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

Percentage Accurate: 99.9% → 99.9%

Time: 7.0s

Alternatives: 4

Speedup: 1.0×

• 25.4% 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, 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 N/A

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

   2. distribute-rgt-in N/A

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

   3. fma-define N/A

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

   4. distribute-lft-neg-out N/A

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

   5. fmm-undef N/A

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

   6. *-lft-identity N/A

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

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

      \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(\left(x \cdot y\right) \cdot x\right)}\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \left(x \cdot \color{blue}{\left(x \cdot y\right)}\right)\right) \]

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

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

   10. *-lowering-*.f64 99.9%

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

4. Applied egg-rr 99.9%

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

Alternative 2: 74.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0 - x \cdot \left(x \cdot y\right)\\ \mathbf{if}\;x \leq -3.5 \cdot 10^{+14}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{+86}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 0.0 (* x (* x y)))))
   (if (<= x -3.5e+14) t_0 (if (<= x 2.4e+86) x t_0))))

C

double code(double x, double y) {
	double t_0 = 0.0 - (x * (x * y));
	double tmp;
	if (x <= -3.5e+14) {
		tmp = t_0;
	} else if (x <= 2.4e+86) {
		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 (x <= (-3.5d+14)) then
        tmp = t_0
    else if (x <= 2.4d+86) 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 (x <= -3.5e+14) {
		tmp = t_0;
	} else if (x <= 2.4e+86) {
		tmp = x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 0.0 - (x * (x * y))
	tmp = 0
	if x <= -3.5e+14:
		tmp = t_0
	elif x <= 2.4e+86:
		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 (x <= -3.5e+14)
		tmp = t_0;
	elseif (x <= 2.4e+86)
		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 (x <= -3.5e+14)
		tmp = t_0;
	elseif (x <= 2.4e+86)
		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[x, -3.5e+14], t$95$0, If[LessEqual[x, 2.4e+86], x, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.5e14 or 2.4e86 < x

   1. Initial program 99.9%

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

   4. Applied egg-rr 56.4%

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

      1. distribute-frac-neg N/A

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

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

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

      3. associate-/l* N/A

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

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

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

      5. +-lowering-+.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

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

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

      9. associate-*l* N/A

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

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

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

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

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

      12. /-lowering-/.f64 N/A

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

      13. +-commutative N/A

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

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

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

      15. *-lowering-*.f64 60.4%

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

   6. Applied egg-rr 60.4%

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

   7. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

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

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

      4. *-lowering-*.f64 81.4%

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

   9. Simplified 81.4%

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

   if -3.5e14 < x < 2.4e86

   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 74.5%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{+14}:\\ \;\;\;\;0 - x \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{+86}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;0 - x \cdot \left(x \cdot y\right)\\ \end{array} \]
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: 50.2% 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 53.3%

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

Reproduce

herbie shell --seed 2024155 
(FPCore (x y)
  :name "Numeric.SpecFunctions:log1p from math-functions-0.1.5.2, A"
  :precision binary64
  (* x (- 1.0 (* x y))))