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

Percentage Accurate: 99.9% → 99.9%

Time: 5.2s

Alternatives: 6

Speedup: 1.0×

• 24.4% 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, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[x \cdot \left(1 - x \cdot y\right) \]

2. Taylor expanded in x around 0 93.2%

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

   1. mul-1-neg 93.2%

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

   2. unsub-neg 93.2%

      \[\leadsto \color{blue}{x - {x}^{2} \cdot y} \]

   3. unpow2 93.2%

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

   4. sqr-neg 93.2%

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

   5. associate-*r* 99.8%

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

   6. *-commutative 99.8%

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

   7. cancel-sign-sub 99.8%

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

   8. +-commutative 99.8%

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

   9. associate-*r* 99.8%

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

   10. fma-def 99.9%

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

4. Simplified 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 99.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[x \cdot \left(1 - x \cdot y\right) \]

2. Taylor expanded in x around 0 99.9%

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

   1. +-commutative 99.9%

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

   2. mul-1-neg 99.9%

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

   3. distribute-rgt-neg-in 99.9%

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

   4. fma-def 99.9%

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

4. Simplified 99.9%

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

5. Final simplification 99.9%

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

Alternative 3: 72.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.15 \cdot 10^{+59} \lor \neg \left(y \leq 4.6 \cdot 10^{+37}\right):\\ \;\;\;\;y \cdot \left(x \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -3.15e+59) (not (<= y 4.6e+37))) (* y (* x (- x))) x))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -3.15e+59) || !(y <= 4.6e+37)) {
		tmp = y * (x * -x);
	} else {
		tmp = 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 <= (-3.15d+59)) .or. (.not. (y <= 4.6d+37))) then
        tmp = y * (x * -x)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -3.15e+59) || !(y <= 4.6e+37)) {
		tmp = y * (x * -x);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -3.15e+59) or not (y <= 4.6e+37):
		tmp = y * (x * -x)
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -3.15e+59) || !(y <= 4.6e+37))
		tmp = Float64(y * Float64(x * Float64(-x)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -3.15e+59) || ~((y <= 4.6e+37)))
		tmp = y * (x * -x);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -3.15e+59], N[Not[LessEqual[y, 4.6e+37]], $MachinePrecision]], N[(y * N[(x * (-x)), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -3.15e59 or 4.60000000000000005e37 < y

   1. Initial program 99.8%

      \[x \cdot \left(1 - x \cdot y\right) \]

   2. Taylor expanded in x around inf 76.1%

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

      1. mul-1-neg 76.1%

         \[\leadsto \color{blue}{-{x}^{2} \cdot y} \]

      2. unpow2 76.1%

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

   4. Simplified 76.1%

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

   if -3.15e59 < y < 4.60000000000000005e37

   1. Initial program 99.9%

      \[x \cdot \left(1 - x \cdot y\right) \]

   2. Taylor expanded in x around 0 71.7%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.15 \cdot 10^{+59} \lor \neg \left(y \leq 4.6 \cdot 10^{+37}\right):\\ \;\;\;\;y \cdot \left(x \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4: 74.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.7 \cdot 10^{+59} \lor \neg \left(y \leq 4.8 \cdot 10^{+37}\right):\\ \;\;\;\;x \cdot \left(y \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -3.7e+59) (not (<= y 4.8e+37))) (* x (* y (- x))) x))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -3.7e+59) || !(y <= 4.8e+37)) {
		tmp = x * (y * -x);
	} else {
		tmp = 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 <= (-3.7d+59)) .or. (.not. (y <= 4.8d+37))) then
        tmp = x * (y * -x)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -3.7e+59) || !(y <= 4.8e+37)) {
		tmp = x * (y * -x);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -3.7e+59) or not (y <= 4.8e+37):
		tmp = x * (y * -x)
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -3.7e+59) || !(y <= 4.8e+37))
		tmp = Float64(x * Float64(y * Float64(-x)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -3.7e+59) || ~((y <= 4.8e+37)))
		tmp = x * (y * -x);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -3.7e+59], N[Not[LessEqual[y, 4.8e+37]], $MachinePrecision]], N[(x * N[(y * (-x)), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -3.69999999999999997e59 or 4.8e37 < y

   1. Initial program 99.8%

      \[x \cdot \left(1 - x \cdot y\right) \]

   2. Taylor expanded in x around inf 82.7%

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

      1. associate-*r* 82.7%

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

      2. neg-mul-1 82.7%

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

      3. *-commutative 82.7%

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

   4. Simplified 82.7%

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

   if -3.69999999999999997e59 < y < 4.8e37

   1. Initial program 99.9%

      \[x \cdot \left(1 - x \cdot y\right) \]

   2. Taylor expanded in x around 0 71.7%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.7 \cdot 10^{+59} \lor \neg \left(y \leq 4.8 \cdot 10^{+37}\right):\\ \;\;\;\;x \cdot \left(y \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5: 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. Final simplification 99.9%

   \[\leadsto x \cdot \left(1 - x \cdot y\right) \]

Alternative 6: 50.8% 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 99.9%

   \[x \cdot \left(1 - x \cdot y\right) \]

2. Taylor expanded in x around 0 50.7%

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

3. Final simplification 50.7%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023275 
(FPCore (x y)
  :name "Numeric.SpecFunctions:log1p from math-functions-0.1.5.2, A"
  :precision binary64
  (* x (- 1.0 (* x y))))