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

Percentage Accurate: 99.9% → 99.9%

Time: 8.5s

Alternatives: 4

Speedup: 1.0×

• 25.6% 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

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

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 2: 72.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.42 \lor \neg \left(y \leq -1.45 \cdot 10^{-149} \lor \neg \left(y \leq -3.2 \cdot 10^{-169}\right) \land y \leq 1.42 \cdot 10^{-121}\right):\\ \;\;\;\;x \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -0.42)
         (not
          (or (<= y -1.45e-149)
              (and (not (<= y -3.2e-169)) (<= y 1.42e-121)))))
   (* x (* x (- y)))
   x))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -0.42) || !((y <= -1.45e-149) || (!(y <= -3.2e-169) && (y <= 1.42e-121)))) {
		tmp = x * (x * -y);
	} 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 ((y <= (-0.42d0)) .or. (.not. (y <= (-1.45d-149)) .or. (.not. (y <= (-3.2d-169))) .and. (y <= 1.42d-121))) then
        tmp = x * (x * -y)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -0.42) || !((y <= -1.45e-149) || (!(y <= -3.2e-169) && (y <= 1.42e-121)))) {
		tmp = x * (x * -y);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -0.42) or not ((y <= -1.45e-149) or (not (y <= -3.2e-169) and (y <= 1.42e-121))):
		tmp = x * (x * -y)
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -0.42) || !((y <= -1.45e-149) || (!(y <= -3.2e-169) && (y <= 1.42e-121))))
		tmp = Float64(x * Float64(x * Float64(-y)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -0.42) || ~(((y <= -1.45e-149) || (~((y <= -3.2e-169)) && (y <= 1.42e-121)))))
		tmp = x * (x * -y);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -0.42], N[Not[Or[LessEqual[y, -1.45e-149], And[N[Not[LessEqual[y, -3.2e-169]], $MachinePrecision], LessEqual[y, 1.42e-121]]]], $MachinePrecision]], N[(x * N[(x * (-y)), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -0.419999999999999984 or -1.45e-149 < y < -3.19999999999999995e-169 or 1.41999999999999991e-121 < y

   1. Initial program 99.9%

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

      1. flip-- 62.9%

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

      2. associate-*r/ 60.0%

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

      3. metadata-eval 60.0%

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

      4. pow2 60.0%

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

      5. +-commutative 60.0%

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

      6. fma-define 60.0%

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

   4. Applied egg-rr 60.0%

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

      1. *-commutative 60.0%

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

      2. associate-/l* 62.8%

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

   6. Simplified 62.8%

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

      1. unpow-prod-down 42.8%

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

      2. unpow2 42.8%

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

      3. associate-*l* 44.5%

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

   8. Applied egg-rr 44.5%

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

      1. cancel-sign-sub-inv 44.5%

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

      2. unpow2 44.5%

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

      3. associate-*l* 58.6%

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

      4. cancel-sign-sub-inv 58.6%

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

      5. *-commutative 58.6%

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

      6. associate-*r/ 55.8%

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

      7. clear-num 55.7%

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

      8. associate-*l* 59.9%

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

      9. *-commutative 59.9%

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

      10. unpow2 59.9%

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

      11. *-commutative 59.9%

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

   10. Applied egg-rr 59.9%

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

   11. Taylor expanded in x around inf 69.1%

       \[\leadsto \frac{1}{\color{blue}{\frac{-1}{{x}^{2} \cdot y}}} \]
   12. Step-by-step derivation

       1. *-commutative 69.1%

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

       2. associate-/r* 69.1%

          \[\leadsto \frac{1}{\color{blue}{\frac{\frac{-1}{y}}{{x}^{2}}}} \]

   13. Simplified 69.1%

       \[\leadsto \frac{1}{\color{blue}{\frac{\frac{-1}{y}}{{x}^{2}}}} \]
   14. Step-by-step derivation

       1. associate-/r/ 69.1%

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

       2. clear-num 69.1%

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

       3. unpow2 69.1%

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

       4. associate-*r* 76.4%

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

       5. clear-num 76.4%

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

       6. frac-2neg 76.4%

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

       7. metadata-eval 76.4%

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

       8. remove-double-div 76.4%

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

       9. distribute-lft-neg-in 76.4%

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

       10. *-commutative 76.4%

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

       11. distribute-rgt-neg-in 76.4%

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

   15. Applied egg-rr 76.4%

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

   if -0.419999999999999984 < y < -1.45e-149 or -3.19999999999999995e-169 < y < 1.41999999999999991e-121

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 79.9%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.42 \lor \neg \left(y \leq -1.45 \cdot 10^{-149} \lor \neg \left(y \leq -3.2 \cdot 10^{-169}\right) \land y \leq 1.42 \cdot 10^{-121}\right):\\ \;\;\;\;x \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 70.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-14} \lor \neg \left(x \leq 2.1 \cdot 10^{-128} \lor \neg \left(x \leq 118\right) \land x \leq 2.75 \cdot 10^{+19}\right):\\ \;\;\;\;y \cdot \left(x \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.5e-14)
         (not (or (<= x 2.1e-128) (and (not (<= x 118.0)) (<= x 2.75e+19)))))
   (* y (* x (- x)))
   x))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.5e-14) || !((x <= 2.1e-128) || (!(x <= 118.0) && (x <= 2.75e+19)))) {
		tmp = y * (x * -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 <= (-1.5d-14)) .or. (.not. (x <= 2.1d-128) .or. (.not. (x <= 118.0d0)) .and. (x <= 2.75d+19))) then
        tmp = y * (x * -x)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.5e-14) || !((x <= 2.1e-128) || (!(x <= 118.0) && (x <= 2.75e+19)))) {
		tmp = y * (x * -x);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.5e-14) or not ((x <= 2.1e-128) or (not (x <= 118.0) and (x <= 2.75e+19))):
		tmp = y * (x * -x)
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.5e-14) || !((x <= 2.1e-128) || (!(x <= 118.0) && (x <= 2.75e+19))))
		tmp = Float64(y * Float64(x * Float64(-x)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.5e-14) || ~(((x <= 2.1e-128) || (~((x <= 118.0)) && (x <= 2.75e+19)))))
		tmp = y * (x * -x);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.5e-14], N[Not[Or[LessEqual[x, 2.1e-128], And[N[Not[LessEqual[x, 118.0]], $MachinePrecision], LessEqual[x, 2.75e+19]]]], $MachinePrecision]], N[(y * N[(x * (-x)), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if x < -1.4999999999999999e-14 or 2.1000000000000001e-128 < x < 118 or 2.75e19 < x

   1. Initial program 99.9%

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

      1. flip-- 60.9%

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

      2. associate-*r/ 57.2%

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

      3. metadata-eval 57.2%

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

      4. pow2 57.2%

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

      5. +-commutative 57.2%

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

      6. fma-define 57.2%

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

   4. Applied egg-rr 57.2%

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

      1. *-commutative 57.2%

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

      2. associate-/l* 60.9%

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

   6. Simplified 60.9%

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

      1. unpow-prod-down 45.0%

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

      2. unpow2 45.0%

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

      3. associate-*l* 53.6%

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

   8. Applied egg-rr 53.6%

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

      1. cancel-sign-sub-inv 53.6%

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

      2. unpow2 53.6%

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

      3. associate-*l* 60.2%

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

      4. cancel-sign-sub-inv 60.2%

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

      5. *-commutative 60.2%

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

      6. associate-*r/ 56.6%

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

      7. clear-num 56.5%

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

      8. associate-*l* 57.1%

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

      9. *-commutative 57.1%

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

      10. unpow2 57.1%

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

      11. *-commutative 57.1%

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

   10. Applied egg-rr 57.1%

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

   11. Taylor expanded in x around inf 73.7%

       \[\leadsto \frac{1}{\color{blue}{\frac{-1}{{x}^{2} \cdot y}}} \]
   12. Step-by-step derivation

       1. *-commutative 73.7%

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

       2. associate-/r* 73.7%

          \[\leadsto \frac{1}{\color{blue}{\frac{\frac{-1}{y}}{{x}^{2}}}} \]

   13. Simplified 73.7%

       \[\leadsto \frac{1}{\color{blue}{\frac{\frac{-1}{y}}{{x}^{2}}}} \]
   14. Step-by-step derivation

       1. *-un-lft-identity 73.7%

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

       2. unpow2 73.7%

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

       3. times-frac 78.4%

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

   15. Applied egg-rr 78.4%

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

       1. frac-times 73.7%

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

       2. *-un-lft-identity 73.7%

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

       3. clear-num 73.8%

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

       4. associate-*r/ 78.5%

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

       5. associate-/r/ 78.5%

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

       6. associate-*r* 73.8%

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

       7. div-inv 73.8%

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

       8. metadata-eval 73.8%

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

   17. Applied egg-rr 73.8%

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

       1. *-commutative 73.8%

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

       2. associate-*r* 78.5%

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

       3. metadata-eval 78.5%

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

       4. div-inv 78.5%

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

       5. associate-/r/ 78.5%

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

       6. clear-num 78.4%

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

       7. div-inv 78.4%

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

       8. associate-/r* 78.5%

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

       9. associate-/r/ 78.5%

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

       10. metadata-eval 78.5%

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

       11. associate-*l* 78.5%

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

       12. *-commutative 78.5%

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

       13. add-sqr-sqrt 38.5%

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

       14. sqrt-unprod 35.7%

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

       15. *-commutative 35.7%

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

       16. *-commutative 35.7%

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

       17. swap-sqr 35.7%

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

       18. metadata-eval 35.7%

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

       19. *-un-lft-identity 35.7%

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

       20. sqrt-unprod 0.7%

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

       21. add-sqr-sqrt 1.1%

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

       22. add-sqr-sqrt 0.7%

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

       23. sqrt-unprod 35.7%

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

       24. *-un-lft-identity 35.7%

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

       25. metadata-eval 35.7%

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

       26. swap-sqr 35.7%

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

       27. *-commutative 35.7%

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

       28. *-commutative 35.7%

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

   19. Applied egg-rr 78.5%

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

       1. div-inv 78.5%

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

       2. *-commutative 78.5%

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

       3. clear-num 78.5%

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

       4. associate-*l* 73.8%

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

       5. clear-num 73.8%

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

       6. frac-2neg 73.8%

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

       7. metadata-eval 73.8%

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

       8. remove-double-div 73.8%

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

   21. Applied egg-rr 73.8%

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

   if -1.4999999999999999e-14 < x < 2.1000000000000001e-128 or 118 < x < 2.75e19

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 77.7%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-14} \lor \neg \left(x \leq 2.1 \cdot 10^{-128} \lor \neg \left(x \leq 118\right) \land x \leq 2.75 \cdot 10^{+19}\right):\\ \;\;\;\;y \cdot \left(x \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. 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 99.9%

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

3. Taylor expanded in x around 0 45.9%

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

Reproduce

herbie shell --seed 2024107 
(FPCore (x y)
  :name "Numeric.SpecFunctions:log1p from math-functions-0.1.5.2, A"
  :precision binary64
  (* x (- 1.0 (* x y))))