Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B

Percentage Accurate: 100.0% → 100.0%

Time: 8.6s

Alternatives: 5

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y + 1, x, y\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (fma (+ y 1.0) x y))

C

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

Julia

function code(x, y)
	return fma(Float64(y + 1.0), x, y)
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. *-commutative N/A

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

   2. distribute-lft1-in N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-+.f64 100.0

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

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y + 1, x, y\right)} \]
5. Add Preprocessing

Alternative 2: 86.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + \left(x + y \cdot x\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;t\_0 \leq 10^{+296}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x (* y x)))))
   (if (<= t_0 (- INFINITY)) (* y x) (if (<= t_0 1e+296) (+ y x) (* y x)))))

C

double code(double x, double y) {
	double t_0 = y + (x + (y * x));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = y * x;
	} else if (t_0 <= 1e+296) {
		tmp = y + x;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = y + (x + (y * x));
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = y * x;
	} else if (t_0 <= 1e+296) {
		tmp = y + x;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y + (x + (y * x))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = y * x
	elif t_0 <= 1e+296:
		tmp = y + x
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y + Float64(x + Float64(y * x)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(y * x);
	elseif (t_0 <= 1e+296)
		tmp = Float64(y + x);
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y + (x + (y * x));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = y * x;
	elseif (t_0 <= 1e+296)
		tmp = y + x;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(y * x), $MachinePrecision], If[LessEqual[t$95$0, 1e+296], N[(y + x), $MachinePrecision], N[(y * x), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (*.f64 x y) x) y) < -inf.0 or 9.99999999999999981e295 < (+.f64 (+.f64 (*.f64 x y) x) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

         \[\leadsto x \cdot y + \color{blue}{x} \]

      4. lower-fma.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around inf

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

      1. lower-*.f64 97.7

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

   8. Simplified 97.7%

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

   if -inf.0 < (+.f64 (+.f64 (*.f64 x y) x) y) < 9.99999999999999981e295

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. distribute-lft1-in N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-+.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0

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

   7. Simplified 90.7%

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. *-lft-identity 90.7

         \[\leadsto y + \color{blue}{x} \]

   9. Applied egg-rr 90.7%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + \left(x + y \cdot x\right) \leq -\infty:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y + \left(x + y \cdot x\right) \leq 10^{+296}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 62.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + \left(x + y \cdot x\right) \leq -2 \cdot 10^{-289}:\\ \;\;\;\;\mathsf{fma}\left(x, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ y (+ x (* y x))) -2e-289) (fma x y x) (fma x y y)))

C

double code(double x, double y) {
	double tmp;
	if ((y + (x + (y * x))) <= -2e-289) {
		tmp = fma(x, y, x);
	} else {
		tmp = fma(x, y, y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y + Float64(x + Float64(y * x))) <= -2e-289)
		tmp = fma(x, y, x);
	else
		tmp = fma(x, y, y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y + N[(x + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-289], N[(x * y + x), $MachinePrecision], N[(x * y + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (*.f64 x y) x) y) < -2e-289

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

         \[\leadsto x \cdot y + \color{blue}{x} \]

      4. lower-fma.f64 64.5

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

   5. Simplified 64.5%

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

   if -2e-289 < (+.f64 (+.f64 (*.f64 x y) x) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

         \[\leadsto \color{blue}{y} + x \cdot y \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot y + y} \]

      4. lower-fma.f64 58.4

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

   5. Simplified 58.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + \left(x + y \cdot x\right) \leq -2 \cdot 10^{-289}:\\ \;\;\;\;\mathsf{fma}\left(x, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -48000000000:\\ \;\;\;\;\mathsf{fma}\left(x, y, x\right)\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{+14}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -48000000000.0) (fma x y x) (if (<= x 3.9e+14) (+ y x) (* y x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -48000000000.0) {
		tmp = fma(x, y, x);
	} else if (x <= 3.9e+14) {
		tmp = y + x;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -48000000000.0)
		tmp = fma(x, y, x);
	elseif (x <= 3.9e+14)
		tmp = Float64(y + x);
	else
		tmp = Float64(y * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -48000000000.0], N[(x * y + x), $MachinePrecision], If[LessEqual[x, 3.9e+14], N[(y + x), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.8e10

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

         \[\leadsto x \cdot y + \color{blue}{x} \]

      4. lower-fma.f64 99.7

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

   5. Simplified 99.7%

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

   if -4.8e10 < x < 3.9e14

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. distribute-lft1-in N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-+.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0

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

   7. Simplified 99.6%

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. *-lft-identity 99.6

         \[\leadsto y + \color{blue}{x} \]

   9. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{y + x} \]

   if 3.9e14 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

         \[\leadsto x \cdot y + \color{blue}{x} \]

      4. lower-fma.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around inf

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

      1. lower-*.f64 44.7

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

   8. Simplified 44.7%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -48000000000:\\ \;\;\;\;\mathsf{fma}\left(x, y, x\right)\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{+14}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ y + x \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ y x))

C

double code(double x, double y) {
	return y + x;
}

Fortran

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

Java

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

Python

def code(x, y):
	return y + x

Julia

function code(x, y)
	return Float64(y + x)
end

MATLAB

function tmp = code(x, y)
	tmp = y + x;
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. *-commutative N/A

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

   2. distribute-lft1-in N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-+.f64 100.0

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in y around 0

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

7. Simplified 76.1%

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

   3. *-lft-identity 76.1

      \[\leadsto y + \color{blue}{x} \]

9. Applied egg-rr 76.1%

   \[\leadsto \color{blue}{y + x} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024212 
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  :precision binary64
  (+ (+ (* x y) x) y))