Numeric.Log:$clog1p from log-domain-0.10.2.1, A

Percentage Accurate: 100.0% → 100.0%

Time: 3.5s

Alternatives: 8

Speedup: 1.2×

• 44.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(x \cdot 2 + x \cdot x\right) + y \cdot y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ (+ (* x 2.0) (* x x)) (* y y)))

C

double code(double x, double y) {
	return ((x * 2.0) + (x * x)) + (y * y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x * 2.0d0) + (x * x)) + (y * y)
end function

Java

public static double code(double x, double y) {
	return ((x * 2.0) + (x * x)) + (y * y);
}

Python

def code(x, y):
	return ((x * 2.0) + (x * x)) + (y * y)

Julia

function code(x, y)
	return Float64(Float64(Float64(x * 2.0) + Float64(x * x)) + Float64(y * y))
end

MATLAB

function tmp = code(x, y)
	tmp = ((x * 2.0) + (x * x)) + (y * y);
end

Wolfram

code[x_, y_] := N[(N[(N[(x * 2.0), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot 2 + x \cdot x\right) + y \cdot y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ (+ (* x 2.0) (* x x)) (* y y)))

C

double code(double x, double y) {
	return ((x * 2.0) + (x * x)) + (y * y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x * 2.0d0) + (x * x)) + (y * y)
end function

Java

public static double code(double x, double y) {
	return ((x * 2.0) + (x * x)) + (y * y);
}

Python

def code(x, y):
	return ((x * 2.0) + (x * x)) + (y * y)

Julia

function code(x, y)
	return Float64(Float64(Float64(x * 2.0) + Float64(x * x)) + Float64(y * y))
end

MATLAB

function tmp = code(x, y)
	tmp = ((x * 2.0) + (x * x)) + (y * y);
end

Wolfram

code[x_, y_] := N[(N[(N[(x * 2.0), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. distribute-lft-out 100.0%

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

3. Simplified 100.0%

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

   1. distribute-lft-in 100.0%

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

   2. +-commutative 100.0%

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

   3. fma-def 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, x \cdot 2\right)} + y \cdot y \]

   4. add-log-exp 45.6%

      \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\log \left(e^{x \cdot 2}\right)}\right) + y \cdot y \]

   5. exp-lft-sqr 45.6%

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

   6. log-prod 45.6%

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

   7. add-log-exp 48.4%

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

   8. add-log-exp 100.0%

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

5. Applied egg-rr 100.0%

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

6. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y, y, x \cdot \left(x + 2\right)\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (fma y y (* x (+ x 2.0))))

C

double code(double x, double y) {
	return fma(y, y, (x * (x + 2.0)));
}

Julia

function code(x, y)
	return fma(y, y, Float64(x * Float64(x + 2.0)))
end

Wolfram

code[x_, y_] := N[(y * y + N[(x * N[(x + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. distribute-lft-out 100.0%

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

3. Simplified 100.0%

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

   1. +-commutative 100.0%

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

   2. fma-def 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, y, x \cdot \left(2 + x\right)\right)} \]

   3. +-commutative 100.0%

      \[\leadsto \mathsf{fma}\left(y, y, x \cdot \color{blue}{\left(x + 2\right)}\right) \]

5. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, y, x \cdot \left(x + 2\right)\right)} \]

6. Final simplification 100.0%

   \[\leadsto \mathsf{fma}\left(y, y, x \cdot \left(x + 2\right)\right) \]

Alternative 3: 64.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 5.5 \cdot 10^{-280}:\\ \;\;\;\;x + x\\ \mathbf{elif}\;y \cdot y \leq 6.8 \cdot 10^{-228}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;y \cdot y \leq 2.2 \cdot 10^{-200}:\\ \;\;\;\;x + x\\ \mathbf{elif}\;y \cdot y \leq 1.36 \cdot 10^{+162}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 5.5e-280)
   (+ x x)
   (if (<= (* y y) 6.8e-228)
     (* x x)
     (if (<= (* y y) 2.2e-200)
       (+ x x)
       (if (<= (* y y) 1.36e+162) (* x x) (* y y))))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 5.5e-280) {
		tmp = x + x;
	} else if ((y * y) <= 6.8e-228) {
		tmp = x * x;
	} else if ((y * y) <= 2.2e-200) {
		tmp = x + x;
	} else if ((y * y) <= 1.36e+162) {
		tmp = x * x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y * y) <= 5.5d-280) then
        tmp = x + x
    else if ((y * y) <= 6.8d-228) then
        tmp = x * x
    else if ((y * y) <= 2.2d-200) then
        tmp = x + x
    else if ((y * y) <= 1.36d+162) then
        tmp = x * x
    else
        tmp = y * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * y) <= 5.5e-280) {
		tmp = x + x;
	} else if ((y * y) <= 6.8e-228) {
		tmp = x * x;
	} else if ((y * y) <= 2.2e-200) {
		tmp = x + x;
	} else if ((y * y) <= 1.36e+162) {
		tmp = x * x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 5.5e-280:
		tmp = x + x
	elif (y * y) <= 6.8e-228:
		tmp = x * x
	elif (y * y) <= 2.2e-200:
		tmp = x + x
	elif (y * y) <= 1.36e+162:
		tmp = x * x
	else:
		tmp = y * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 5.5e-280)
		tmp = Float64(x + x);
	elseif (Float64(y * y) <= 6.8e-228)
		tmp = Float64(x * x);
	elseif (Float64(y * y) <= 2.2e-200)
		tmp = Float64(x + x);
	elseif (Float64(y * y) <= 1.36e+162)
		tmp = Float64(x * x);
	else
		tmp = Float64(y * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 5.5e-280)
		tmp = x + x;
	elseif ((y * y) <= 6.8e-228)
		tmp = x * x;
	elseif ((y * y) <= 2.2e-200)
		tmp = x + x;
	elseif ((y * y) <= 1.36e+162)
		tmp = x * x;
	else
		tmp = y * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 5.5e-280], N[(x + x), $MachinePrecision], If[LessEqual[N[(y * y), $MachinePrecision], 6.8e-228], N[(x * x), $MachinePrecision], If[LessEqual[N[(y * y), $MachinePrecision], 2.2e-200], N[(x + x), $MachinePrecision], If[LessEqual[N[(y * y), $MachinePrecision], 1.36e+162], N[(x * x), $MachinePrecision], N[(y * y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 y y) < 5.50000000000000001e-280 or 6.79999999999999981e-228 < (*.f64 y y) < 2.20000000000000013e-200

   1. Initial program 99.9%

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

      1. distribute-lft-out 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 100.0%

      \[\leadsto \color{blue}{\left(2 + x\right) \cdot x} \]

   5. Taylor expanded in x around 0 65.9%

      \[\leadsto \color{blue}{2 \cdot x} \]

   7. Simplified 65.9%

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

   if 5.50000000000000001e-280 < (*.f64 y y) < 6.79999999999999981e-228 or 2.20000000000000013e-200 < (*.f64 y y) < 1.35999999999999997e162

   1. Initial program 100.0%

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

      1. distribute-lft-out 100.0%

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

   3. Simplified 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. +-commutative 100.0%

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

      3. fma-def 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, x \cdot 2\right)} + y \cdot y \]

      4. add-log-exp 51.0%

         \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\log \left(e^{x \cdot 2}\right)}\right) + y \cdot y \]

      5. exp-lft-sqr 51.0%

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

      6. log-prod 51.0%

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

      7. add-log-exp 52.0%

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

      8. add-log-exp 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around inf 59.5%

      \[\leadsto \color{blue}{{x}^{2}} \]

   8. Simplified 59.5%

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

   if 1.35999999999999997e162 < (*.f64 y y)

   1. Initial program 100.0%

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

      1. distribute-lft-out 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 89.5%

      \[\leadsto \color{blue}{{y}^{2}} \]
   5. Step-by-step derivation

      1. unpow2 89.5%

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

   6. Simplified 89.5%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 5.5 \cdot 10^{-280}:\\ \;\;\;\;x + x\\ \mathbf{elif}\;y \cdot y \leq 6.8 \cdot 10^{-228}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;y \cdot y \leq 2.2 \cdot 10^{-200}:\\ \;\;\;\;x + x\\ \mathbf{elif}\;y \cdot y \leq 1.36 \cdot 10^{+162}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 4: 71.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{+47} \lor \neg \left(x \leq 38000\right) \land \left(x \leq 1.9 \cdot 10^{+65} \lor \neg \left(x \leq 5.7 \cdot 10^{+103}\right)\right):\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -2.9e+47)
         (and (not (<= x 38000.0)) (or (<= x 1.9e+65) (not (<= x 5.7e+103)))))
   (* x x)
   (* y y)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -2.9e+47) || (!(x <= 38000.0) && ((x <= 1.9e+65) || !(x <= 5.7e+103)))) {
		tmp = x * x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-2.9d+47)) .or. (.not. (x <= 38000.0d0)) .and. (x <= 1.9d+65) .or. (.not. (x <= 5.7d+103))) then
        tmp = x * x
    else
        tmp = y * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -2.9e+47) || (!(x <= 38000.0) && ((x <= 1.9e+65) || !(x <= 5.7e+103)))) {
		tmp = x * x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -2.9e+47) or (not (x <= 38000.0) and ((x <= 1.9e+65) or not (x <= 5.7e+103))):
		tmp = x * x
	else:
		tmp = y * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -2.9e+47) || (!(x <= 38000.0) && ((x <= 1.9e+65) || !(x <= 5.7e+103))))
		tmp = Float64(x * x);
	else
		tmp = Float64(y * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -2.9e+47) || (~((x <= 38000.0)) && ((x <= 1.9e+65) || ~((x <= 5.7e+103)))))
		tmp = x * x;
	else
		tmp = y * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -2.9e+47], And[N[Not[LessEqual[x, 38000.0]], $MachinePrecision], Or[LessEqual[x, 1.9e+65], N[Not[LessEqual[x, 5.7e+103]], $MachinePrecision]]]], N[(x * x), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.8999999999999998e47 or 38000 < x < 1.90000000000000006e65 or 5.70000000000000033e103 < x

   1. Initial program 100.0%

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

      1. distribute-lft-out 100.0%

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

   3. Simplified 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. +-commutative 100.0%

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

      3. fma-def 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, x \cdot 2\right)} + y \cdot y \]

      4. add-log-exp 30.3%

         \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\log \left(e^{x \cdot 2}\right)}\right) + y \cdot y \]

      5. exp-lft-sqr 30.3%

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

      6. log-prod 30.3%

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

      7. add-log-exp 30.3%

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

      8. add-log-exp 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around inf 85.8%

      \[\leadsto \color{blue}{{x}^{2}} \]

   8. Simplified 85.8%

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

   if -2.8999999999999998e47 < x < 38000 or 1.90000000000000006e65 < x < 5.70000000000000033e103

   1. Initial program 100.0%

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

      1. distribute-lft-out 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 60.9%

      \[\leadsto \color{blue}{{y}^{2}} \]
   5. Step-by-step derivation

      1. unpow2 60.9%

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

   6. Simplified 60.9%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{+47} \lor \neg \left(x \leq 38000\right) \land \left(x \leq 1.9 \cdot 10^{+65} \lor \neg \left(x \leq 5.7 \cdot 10^{+103}\right)\right):\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 5: 96.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 1.25 \cdot 10^{-91}:\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y + x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 1.25e-91) (* x (+ x 2.0)) (+ (* y y) (* x x))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 1.25e-91) {
		tmp = x * (x + 2.0);
	} else {
		tmp = (y * y) + (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y * y) <= 1.25d-91) then
        tmp = x * (x + 2.0d0)
    else
        tmp = (y * y) + (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * y) <= 1.25e-91) {
		tmp = x * (x + 2.0);
	} else {
		tmp = (y * y) + (x * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 1.25e-91:
		tmp = x * (x + 2.0)
	else:
		tmp = (y * y) + (x * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 1.25e-91)
		tmp = Float64(x * Float64(x + 2.0));
	else
		tmp = Float64(Float64(y * y) + Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 1.25e-91)
		tmp = x * (x + 2.0);
	else
		tmp = (y * y) + (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 1.25e-91], N[(x * N[(x + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(y * y), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 1.24999999999999999e-91

   1. Initial program 99.9%

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

      1. distribute-lft-out 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 94.6%

      \[\leadsto \color{blue}{\left(2 + x\right) \cdot x} \]

   if 1.24999999999999999e-91 < (*.f64 y y)

   1. Initial program 100.0%

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

      1. distribute-lft-out 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{{x}^{2}} + y \cdot y \]
   5. Step-by-step derivation

      1. unpow2 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 1.25 \cdot 10^{-91}:\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y + x \cdot x\\ \end{array} \]

Alternative 6: 82.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 1.1 \cdot 10^{+162}:\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 1.1e+162) (* x (+ x 2.0)) (* y y)))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 1.1e+162) {
		tmp = x * (x + 2.0);
	} else {
		tmp = y * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y * y) <= 1.1d+162) then
        tmp = x * (x + 2.0d0)
    else
        tmp = y * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * y) <= 1.1e+162) {
		tmp = x * (x + 2.0);
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 1.1e+162:
		tmp = x * (x + 2.0)
	else:
		tmp = y * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 1.1e+162)
		tmp = Float64(x * Float64(x + 2.0));
	else
		tmp = Float64(y * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 1.1e+162)
		tmp = x * (x + 2.0);
	else
		tmp = y * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 1.1e+162], N[(x * N[(x + 2.0), $MachinePrecision]), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 1.1000000000000001e162

   1. Initial program 100.0%

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

      1. distribute-lft-out 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 83.0%

      \[\leadsto \color{blue}{\left(2 + x\right) \cdot x} \]

   if 1.1000000000000001e162 < (*.f64 y y)

   1. Initial program 100.0%

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

      1. distribute-lft-out 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 89.5%

      \[\leadsto \color{blue}{{y}^{2}} \]
   5. Step-by-step derivation

      1. unpow2 89.5%

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

   6. Simplified 89.5%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 1.1 \cdot 10^{+162}:\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 7: 100.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ y \cdot y + x \cdot \left(x + 2\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ (* y y) (* x (+ x 2.0))))

C

double code(double x, double y) {
	return (y * y) + (x * (x + 2.0));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (y * y) + (x * (x + 2.0d0))
end function

Java

public static double code(double x, double y) {
	return (y * y) + (x * (x + 2.0));
}

Python

def code(x, y):
	return (y * y) + (x * (x + 2.0))

Julia

function code(x, y)
	return Float64(Float64(y * y) + Float64(x * Float64(x + 2.0)))
end

MATLAB

function tmp = code(x, y)
	tmp = (y * y) + (x * (x + 2.0));
end

Wolfram

code[x_, y_] := N[(N[(y * y), $MachinePrecision] + N[(x * N[(x + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. distribute-lft-out 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto y \cdot y + x \cdot \left(x + 2\right) \]

Alternative 8: 42.8% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. distribute-lft-out 100.0%

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

3. Simplified 100.0%

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

   1. distribute-lft-in 100.0%

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

   2. +-commutative 100.0%

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

   3. fma-def 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, x \cdot 2\right)} + y \cdot y \]

   4. add-log-exp 45.6%

      \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\log \left(e^{x \cdot 2}\right)}\right) + y \cdot y \]

   5. exp-lft-sqr 45.6%

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

   6. log-prod 45.6%

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

   7. add-log-exp 48.4%

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

   8. add-log-exp 100.0%

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

5. Applied egg-rr 100.0%

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

6. Taylor expanded in x around inf 38.9%

   \[\leadsto \color{blue}{{x}^{2}} \]

8. Simplified 38.9%

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

9. Final simplification 38.9%

   \[\leadsto x \cdot x \]

Developer target: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ y \cdot y + \left(2 \cdot x + x \cdot x\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ (* y y) (+ (* 2.0 x) (* x x))))

C

double code(double x, double y) {
	return (y * y) + ((2.0 * x) + (x * x));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (y * y) + ((2.0d0 * x) + (x * x))
end function

Java

public static double code(double x, double y) {
	return (y * y) + ((2.0 * x) + (x * x));
}

Python

def code(x, y):
	return (y * y) + ((2.0 * x) + (x * x))

Julia

function code(x, y)
	return Float64(Float64(y * y) + Float64(Float64(2.0 * x) + Float64(x * x)))
end

MATLAB

function tmp = code(x, y)
	tmp = (y * y) + ((2.0 * x) + (x * x));
end

Wolfram

code[x_, y_] := N[(N[(y * y), $MachinePrecision] + N[(N[(2.0 * x), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2023258 
(FPCore (x y)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, A"
  :precision binary64

  :herbie-target
  (+ (* y y) (+ (* 2.0 x) (* x x)))

  (+ (+ (* x 2.0) (* x x)) (* y y)))