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

Percentage Accurate: 99.9% → 99.9%

Time: 5.8s

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. Final simplification 99.9%

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

Alternative 2: 73.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{+128} \lor \neg \left(x \leq -3.75 \cdot 10^{+93}\right) \land \left(x \leq -5.4 \cdot 10^{-72} \lor \neg \left(x \leq 1.05 \cdot 10^{-30}\right)\right):\\ \;\;\;\;x \cdot \left(y \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.2e+128)
         (and (not (<= x -3.75e+93))
              (or (<= x -5.4e-72) (not (<= x 1.05e-30)))))
   (* x (* y (- x)))
   x))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.2e+128) || (!(x <= -3.75e+93) && ((x <= -5.4e-72) || !(x <= 1.05e-30)))) {
		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 ((x <= (-1.2d+128)) .or. (.not. (x <= (-3.75d+93))) .and. (x <= (-5.4d-72)) .or. (.not. (x <= 1.05d-30))) 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 ((x <= -1.2e+128) || (!(x <= -3.75e+93) && ((x <= -5.4e-72) || !(x <= 1.05e-30)))) {
		tmp = x * (y * -x);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.2e+128) or (not (x <= -3.75e+93) and ((x <= -5.4e-72) or not (x <= 1.05e-30))):
		tmp = x * (y * -x)
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.2e+128) || (!(x <= -3.75e+93) && ((x <= -5.4e-72) || !(x <= 1.05e-30))))
		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 ((x <= -1.2e+128) || (~((x <= -3.75e+93)) && ((x <= -5.4e-72) || ~((x <= 1.05e-30)))))
		tmp = x * (y * -x);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.2e+128], And[N[Not[LessEqual[x, -3.75e+93]], $MachinePrecision], Or[LessEqual[x, -5.4e-72], N[Not[LessEqual[x, 1.05e-30]], $MachinePrecision]]]], N[(x * N[(y * (-x)), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if x < -1.2000000000000001e128 or -3.7500000000000001e93 < x < -5.4e-72 or 1.0500000000000001e-30 < x

   1. Initial program 99.9%

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

      1. add-cube-cbrt 99.0%

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

      2. pow3 98.9%

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

      3. sub-neg 98.9%

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

      4. +-commutative 98.9%

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

      5. distribute-rgt-neg-in 98.9%

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

      6. fma-define 98.9%

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

   4. Applied egg-rr 98.9%

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

   5. Taylor expanded in y around -inf 74.1%

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

      1. mul-1-neg 74.1%

         \[\leadsto {\color{blue}{\left(-\sqrt[3]{{x}^{2} \cdot y}\right)}}^{3} \]

   7. Simplified 74.1%

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

      1. cube-neg 74.1%

         \[\leadsto \color{blue}{-{\left(\sqrt[3]{{x}^{2} \cdot y}\right)}^{3}} \]

      2. rem-cube-cbrt 74.5%

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

      3. unpow2 74.5%

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

      4. add-sqr-sqrt 36.2%

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

      5. swap-sqr 40.0%

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

      6. distribute-lft-neg-out 40.0%

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

      7. distribute-lft-neg-in 40.0%

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

      8. associate-*l* 40.0%

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

      9. *-commutative 40.0%

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

      10. associate-*l* 40.0%

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

      11. add-sqr-sqrt 81.2%

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

      12. *-commutative 81.2%

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

   9. Applied egg-rr 81.2%

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

   if -1.2000000000000001e128 < x < -3.7500000000000001e93 or -5.4e-72 < x < 1.0500000000000001e-30

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 83.2%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{+128} \lor \neg \left(x \leq -3.75 \cdot 10^{+93}\right) \land \left(x \leq -5.4 \cdot 10^{-72} \lor \neg \left(x \leq 1.05 \cdot 10^{-30}\right)\right):\\ \;\;\;\;x \cdot \left(y \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 49.8% 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. Add Preprocessing

3. Taylor expanded in x around 0 50.8%

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

4. Final simplification 50.8%

   \[\leadsto x \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024055 
(FPCore (x y)
  :name "Numeric.SpecFunctions:log1p from math-functions-0.1.5.2, A"
  :precision binary64
  (* x (- 1.0 (* x y))))