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

Percentage Accurate: 99.9% → 99.9%

Time: 8.9s

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. *-lft-identity N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. *-commutative N/A

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

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

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

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

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

   9. neg-lowering-neg.f64 99.9

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

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

   \[\leadsto \mathsf{fma}\left(-y \cdot x, x, x\right) \]
6. Add Preprocessing

Alternative 2: 82.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(1 - y \cdot x\right)\\ t_1 := \left(-x\right) \cdot \left(y \cdot x\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+14}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{+123}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (- 1.0 (* y x)))) (t_1 (* (- x) (* y x))))
   (if (<= t_0 -2e+14) t_1 (if (<= t_0 1e+123) x t_1))))

C

double code(double x, double y) {
	double t_0 = x * (1.0 - (y * x));
	double t_1 = -x * (y * x);
	double tmp;
	if (t_0 <= -2e+14) {
		tmp = t_1;
	} else if (t_0 <= 1e+123) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = x * (1.0d0 - (y * x))
    t_1 = -x * (y * x)
    if (t_0 <= (-2d+14)) then
        tmp = t_1
    else if (t_0 <= 1d+123) then
        tmp = x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x * (1.0 - (y * x));
	double t_1 = -x * (y * x);
	double tmp;
	if (t_0 <= -2e+14) {
		tmp = t_1;
	} else if (t_0 <= 1e+123) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * (1.0 - (y * x))
	t_1 = -x * (y * x)
	tmp = 0
	if t_0 <= -2e+14:
		tmp = t_1
	elif t_0 <= 1e+123:
		tmp = x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x * Float64(1.0 - Float64(y * x)))
	t_1 = Float64(Float64(-x) * Float64(y * x))
	tmp = 0.0
	if (t_0 <= -2e+14)
		tmp = t_1;
	elseif (t_0 <= 1e+123)
		tmp = x;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x * (1.0 - (y * x));
	t_1 = -x * (y * x);
	tmp = 0.0;
	if (t_0 <= -2e+14)
		tmp = t_1;
	elseif (t_0 <= 1e+123)
		tmp = x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(1.0 - N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-x) * N[(y * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+14], t$95$1, If[LessEqual[t$95$0, 1e+123], x, t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (-.f64 #s(literal 1 binary64) (*.f64 x y))) < -2e14 or 9.99999999999999978e122 < (*.f64 x (-.f64 #s(literal 1 binary64) (*.f64 x y)))

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left({x}^{2} \cdot y\right)} \]

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

      5. mul-1-neg N/A

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

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

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

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

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

      10. mul-1-neg N/A

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

      11. neg-lowering-neg.f64 90.0

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

   5. Simplified 90.0%

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

   if -2e14 < (*.f64 x (-.f64 #s(literal 1 binary64) (*.f64 x y))) < 9.99999999999999978e122

   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 82.5%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \left(1 - y \cdot x\right) \leq -2 \cdot 10^{+14}:\\ \;\;\;\;\left(-x\right) \cdot \left(y \cdot x\right)\\ \mathbf{elif}\;x \cdot \left(1 - y \cdot x\right) \leq 10^{+123}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) \cdot \left(y \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 - y \cdot x\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x (- 1.0 (* y x))))

C

double code(double x, double y) {
	return x * (1.0 - (y * x));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * (1.0d0 - (y * x))
end function

Java

public static double code(double x, double y) {
	return x * (1.0 - (y * x));
}

Python

def code(x, y):
	return x * (1.0 - (y * x))

Julia

function code(x, y)
	return Float64(x * Float64(1.0 - Float64(y * x)))
end

MATLAB

function tmp = code(x, y)
	tmp = x * (1.0 - (y * x));
end

Wolfram

code[x_, y_] := N[(x * N[(1.0 - N[(y * x), $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%

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

Alternative 4: 50.7% accurate, 14.0× 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 49.6%

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

Reproduce

herbie shell --seed 2024204 
(FPCore (x y)
  :name "Numeric.SpecFunctions:log1p from math-functions-0.1.5.2, A"
  :precision binary64
  (* x (- 1.0 (* x y))))