Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B

Percentage Accurate: 100.0% → 100.0%

Time: 3.4s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. +-commutative 100.0%

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

   2. fma-def 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 60.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+276}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -2.05 \cdot 10^{+173}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq -3.8 \cdot 10^{+41}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -6.8 \cdot 10^{+14}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq -1.45 \cdot 10^{-130}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.35e+276)
   x
   (if (<= x -2.05e+173)
     (* y x)
     (if (<= x -3.8e+41)
       x
       (if (<= x -6.8e+14)
         (* y x)
         (if (<= x -1.45e-130) x (if (<= x 1.0) y (* y x))))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.35e+276) {
		tmp = x;
	} else if (x <= -2.05e+173) {
		tmp = y * x;
	} else if (x <= -3.8e+41) {
		tmp = x;
	} else if (x <= -6.8e+14) {
		tmp = y * x;
	} else if (x <= -1.45e-130) {
		tmp = x;
	} else if (x <= 1.0) {
		tmp = y;
	} else {
		tmp = y * 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 <= (-1.35d+276)) then
        tmp = x
    else if (x <= (-2.05d+173)) then
        tmp = y * x
    else if (x <= (-3.8d+41)) then
        tmp = x
    else if (x <= (-6.8d+14)) then
        tmp = y * x
    else if (x <= (-1.45d-130)) then
        tmp = x
    else if (x <= 1.0d0) then
        tmp = y
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.35e+276) {
		tmp = x;
	} else if (x <= -2.05e+173) {
		tmp = y * x;
	} else if (x <= -3.8e+41) {
		tmp = x;
	} else if (x <= -6.8e+14) {
		tmp = y * x;
	} else if (x <= -1.45e-130) {
		tmp = x;
	} else if (x <= 1.0) {
		tmp = y;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.35e+276:
		tmp = x
	elif x <= -2.05e+173:
		tmp = y * x
	elif x <= -3.8e+41:
		tmp = x
	elif x <= -6.8e+14:
		tmp = y * x
	elif x <= -1.45e-130:
		tmp = x
	elif x <= 1.0:
		tmp = y
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.35e+276)
		tmp = x;
	elseif (x <= -2.05e+173)
		tmp = Float64(y * x);
	elseif (x <= -3.8e+41)
		tmp = x;
	elseif (x <= -6.8e+14)
		tmp = Float64(y * x);
	elseif (x <= -1.45e-130)
		tmp = x;
	elseif (x <= 1.0)
		tmp = y;
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.35e+276)
		tmp = x;
	elseif (x <= -2.05e+173)
		tmp = y * x;
	elseif (x <= -3.8e+41)
		tmp = x;
	elseif (x <= -6.8e+14)
		tmp = y * x;
	elseif (x <= -1.45e-130)
		tmp = x;
	elseif (x <= 1.0)
		tmp = y;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.35e+276], x, If[LessEqual[x, -2.05e+173], N[(y * x), $MachinePrecision], If[LessEqual[x, -3.8e+41], x, If[LessEqual[x, -6.8e+14], N[(y * x), $MachinePrecision], If[LessEqual[x, -1.45e-130], x, If[LessEqual[x, 1.0], y, N[(y * x), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.34999999999999998e276 or -2.04999999999999988e173 < x < -3.8000000000000001e41 or -6.8e14 < x < -1.45e-130

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 62.5%

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

   if -1.34999999999999998e276 < x < -2.04999999999999988e173 or -3.8000000000000001e41 < x < -6.8e14 or 1 < x

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 100.0%

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

   5. Taylor expanded in y around inf 57.0%

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

   if -1.45e-130 < x < 1

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 74.4%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+276}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -2.05 \cdot 10^{+173}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq -3.8 \cdot 10^{+41}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -6.8 \cdot 10^{+14}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq -1.45 \cdot 10^{-130}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 3: 75.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.6 \cdot 10^{+275}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -3.45 \cdot 10^{+174}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq -2.05 \cdot 10^{+40}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -2.3 \cdot 10^{+15}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 12500000000:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -6.6e+275)
   x
   (if (<= x -3.45e+174)
     (* y x)
     (if (<= x -2.05e+40)
       x
       (if (<= x -2.3e+15)
         (* y x)
         (if (<= x 12500000000.0) (+ y x) (* y x)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -6.6e+275) {
		tmp = x;
	} else if (x <= -3.45e+174) {
		tmp = y * x;
	} else if (x <= -2.05e+40) {
		tmp = x;
	} else if (x <= -2.3e+15) {
		tmp = y * x;
	} else if (x <= 12500000000.0) {
		tmp = y + x;
	} else {
		tmp = y * 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 <= (-6.6d+275)) then
        tmp = x
    else if (x <= (-3.45d+174)) then
        tmp = y * x
    else if (x <= (-2.05d+40)) then
        tmp = x
    else if (x <= (-2.3d+15)) then
        tmp = y * x
    else if (x <= 12500000000.0d0) then
        tmp = y + x
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -6.6e+275) {
		tmp = x;
	} else if (x <= -3.45e+174) {
		tmp = y * x;
	} else if (x <= -2.05e+40) {
		tmp = x;
	} else if (x <= -2.3e+15) {
		tmp = y * x;
	} else if (x <= 12500000000.0) {
		tmp = y + x;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -6.6e+275:
		tmp = x
	elif x <= -3.45e+174:
		tmp = y * x
	elif x <= -2.05e+40:
		tmp = x
	elif x <= -2.3e+15:
		tmp = y * x
	elif x <= 12500000000.0:
		tmp = y + x
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -6.6e+275)
		tmp = x;
	elseif (x <= -3.45e+174)
		tmp = Float64(y * x);
	elseif (x <= -2.05e+40)
		tmp = x;
	elseif (x <= -2.3e+15)
		tmp = Float64(y * x);
	elseif (x <= 12500000000.0)
		tmp = Float64(y + x);
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -6.6e+275)
		tmp = x;
	elseif (x <= -3.45e+174)
		tmp = y * x;
	elseif (x <= -2.05e+40)
		tmp = x;
	elseif (x <= -2.3e+15)
		tmp = y * x;
	elseif (x <= 12500000000.0)
		tmp = y + x;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -6.6e+275], x, If[LessEqual[x, -3.45e+174], N[(y * x), $MachinePrecision], If[LessEqual[x, -2.05e+40], x, If[LessEqual[x, -2.3e+15], N[(y * x), $MachinePrecision], If[LessEqual[x, 12500000000.0], N[(y + x), $MachinePrecision], N[(y * x), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.60000000000000044e275 or -3.4500000000000001e174 < x < -2.0500000000000001e40

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 62.6%

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

   if -6.60000000000000044e275 < x < -3.4500000000000001e174 or -2.0500000000000001e40 < x < -2.3e15 or 1.25e10 < x

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 100.0%

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

   5. Taylor expanded in y around inf 57.6%

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

   if -2.3e15 < x < 1.25e10

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-def 100.0%

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

   3. Simplified 100.0%

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

      1. fma-def 100.0%

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

      2. +-commutative 100.0%

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

      3. add-sqr-sqrt 51.9%

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

      4. pow2 51.9%

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

      5. +-commutative 51.9%

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

      6. *-commutative 51.9%

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

      7. fma-def 51.9%

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

   5. Applied egg-rr 51.9%

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

   6. Taylor expanded in y around -inf 51.9%

      \[\leadsto {\left(\sqrt{\color{blue}{x + -1 \cdot \left(\left(-1 \cdot x - 1\right) \cdot y\right)}}\right)}^{2} \]
   7. Step-by-step derivation

      1. mul-1-neg 51.9%

         \[\leadsto {\left(\sqrt{x + \color{blue}{\left(-\left(-1 \cdot x - 1\right) \cdot y\right)}}\right)}^{2} \]

      2. unsub-neg 51.9%

         \[\leadsto {\left(\sqrt{\color{blue}{x - \left(-1 \cdot x - 1\right) \cdot y}}\right)}^{2} \]

      3. *-commutative 51.9%

         \[\leadsto {\left(\sqrt{x - \color{blue}{y \cdot \left(-1 \cdot x - 1\right)}}\right)}^{2} \]

      4. sub-neg 51.9%

         \[\leadsto {\left(\sqrt{x - y \cdot \color{blue}{\left(-1 \cdot x + \left(-1\right)\right)}}\right)}^{2} \]

      5. metadata-eval 51.9%

         \[\leadsto {\left(\sqrt{x - y \cdot \left(-1 \cdot x + \color{blue}{-1}\right)}\right)}^{2} \]

      6. +-commutative 51.9%

         \[\leadsto {\left(\sqrt{x - y \cdot \color{blue}{\left(-1 + -1 \cdot x\right)}}\right)}^{2} \]

      7. mul-1-neg 51.9%

         \[\leadsto {\left(\sqrt{x - y \cdot \left(-1 + \color{blue}{\left(-x\right)}\right)}\right)}^{2} \]

      8. unsub-neg 51.9%

         \[\leadsto {\left(\sqrt{x - y \cdot \color{blue}{\left(-1 - x\right)}}\right)}^{2} \]

   8. Simplified 51.9%

      \[\leadsto {\left(\sqrt{\color{blue}{x - y \cdot \left(-1 - x\right)}}\right)}^{2} \]

   9. Taylor expanded in x around 0 51.4%

      \[\leadsto {\left(\sqrt{x - \color{blue}{-1 \cdot y}}\right)}^{2} \]
   10. Step-by-step derivation

       1. neg-mul-1 51.4%

          \[\leadsto {\left(\sqrt{x - \color{blue}{\left(-y\right)}}\right)}^{2} \]

   11. Simplified 51.4%

       \[\leadsto {\left(\sqrt{x - \color{blue}{\left(-y\right)}}\right)}^{2} \]

   12. Taylor expanded in x around 0 98.8%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.6 \cdot 10^{+275}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -3.45 \cdot 10^{+174}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq -2.05 \cdot 10^{+40}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -2.3 \cdot 10^{+15}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 12500000000:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 4: 84.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1950000000000:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-57}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;y + y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1950000000000.0)
   (* x (+ y 1.0))
   (if (<= x 7e-57) (+ y x) (+ y (* y x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1950000000000.0) {
		tmp = x * (y + 1.0);
	} else if (x <= 7e-57) {
		tmp = y + x;
	} else {
		tmp = y + (y * 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 <= (-1950000000000.0d0)) then
        tmp = x * (y + 1.0d0)
    else if (x <= 7d-57) then
        tmp = y + x
    else
        tmp = y + (y * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1950000000000.0) {
		tmp = x * (y + 1.0);
	} else if (x <= 7e-57) {
		tmp = y + x;
	} else {
		tmp = y + (y * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1950000000000.0:
		tmp = x * (y + 1.0)
	elif x <= 7e-57:
		tmp = y + x
	else:
		tmp = y + (y * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1950000000000.0)
		tmp = Float64(x * Float64(y + 1.0));
	elseif (x <= 7e-57)
		tmp = Float64(y + x);
	else
		tmp = Float64(y + Float64(y * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1950000000000.0)
		tmp = x * (y + 1.0);
	elseif (x <= 7e-57)
		tmp = y + x;
	else
		tmp = y + (y * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1950000000000.0], N[(x * N[(y + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7e-57], N[(y + x), $MachinePrecision], N[(y + N[(y * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.95e12

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 100.0%

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

   if -1.95e12 < x < 6.99999999999999983e-57

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-def 100.0%

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

   3. Simplified 100.0%

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

      1. fma-def 100.0%

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

      2. +-commutative 100.0%

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

      3. add-sqr-sqrt 49.6%

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

      4. pow2 49.6%

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

      5. +-commutative 49.6%

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

      6. *-commutative 49.6%

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

      7. fma-def 49.6%

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

   5. Applied egg-rr 49.6%

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

   6. Taylor expanded in y around -inf 49.6%

      \[\leadsto {\left(\sqrt{\color{blue}{x + -1 \cdot \left(\left(-1 \cdot x - 1\right) \cdot y\right)}}\right)}^{2} \]
   7. Step-by-step derivation

      1. mul-1-neg 49.6%

         \[\leadsto {\left(\sqrt{x + \color{blue}{\left(-\left(-1 \cdot x - 1\right) \cdot y\right)}}\right)}^{2} \]

      2. unsub-neg 49.6%

         \[\leadsto {\left(\sqrt{\color{blue}{x - \left(-1 \cdot x - 1\right) \cdot y}}\right)}^{2} \]

      3. *-commutative 49.6%

         \[\leadsto {\left(\sqrt{x - \color{blue}{y \cdot \left(-1 \cdot x - 1\right)}}\right)}^{2} \]

      4. sub-neg 49.6%

         \[\leadsto {\left(\sqrt{x - y \cdot \color{blue}{\left(-1 \cdot x + \left(-1\right)\right)}}\right)}^{2} \]

      5. metadata-eval 49.6%

         \[\leadsto {\left(\sqrt{x - y \cdot \left(-1 \cdot x + \color{blue}{-1}\right)}\right)}^{2} \]

      6. +-commutative 49.6%

         \[\leadsto {\left(\sqrt{x - y \cdot \color{blue}{\left(-1 + -1 \cdot x\right)}}\right)}^{2} \]

      7. mul-1-neg 49.6%

         \[\leadsto {\left(\sqrt{x - y \cdot \left(-1 + \color{blue}{\left(-x\right)}\right)}\right)}^{2} \]

      8. unsub-neg 49.6%

         \[\leadsto {\left(\sqrt{x - y \cdot \color{blue}{\left(-1 - x\right)}}\right)}^{2} \]

   8. Simplified 49.6%

      \[\leadsto {\left(\sqrt{\color{blue}{x - y \cdot \left(-1 - x\right)}}\right)}^{2} \]

   9. Taylor expanded in x around 0 49.6%

      \[\leadsto {\left(\sqrt{x - \color{blue}{-1 \cdot y}}\right)}^{2} \]
   10. Step-by-step derivation

       1. neg-mul-1 49.6%

          \[\leadsto {\left(\sqrt{x - \color{blue}{\left(-y\right)}}\right)}^{2} \]

   11. Simplified 49.6%

       \[\leadsto {\left(\sqrt{x - \color{blue}{\left(-y\right)}}\right)}^{2} \]

   12. Taylor expanded in x around 0 99.6%

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

   if 6.99999999999999983e-57 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 51.9%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1950000000000:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-57}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;y + y \cdot x\\ \end{array} \]

Alternative 5: 87.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1950000000000:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{elif}\;x \leq 1900000000000:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1950000000000.0)
   (* x (+ y 1.0))
   (if (<= x 1900000000000.0) (+ y x) (* y x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1950000000000.0) {
		tmp = x * (y + 1.0);
	} else if (x <= 1900000000000.0) {
		tmp = y + x;
	} else {
		tmp = y * 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 <= (-1950000000000.0d0)) then
        tmp = x * (y + 1.0d0)
    else if (x <= 1900000000000.0d0) then
        tmp = y + x
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1950000000000.0) {
		tmp = x * (y + 1.0);
	} else if (x <= 1900000000000.0) {
		tmp = y + x;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1950000000000.0:
		tmp = x * (y + 1.0)
	elif x <= 1900000000000.0:
		tmp = y + x
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1950000000000.0)
		tmp = Float64(x * Float64(y + 1.0));
	elseif (x <= 1900000000000.0)
		tmp = Float64(y + x);
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1950000000000.0)
		tmp = x * (y + 1.0);
	elseif (x <= 1900000000000.0)
		tmp = y + x;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1950000000000.0], N[(x * N[(y + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1900000000000.0], N[(y + x), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.95e12

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 100.0%

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

   if -1.95e12 < x < 1.9e12

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-def 100.0%

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

   3. Simplified 100.0%

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

      1. fma-def 100.0%

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

      2. +-commutative 100.0%

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

      3. add-sqr-sqrt 51.9%

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

      4. pow2 51.9%

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

      5. +-commutative 51.9%

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

      6. *-commutative 51.9%

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

      7. fma-def 51.9%

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

   5. Applied egg-rr 51.9%

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

   6. Taylor expanded in y around -inf 51.9%

      \[\leadsto {\left(\sqrt{\color{blue}{x + -1 \cdot \left(\left(-1 \cdot x - 1\right) \cdot y\right)}}\right)}^{2} \]
   7. Step-by-step derivation

      1. mul-1-neg 51.9%

         \[\leadsto {\left(\sqrt{x + \color{blue}{\left(-\left(-1 \cdot x - 1\right) \cdot y\right)}}\right)}^{2} \]

      2. unsub-neg 51.9%

         \[\leadsto {\left(\sqrt{\color{blue}{x - \left(-1 \cdot x - 1\right) \cdot y}}\right)}^{2} \]

      3. *-commutative 51.9%

         \[\leadsto {\left(\sqrt{x - \color{blue}{y \cdot \left(-1 \cdot x - 1\right)}}\right)}^{2} \]

      4. sub-neg 51.9%

         \[\leadsto {\left(\sqrt{x - y \cdot \color{blue}{\left(-1 \cdot x + \left(-1\right)\right)}}\right)}^{2} \]

      5. metadata-eval 51.9%

         \[\leadsto {\left(\sqrt{x - y \cdot \left(-1 \cdot x + \color{blue}{-1}\right)}\right)}^{2} \]

      6. +-commutative 51.9%

         \[\leadsto {\left(\sqrt{x - y \cdot \color{blue}{\left(-1 + -1 \cdot x\right)}}\right)}^{2} \]

      7. mul-1-neg 51.9%

         \[\leadsto {\left(\sqrt{x - y \cdot \left(-1 + \color{blue}{\left(-x\right)}\right)}\right)}^{2} \]

      8. unsub-neg 51.9%

         \[\leadsto {\left(\sqrt{x - y \cdot \color{blue}{\left(-1 - x\right)}}\right)}^{2} \]

   8. Simplified 51.9%

      \[\leadsto {\left(\sqrt{\color{blue}{x - y \cdot \left(-1 - x\right)}}\right)}^{2} \]

   9. Taylor expanded in x around 0 51.4%

      \[\leadsto {\left(\sqrt{x - \color{blue}{-1 \cdot y}}\right)}^{2} \]
   10. Step-by-step derivation

       1. neg-mul-1 51.4%

          \[\leadsto {\left(\sqrt{x - \color{blue}{\left(-y\right)}}\right)}^{2} \]

   11. Simplified 51.4%

       \[\leadsto {\left(\sqrt{x - \color{blue}{\left(-y\right)}}\right)}^{2} \]

   12. Taylor expanded in x around 0 98.8%

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

   if 1.9e12 < x

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 100.0%

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

   5. Taylor expanded in y around inf 52.0%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1950000000000:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{elif}\;x \leq 1900000000000:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 6: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Final simplification 100.0%

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

Alternative 7: 49.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-129}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x -2.2e-129) x y))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.2e-129) {
		tmp = x;
	} else {
		tmp = 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 <= (-2.2d-129)) then
        tmp = x
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.2e-129) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.2e-129:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.2e-129)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.2e-129)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.2e-129], x, y]

Derivation

1. Split input into 2 regimes

2. if x < -2.20000000000000003e-129

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 52.7%

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

   if -2.20000000000000003e-129 < x

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 47.1%

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

4. Final simplification 49.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-129}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 8: 38.8% accurate, 7.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

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

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 100.0%

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

   1. +-commutative 100.0%

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

   2. fma-def 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in y around 0 42.5%

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

5. Final simplification 42.5%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023253 
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  :precision binary64
  (+ (+ (* x y) x) y))