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

Percentage Accurate: 99.9% → 99.9%

Time: 4.6s

Alternatives: 5

Speedup: 1.0×

• 25.3% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   4. fp-cancel-sub-sign-inv N/A

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

   5. +-commutative N/A

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

   6. distribute-rgt-in N/A

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

   7. *-lft-identity N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-neg.f64 99.9

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

4. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(-x\right) \cdot y, x, x\right)} \]
5. Step-by-step derivation

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. fp-cancel-sign-sub-inv N/A

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

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

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

   5. lower--.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. associate-*r* N/A

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

   10. *-commutative N/A

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

   11. lift-*.f64 N/A

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

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

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

   13. lift-neg.f64 N/A

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

   14. remove-double-neg N/A

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

   15. lower-*.f64 99.9

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

6. Applied rewrites 99.9%

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

Alternative 2: 82.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(1 - x \cdot y\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+83} \lor \neg \left(t\_0 \leq 10^{-6}\right):\\ \;\;\;\;x \cdot \left(\left(-x\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (- 1.0 (* x y)))))
   (if (or (<= t_0 -2e+83) (not (<= t_0 1e-6))) (* x (* (- x) y)) (* x 1.0))))

C

double code(double x, double y) {
	double t_0 = x * (1.0 - (x * y));
	double tmp;
	if ((t_0 <= -2e+83) || !(t_0 <= 1e-6)) {
		tmp = x * (-x * y);
	} else {
		tmp = x * 1.0;
	}
	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 = x * (1.0d0 - (x * y))
    if ((t_0 <= (-2d+83)) .or. (.not. (t_0 <= 1d-6))) then
        tmp = x * (-x * y)
    else
        tmp = x * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x * (1.0 - (x * y));
	double tmp;
	if ((t_0 <= -2e+83) || !(t_0 <= 1e-6)) {
		tmp = x * (-x * y);
	} else {
		tmp = x * 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * (1.0 - (x * y))
	tmp = 0
	if (t_0 <= -2e+83) or not (t_0 <= 1e-6):
		tmp = x * (-x * y)
	else:
		tmp = x * 1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x * Float64(1.0 - Float64(x * y)))
	tmp = 0.0
	if ((t_0 <= -2e+83) || !(t_0 <= 1e-6))
		tmp = Float64(x * Float64(Float64(-x) * y));
	else
		tmp = Float64(x * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x * (1.0 - (x * y));
	tmp = 0.0;
	if ((t_0 <= -2e+83) || ~((t_0 <= 1e-6)))
		tmp = x * (-x * y);
	else
		tmp = x * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(1.0 - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -2e+83], N[Not[LessEqual[t$95$0, 1e-6]], $MachinePrecision]], N[(x * N[((-x) * y), $MachinePrecision]), $MachinePrecision], N[(x * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (-.f64 #s(literal 1 binary64) (*.f64 x y))) < -2.00000000000000006e83 or 9.99999999999999955e-7 < (*.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

      \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \left(x \cdot y\right)\right)} \]
   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 86.7

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

   5. Applied rewrites 86.7%

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

   if -2.00000000000000006e83 < (*.f64 x (-.f64 #s(literal 1 binary64) (*.f64 x y))) < 9.99999999999999955e-7

   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 85.5%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \left(1 - x \cdot y\right) \leq -2 \cdot 10^{+83} \lor \neg \left(x \cdot \left(1 - x \cdot y\right) \leq 10^{-6}\right):\\ \;\;\;\;x \cdot \left(\left(-x\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(1 - x \cdot y\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+83} \lor \neg \left(t\_0 \leq 5 \cdot 10^{+56}\right):\\ \;\;\;\;\left(-y\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (- 1.0 (* x y)))))
   (if (or (<= t_0 -2e+83) (not (<= t_0 5e+56))) (* (- y) (* x x)) (* x 1.0))))

C

double code(double x, double y) {
	double t_0 = x * (1.0 - (x * y));
	double tmp;
	if ((t_0 <= -2e+83) || !(t_0 <= 5e+56)) {
		tmp = -y * (x * x);
	} else {
		tmp = x * 1.0;
	}
	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 = x * (1.0d0 - (x * y))
    if ((t_0 <= (-2d+83)) .or. (.not. (t_0 <= 5d+56))) then
        tmp = -y * (x * x)
    else
        tmp = x * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x * (1.0 - (x * y));
	double tmp;
	if ((t_0 <= -2e+83) || !(t_0 <= 5e+56)) {
		tmp = -y * (x * x);
	} else {
		tmp = x * 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * (1.0 - (x * y))
	tmp = 0
	if (t_0 <= -2e+83) or not (t_0 <= 5e+56):
		tmp = -y * (x * x)
	else:
		tmp = x * 1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x * Float64(1.0 - Float64(x * y)))
	tmp = 0.0
	if ((t_0 <= -2e+83) || !(t_0 <= 5e+56))
		tmp = Float64(Float64(-y) * Float64(x * x));
	else
		tmp = Float64(x * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x * (1.0 - (x * y));
	tmp = 0.0;
	if ((t_0 <= -2e+83) || ~((t_0 <= 5e+56)))
		tmp = -y * (x * x);
	else
		tmp = x * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(1.0 - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -2e+83], N[Not[LessEqual[t$95$0, 5e+56]], $MachinePrecision]], N[((-y) * N[(x * x), $MachinePrecision]), $MachinePrecision], N[(x * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (-.f64 #s(literal 1 binary64) (*.f64 x y))) < -2.00000000000000006e83 or 5.00000000000000024e56 < (*.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. Step-by-step derivation

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow1 N/A

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

      4. pow2 N/A

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

      5. sqrt-prod N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 50.2

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 50.2

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

   4. Applied rewrites 50.2%

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

   5. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 82.6

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

   7. Applied rewrites 82.6%

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

   if -2.00000000000000006e83 < (*.f64 x (-.f64 #s(literal 1 binary64) (*.f64 x y))) < 5.00000000000000024e56

   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 84.5%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \left(1 - x \cdot y\right) \leq -2 \cdot 10^{+83} \lor \neg \left(x \cdot \left(1 - x \cdot y\right) \leq 5 \cdot 10^{+56}\right):\\ \;\;\;\;\left(-y\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 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 5: 49.8% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(x * 1.0), $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 54.4%

   \[\leadsto x \cdot \color{blue}{1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024326 
(FPCore (x y)
  :name "Numeric.SpecFunctions:log1p from math-functions-0.1.5.2, A"
  :precision binary64
  (* x (- 1.0 (* x y))))