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

Percentage Accurate: 99.9% → 99.9%

Time: 8.4s

Alternatives: 3

Speedup: 1.0×

• 25.5% 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: 75.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0 - x \cdot \left(x \cdot y\right)\\ \mathbf{if}\;y \leq -5.5 \cdot 10^{+36}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-46}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 0.0 (* x (* x y)))))
   (if (<= y -5.5e+36) t_0 (if (<= y 7.5e-46) x t_0))))

C

double code(double x, double y) {
	double t_0 = 0.0 - (x * (x * y));
	double tmp;
	if (y <= -5.5e+36) {
		tmp = t_0;
	} else if (y <= 7.5e-46) {
		tmp = x;
	} else {
		tmp = t_0;
	}
	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 = 0.0d0 - (x * (x * y))
    if (y <= (-5.5d+36)) then
        tmp = t_0
    else if (y <= 7.5d-46) then
        tmp = x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 0.0 - (x * (x * y));
	double tmp;
	if (y <= -5.5e+36) {
		tmp = t_0;
	} else if (y <= 7.5e-46) {
		tmp = x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 0.0 - (x * (x * y))
	tmp = 0
	if y <= -5.5e+36:
		tmp = t_0
	elif y <= 7.5e-46:
		tmp = x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(0.0 - Float64(x * Float64(x * y)))
	tmp = 0.0
	if (y <= -5.5e+36)
		tmp = t_0;
	elseif (y <= 7.5e-46)
		tmp = x;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 0.0 - (x * (x * y));
	tmp = 0.0;
	if (y <= -5.5e+36)
		tmp = t_0;
	elseif (y <= 7.5e-46)
		tmp = x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.0 - N[(x * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -5.5e+36], t$95$0, If[LessEqual[y, 7.5e-46], x, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.5000000000000002e36 or 7.50000000000000027e-46 < y

   1. Initial program 99.8%

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

   4. Applied egg-rr 71.8%

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

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{1 + x \cdot y}{\mathsf{neg}\left(\left(-1 + y \cdot \left(x \cdot \left(x \cdot y\right)\right)\right) \cdot x\right)}\right)}\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1}{\color{blue}{\frac{\mathsf{neg}\left(\left(-1 + y \cdot \left(x \cdot \left(x \cdot y\right)\right)\right) \cdot x\right)}{1 + x \cdot y}}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\mathsf{neg}\left(\left(-1 + y \cdot \left(x \cdot \left(x \cdot y\right)\right)\right) \cdot x\right)}{1 + x \cdot y}\right)}\right)\right) \]

      5. neg-mul-1 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \left(\frac{-1 \cdot \left(\left(-1 + y \cdot \left(x \cdot \left(x \cdot y\right)\right)\right) \cdot x\right)}{\color{blue}{1} + x \cdot y}\right)\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \left(\frac{\left(-1 \cdot \left(-1 + y \cdot \left(x \cdot \left(x \cdot y\right)\right)\right)\right) \cdot x}{\color{blue}{1} + x \cdot y}\right)\right)\right) \]

      7. neg-mul-1 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \left(\frac{\left(\mathsf{neg}\left(\left(-1 + y \cdot \left(x \cdot \left(x \cdot y\right)\right)\right)\right)\right) \cdot x}{1 + x \cdot y}\right)\right)\right) \]

      8. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \left(\frac{\left(\left(\mathsf{neg}\left(-1\right)\right) + \left(\mathsf{neg}\left(y \cdot \left(x \cdot \left(x \cdot y\right)\right)\right)\right)\right) \cdot x}{1 + x \cdot y}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \left(\frac{\left(1 + \left(\mathsf{neg}\left(y \cdot \left(x \cdot \left(x \cdot y\right)\right)\right)\right)\right) \cdot x}{1 + x \cdot y}\right)\right)\right) \]

      10. /-rgt-identity N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \left(\frac{\left(1 + \left(\mathsf{neg}\left(\frac{y \cdot \left(x \cdot \left(x \cdot y\right)\right)}{1}\right)\right)\right) \cdot x}{1 + x \cdot y}\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \left(\frac{\left(1 - \frac{y \cdot \left(x \cdot \left(x \cdot y\right)\right)}{1}\right) \cdot x}{1 + x \cdot y}\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \left(\frac{x \cdot \left(1 - \frac{y \cdot \left(x \cdot \left(x \cdot y\right)\right)}{1}\right)}{\color{blue}{1} + x \cdot y}\right)\right)\right) \]

      13. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \left(x \cdot \color{blue}{\frac{1 - \frac{y \cdot \left(x \cdot \left(x \cdot y\right)\right)}{1}}{1 + x \cdot y}}\right)\right)\right) \]

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{-1}{{x}^{2} \cdot y}\right)}\right) \]
   8. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\frac{-1}{{x}^{2}}}{\color{blue}{y}}\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\frac{\mathsf{neg}\left(1\right)}{{x}^{2}}}{y}\right)\right) \]

      3. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{{x}^{2}}\right)}{y}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{{x}^{2}}\right)\right), \color{blue}{y}\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{\mathsf{neg}\left(1\right)}{{x}^{2}}\right), y\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{-1}{{x}^{2}}\right), y\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, \left({x}^{2}\right)\right), y\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, \left(x \cdot x\right)\right), y\right)\right) \]

      9. *-lowering-*.f64 66.4%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, x\right)\right), y\right)\right) \]

   9. Simplified 66.4%

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

       1. associate-/l/ N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. remove-double-div N/A

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

       5. div-inv N/A

          \[\leadsto \frac{1}{\frac{-1}{\frac{x \cdot y}{\color{blue}{\frac{1}{x}}}}} \]

       6. associate-/r/ N/A

          \[\leadsto \frac{1}{-1} \cdot \color{blue}{\frac{x \cdot y}{\frac{1}{x}}} \]

       7. metadata-eval N/A

          \[\leadsto -1 \cdot \frac{\color{blue}{x \cdot y}}{\frac{1}{x}} \]

       8. neg-mul-1 N/A

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

       9. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{x \cdot y}{\frac{1}{x}}\right)\right) \]

       10. associate-/l* N/A

           \[\leadsto \mathsf{neg.f64}\left(\left(x \cdot \frac{y}{\frac{1}{x}}\right)\right) \]

       11. un-div-inv N/A

           \[\leadsto \mathsf{neg.f64}\left(\left(x \cdot \left(y \cdot \frac{1}{\frac{1}{x}}\right)\right)\right) \]

       12. remove-double-div N/A

           \[\leadsto \mathsf{neg.f64}\left(\left(x \cdot \left(y \cdot x\right)\right)\right) \]

       13. *-commutative N/A

           \[\leadsto \mathsf{neg.f64}\left(\left(x \cdot \left(x \cdot y\right)\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot y\right)\right)\right) \]

       15. *-lowering-*.f64 73.0%

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, y\right)\right)\right) \]

   11. Applied egg-rr 73.0%

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

   if -5.5000000000000002e36 < y < 7.50000000000000027e-46

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 79.4%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+36}:\\ \;\;\;\;0 - x \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-46}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;0 - x \cdot \left(x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 51.4% 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

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

5. Simplified 55.2%

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

Reproduce

herbie shell --seed 2024160 
(FPCore (x y)
  :name "Numeric.SpecFunctions:log1p from math-functions-0.1.5.2, A"
  :precision binary64
  (* x (- 1.0 (* x y))))