x / (x^2 + 1)

Percentage Accurate: 76.5% → 99.8%

Time: 3.7s

Alternatives: 3

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: 76.5% 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: 99.8% 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 72.7%

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

   1. clear-num 72.7%

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

   2. associate-/r/ 72.7%

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

   3. add-sqr-sqrt 72.6%

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

   4. pow2 72.6%

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

   5. pow-flip 72.7%

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

   6. +-commutative 72.7%

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

   7. hypot-1-def 72.9%

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

   8. metadata-eval 72.9%

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

4. Applied egg-rr 72.9%

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

   1. sqr-pow 72.9%

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

   2. pow2 72.9%

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

   3. add-sqr-sqrt 35.2%

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

   4. sqrt-unprod 25.0%

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

   5. sqr-neg 25.0%

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

   6. sqrt-unprod 1.9%

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

   7. pow2 1.9%

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

   8. unpow-prod-down 1.8%

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

6. Applied egg-rr 72.7%

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

7. Taylor expanded in x around 0 72.7%

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

   1. +-commutative 72.7%

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

   2. unpow2 72.7%

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

   3. fma-undefine 72.7%

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

   4. *-lft-identity 72.7%

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

   5. associate-*l/ 72.6%

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

   6. fma-undefine 72.6%

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

   7. unpow2 72.6%

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

   8. distribute-rgt-in 72.6%

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

   9. unpow2 72.6%

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

   10. remove-double-div 72.6%

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

   11. unpow-1 72.6%

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

   12. remove-double-div 72.6%

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

   13. unpow-1 72.6%

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

   14. pow-sqr 72.7%

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

   15. metadata-eval 72.7%

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

   16. pow-plus 99.9%

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

   17. metadata-eval 99.9%

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

   18. unpow-1 99.9%

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

   19. remove-double-div 99.9%

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

   20. *-lft-identity 99.9%

       \[\leadsto \frac{1}{x + \color{blue}{\frac{1}{x}}} \]

9. Simplified 99.9%

   \[\leadsto \frac{1}{\color{blue}{x + \frac{1}{x}}} \]

10. Final simplification 99.9%

    \[\leadsto \frac{1}{x + \frac{1}{x}} \]
11. Add Preprocessing

Alternative 2: 74.6% 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 85.8%

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

   3. Taylor expanded in x around 0 67.0%

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

   if 1 < x

   1. Initial program 42.8%

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

   3. Taylor expanded in x around inf 100.0%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 51.0% 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 72.7%

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

3. Taylor expanded in x around 0 47.5%

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

4. Final simplification 47.5%

   \[\leadsto x \]
5. Add Preprocessing

Developer target: 99.8% 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 2024112 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

  :alt
  (/ 1.0 (+ x (/ 1.0 x)))

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