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

Percentage Accurate: 99.9% → 99.9%

Time: 6.8s

Alternatives: 4

Speedup: 1.0×

• 24.1% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. sub-neg 99.9%

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

   2. distribute-rgt-in 99.9%

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

   3. *-un-lft-identity 99.9%

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

   4. distribute-rgt-neg-in 99.9%

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

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

   \[\leadsto x - x \cdot \left(x \cdot y\right) \]
6. Add Preprocessing

Alternative 2: 73.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{-66} \lor \neg \left(y \leq 2.1 \cdot 10^{-100}\right):\\ \;\;\;\;x \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -3.5e-66) (not (<= y 2.1e-100))) (* x (* x (- y))) x))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -3.5e-66) || !(y <= 2.1e-100)) {
		tmp = x * (x * -y);
	} 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 <= (-3.5d-66)) .or. (.not. (y <= 2.1d-100))) then
        tmp = x * (x * -y)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -3.5e-66) || !(y <= 2.1e-100)) {
		tmp = x * (x * -y);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -3.5e-66) or not (y <= 2.1e-100):
		tmp = x * (x * -y)
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -3.5e-66) || !(y <= 2.1e-100))
		tmp = Float64(x * Float64(x * Float64(-y)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -3.5e-66) || ~((y <= 2.1e-100)))
		tmp = x * (x * -y);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -3.5e-66], N[Not[LessEqual[y, 2.1e-100]], $MachinePrecision]], N[(x * N[(x * (-y)), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -3.5e-66 or 2.10000000000000009e-100 < y

   1. Initial program 99.8%

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

      1. flip3-- 47.3%

         \[\leadsto x \cdot \color{blue}{\frac{{1}^{3} - {\left(x \cdot y\right)}^{3}}{1 \cdot 1 + \left(\left(x \cdot y\right) \cdot \left(x \cdot y\right) + 1 \cdot \left(x \cdot y\right)\right)}} \]

      2. associate-*r/ 46.7%

         \[\leadsto \color{blue}{\frac{x \cdot \left({1}^{3} - {\left(x \cdot y\right)}^{3}\right)}{1 \cdot 1 + \left(\left(x \cdot y\right) \cdot \left(x \cdot y\right) + 1 \cdot \left(x \cdot y\right)\right)}} \]

      3. metadata-eval 46.7%

         \[\leadsto \frac{x \cdot \left(\color{blue}{1} - {\left(x \cdot y\right)}^{3}\right)}{1 \cdot 1 + \left(\left(x \cdot y\right) \cdot \left(x \cdot y\right) + 1 \cdot \left(x \cdot y\right)\right)} \]

      4. metadata-eval 46.7%

         \[\leadsto \frac{x \cdot \left(1 - {\left(x \cdot y\right)}^{3}\right)}{\color{blue}{1} + \left(\left(x \cdot y\right) \cdot \left(x \cdot y\right) + 1 \cdot \left(x \cdot y\right)\right)} \]

      5. +-commutative 46.7%

         \[\leadsto \frac{x \cdot \left(1 - {\left(x \cdot y\right)}^{3}\right)}{\color{blue}{\left(\left(x \cdot y\right) \cdot \left(x \cdot y\right) + 1 \cdot \left(x \cdot y\right)\right) + 1}} \]

      6. distribute-rgt-out 46.7%

         \[\leadsto \frac{x \cdot \left(1 - {\left(x \cdot y\right)}^{3}\right)}{\color{blue}{\left(x \cdot y\right) \cdot \left(x \cdot y + 1\right)} + 1} \]

      7. +-commutative 46.7%

         \[\leadsto \frac{x \cdot \left(1 - {\left(x \cdot y\right)}^{3}\right)}{\left(x \cdot y\right) \cdot \color{blue}{\left(1 + x \cdot y\right)} + 1} \]

      8. fma-define 46.7%

         \[\leadsto \frac{x \cdot \left(1 - {\left(x \cdot y\right)}^{3}\right)}{\color{blue}{\mathsf{fma}\left(x \cdot y, 1 + x \cdot y, 1\right)}} \]

      9. +-commutative 46.7%

         \[\leadsto \frac{x \cdot \left(1 - {\left(x \cdot y\right)}^{3}\right)}{\mathsf{fma}\left(x \cdot y, \color{blue}{x \cdot y + 1}, 1\right)} \]

      10. fma-define 46.7%

          \[\leadsto \frac{x \cdot \left(1 - {\left(x \cdot y\right)}^{3}\right)}{\mathsf{fma}\left(x \cdot y, \color{blue}{\mathsf{fma}\left(x, y, 1\right)}, 1\right)} \]

   4. Applied egg-rr 46.7%

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

      1. associate-/l* 47.3%

         \[\leadsto \color{blue}{x \cdot \frac{1 - {\left(x \cdot y\right)}^{3}}{\mathsf{fma}\left(x \cdot y, \mathsf{fma}\left(x, y, 1\right), 1\right)}} \]

   6. Simplified 47.3%

      \[\leadsto \color{blue}{x \cdot \frac{1 - {\left(x \cdot y\right)}^{3}}{\mathsf{fma}\left(x \cdot y, \mathsf{fma}\left(x, y, 1\right), 1\right)}} \]

   7. Taylor expanded in x around inf 75.4%

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

      1. associate-*r* 75.4%

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

      2. mul-1-neg 75.4%

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

   9. Simplified 75.4%

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

   if -3.5e-66 < y < 2.10000000000000009e-100

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 80.6%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{-66} \lor \neg \left(y \leq 2.1 \cdot 10^{-100}\right):\\ \;\;\;\;x \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 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 4: 51.5% 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. Add Preprocessing

3. Taylor expanded in x around 0 49.5%

   \[\leadsto \color{blue}{x} \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2024086 
(FPCore (x y)
  :name "Numeric.SpecFunctions:log1p from math-functions-0.1.5.2, A"
  :precision binary64
  (* x (- 1.0 (* x y))))