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

Percentage Accurate: 100.0% → 100.0%

Time: 4.1s

Alternatives: 7

Speedup: 1.2×

• 42.5% 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 51.4%

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

   5. exp-lft-sqr 51.3%

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

   6. log-prod 51.4%

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

   7. add-log-exp 53.7%

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

   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: 83.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 2.2 \cdot 10^{-30} \lor \neg \left(y \cdot y \leq 6.9 \cdot 10^{+102}\right) \land y \cdot y \leq 2.7 \cdot 10^{+174}:\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (* y y) 2.2e-30)
         (and (not (<= (* y y) 6.9e+102)) (<= (* y y) 2.7e+174)))
   (* x (+ x 2.0))
   (* y y)))

C

double code(double x, double y) {
	double tmp;
	if (((y * y) <= 2.2e-30) || (!((y * y) <= 6.9e+102) && ((y * y) <= 2.7e+174))) {
		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) <= 2.2d-30) .or. (.not. ((y * y) <= 6.9d+102)) .and. ((y * y) <= 2.7d+174)) 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) <= 2.2e-30) || (!((y * y) <= 6.9e+102) && ((y * y) <= 2.7e+174))) {
		tmp = x * (x + 2.0);
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((y * y) <= 2.2e-30) or (not ((y * y) <= 6.9e+102) and ((y * y) <= 2.7e+174)):
		tmp = x * (x + 2.0)
	else:
		tmp = y * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((Float64(y * y) <= 2.2e-30) || (!(Float64(y * y) <= 6.9e+102) && (Float64(y * y) <= 2.7e+174)))
		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) <= 2.2e-30) || (~(((y * y) <= 6.9e+102)) && ((y * y) <= 2.7e+174)))
		tmp = x * (x + 2.0);
	else
		tmp = y * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[N[(y * y), $MachinePrecision], 2.2e-30], And[N[Not[LessEqual[N[(y * y), $MachinePrecision], 6.9e+102]], $MachinePrecision], LessEqual[N[(y * y), $MachinePrecision], 2.7e+174]]], N[(x * N[(x + 2.0), $MachinePrecision]), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 2.19999999999999983e-30 or 6.89999999999999966e102 < (*.f64 y y) < 2.6999999999999999e174

   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 87.6%

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

   if 2.19999999999999983e-30 < (*.f64 y y) < 6.89999999999999966e102 or 2.6999999999999999e174 < (*.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 83.5%

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

      1. unpow2 83.5%

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

   6. Simplified 83.5%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 2.2 \cdot 10^{-30} \lor \neg \left(y \cdot y \leq 6.9 \cdot 10^{+102}\right) \land y \cdot y \leq 2.7 \cdot 10^{+174}:\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 3: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{+29} \lor \neg \left(x \leq 2\right):\\ \;\;\;\;y \cdot y + x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(x + x\right) + y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.8e+29) (not (<= x 2.0)))
   (+ (* y y) (* x x))
   (+ (+ x x) (* y y))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.8e+29) || !(x <= 2.0)) {
		tmp = (y * y) + (x * x);
	} else {
		tmp = (x + x) + (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 <= (-1.8d+29)) .or. (.not. (x <= 2.0d0))) then
        tmp = (y * y) + (x * x)
    else
        tmp = (x + x) + (y * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.8e+29) || !(x <= 2.0)) {
		tmp = (y * y) + (x * x);
	} else {
		tmp = (x + x) + (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.8e+29) or not (x <= 2.0):
		tmp = (y * y) + (x * x)
	else:
		tmp = (x + x) + (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.8e+29) || !(x <= 2.0))
		tmp = Float64(Float64(y * y) + Float64(x * x));
	else
		tmp = Float64(Float64(x + x) + Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.8e+29) || ~((x <= 2.0)))
		tmp = (y * y) + (x * x);
	else
		tmp = (x + x) + (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.8e+29], N[Not[LessEqual[x, 2.0]], $MachinePrecision]], N[(N[(y * y), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(x + x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.79999999999999988e29 or 2 < 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. Taylor expanded in x around inf 99.6%

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

      1. unpow2 99.6%

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

   6. Simplified 99.6%

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

   if -1.79999999999999988e29 < x < 2

   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 98.5%

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

      1. count-2 98.5%

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

   6. Simplified 98.5%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{+29} \lor \neg \left(x \leq 2\right):\\ \;\;\;\;y \cdot y + x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(x + x\right) + y \cdot y\\ \end{array} \]

Alternative 4: 96.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 7 \cdot 10^{-182}:\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y + x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 7e-182) (* x (+ x 2.0)) (+ (* y y) (* x x))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 7e-182) {
		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) <= 7d-182) 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) <= 7e-182) {
		tmp = x * (x + 2.0);
	} else {
		tmp = (y * y) + (x * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 7e-182:
		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) <= 7e-182)
		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) <= 7e-182)
		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], 7e-182], 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) < 6.99999999999999966e-182

   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 0 100.0%

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

   if 6.99999999999999966e-182 < (*.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 95.8%

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

      1. unpow2 95.8%

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

   6. Simplified 95.8%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 7 \cdot 10^{-182}:\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y + x \cdot x\\ \end{array} \]

Alternative 5: 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 6: 72.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{+29}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-5}:\\ \;\;\;\;y \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.7e+29) (* x x) (if (<= x 7.5e-5) (* y y) (* x x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.7e+29) {
		tmp = x * x;
	} else if (x <= 7.5e-5) {
		tmp = y * y;
	} else {
		tmp = 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 (x <= (-2.7d+29)) then
        tmp = x * x
    else if (x <= 7.5d-5) then
        tmp = y * y
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.7e+29) {
		tmp = x * x;
	} else if (x <= 7.5e-5) {
		tmp = y * y;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.7e+29:
		tmp = x * x
	elif x <= 7.5e-5:
		tmp = y * y
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.7e+29)
		tmp = Float64(x * x);
	elseif (x <= 7.5e-5)
		tmp = Float64(y * y);
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.7e+29)
		tmp = x * x;
	elseif (x <= 7.5e-5)
		tmp = y * y;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.7e+29], N[(x * x), $MachinePrecision], If[LessEqual[x, 7.5e-5], N[(y * y), $MachinePrecision], N[(x * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.7e29 or 7.49999999999999934e-5 < 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 35.5%

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

      5. exp-lft-sqr 35.5%

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

      6. log-prod 35.5%

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

      7. add-log-exp 35.6%

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

      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 87.7%

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

   8. Simplified 87.7%

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

   if -2.7e29 < x < 7.49999999999999934e-5

   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 66.4%

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

      1. unpow2 66.4%

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

   6. Simplified 66.4%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{+29}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-5}:\\ \;\;\;\;y \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 7: 41.8% 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 51.4%

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

   5. exp-lft-sqr 51.3%

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

   6. log-prod 51.4%

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

   7. add-log-exp 53.7%

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

   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 43.7%

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

8. Simplified 43.7%

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

9. Final simplification 43.7%

   \[\leadsto x \cdot x \]

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 2023218 
(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)))