x / (x^2 + 1)

Percentage Accurate: 77.1% → 99.9%

Time: 3.9s

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: 77.1% 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 78.5%

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

   1. clear-num 78.4%

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

   2. inv-pow 78.4%

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

   3. fma-def 78.4%

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

3. Applied egg-rr 78.4%

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

4. Taylor expanded in x around 0 99.8%

   \[\leadsto {\color{blue}{\left(x + \frac{1}{x}\right)}}^{-1} \]
5. 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-udef 7.6%

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

   3. sub-neg 7.6%

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

   4. unpow-1 7.6%

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

   5. metadata-eval 7.6%

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

6. Applied egg-rr 7.6%

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

   1. metadata-eval 7.6%

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

   2. sub-neg 7.6%

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

   3. expm1-def 99.8%

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

   4. expm1-log1p 99.8%

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

8. Simplified 99.8%

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

9. Final simplification 99.8%

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

Alternative 2: 99.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 1.0))) (/ 1.0 x) x))

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = 1.0d0 / x
    else
        tmp = x
    end if
    code = tmp
end function

Java

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

Python

def code(x):
	tmp = 0
	if (x <= -1.0) or not (x <= 1.0):
		tmp = 1.0 / x
	else:
		tmp = x
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(1.0 / x), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 54.5%

      \[\frac{x}{x \cdot x + 1} \]

   2. Taylor expanded in x around inf 99.3%

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

   if -1 < x < 1

   1. Initial program 100.0%

      \[\frac{x}{x \cdot x + 1} \]

   2. Taylor expanded in x around 0 99.1%

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

4. Final simplification 99.2%

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

Alternative 3: 3.8% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 -1.0)

C

double code(double x) {
	return -1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return -1.0

Julia

function code(x)
	return -1.0
end

MATLAB

function tmp = code(x)
	tmp = -1.0;
end

Wolfram

code[x_] := -1.0

Derivation

1. Initial program 78.5%

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

   1. clear-num 78.4%

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

   2. inv-pow 78.4%

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

   3. fma-def 78.4%

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

3. Applied egg-rr 78.4%

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

5. Applied egg-rr 0.8%

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

   1. distribute-frac-neg 0.8%

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

   2. *-inverses 3.8%

      \[\leadsto -\color{blue}{1} \]

   3. metadata-eval 3.8%

      \[\leadsto \color{blue}{-1} \]

7. Simplified 3.8%

   \[\leadsto \color{blue}{-1} \]

8. Final simplification 3.8%

   \[\leadsto -1 \]

Alternative 4: 52.6% 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 78.5%

   \[\frac{x}{x \cdot x + 1} \]

2. Taylor expanded in x around 0 54.0%

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

3. Final simplification 54.0%

   \[\leadsto x \]

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 2023305 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

  :herbie-target
  (/ 1.0 (+ x (/ 1.0 x)))

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