Numeric.SpecFunctions:log1p from math-functions-0.1.5.2, A

Percentage Accurate: 99.9% → 99.9%

Time: 8.1s

Alternatives: 8

Speedup: 1.0×

• 25.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ x \cdot \left(1 - x \cdot y\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x (- 1.0 (* x y))))

C

double code(double x, double y) {
	return x * (1.0 - (x * y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * (1.0d0 - (x * y))
end function

Java

public static double code(double x, double y) {
	return x * (1.0 - (x * y));
}

Python

def code(x, y):
	return x * (1.0 - (x * y))

Julia

function code(x, y)
	return Float64(x * Float64(1.0 - Float64(x * y)))
end

MATLAB

function tmp = code(x, y)
	tmp = x * (1.0 - (x * y));
end

Wolfram

code[x_, y_] := N[(x * N[(1.0 - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 - x \cdot y\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x (- 1.0 (* x y))))

C

double code(double x, double y) {
	return x * (1.0 - (x * y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * (1.0d0 - (x * y))
end function

Java

public static double code(double x, double y) {
	return x * (1.0 - (x * y));
}

Python

def code(x, y):
	return x * (1.0 - (x * y))

Julia

function code(x, y)
	return Float64(x * Float64(1.0 - Float64(x * y)))
end

MATLAB

function tmp = code(x, y)
	tmp = x * (1.0 - (x * y));
end

Wolfram

code[x_, y_] := N[(x * N[(1.0 - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 - x \cdot y\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x (- 1.0 (* x y))))

C

double code(double x, double y) {
	return x * (1.0 - (x * y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * (1.0d0 - (x * y))
end function

Java

public static double code(double x, double y) {
	return x * (1.0 - (x * y));
}

Python

def code(x, y):
	return x * (1.0 - (x * y))

Julia

function code(x, y)
	return Float64(x * Float64(1.0 - Float64(x * y)))
end

MATLAB

function tmp = code(x, y)
	tmp = x * (1.0 - (x * y));
end

Wolfram

code[x_, y_] := N[(x * N[(1.0 - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[x \cdot \left(1 - x \cdot y\right) \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 80.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(1 - x \cdot y\right)\\ t_1 := x \cdot \left(y \cdot \left(-x\right)\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+262}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{+103}:\\ \;\;\;\;x \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (- 1.0 (* x y)))) (t_1 (* x (* y (- x)))))
   (if (<= t_0 -2e+262) t_1 (if (<= t_0 1e+103) (* x 1.0) t_1))))

C

double code(double x, double y) {
	double t_0 = x * (1.0 - (x * y));
	double t_1 = x * (y * -x);
	double tmp;
	if (t_0 <= -2e+262) {
		tmp = t_1;
	} else if (t_0 <= 1e+103) {
		tmp = x * 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x * (1.0d0 - (x * y))
    t_1 = x * (y * -x)
    if (t_0 <= (-2d+262)) then
        tmp = t_1
    else if (t_0 <= 1d+103) then
        tmp = x * 1.0d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x * (1.0 - (x * y));
	double t_1 = x * (y * -x);
	double tmp;
	if (t_0 <= -2e+262) {
		tmp = t_1;
	} else if (t_0 <= 1e+103) {
		tmp = x * 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * (1.0 - (x * y))
	t_1 = x * (y * -x)
	tmp = 0
	if t_0 <= -2e+262:
		tmp = t_1
	elif t_0 <= 1e+103:
		tmp = x * 1.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x * Float64(1.0 - Float64(x * y)))
	t_1 = Float64(x * Float64(y * Float64(-x)))
	tmp = 0.0
	if (t_0 <= -2e+262)
		tmp = t_1;
	elseif (t_0 <= 1e+103)
		tmp = Float64(x * 1.0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x * (1.0 - (x * y));
	t_1 = x * (y * -x);
	tmp = 0.0;
	if (t_0 <= -2e+262)
		tmp = t_1;
	elseif (t_0 <= 1e+103)
		tmp = x * 1.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(1.0 - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(y * (-x)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+262], t$95$1, If[LessEqual[t$95$0, 1e+103], N[(x * 1.0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (-.f64 #s(literal 1 binary64) (*.f64 x y))) < -2e262 or 1e103 < (*.f64 x (-.f64 #s(literal 1 binary64) (*.f64 x y)))

   1. Initial program 99.9%

      \[x \cdot \left(1 - x \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 92.6

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

   5. Applied rewrites 92.6%

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

   if -2e262 < (*.f64 x (-.f64 #s(literal 1 binary64) (*.f64 x y))) < 1e103

   1. Initial program 99.9%

      \[x \cdot \left(1 - x \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto x \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 84.1%

      \[\leadsto x \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \left(1 - x \cdot y\right) \leq -2 \cdot 10^{+262}:\\ \;\;\;\;x \cdot \left(y \cdot \left(-x\right)\right)\\ \mathbf{elif}\;x \cdot \left(1 - x \cdot y\right) \leq 10^{+103}:\\ \;\;\;\;x \cdot 1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \left(-x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 62.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(1 - x \cdot y\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+266}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+222}:\\ \;\;\;\;x \cdot 1\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (- 1.0 (* x y)))))
   (if (<= t_0 -5e+266) (* y (- x)) (if (<= t_0 5e+222) (* x 1.0) (* x x)))))

C

double code(double x, double y) {
	double t_0 = x * (1.0 - (x * y));
	double tmp;
	if (t_0 <= -5e+266) {
		tmp = y * -x;
	} else if (t_0 <= 5e+222) {
		tmp = x * 1.0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = x * (1.0d0 - (x * y))
    if (t_0 <= (-5d+266)) then
        tmp = y * -x
    else if (t_0 <= 5d+222) then
        tmp = x * 1.0d0
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x * (1.0 - (x * y));
	double tmp;
	if (t_0 <= -5e+266) {
		tmp = y * -x;
	} else if (t_0 <= 5e+222) {
		tmp = x * 1.0;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * (1.0 - (x * y))
	tmp = 0
	if t_0 <= -5e+266:
		tmp = y * -x
	elif t_0 <= 5e+222:
		tmp = x * 1.0
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x * Float64(1.0 - Float64(x * y)))
	tmp = 0.0
	if (t_0 <= -5e+266)
		tmp = Float64(y * Float64(-x));
	elseif (t_0 <= 5e+222)
		tmp = Float64(x * 1.0);
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x * (1.0 - (x * y));
	tmp = 0.0;
	if (t_0 <= -5e+266)
		tmp = y * -x;
	elseif (t_0 <= 5e+222)
		tmp = x * 1.0;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(1.0 - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+266], N[(y * (-x)), $MachinePrecision], If[LessEqual[t$95$0, 5e+222], N[(x * 1.0), $MachinePrecision], N[(x * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x (-.f64 #s(literal 1 binary64) (*.f64 x y))) < -4.9999999999999999e266

   1. Initial program 99.9%

      \[x \cdot \left(1 - x \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 23.8

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

   5. Applied rewrites 23.8%

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

   if -4.9999999999999999e266 < (*.f64 x (-.f64 #s(literal 1 binary64) (*.f64 x y))) < 5.00000000000000023e222

   1. Initial program 99.9%

      \[x \cdot \left(1 - x \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto x \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 79.8%

      \[\leadsto x \cdot \color{blue}{1} \]

   if 5.00000000000000023e222 < (*.f64 x (-.f64 #s(literal 1 binary64) (*.f64 x y)))

   1. Initial program 100.0%

      \[x \cdot \left(1 - x \cdot y\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-lft-neg-in N/A

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

      8. associate-*r* N/A

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

      9. *-rgt-identity N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 95.6

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

   4. Applied rewrites 95.6%

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

   6. Applied rewrites 0.6%

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

   7. Taylor expanded in y around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 63.6

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

   9. Applied rewrites 63.6%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \left(1 - x \cdot y\right) \leq -5 \cdot 10^{+266}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;x \cdot \left(1 - x \cdot y\right) \leq 5 \cdot 10^{+222}:\\ \;\;\;\;x \cdot 1\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(1 - x \cdot y\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+302}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+222}:\\ \;\;\;\;x \cdot 1\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (- 1.0 (* x y)))))
   (if (<= t_0 -2e+302) (* x y) (if (<= t_0 5e+222) (* x 1.0) (* x x)))))

C

double code(double x, double y) {
	double t_0 = x * (1.0 - (x * y));
	double tmp;
	if (t_0 <= -2e+302) {
		tmp = x * y;
	} else if (t_0 <= 5e+222) {
		tmp = x * 1.0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = x * (1.0d0 - (x * y))
    if (t_0 <= (-2d+302)) then
        tmp = x * y
    else if (t_0 <= 5d+222) then
        tmp = x * 1.0d0
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x * (1.0 - (x * y));
	double tmp;
	if (t_0 <= -2e+302) {
		tmp = x * y;
	} else if (t_0 <= 5e+222) {
		tmp = x * 1.0;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * (1.0 - (x * y))
	tmp = 0
	if t_0 <= -2e+302:
		tmp = x * y
	elif t_0 <= 5e+222:
		tmp = x * 1.0
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x * Float64(1.0 - Float64(x * y)))
	tmp = 0.0
	if (t_0 <= -2e+302)
		tmp = Float64(x * y);
	elseif (t_0 <= 5e+222)
		tmp = Float64(x * 1.0);
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x * (1.0 - (x * y));
	tmp = 0.0;
	if (t_0 <= -2e+302)
		tmp = x * y;
	elseif (t_0 <= 5e+222)
		tmp = x * 1.0;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(1.0 - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+302], N[(x * y), $MachinePrecision], If[LessEqual[t$95$0, 5e+222], N[(x * 1.0), $MachinePrecision], N[(x * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x (-.f64 #s(literal 1 binary64) (*.f64 x y))) < -2.0000000000000002e302

   1. Initial program 100.0%

      \[x \cdot \left(1 - x \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 25.6

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

   5. Applied rewrites 25.6%

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

   7. Applied rewrites 20.9%

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

   if -2.0000000000000002e302 < (*.f64 x (-.f64 #s(literal 1 binary64) (*.f64 x y))) < 5.00000000000000023e222

   1. Initial program 99.9%

      \[x \cdot \left(1 - x \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto x \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 77.6%

      \[\leadsto x \cdot \color{blue}{1} \]

   if 5.00000000000000023e222 < (*.f64 x (-.f64 #s(literal 1 binary64) (*.f64 x y)))

   1. Initial program 100.0%

      \[x \cdot \left(1 - x \cdot y\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-lft-neg-in N/A

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

      8. associate-*r* N/A

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

      9. *-rgt-identity N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 95.6

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

   4. Applied rewrites 95.6%

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

   6. Applied rewrites 0.6%

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

   7. Taylor expanded in y around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 63.6

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

   9. Applied rewrites 63.6%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \left(1 - x \cdot y\right) \leq -2 \cdot 10^{+302}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot \left(1 - x \cdot y\right) \leq 5 \cdot 10^{+222}:\\ \;\;\;\;x \cdot 1\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 17.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot \left(1 - x \cdot y\right) \leq -2 \cdot 10^{-161}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x (- 1.0 (* x y))) -2e-161) (* x y) (* x x)))

C

double code(double x, double y) {
	double tmp;
	if ((x * (1.0 - (x * y))) <= -2e-161) {
		tmp = x * 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 * (1.0d0 - (x * y))) <= (-2d-161)) then
        tmp = x * y
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x * (1.0 - (x * y))) <= -2e-161) {
		tmp = x * y;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x * (1.0 - (x * y))) <= -2e-161:
		tmp = x * y
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * Float64(1.0 - Float64(x * y))) <= -2e-161)
		tmp = Float64(x * y);
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x * (1.0 - (x * y))) <= -2e-161)
		tmp = x * y;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * N[(1.0 - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-161], N[(x * y), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (-.f64 #s(literal 1 binary64) (*.f64 x y))) < -2.00000000000000006e-161

   1. Initial program 99.9%

      \[x \cdot \left(1 - x \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 12.7

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

   5. Applied rewrites 12.7%

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

   7. Applied rewrites 10.7%

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

   if -2.00000000000000006e-161 < (*.f64 x (-.f64 #s(literal 1 binary64) (*.f64 x y)))

   1. Initial program 99.9%

      \[x \cdot \left(1 - x \cdot y\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-lft-neg-in N/A

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

      8. associate-*r* N/A

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

      9. *-rgt-identity N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 95.4

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

   4. Applied rewrites 95.4%

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

   6. Applied rewrites 27.1%

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

   7. Taylor expanded in y around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 21.9

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

   9. Applied rewrites 21.9%

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

4. Final simplification 17.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \left(1 - x \cdot y\right) \leq -2 \cdot 10^{-161}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 10.1% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[x \cdot \left(1 - x \cdot y\right) \]
2. Add Preprocessing

3. Taylor expanded in x around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-*.f64 10.4

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

5. Applied rewrites 10.4%

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

7. Applied rewrites 9.6%

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

8. Final simplification 9.6%

   \[\leadsto x \cdot y \]
9. Add Preprocessing

Alternative 7: 4.4% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ -y \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return -y

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := (-y)

Derivation

1. Initial program 99.9%

   \[x \cdot \left(1 - x \cdot y\right) \]
2. Add Preprocessing

4. Applied rewrites 69.3%

   \[\leadsto \color{blue}{\left(-x \cdot \mathsf{fma}\left(x, x \cdot \left(y \cdot y\right), 1\right)\right) \cdot \frac{1}{\mathsf{fma}\left(x, y, -1\right)}} \]

5. Taylor expanded in x around -inf

   \[\leadsto \color{blue}{-1 \cdot y} \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(y\right)} \]

   2. lower-neg.f64 4.0

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

7. Applied rewrites 4.0%

   \[\leadsto \color{blue}{-y} \]
8. Add Preprocessing

Alternative 8: 3.1% accurate, 14.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 1.0)

C

double code(double x, double y) {
	return 1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0
end function

Java

public static double code(double x, double y) {
	return 1.0;
}

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0;
end

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 99.9%

   \[x \cdot \left(1 - x \cdot y\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \]
4. Step-by-step derivation

5. Applied rewrites 3.1%

   \[\leadsto \color{blue}{1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024223 
(FPCore (x y)
  :name "Numeric.SpecFunctions:log1p from math-functions-0.1.5.2, A"
  :precision binary64
  (* x (- 1.0 (* x y))))