x / (x^2 + 1)

Percentage Accurate: 77.1% → 100.0%

Time: 4.1s

Alternatives: 7

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: 100.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{{x}^{4} + -1}\\ \mathbf{if}\;x \leq -4000 \lor \neg \left(x \leq 500\right):\\ \;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) + \frac{-1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(x \cdot x\right) - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ x (+ (pow x 4.0) -1.0))))
   (if (or (<= x -4000.0) (not (<= x 500.0)))
     (+ (+ (/ 1.0 (pow x 5.0)) (/ 1.0 x)) (/ -1.0 (pow x 3.0)))
     (- (* t_0 (* x x)) t_0))))

C

double code(double x) {
	double t_0 = x / (pow(x, 4.0) + -1.0);
	double tmp;
	if ((x <= -4000.0) || !(x <= 500.0)) {
		tmp = ((1.0 / pow(x, 5.0)) + (1.0 / x)) + (-1.0 / pow(x, 3.0));
	} else {
		tmp = (t_0 * (x * x)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / ((x ** 4.0d0) + (-1.0d0))
    if ((x <= (-4000.0d0)) .or. (.not. (x <= 500.0d0))) then
        tmp = ((1.0d0 / (x ** 5.0d0)) + (1.0d0 / x)) + ((-1.0d0) / (x ** 3.0d0))
    else
        tmp = (t_0 * (x * x)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = x / (Math.pow(x, 4.0) + -1.0);
	double tmp;
	if ((x <= -4000.0) || !(x <= 500.0)) {
		tmp = ((1.0 / Math.pow(x, 5.0)) + (1.0 / x)) + (-1.0 / Math.pow(x, 3.0));
	} else {
		tmp = (t_0 * (x * x)) - t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = x / (math.pow(x, 4.0) + -1.0)
	tmp = 0
	if (x <= -4000.0) or not (x <= 500.0):
		tmp = ((1.0 / math.pow(x, 5.0)) + (1.0 / x)) + (-1.0 / math.pow(x, 3.0))
	else:
		tmp = (t_0 * (x * x)) - t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(x / Float64((x ^ 4.0) + -1.0))
	tmp = 0.0
	if ((x <= -4000.0) || !(x <= 500.0))
		tmp = Float64(Float64(Float64(1.0 / (x ^ 5.0)) + Float64(1.0 / x)) + Float64(-1.0 / (x ^ 3.0)));
	else
		tmp = Float64(Float64(t_0 * Float64(x * x)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x / ((x ^ 4.0) + -1.0);
	tmp = 0.0;
	if ((x <= -4000.0) || ~((x <= 500.0)))
		tmp = ((1.0 / (x ^ 5.0)) + (1.0 / x)) + (-1.0 / (x ^ 3.0));
	else
		tmp = (t_0 * (x * x)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x / N[(N[Power[x, 4.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -4000.0], N[Not[LessEqual[x, 500.0]], $MachinePrecision]], N[(N[(N[(1.0 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[(x * x), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -4e3 or 500 < x

   1. Initial program 53.4%

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

   2. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{{x}^{3}}} \]

   if -4e3 < x < 500

   1. Initial program 100.0%

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

      1. flip-+ 100.0%

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

      2. associate-/r/ 100.0%

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

      3. metadata-eval 100.0%

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

      4. sub-neg 100.0%

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

      5. pow2 100.0%

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

      6. pow2 100.0%

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

      7. pow-prod-up 100.0%

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

      8. metadata-eval 100.0%

         \[\leadsto \frac{x}{{x}^{\color{blue}{4}} + \left(-1\right)} \cdot \left(x \cdot x - 1\right) \]

      9. metadata-eval 100.0%

         \[\leadsto \frac{x}{{x}^{4} + \color{blue}{-1}} \cdot \left(x \cdot x - 1\right) \]

      10. fma-neg 100.0%

          \[\leadsto \frac{x}{{x}^{4} + -1} \cdot \color{blue}{\mathsf{fma}\left(x, x, -1\right)} \]

      11. metadata-eval 100.0%

          \[\leadsto \frac{x}{{x}^{4} + -1} \cdot \mathsf{fma}\left(x, x, \color{blue}{-1}\right) \]

   3. Applied egg-rr 100.0%

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

      1. fma-udef 100.0%

         \[\leadsto \frac{x}{{x}^{4} + -1} \cdot \color{blue}{\left(x \cdot x + -1\right)} \]

      2. distribute-lft-in 100.0%

         \[\leadsto \color{blue}{\frac{x}{{x}^{4} + -1} \cdot \left(x \cdot x\right) + \frac{x}{{x}^{4} + -1} \cdot -1} \]

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{x}{{x}^{4} + -1} \cdot \left(x \cdot x\right) + \frac{x}{{x}^{4} + -1} \cdot -1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4000 \lor \neg \left(x \leq 500\right):\\ \;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) + \frac{-1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{{x}^{4} + -1} \cdot \left(x \cdot x\right) - \frac{x}{{x}^{4} + -1}\\ \end{array} \]

Alternative 2: 100.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -20000000 \lor \neg \left(x \leq 10000\right):\\ \;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) + \frac{-1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - {x}^{3}}{1 - {x}^{4}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -20000000.0) (not (<= x 10000.0)))
   (+ (+ (/ 1.0 (pow x 5.0)) (/ 1.0 x)) (/ -1.0 (pow x 3.0)))
   (/ (- x (pow x 3.0)) (- 1.0 (pow x 4.0)))))

C

double code(double x) {
	double tmp;
	if ((x <= -20000000.0) || !(x <= 10000.0)) {
		tmp = ((1.0 / pow(x, 5.0)) + (1.0 / x)) + (-1.0 / pow(x, 3.0));
	} else {
		tmp = (x - pow(x, 3.0)) / (1.0 - pow(x, 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-20000000.0d0)) .or. (.not. (x <= 10000.0d0))) then
        tmp = ((1.0d0 / (x ** 5.0d0)) + (1.0d0 / x)) + ((-1.0d0) / (x ** 3.0d0))
    else
        tmp = (x - (x ** 3.0d0)) / (1.0d0 - (x ** 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -20000000.0) || !(x <= 10000.0)) {
		tmp = ((1.0 / Math.pow(x, 5.0)) + (1.0 / x)) + (-1.0 / Math.pow(x, 3.0));
	} else {
		tmp = (x - Math.pow(x, 3.0)) / (1.0 - Math.pow(x, 4.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -20000000.0) or not (x <= 10000.0):
		tmp = ((1.0 / math.pow(x, 5.0)) + (1.0 / x)) + (-1.0 / math.pow(x, 3.0))
	else:
		tmp = (x - math.pow(x, 3.0)) / (1.0 - math.pow(x, 4.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -20000000.0) || !(x <= 10000.0))
		tmp = Float64(Float64(Float64(1.0 / (x ^ 5.0)) + Float64(1.0 / x)) + Float64(-1.0 / (x ^ 3.0)));
	else
		tmp = Float64(Float64(x - (x ^ 3.0)) / Float64(1.0 - (x ^ 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -20000000.0) || ~((x <= 10000.0)))
		tmp = ((1.0 / (x ^ 5.0)) + (1.0 / x)) + (-1.0 / (x ^ 3.0));
	else
		tmp = (x - (x ^ 3.0)) / (1.0 - (x ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -20000000.0], N[Not[LessEqual[x, 10000.0]], $MachinePrecision]], N[(N[(N[(1.0 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x - N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2e7 or 1e4 < x

   1. Initial program 52.7%

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

   2. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{{x}^{3}}} \]

   if -2e7 < x < 1e4

   1. Initial program 99.9%

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

      1. flip-+ 100.0%

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

      2. associate-/r/ 100.0%

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

      3. metadata-eval 100.0%

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

      4. sub-neg 100.0%

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

      5. pow2 100.0%

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

      6. pow2 100.0%

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

      7. pow-prod-up 100.0%

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

      8. metadata-eval 100.0%

         \[\leadsto \frac{x}{{x}^{\color{blue}{4}} + \left(-1\right)} \cdot \left(x \cdot x - 1\right) \]

      9. metadata-eval 100.0%

         \[\leadsto \frac{x}{{x}^{4} + \color{blue}{-1}} \cdot \left(x \cdot x - 1\right) \]

      10. fma-neg 100.0%

          \[\leadsto \frac{x}{{x}^{4} + -1} \cdot \color{blue}{\mathsf{fma}\left(x, x, -1\right)} \]

      11. metadata-eval 100.0%

          \[\leadsto \frac{x}{{x}^{4} + -1} \cdot \mathsf{fma}\left(x, x, \color{blue}{-1}\right) \]

   3. Applied egg-rr 100.0%

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

      1. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{x \cdot \mathsf{fma}\left(x, x, -1\right)}{{x}^{4} + -1}} \]

      2. frac-2neg 100.0%

         \[\leadsto \color{blue}{\frac{-x \cdot \mathsf{fma}\left(x, x, -1\right)}{-\left({x}^{4} + -1\right)}} \]

      3. +-commutative 100.0%

         \[\leadsto \frac{-x \cdot \mathsf{fma}\left(x, x, -1\right)}{-\color{blue}{\left(-1 + {x}^{4}\right)}} \]

      4. distribute-neg-in 100.0%

         \[\leadsto \frac{-x \cdot \mathsf{fma}\left(x, x, -1\right)}{\color{blue}{\left(--1\right) + \left(-{x}^{4}\right)}} \]

      5. metadata-eval 100.0%

         \[\leadsto \frac{-x \cdot \mathsf{fma}\left(x, x, -1\right)}{\color{blue}{1} + \left(-{x}^{4}\right)} \]

   5. Applied egg-rr 100.0%

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

      1. distribute-lft-neg-in 100.0%

         \[\leadsto \frac{\color{blue}{\left(-x\right) \cdot \mathsf{fma}\left(x, x, -1\right)}}{1 + \left(-{x}^{4}\right)} \]

      2. fma-udef 100.0%

         \[\leadsto \frac{\left(-x\right) \cdot \color{blue}{\left(x \cdot x + -1\right)}}{1 + \left(-{x}^{4}\right)} \]

      3. distribute-rgt-in 100.0%

         \[\leadsto \frac{\color{blue}{\left(x \cdot x\right) \cdot \left(-x\right) + -1 \cdot \left(-x\right)}}{1 + \left(-{x}^{4}\right)} \]

      4. distribute-rgt-neg-in 100.0%

         \[\leadsto \frac{\color{blue}{\left(-\left(x \cdot x\right) \cdot x\right)} + -1 \cdot \left(-x\right)}{1 + \left(-{x}^{4}\right)} \]

      5. unpow3 100.0%

         \[\leadsto \frac{\left(-\color{blue}{{x}^{3}}\right) + -1 \cdot \left(-x\right)}{1 + \left(-{x}^{4}\right)} \]

      6. neg-mul-1 100.0%

         \[\leadsto \frac{\left(-{x}^{3}\right) + \color{blue}{\left(-\left(-x\right)\right)}}{1 + \left(-{x}^{4}\right)} \]

      7. remove-double-neg 100.0%

         \[\leadsto \frac{\left(-{x}^{3}\right) + \color{blue}{x}}{1 + \left(-{x}^{4}\right)} \]

      8. +-commutative 100.0%

         \[\leadsto \frac{\color{blue}{x + \left(-{x}^{3}\right)}}{1 + \left(-{x}^{4}\right)} \]

      9. sub-neg 100.0%

         \[\leadsto \frac{\color{blue}{x - {x}^{3}}}{1 + \left(-{x}^{4}\right)} \]

      10. unsub-neg 100.0%

          \[\leadsto \frac{x - {x}^{3}}{\color{blue}{1 - {x}^{4}}} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{x - {x}^{3}}{1 - {x}^{4}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -20000000 \lor \neg \left(x \leq 10000\right):\\ \;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) + \frac{-1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - {x}^{3}}{1 - {x}^{4}}\\ \end{array} \]

Alternative 3: 100.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+38}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 500000:\\ \;\;\;\;\frac{x - {x}^{3}}{1 - {x}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5e+38)
   (/ 1.0 x)
   (if (<= x 500000.0) (/ (- x (pow x 3.0)) (- 1.0 (pow x 4.0))) (/ 1.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= -5e+38) {
		tmp = 1.0 / x;
	} else if (x <= 500000.0) {
		tmp = (x - pow(x, 3.0)) / (1.0 - pow(x, 4.0));
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-5d+38)) then
        tmp = 1.0d0 / x
    else if (x <= 500000.0d0) then
        tmp = (x - (x ** 3.0d0)) / (1.0d0 - (x ** 4.0d0))
    else
        tmp = 1.0d0 / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -5e+38) {
		tmp = 1.0 / x;
	} else if (x <= 500000.0) {
		tmp = (x - Math.pow(x, 3.0)) / (1.0 - Math.pow(x, 4.0));
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -5e+38:
		tmp = 1.0 / x
	elif x <= 500000.0:
		tmp = (x - math.pow(x, 3.0)) / (1.0 - math.pow(x, 4.0))
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -5e+38)
		tmp = Float64(1.0 / x);
	elseif (x <= 500000.0)
		tmp = Float64(Float64(x - (x ^ 3.0)) / Float64(1.0 - (x ^ 4.0)));
	else
		tmp = Float64(1.0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -5e+38)
		tmp = 1.0 / x;
	elseif (x <= 500000.0)
		tmp = (x - (x ^ 3.0)) / (1.0 - (x ^ 4.0));
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -5e+38], N[(1.0 / x), $MachinePrecision], If[LessEqual[x, 500000.0], N[(N[(x - N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.9999999999999997e38 or 5e5 < x

   1. Initial program 49.7%

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

   2. Taylor expanded in x around inf 100.0%

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

   if -4.9999999999999997e38 < x < 5e5

   1. Initial program 100.0%

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

      1. flip-+ 100.0%

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

      2. associate-/r/ 99.9%

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

      3. metadata-eval 99.9%

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

      4. sub-neg 99.9%

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

      5. pow2 99.9%

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

      6. pow2 99.9%

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

      7. pow-prod-up 99.9%

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

      8. metadata-eval 99.9%

         \[\leadsto \frac{x}{{x}^{\color{blue}{4}} + \left(-1\right)} \cdot \left(x \cdot x - 1\right) \]

      9. metadata-eval 99.9%

         \[\leadsto \frac{x}{{x}^{4} + \color{blue}{-1}} \cdot \left(x \cdot x - 1\right) \]

      10. fma-neg 99.9%

          \[\leadsto \frac{x}{{x}^{4} + -1} \cdot \color{blue}{\mathsf{fma}\left(x, x, -1\right)} \]

      11. metadata-eval 99.9%

          \[\leadsto \frac{x}{{x}^{4} + -1} \cdot \mathsf{fma}\left(x, x, \color{blue}{-1}\right) \]

   3. Applied egg-rr 99.9%

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

      1. associate-*l/ 99.9%

         \[\leadsto \color{blue}{\frac{x \cdot \mathsf{fma}\left(x, x, -1\right)}{{x}^{4} + -1}} \]

      2. frac-2neg 99.9%

         \[\leadsto \color{blue}{\frac{-x \cdot \mathsf{fma}\left(x, x, -1\right)}{-\left({x}^{4} + -1\right)}} \]

      3. +-commutative 99.9%

         \[\leadsto \frac{-x \cdot \mathsf{fma}\left(x, x, -1\right)}{-\color{blue}{\left(-1 + {x}^{4}\right)}} \]

      4. distribute-neg-in 99.9%

         \[\leadsto \frac{-x \cdot \mathsf{fma}\left(x, x, -1\right)}{\color{blue}{\left(--1\right) + \left(-{x}^{4}\right)}} \]

      5. metadata-eval 99.9%

         \[\leadsto \frac{-x \cdot \mathsf{fma}\left(x, x, -1\right)}{\color{blue}{1} + \left(-{x}^{4}\right)} \]

   5. Applied egg-rr 99.9%

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

      1. distribute-lft-neg-in 99.9%

         \[\leadsto \frac{\color{blue}{\left(-x\right) \cdot \mathsf{fma}\left(x, x, -1\right)}}{1 + \left(-{x}^{4}\right)} \]

      2. fma-udef 99.9%

         \[\leadsto \frac{\left(-x\right) \cdot \color{blue}{\left(x \cdot x + -1\right)}}{1 + \left(-{x}^{4}\right)} \]

      3. distribute-rgt-in 99.9%

         \[\leadsto \frac{\color{blue}{\left(x \cdot x\right) \cdot \left(-x\right) + -1 \cdot \left(-x\right)}}{1 + \left(-{x}^{4}\right)} \]

      4. distribute-rgt-neg-in 99.9%

         \[\leadsto \frac{\color{blue}{\left(-\left(x \cdot x\right) \cdot x\right)} + -1 \cdot \left(-x\right)}{1 + \left(-{x}^{4}\right)} \]

      5. unpow3 100.0%

         \[\leadsto \frac{\left(-\color{blue}{{x}^{3}}\right) + -1 \cdot \left(-x\right)}{1 + \left(-{x}^{4}\right)} \]

      6. neg-mul-1 100.0%

         \[\leadsto \frac{\left(-{x}^{3}\right) + \color{blue}{\left(-\left(-x\right)\right)}}{1 + \left(-{x}^{4}\right)} \]

      7. remove-double-neg 100.0%

         \[\leadsto \frac{\left(-{x}^{3}\right) + \color{blue}{x}}{1 + \left(-{x}^{4}\right)} \]

      8. +-commutative 100.0%

         \[\leadsto \frac{\color{blue}{x + \left(-{x}^{3}\right)}}{1 + \left(-{x}^{4}\right)} \]

      9. sub-neg 100.0%

         \[\leadsto \frac{\color{blue}{x - {x}^{3}}}{1 + \left(-{x}^{4}\right)} \]

      10. unsub-neg 100.0%

          \[\leadsto \frac{x - {x}^{3}}{\color{blue}{1 - {x}^{4}}} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{x - {x}^{3}}{1 - {x}^{4}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+38}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 500000:\\ \;\;\;\;\frac{x - {x}^{3}}{1 - {x}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternative 4: 99.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.86:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 0.85:\\ \;\;\;\;x \cdot \left(1 - x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.86) (/ 1.0 x) (if (<= x 0.85) (* x (- 1.0 (* x x))) (/ 1.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= -0.86) {
		tmp = 1.0 / x;
	} else if (x <= 0.85) {
		tmp = x * (1.0 - (x * x));
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-0.86d0)) then
        tmp = 1.0d0 / x
    else if (x <= 0.85d0) then
        tmp = x * (1.0d0 - (x * x))
    else
        tmp = 1.0d0 / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.86) {
		tmp = 1.0 / x;
	} else if (x <= 0.85) {
		tmp = x * (1.0 - (x * x));
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.86:
		tmp = 1.0 / x
	elif x <= 0.85:
		tmp = x * (1.0 - (x * x))
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.86)
		tmp = Float64(1.0 / x);
	elseif (x <= 0.85)
		tmp = Float64(x * Float64(1.0 - Float64(x * x)));
	else
		tmp = Float64(1.0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -0.86)
		tmp = 1.0 / x;
	elseif (x <= 0.85)
		tmp = x * (1.0 - (x * x));
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -0.86], N[(1.0 / x), $MachinePrecision], If[LessEqual[x, 0.85], N[(x * N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.859999999999999987 or 0.849999999999999978 < x

   1. Initial program 53.4%

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

   2. Taylor expanded in x around inf 99.1%

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

   if -0.859999999999999987 < x < 0.849999999999999978

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.9%

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

      1. +-commutative 98.9%

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

      2. mul-1-neg 98.9%

         \[\leadsto x + \color{blue}{\left(-{x}^{3}\right)} \]

      3. unsub-neg 98.9%

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

   4. Simplified 98.9%

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

      1. *-un-lft-identity 98.9%

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

      2. unpow3 98.9%

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

      3. distribute-rgt-out-- 98.9%

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

   6. Applied egg-rr 98.9%

      \[\leadsto \color{blue}{x \cdot \left(1 - x \cdot x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.86:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 0.85:\\ \;\;\;\;x \cdot \left(1 - x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternative 5: 100.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+38}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 500000:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5e+38)
   (/ 1.0 x)
   (if (<= x 500000.0) (/ x (+ 1.0 (* x x))) (/ 1.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= -5e+38) {
		tmp = 1.0 / x;
	} else if (x <= 500000.0) {
		tmp = x / (1.0 + (x * x));
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-5d+38)) then
        tmp = 1.0d0 / x
    else if (x <= 500000.0d0) then
        tmp = x / (1.0d0 + (x * x))
    else
        tmp = 1.0d0 / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -5e+38) {
		tmp = 1.0 / x;
	} else if (x <= 500000.0) {
		tmp = x / (1.0 + (x * x));
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -5e+38:
		tmp = 1.0 / x
	elif x <= 500000.0:
		tmp = x / (1.0 + (x * x))
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -5e+38)
		tmp = Float64(1.0 / x);
	elseif (x <= 500000.0)
		tmp = Float64(x / Float64(1.0 + Float64(x * x)));
	else
		tmp = Float64(1.0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -5e+38)
		tmp = 1.0 / x;
	elseif (x <= 500000.0)
		tmp = x / (1.0 + (x * x));
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -5e+38], N[(1.0 / x), $MachinePrecision], If[LessEqual[x, 500000.0], N[(x / N[(1.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.9999999999999997e38 or 5e5 < x

   1. Initial program 49.7%

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

   2. Taylor expanded in x around inf 100.0%

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

   if -4.9999999999999997e38 < x < 5e5

   1. Initial program 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+38}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 500000:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternative 6: 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 53.4%

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

   2. Taylor expanded in x around inf 99.1%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.3%

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

4. Final simplification 98.7%

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

Alternative 7: 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 75.6%

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

2. Taylor expanded in x around 0 48.9%

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

3. Final simplification 48.9%

   \[\leadsto x \]

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 2023250 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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