x / (x^2 + 1)

Percentage Accurate: 76.7% → 99.9%

Time: 3.6s

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: 76.7% 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 75.6%

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

   1. frac-2neg 75.6%

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

   2. div-inv 75.5%

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

   3. fma-define 75.5%

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

4. Applied egg-rr 75.5%

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

   1. un-div-inv 75.6%

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

   2. clear-num 75.5%

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

   3. add-sqr-sqrt 0.0%

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

   4. sqrt-unprod 4.3%

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

   5. sqr-neg 4.3%

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

   6. sqrt-unprod 4.1%

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

   7. add-sqr-sqrt 4.1%

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

   8. add-sqr-sqrt 1.9%

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

   9. sqrt-unprod 30.4%

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

   10. sqr-neg 30.4%

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

   11. sqrt-unprod 40.4%

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

   12. add-sqr-sqrt 75.5%

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

6. Applied egg-rr 75.5%

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

7. Taylor expanded in x around 0 75.5%

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

   1. +-commutative 75.5%

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

   2. unpow2 75.5%

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

   3. fma-undefine 75.5%

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

   4. *-lft-identity 75.5%

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

   5. associate-*l/ 75.5%

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

   6. fma-undefine 75.5%

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

   7. unpow2 75.5%

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

   8. distribute-rgt-in 75.4%

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

   9. unpow2 75.4%

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

   10. remove-double-div 75.5%

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

   11. unpow-1 75.5%

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

   12. remove-double-div 75.5%

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

   13. unpow-1 75.5%

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

   14. pow-sqr 75.5%

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

   15. metadata-eval 75.5%

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

   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: 75.0% 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 83.4%

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

   3. Taylor expanded in x around 0 65.7%

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

   if 1 < x

   1. Initial program 52.1%

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

   3. Taylor expanded in x around inf 98.1%

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

4. Final simplification 73.8%

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

Alternative 3: 51.1% 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 75.6%

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

3. Taylor expanded in x around 0 50.2%

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

4. Final simplification 50.2%

   \[\leadsto x \]
5. Add Preprocessing

Developer target: 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 2024067 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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