x / (x^2 + 1)

Percentage Accurate: 76.4% → 99.9%

Time: 5.6s

Alternatives: 4

Speedup: 1.0×

Specification

Math

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

FPCore

(FPCore (x) :precision binary64 (/ x (+ (* x x) 1.0)))

C

double code(double x) {
	return x / ((x * x) + 1.0);
}

Fortran

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

Java

public static double code(double x) {
	return x / ((x * x) + 1.0);
}

Python

def code(x):
	return x / ((x * x) + 1.0)

Julia

function code(x)
	return Float64(x / Float64(Float64(x * x) + 1.0))
end

MATLAB

function tmp = code(x)
	tmp = x / ((x * x) + 1.0);
end

Wolfram

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

Initial Program: 76.4% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (/ x (+ (* x x) 1.0)))

C

double code(double x) {
	return x / ((x * x) + 1.0);
}

Fortran

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

Java

public static double code(double x) {
	return x / ((x * x) + 1.0);
}

Python

def code(x):
	return x / ((x * x) + 1.0)

Julia

function code(x)
	return Float64(x / Float64(Float64(x * x) + 1.0))
end

MATLAB

function tmp = code(x)
	tmp = x / ((x * x) + 1.0);
end

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x + \frac{1}{x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (+ x (/ 1.0 x))))

C

double code(double x) {
	return 1.0 / (x + (1.0 / x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / (x + (1.0d0 / x))
end function

Java

public static double code(double x) {
	return 1.0 / (x + (1.0 / x));
}

Python

def code(x):
	return 1.0 / (x + (1.0 / x))

Julia

function code(x)
	return Float64(1.0 / Float64(x + Float64(1.0 / x)))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / (x + (1.0 / x));
end

Wolfram

code[x_] := N[(1.0 / N[(x + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.6%

   \[\frac{x}{x \cdot x + 1} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. flip-+ N/A

      \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 1 \cdot 1}{\color{blue}{x \cdot x - 1}}\right)\right) \]

   2. difference-of-sqr-1 N/A

      \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 1 \cdot 1}{\left(x + 1\right) \cdot \color{blue}{\left(x - 1\right)}}\right)\right) \]

   3. associate-/r* N/A

      \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 1 \cdot 1}{x + 1}}{\color{blue}{x - 1}}\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\left(\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 1 \cdot 1}{x + 1}\right), \color{blue}{\left(x - 1\right)}\right)\right) \]

4. Applied egg-rr 63.3%

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

   1. associate-/r/ N/A

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

   2. flip-+ N/A

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

   5. associate-*r/ N/A

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

   6. clear-num N/A

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

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{x + 1}{\frac{x}{\frac{x \cdot \left(x \cdot \left(x \cdot x\right)\right) + -1}{x + 1}} \cdot \left(x \cdot x - -1 \cdot -1\right)}\right)}\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(x + 1\right), \color{blue}{\left(\frac{x}{\frac{x \cdot \left(x \cdot \left(x \cdot x\right)\right) + -1}{x + 1}} \cdot \left(x \cdot x - -1 \cdot -1\right)\right)}\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, 1\right), \left(\color{blue}{\frac{x}{\frac{x \cdot \left(x \cdot \left(x \cdot x\right)\right) + -1}{x + 1}}} \cdot \left(x \cdot x - -1 \cdot -1\right)\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, 1\right), \left(\frac{x}{\frac{x \cdot \left(x \cdot \left(x \cdot x\right)\right) + -1}{x + 1}} \cdot \left(x \cdot x - 1\right)\right)\right)\right) \]

   11. difference-of-sqr-1 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, 1\right), \left(\frac{x}{\frac{x \cdot \left(x \cdot \left(x \cdot x\right)\right) + -1}{x + 1}} \cdot \left(\left(x + 1\right) \cdot \color{blue}{\left(x - 1\right)}\right)\right)\right)\right) \]

   12. sub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, 1\right), \left(\frac{x}{\frac{x \cdot \left(x \cdot \left(x \cdot x\right)\right) + -1}{x + 1}} \cdot \left(\left(x + 1\right) \cdot \left(x + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right)\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, 1\right), \left(\frac{x}{\frac{x \cdot \left(x \cdot \left(x \cdot x\right)\right) + -1}{x + 1}} \cdot \left(\left(x + 1\right) \cdot \left(x + -1\right)\right)\right)\right)\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, 1\right), \left(\frac{x}{\frac{x \cdot \left(x \cdot \left(x \cdot x\right)\right) + -1}{x + 1}} \cdot \left(\left(x + -1\right) \cdot \color{blue}{\left(x + 1\right)}\right)\right)\right)\right) \]

6. Applied egg-rr 72.5%

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

7. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{1 + {x}^{2}}{x}\right)}\right) \]
8. Step-by-step derivation

   1. rgt-mult-inverse N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{{x}^{2} \cdot \frac{1}{{x}^{2}} + {x}^{2}}{x}\right)\right) \]

   2. *-rgt-identity N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{{x}^{2} \cdot \frac{1}{{x}^{2}} + {x}^{2} \cdot 1}{x}\right)\right) \]

   3. distribute-lft-in N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{{x}^{2} \cdot \left(\frac{1}{{x}^{2}} + 1\right)}{x}\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{{x}^{2} \cdot \left(1 + \frac{1}{{x}^{2}}\right)}{x}\right)\right) \]

   5. associate-*l/ N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{{x}^{2}}{x} \cdot \color{blue}{\left(1 + \frac{1}{{x}^{2}}\right)}\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{x \cdot x}{x} \cdot \left(1 + \frac{1}{{x}^{2}}\right)\right)\right) \]

   7. associate-/l* N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\left(x \cdot \frac{x}{x}\right) \cdot \left(\color{blue}{1} + \frac{1}{{x}^{2}}\right)\right)\right) \]

   8. *-rgt-identity N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\left(x \cdot \frac{x \cdot 1}{x}\right) \cdot \left(1 + \frac{1}{{x}^{2}}\right)\right)\right) \]

   9. associate-*r/ N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\left(x \cdot \left(x \cdot \frac{1}{x}\right)\right) \cdot \left(1 + \frac{1}{{x}^{2}}\right)\right)\right) \]

   10. rgt-mult-inverse N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\left(x \cdot 1\right) \cdot \left(1 + \frac{1}{{x}^{2}}\right)\right)\right) \]

   11. *-rgt-identity N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(x \cdot \left(\color{blue}{1} + \frac{1}{{x}^{2}}\right)\right)\right) \]

   12. distribute-lft-in N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(x \cdot 1 + \color{blue}{x \cdot \frac{1}{{x}^{2}}}\right)\right) \]

   13. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(x \cdot 1 + x \cdot \frac{1}{x \cdot \color{blue}{x}}\right)\right) \]

   14. associate-/r* N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(x \cdot 1 + x \cdot \frac{\frac{1}{x}}{\color{blue}{x}}\right)\right) \]

   15. associate-*r/ N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(x \cdot 1 + \frac{x \cdot \frac{1}{x}}{\color{blue}{x}}\right)\right) \]

   16. rgt-mult-inverse N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(x \cdot 1 + \frac{1}{x}\right)\right) \]

   17. *-rgt-identity N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(x + \frac{\color{blue}{1}}{x}\right)\right) \]

   18. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, \color{blue}{\left(\frac{1}{x}\right)}\right)\right) \]

   19. /-lowering-/.f64 99.9%

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(1, \color{blue}{x}\right)\right)\right) \]

9. Simplified 99.9%

   \[\leadsto \frac{1}{\color{blue}{x + \frac{1}{x}}} \]
10. Add Preprocessing

Alternative 2: 74.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.86:\\ \;\;\;\;x \cdot \left(1 - x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.86) (* x (- 1.0 (* x x))) (/ 1.0 x)))

C

double code(double x) {
	double tmp;
	if (x <= 0.86) {
		tmp = x * (1.0 - (x * x));
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.86d0) then
        tmp = x * (1.0d0 - (x * x))
    else
        tmp = 1.0d0 / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.86) {
		tmp = x * (1.0 - (x * x));
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.86:
		tmp = x * (1.0 - (x * x))
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.86)
		tmp = Float64(x * Float64(1.0 - Float64(x * x)));
	else
		tmp = Float64(1.0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.86)
		tmp = x * (1.0 - (x * x));
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.86], N[(x * N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.859999999999999987

   1. Initial program 80.9%

      \[\frac{x}{x \cdot x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + -1 \cdot {x}^{2}\right)}\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(1 + \left(\mathsf{neg}\left({x}^{2}\right)\right)\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(1 - \color{blue}{{x}^{2}}\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \color{blue}{\left({x}^{2}\right)}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \left(x \cdot \color{blue}{x}\right)\right)\right) \]

      6. *-lowering-*.f64 66.8%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

   5. Simplified 66.8%

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

   if 0.859999999999999987 < x

   1. Initial program 48.9%

      \[\frac{x}{x \cdot x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{x}\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 75.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.0) x (/ 1.0 x)))

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = x;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.0d0) then
        tmp = x
    else
        tmp = 1.0d0 / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = x;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = x
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = x;
	else
		tmp = Float64(1.0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = x;
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.0], x, N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 80.9%

      \[\frac{x}{x \cdot x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Simplified 67.4%

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

   if 1 < x

   1. Initial program 48.9%

      \[\frac{x}{x \cdot x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{x}\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 51.5% accurate, 7.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 x)

C

double code(double x) {
	return x;
}

Fortran

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

Java

public static double code(double x) {
	return x;
}

Python

def code(x):
	return x

Julia

function code(x)
	return x
end

MATLAB

function tmp = code(x)
	tmp = x;
end

Wolfram

code[x_] := x

Derivation

1. Initial program 72.6%

   \[\frac{x}{x \cdot x + 1} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

5. Simplified 50.9%

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

Developer Target 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x + \frac{1}{x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (+ x (/ 1.0 x))))

C

double code(double x) {
	return 1.0 / (x + (1.0 / x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / (x + (1.0d0 / x))
end function

Java

public static double code(double x) {
	return 1.0 / (x + (1.0 / x));
}

Python

def code(x):
	return 1.0 / (x + (1.0 / x))

Julia

function code(x)
	return Float64(1.0 / Float64(x + Float64(1.0 / x)))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / (x + (1.0 / x));
end

Wolfram

code[x_] := N[(1.0 / N[(x + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024158 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

  :alt
  (! :herbie-platform default (/ 1 (+ x (/ 1 x))))

  (/ x (+ (* x x) 1.0)))