fabs fraction 2

Percentage Accurate: 100.0% → 100.0%

Time: 5.5s

Alternatives: 3

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \frac{\left|a - b\right|}{2} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (/ (fabs (- a b)) 2.0))

C

double code(double a, double b) {
	return fabs((a - b)) / 2.0;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = abs((a - b)) / 2.0d0
end function

Java

public static double code(double a, double b) {
	return Math.abs((a - b)) / 2.0;
}

Python

def code(a, b):
	return math.fabs((a - b)) / 2.0

Julia

function code(a, b)
	return Float64(abs(Float64(a - b)) / 2.0)
end

MATLAB

function tmp = code(a, b)
	tmp = abs((a - b)) / 2.0;
end

Wolfram

code[a_, b_] := N[(N[Abs[N[(a - b), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left|a - b\right|}{2} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (/ (fabs (- a b)) 2.0))

C

double code(double a, double b) {
	return fabs((a - b)) / 2.0;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = abs((a - b)) / 2.0d0
end function

Java

public static double code(double a, double b) {
	return Math.abs((a - b)) / 2.0;
}

Python

def code(a, b):
	return math.fabs((a - b)) / 2.0

Julia

function code(a, b)
	return Float64(abs(Float64(a - b)) / 2.0)
end

MATLAB

function tmp = code(a, b)
	tmp = abs((a - b)) / 2.0;
end

Wolfram

code[a_, b_] := N[(N[Abs[N[(a - b), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left|a - b\right|}{2} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (/ (fabs (- a b)) 2.0))

C

double code(double a, double b) {
	return fabs((a - b)) / 2.0;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = abs((a - b)) / 2.0d0
end function

Java

public static double code(double a, double b) {
	return Math.abs((a - b)) / 2.0;
}

Python

def code(a, b):
	return math.fabs((a - b)) / 2.0

Julia

function code(a, b)
	return Float64(abs(Float64(a - b)) / 2.0)
end

MATLAB

function tmp = code(a, b)
	tmp = abs((a - b)) / 2.0;
end

Wolfram

code[a_, b_] := N[(N[Abs[N[(a - b), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{\left|a - b\right|}{2} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 51.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left|a\right|}{2} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (/ (fabs a) 2.0))

C

double code(double a, double b) {
	return fabs(a) / 2.0;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = abs(a) / 2.0d0
end function

Java

public static double code(double a, double b) {
	return Math.abs(a) / 2.0;
}

Python

def code(a, b):
	return math.fabs(a) / 2.0

Julia

function code(a, b)
	return Float64(abs(a) / 2.0)
end

MATLAB

function tmp = code(a, b)
	tmp = abs(a) / 2.0;
end

Wolfram

code[a_, b_] := N[(N[Abs[a], $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{\left|a - b\right|}{2} \]
2. Add Preprocessing

3. Taylor expanded in a around inf

   \[\leadsto \mathsf{/.f64}\left(\mathsf{fabs.f64}\left(\color{blue}{a}\right), 2\right) \]
4. Step-by-step derivation

5. Simplified 48.8%

   \[\leadsto \frac{\left|\color{blue}{a}\right|}{2} \]
6. Add Preprocessing

Alternative 3: 26.6% accurate, 35.0× speedup

Math

\[\begin{array}{l} \\ \frac{a}{-2} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (/ a -2.0))

C

double code(double a, double b) {
	return a / -2.0;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = a / (-2.0d0)
end function

Java

public static double code(double a, double b) {
	return a / -2.0;
}

Python

def code(a, b):
	return a / -2.0

Julia

function code(a, b)
	return Float64(a / -2.0)
end

MATLAB

function tmp = code(a, b)
	tmp = a / -2.0;
end

Wolfram

code[a_, b_] := N[(a / -2.0), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{\left|a - b\right|}{2} \]
2. Add Preprocessing

3. Taylor expanded in a around inf

   \[\leadsto \mathsf{/.f64}\left(\mathsf{fabs.f64}\left(\color{blue}{a}\right), 2\right) \]
4. Step-by-step derivation

5. Simplified 48.8%

   \[\leadsto \frac{\left|\color{blue}{a}\right|}{2} \]
6. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{2}{\left|a\right|}}} \]

   2. inv-pow N/A

      \[\leadsto {\left(\frac{2}{\left|a\right|}\right)}^{\color{blue}{-1}} \]

   3. sqr-pow N/A

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

   4. remove-double-neg N/A

      \[\leadsto {\left(\frac{2}{\left|a\right|}\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{2}{\left|a\right|}\right)\right)\right)\right)}^{\left(\frac{\color{blue}{-1}}{2}\right)} \]

   5. neg-mul-1 N/A

      \[\leadsto {\left(\frac{2}{\left|a\right|}\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(-1 \cdot \left(\mathsf{neg}\left(\frac{2}{\left|a\right|}\right)\right)\right)}^{\left(\frac{\color{blue}{-1}}{2}\right)} \]

   6. metadata-eval N/A

      \[\leadsto {\left(\frac{2}{\left|a\right|}\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(-1 \cdot \left(\mathsf{neg}\left(\frac{2}{\left|a\right|}\right)\right)\right)}^{\frac{-1}{2}} \]

   7. metadata-eval N/A

      \[\leadsto {\left(\frac{2}{\left|a\right|}\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(-1 \cdot \left(\mathsf{neg}\left(\frac{2}{\left|a\right|}\right)\right)\right)}^{\left(\frac{1}{\color{blue}{-2}}\right)} \]

   8. metadata-eval N/A

      \[\leadsto {\left(\frac{2}{\left|a\right|}\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(-1 \cdot \left(\mathsf{neg}\left(\frac{2}{\left|a\right|}\right)\right)\right)}^{\left(\frac{1}{\mathsf{neg}\left(2\right)}\right)} \]

   9. unpow-prod-down N/A

      \[\leadsto {\left(\frac{2}{\left|a\right|}\right)}^{\left(\frac{-1}{2}\right)} \cdot \left({-1}^{\left(\frac{1}{\mathsf{neg}\left(2\right)}\right)} \cdot \color{blue}{{\left(\mathsf{neg}\left(\frac{2}{\left|a\right|}\right)\right)}^{\left(\frac{1}{\mathsf{neg}\left(2\right)}\right)}}\right) \]

   10. associate-*r* N/A

       \[\leadsto \left({\left(\frac{2}{\left|a\right|}\right)}^{\left(\frac{-1}{2}\right)} \cdot {-1}^{\left(\frac{1}{\mathsf{neg}\left(2\right)}\right)}\right) \cdot \color{blue}{{\left(\mathsf{neg}\left(\frac{2}{\left|a\right|}\right)\right)}^{\left(\frac{1}{\mathsf{neg}\left(2\right)}\right)}} \]

   11. metadata-eval N/A

       \[\leadsto \left({\left(\frac{2}{\left|a\right|}\right)}^{\frac{-1}{2}} \cdot {-1}^{\left(\frac{1}{\mathsf{neg}\left(2\right)}\right)}\right) \cdot {\left(\mathsf{neg}\left(\frac{2}{\color{blue}{\left|a\right|}}\right)\right)}^{\left(\frac{1}{\mathsf{neg}\left(2\right)}\right)} \]

   12. metadata-eval N/A

       \[\leadsto \left({\left(\frac{2}{\left|a\right|}\right)}^{\left(\frac{1}{-2}\right)} \cdot {-1}^{\left(\frac{1}{\mathsf{neg}\left(2\right)}\right)}\right) \cdot {\left(\mathsf{neg}\left(\frac{2}{\color{blue}{\left|a\right|}}\right)\right)}^{\left(\frac{1}{\mathsf{neg}\left(2\right)}\right)} \]

   13. metadata-eval N/A

       \[\leadsto \left({\left(\frac{2}{\left|a\right|}\right)}^{\left(\frac{1}{\mathsf{neg}\left(2\right)}\right)} \cdot {-1}^{\left(\frac{1}{\mathsf{neg}\left(2\right)}\right)}\right) \cdot {\left(\mathsf{neg}\left(\frac{2}{\left|a\right|}\right)\right)}^{\left(\frac{1}{\mathsf{neg}\left(2\right)}\right)} \]

   14. unpow-prod-down N/A

       \[\leadsto {\left(\frac{2}{\left|a\right|} \cdot -1\right)}^{\left(\frac{1}{\mathsf{neg}\left(2\right)}\right)} \cdot {\color{blue}{\left(\mathsf{neg}\left(\frac{2}{\left|a\right|}\right)\right)}}^{\left(\frac{1}{\mathsf{neg}\left(2\right)}\right)} \]

   15. *-commutative N/A

       \[\leadsto {\left(-1 \cdot \frac{2}{\left|a\right|}\right)}^{\left(\frac{1}{\mathsf{neg}\left(2\right)}\right)} \cdot {\left(\mathsf{neg}\left(\color{blue}{\frac{2}{\left|a\right|}}\right)\right)}^{\left(\frac{1}{\mathsf{neg}\left(2\right)}\right)} \]

   16. neg-mul-1 N/A

       \[\leadsto {\left(\mathsf{neg}\left(\frac{2}{\left|a\right|}\right)\right)}^{\left(\frac{1}{\mathsf{neg}\left(2\right)}\right)} \cdot {\left(\mathsf{neg}\left(\color{blue}{\frac{2}{\left|a\right|}}\right)\right)}^{\left(\frac{1}{\mathsf{neg}\left(2\right)}\right)} \]

   17. metadata-eval N/A

       \[\leadsto {\left(\mathsf{neg}\left(\frac{2}{\left|a\right|}\right)\right)}^{\left(\frac{1}{-2}\right)} \cdot {\left(\mathsf{neg}\left(\frac{2}{\left|a\right|}\right)\right)}^{\left(\frac{1}{\mathsf{neg}\left(2\right)}\right)} \]

   18. metadata-eval N/A

       \[\leadsto {\left(\mathsf{neg}\left(\frac{2}{\left|a\right|}\right)\right)}^{\frac{-1}{2}} \cdot {\left(\mathsf{neg}\left(\frac{2}{\left|a\right|}\right)\right)}^{\left(\frac{1}{\mathsf{neg}\left(2\right)}\right)} \]

   19. metadata-eval N/A

       \[\leadsto {\left(\mathsf{neg}\left(\frac{2}{\left|a\right|}\right)\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\mathsf{neg}\left(\frac{2}{\left|a\right|}\right)\right)}^{\left(\frac{1}{\mathsf{neg}\left(2\right)}\right)} \]

7. Applied egg-rr 27.5%

   \[\leadsto \color{blue}{\frac{a}{-2}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024149 
(FPCore (a b)
  :name "fabs fraction 2"
  :precision binary64
  (/ (fabs (- a b)) 2.0))