1. associate-/l* 66.8%

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

      2. frac-2neg 66.8%

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

      3. *-inverses 99.7%

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

      4. associate-/l/ 99.7%

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

      5. *-un-lft-identity 99.7%

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

      6. distribute-frac-neg 99.7%

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

      7. +-commutative 99.7%

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

      8. distribute-neg-frac 99.7%

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

      9. metadata-eval 99.7%

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

   9. Applied egg-rr 99.7%

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

   10. Taylor expanded in x around 0 51.8%

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

   if -3.3999999999999999e-76 < y < 2.50000000000000014e-33

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 82.6%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{-76} \lor \neg \left(y \leq 2.5 \cdot 10^{-33}\right):\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6: 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 7: 36.1% 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 39.1%

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

3. Final simplification 39.1%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023311 
(FPCore (x y z)
  :name "Main:bigenough2 from A"
  :precision binary64
  (+ x (* y (+ z x))))