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

Percentage Accurate: 99.9% → 99.9%

Time: 6.1s

Alternatives: 5

Speedup: 1.0×

• 24.6% 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

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(N[((-x) * y), $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. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-neg N/A

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

   4. +-commutative N/A

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

   5. distribute-rgt-in N/A

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

   6. *-lft-identity N/A

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

   7. lower-fma.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. distribute-lft-neg-in N/A

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

   11. lower-*.f64 N/A

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

   12. lower-neg.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 82.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - x \cdot y\right) \cdot x\\ t_1 := \left(\left(-x\right) \cdot y\right) \cdot x\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+127}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+38}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (- 1.0 (* x y)) x)) (t_1 (* (* (- x) y) x)))
   (if (<= t_0 -2e+127) t_1 (if (<= t_0 4e+38) (* 1.0 x) t_1))))

C

double code(double x, double y) {
	double t_0 = (1.0 - (x * y)) * x;
	double t_1 = (-x * y) * x;
	double tmp;
	if (t_0 <= -2e+127) {
		tmp = t_1;
	} else if (t_0 <= 4e+38) {
		tmp = 1.0 * 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 = (1.0d0 - (x * y)) * x
    t_1 = (-x * y) * x
    if (t_0 <= (-2d+127)) then
        tmp = t_1
    else if (t_0 <= 4d+38) then
        tmp = 1.0d0 * x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (1.0 - (x * y)) * x;
	double t_1 = (-x * y) * x;
	double tmp;
	if (t_0 <= -2e+127) {
		tmp = t_1;
	} else if (t_0 <= 4e+38) {
		tmp = 1.0 * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (1.0 - (x * y)) * x
	t_1 = (-x * y) * x
	tmp = 0
	if t_0 <= -2e+127:
		tmp = t_1
	elif t_0 <= 4e+38:
		tmp = 1.0 * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(1.0 - Float64(x * y)) * x)
	t_1 = Float64(Float64(Float64(-x) * y) * x)
	tmp = 0.0
	if (t_0 <= -2e+127)
		tmp = t_1;
	elseif (t_0 <= 4e+38)
		tmp = Float64(1.0 * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (1.0 - (x * y)) * x;
	t_1 = (-x * y) * x;
	tmp = 0.0;
	if (t_0 <= -2e+127)
		tmp = t_1;
	elseif (t_0 <= 4e+38)
		tmp = 1.0 * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(1.0 - N[(x * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$1 = N[(N[((-x) * y), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+127], t$95$1, If[LessEqual[t$95$0, 4e+38], N[(1.0 * x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (-.f64 #s(literal 1 binary64) (*.f64 x y))) < -1.99999999999999991e127 or 3.99999999999999991e38 < (*.f64 x (-.f64 #s(literal 1 binary64) (*.f64 x y)))

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 90.8

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

   5. Applied rewrites 90.8%

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

   if -1.99999999999999991e127 < (*.f64 x (-.f64 #s(literal 1 binary64) (*.f64 x y))) < 3.99999999999999991e38

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 85.6%

      \[\leadsto x \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - x \cdot y\right) \cdot x \leq -2 \cdot 10^{+127}:\\ \;\;\;\;\left(\left(-x\right) \cdot y\right) \cdot x\\ \mathbf{elif}\;\left(1 - x \cdot y\right) \cdot x \leq 4 \cdot 10^{+38}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-x\right) \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - x \cdot y\right) \cdot x\\ t_1 := \left(\left(-x\right) \cdot x\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+127}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+38}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (- 1.0 (* x y)) x)) (t_1 (* (* (- x) x) y)))
   (if (<= t_0 -2e+127) t_1 (if (<= t_0 4e+38) (* 1.0 x) t_1))))

C

double code(double x, double y) {
	double t_0 = (1.0 - (x * y)) * x;
	double t_1 = (-x * x) * y;
	double tmp;
	if (t_0 <= -2e+127) {
		tmp = t_1;
	} else if (t_0 <= 4e+38) {
		tmp = 1.0 * 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 = (1.0d0 - (x * y)) * x
    t_1 = (-x * x) * y
    if (t_0 <= (-2d+127)) then
        tmp = t_1
    else if (t_0 <= 4d+38) then
        tmp = 1.0d0 * x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (1.0 - (x * y)) * x;
	double t_1 = (-x * x) * y;
	double tmp;
	if (t_0 <= -2e+127) {
		tmp = t_1;
	} else if (t_0 <= 4e+38) {
		tmp = 1.0 * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (1.0 - (x * y)) * x
	t_1 = (-x * x) * y
	tmp = 0
	if t_0 <= -2e+127:
		tmp = t_1
	elif t_0 <= 4e+38:
		tmp = 1.0 * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(1.0 - Float64(x * y)) * x)
	t_1 = Float64(Float64(Float64(-x) * x) * y)
	tmp = 0.0
	if (t_0 <= -2e+127)
		tmp = t_1;
	elseif (t_0 <= 4e+38)
		tmp = Float64(1.0 * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (1.0 - (x * y)) * x;
	t_1 = (-x * x) * y;
	tmp = 0.0;
	if (t_0 <= -2e+127)
		tmp = t_1;
	elseif (t_0 <= 4e+38)
		tmp = 1.0 * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(1.0 - N[(x * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$1 = N[(N[((-x) * x), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+127], t$95$1, If[LessEqual[t$95$0, 4e+38], N[(1.0 * x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (-.f64 #s(literal 1 binary64) (*.f64 x y))) < -1.99999999999999991e127 or 3.99999999999999991e38 < (*.f64 x (-.f64 #s(literal 1 binary64) (*.f64 x y)))

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. *-lft-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-neg-in N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around inf

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

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot {x}^{2}\right) \cdot y} \]

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 86.4

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

   7. Applied rewrites 86.4%

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

   if -1.99999999999999991e127 < (*.f64 x (-.f64 #s(literal 1 binary64) (*.f64 x y))) < 3.99999999999999991e38

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 85.6%

      \[\leadsto x \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - x \cdot y\right) \cdot x \leq -2 \cdot 10^{+127}:\\ \;\;\;\;\left(\left(-x\right) \cdot x\right) \cdot y\\ \mathbf{elif}\;\left(1 - x \cdot y\right) \cdot x \leq 4 \cdot 10^{+38}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-x\right) \cdot x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 5: 52.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ 1 \cdot x \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(1.0 * x), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 51.0%

   \[\leadsto x \cdot \color{blue}{1} \]

6. Final simplification 51.0%

   \[\leadsto 1 \cdot x \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024295 
(FPCore (x y)
  :name "Numeric.SpecFunctions:log1p from math-functions-0.1.5.2, A"
  :precision binary64
  (* x (- 1.0 (* x y))))