x / (x^2 + 1)

Percentage Accurate: 76.3% → 100.0%

Time: 3.9s

Alternatives: 6

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: 76.3% 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: 100.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -20000000000:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 200000:\\ \;\;\;\;\left(-x\right) \cdot \frac{-1}{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - {x}^{-3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -20000000000.0)
   (/ 1.0 x)
   (if (<= x 200000.0)
     (* (- x) (/ -1.0 (fma x x 1.0)))
     (- (/ 1.0 x) (pow x -3.0)))))

C

double code(double x) {
	double tmp;
	if (x <= -20000000000.0) {
		tmp = 1.0 / x;
	} else if (x <= 200000.0) {
		tmp = -x * (-1.0 / fma(x, x, 1.0));
	} else {
		tmp = (1.0 / x) - pow(x, -3.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -20000000000.0)
		tmp = Float64(1.0 / x);
	elseif (x <= 200000.0)
		tmp = Float64(Float64(-x) * Float64(-1.0 / fma(x, x, 1.0)));
	else
		tmp = Float64(Float64(1.0 / x) - (x ^ -3.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -20000000000.0], N[(1.0 / x), $MachinePrecision], If[LessEqual[x, 200000.0], N[((-x) * N[(-1.0 / N[(x * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] - N[Power[x, -3.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2e10

   1. Initial program 49.4%

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

   2. Taylor expanded in x around inf 100.0%

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

   if -2e10 < x < 2e5

   1. Initial program 100.0%

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

      1. frac-2neg 100.0%

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

      2. div-inv 100.0%

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

      3. fma-def 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. frac-2neg 100.0%

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

      2. metadata-eval 100.0%

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

      3. div-inv 100.0%

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

      4. remove-double-neg 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   7. Simplified 100.0%

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

   if 2e5 < x

   1. Initial program 59.7%

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

      1. frac-2neg 59.7%

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

      2. div-inv 59.6%

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

      3. fma-def 59.6%

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

   3. Applied egg-rr 59.6%

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

   4. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{\frac{1}{x} - \frac{1}{{x}^{3}}} \]
   5. Step-by-step derivation

      1. exp-to-pow 100.0%

         \[\leadsto \frac{1}{x} - \frac{1}{\color{blue}{e^{\log x \cdot 3}}} \]

      2. *-commutative 100.0%

         \[\leadsto \frac{1}{x} - \frac{1}{e^{\color{blue}{3 \cdot \log x}}} \]

      3. rec-exp 100.0%

         \[\leadsto \frac{1}{x} - \color{blue}{e^{-3 \cdot \log x}} \]

      4. mul-1-neg 100.0%

         \[\leadsto \frac{1}{x} - e^{\color{blue}{-1 \cdot \left(3 \cdot \log x\right)}} \]

      5. associate-*r* 100.0%

         \[\leadsto \frac{1}{x} - e^{\color{blue}{\left(-1 \cdot 3\right) \cdot \log x}} \]

      6. metadata-eval 100.0%

         \[\leadsto \frac{1}{x} - e^{\color{blue}{-3} \cdot \log x} \]

      7. *-commutative 100.0%

         \[\leadsto \frac{1}{x} - e^{\color{blue}{\log x \cdot -3}} \]

      8. exp-to-pow 100.0%

         \[\leadsto \frac{1}{x} - \color{blue}{{x}^{-3}} \]

   6. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -20000000000:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 200000:\\ \;\;\;\;\left(-x\right) \cdot \frac{-1}{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - {x}^{-3}\\ \end{array} \]

Alternative 2: 100.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{x}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)} \end{array} \]

FPCore

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

C

double code(double x) {
	return (x / hypot(1.0, x)) / hypot(1.0, x);
}

Java

public static double code(double x) {
	return (x / Math.hypot(1.0, x)) / Math.hypot(1.0, x);
}

Python

def code(x):
	return (x / math.hypot(1.0, x)) / math.hypot(1.0, x)

Julia

function code(x)
	return Float64(Float64(x / hypot(1.0, x)) / hypot(1.0, x))
end

MATLAB

function tmp = code(x)
	tmp = (x / hypot(1.0, x)) / hypot(1.0, x);
end

Wolfram

code[x_] := N[(N[(x / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.9%

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

   1. frac-2neg 78.9%

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

   2. div-inv 78.8%

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

   3. fma-def 78.8%

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

3. Applied egg-rr 78.8%

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

   1. frac-2neg 78.8%

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

   2. metadata-eval 78.8%

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

   3. div-inv 78.8%

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

   4. remove-double-neg 78.8%

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

5. Applied egg-rr 78.8%

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

   1. associate-*r/ 78.8%

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

   2. metadata-eval 78.8%

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

7. Simplified 78.8%

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

   1. associate-*r/ 78.9%

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

   2. add-sqr-sqrt 78.9%

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

   3. associate-/r* 78.9%

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

   4. add-sqr-sqrt 39.6%

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

   5. sqrt-unprod 27.5%

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

   6. sqr-neg 27.5%

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

   7. sqrt-unprod 1.9%

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

   8. add-sqr-sqrt 4.0%

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

   9. *-commutative 4.0%

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

   10. neg-mul-1 4.0%

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

   11. add-sqr-sqrt 2.1%

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

   12. sqrt-unprod 27.6%

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

   13. sqr-neg 27.6%

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

   14. sqrt-unprod 38.7%

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

   15. add-sqr-sqrt 78.9%

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

   16. fma-udef 78.9%

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

   17. +-commutative 78.9%

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

   18. hypot-1-def 78.9%

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

   19. fma-udef 78.9%

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

   20. +-commutative 78.9%

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

   21. hypot-1-def 100.0%

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

9. Applied egg-rr 100.0%

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

10. Final simplification 100.0%

    \[\leadsto \frac{\frac{x}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)} \]

Alternative 3: 100.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -500000 \lor \neg \left(x \leq 50000000\right):\\ \;\;\;\;\frac{1}{x} - {x}^{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -500000.0) (not (<= x 50000000.0)))
   (- (/ 1.0 x) (pow x -3.0))
   (/ x (+ 1.0 (* x x)))))

C

double code(double x) {
	double tmp;
	if ((x <= -500000.0) || !(x <= 50000000.0)) {
		tmp = (1.0 / x) - pow(x, -3.0);
	} else {
		tmp = x / (1.0 + (x * x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-500000.0d0)) .or. (.not. (x <= 50000000.0d0))) then
        tmp = (1.0d0 / x) - (x ** (-3.0d0))
    else
        tmp = x / (1.0d0 + (x * x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -500000.0) || !(x <= 50000000.0)) {
		tmp = (1.0 / x) - Math.pow(x, -3.0);
	} else {
		tmp = x / (1.0 + (x * x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -500000.0) or not (x <= 50000000.0):
		tmp = (1.0 / x) - math.pow(x, -3.0)
	else:
		tmp = x / (1.0 + (x * x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -500000.0) || !(x <= 50000000.0))
		tmp = Float64(Float64(1.0 / x) - (x ^ -3.0));
	else
		tmp = Float64(x / Float64(1.0 + Float64(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -500000.0) || ~((x <= 50000000.0)))
		tmp = (1.0 / x) - (x ^ -3.0);
	else
		tmp = x / (1.0 + (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -500000.0], N[Not[LessEqual[x, 50000000.0]], $MachinePrecision]], N[(N[(1.0 / x), $MachinePrecision] - N[Power[x, -3.0], $MachinePrecision]), $MachinePrecision], N[(x / N[(1.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5e5 or 5e7 < x

   1. Initial program 55.4%

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

      1. frac-2neg 55.4%

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

      2. div-inv 55.2%

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

      3. fma-def 55.2%

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

   3. Applied egg-rr 55.2%

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

   4. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{\frac{1}{x} - \frac{1}{{x}^{3}}} \]
   5. Step-by-step derivation

      1. exp-to-pow 53.7%

         \[\leadsto \frac{1}{x} - \frac{1}{\color{blue}{e^{\log x \cdot 3}}} \]

      2. *-commutative 53.7%

         \[\leadsto \frac{1}{x} - \frac{1}{e^{\color{blue}{3 \cdot \log x}}} \]

      3. rec-exp 53.7%

         \[\leadsto \frac{1}{x} - \color{blue}{e^{-3 \cdot \log x}} \]

      4. mul-1-neg 53.7%

         \[\leadsto \frac{1}{x} - e^{\color{blue}{-1 \cdot \left(3 \cdot \log x\right)}} \]

      5. associate-*r* 53.7%

         \[\leadsto \frac{1}{x} - e^{\color{blue}{\left(-1 \cdot 3\right) \cdot \log x}} \]

      6. metadata-eval 53.7%

         \[\leadsto \frac{1}{x} - e^{\color{blue}{-3} \cdot \log x} \]

      7. *-commutative 53.7%

         \[\leadsto \frac{1}{x} - e^{\color{blue}{\log x \cdot -3}} \]

      8. exp-to-pow 100.0%

         \[\leadsto \frac{1}{x} - \color{blue}{{x}^{-3}} \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{x} - {x}^{-3}} \]

   if -5e5 < x < 5e7

   1. Initial program 100.0%

      \[\frac{x}{x \cdot x + 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -500000 \lor \neg \left(x \leq 50000000\right):\\ \;\;\;\;\frac{1}{x} - {x}^{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \end{array} \]

Alternative 4: 100.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -20000000000 \lor \neg \left(x \leq 100000000\right):\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -20000000000.0) (not (<= x 100000000.0)))
   (/ 1.0 x)
   (/ x (+ 1.0 (* x x)))))

C

double code(double x) {
	double tmp;
	if ((x <= -20000000000.0) || !(x <= 100000000.0)) {
		tmp = 1.0 / x;
	} else {
		tmp = x / (1.0 + (x * x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-20000000000.0d0)) .or. (.not. (x <= 100000000.0d0))) then
        tmp = 1.0d0 / x
    else
        tmp = x / (1.0d0 + (x * x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -20000000000.0) || !(x <= 100000000.0)) {
		tmp = 1.0 / x;
	} else {
		tmp = x / (1.0 + (x * x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -20000000000.0) or not (x <= 100000000.0):
		tmp = 1.0 / x
	else:
		tmp = x / (1.0 + (x * x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -20000000000.0) || !(x <= 100000000.0))
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(x / Float64(1.0 + Float64(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -20000000000.0) || ~((x <= 100000000.0)))
		tmp = 1.0 / x;
	else
		tmp = x / (1.0 + (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -20000000000.0], N[Not[LessEqual[x, 100000000.0]], $MachinePrecision]], N[(1.0 / x), $MachinePrecision], N[(x / N[(1.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2e10 or 1e8 < x

   1. Initial program 55.0%

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

   2. Taylor expanded in x around inf 100.0%

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

   if -2e10 < x < 1e8

   1. Initial program 100.0%

      \[\frac{x}{x \cdot x + 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -20000000000 \lor \neg \left(x \leq 100000000\right):\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \end{array} \]

Alternative 5: 98.9% 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 56.8%

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

   2. Taylor expanded in x around inf 98.2%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.5%

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

4. Final simplification 98.9%

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

Alternative 6: 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 78.9%

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

2. Taylor expanded in x around 0 52.9%

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

3. Final simplification 52.9%

   \[\leadsto x \]

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 2023314 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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