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×

• 24.8% 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 99.8%

   \[x \cdot \left(1 - x \cdot y\right) \]
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: 74.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0 - x \cdot \left(x \cdot y\right)\\ \mathbf{if}\;x \leq -1.95 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{-44}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 0.0 (* x (* x y)))))
   (if (<= x -1.95e-5) t_0 (if (<= x 8.8e-44) x t_0))))

C

double code(double x, double y) {
	double t_0 = 0.0 - (x * (x * y));
	double tmp;
	if (x <= -1.95e-5) {
		tmp = t_0;
	} else if (x <= 8.8e-44) {
		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 <= (-1.95d-5)) then
        tmp = t_0
    else if (x <= 8.8d-44) 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 <= -1.95e-5) {
		tmp = t_0;
	} else if (x <= 8.8e-44) {
		tmp = x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 0.0 - (x * (x * y))
	tmp = 0
	if x <= -1.95e-5:
		tmp = t_0
	elif x <= 8.8e-44:
		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 <= -1.95e-5)
		tmp = t_0;
	elseif (x <= 8.8e-44)
		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 <= -1.95e-5)
		tmp = t_0;
	elseif (x <= 8.8e-44)
		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, -1.95e-5], t$95$0, If[LessEqual[x, 8.8e-44], x, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.95e-5 or 8.80000000000000048e-44 < x

   1. Initial program 99.8%

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

   4. Applied egg-rr 65.6%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 79.3%

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

   9. Simplified 79.3%

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

       1. div-inv N/A

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

       2. frac-2neg N/A

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

       3. metadata-eval N/A

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

       4. remove-double-div N/A

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

       5. distribute-rgt-neg-in N/A

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

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

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

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

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

       8. *-lowering-*.f64 79.3%

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

   11. Applied egg-rr 79.3%

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

   if -1.95e-5 < x < 8.80000000000000048e-44

   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 78.1%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{-5}:\\ \;\;\;\;0 - x \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{-44}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;0 - x \cdot \left(x \cdot y\right)\\ \end{array} \]
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 99.8%

   \[x \cdot \left(1 - x \cdot y\right) \]
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 99.8%

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

3. Taylor expanded in x around 0

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

5. Simplified 48.9%

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

Reproduce

herbie shell --seed 2024158 
(FPCore (x y)
  :name "Numeric.SpecFunctions:log1p from math-functions-0.1.5.2, A"
  :precision binary64
  (* x (- 1.0 (* x y))))