x / (x^2 + 1)

Percentage Accurate: 76.4% → 99.8%

Time: 4.5s

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: 76.4% 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: 99.8% 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 75.3%

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

   1. clear-num 75.2%

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

   2. inv-pow 75.2%

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

   3. fma-define 75.2%

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

4. Applied egg-rr 75.2%

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

5. Taylor expanded in x around inf 75.2%

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

   1. distribute-rgt-in 75.2%

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

   2. *-lft-identity 75.2%

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

   3. exp-to-pow 38.9%

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

   4. *-commutative 38.9%

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

   5. rec-exp 38.9%

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

   6. mul-1-neg 38.9%

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

   7. associate-*r* 38.9%

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

   8. metadata-eval 38.9%

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

   9. *-commutative 38.9%

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

   10. exp-to-pow 75.2%

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

   11. pow-plus 99.8%

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

   12. metadata-eval 99.8%

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

   13. exp-to-pow 50.1%

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

   14. *-commutative 50.1%

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

   15. neg-mul-1 50.1%

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

   16. rec-exp 50.1%

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

   17. rem-exp-log 99.8%

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

7. Simplified 99.8%

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

   1. expm1-log1p-u 99.8%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(x + \frac{1}{x}\right)}^{-1}\right)\right)} \]

   2. expm1-undefine 9.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(x + \frac{1}{x}\right)}^{-1}\right)} - 1} \]

   3. unpow-1 9.2%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{1}{x + \frac{1}{x}}}\right)} - 1 \]

9. Applied egg-rr 9.2%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{x + \frac{1}{x}}\right)} - 1} \]
10. Step-by-step derivation

    1. sub-neg 9.2%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{x + \frac{1}{x}}\right)} + \left(-1\right)} \]

    2. metadata-eval 9.2%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{x + \frac{1}{x}}\right)} + \color{blue}{-1} \]

    3. +-commutative 9.2%

       \[\leadsto \color{blue}{-1 + e^{\mathsf{log1p}\left(\frac{1}{x + \frac{1}{x}}\right)}} \]

    4. log1p-undefine 9.2%

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

    5. rem-exp-log 9.2%

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

    6. associate-+r+ 99.8%

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

    7. metadata-eval 99.8%

       \[\leadsto \color{blue}{0} + \frac{1}{x + \frac{1}{x}} \]

    8. metadata-eval 99.8%

       \[\leadsto 0 + \frac{\color{blue}{--1}}{x + \frac{1}{x}} \]

    9. distribute-neg-frac 99.8%

       \[\leadsto 0 + \color{blue}{\left(-\frac{-1}{x + \frac{1}{x}}\right)} \]

    10. sub-neg 99.8%

        \[\leadsto \color{blue}{0 - \frac{-1}{x + \frac{1}{x}}} \]

    11. neg-sub0 99.8%

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

    12. distribute-neg-frac 99.8%

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

    13. metadata-eval 99.8%

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

11. Simplified 99.8%

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

12. Final simplification 99.8%

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

Alternative 2: 75.7% 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 86.9%

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

   3. Taylor expanded in x around 0 69.1%

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

   if 1 < x

   1. Initial program 46.3%

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

   3. Taylor expanded in x around inf 97.8%

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

4. Final simplification 77.3%

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

Alternative 3: 51.7% 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 75.3%

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

3. Taylor expanded in x around 0 50.5%

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

4. Final simplification 50.5%

   \[\leadsto x \]
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 2024077 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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