x / (x^2 + 1)

Percentage Accurate: 76.1% → 100.0%

Time: 4.1s

Alternatives: 6

Speedup: 0.9×

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: 76.1% 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

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 8000:\\ \;\;\;\;\frac{x\_m + \left({x\_m}^{5} - {x\_m}^{3}\right)}{1 + {x\_m}^{6}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x\_m} - {x\_m}^{-3}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 8000.0)
    (/ (+ x_m (- (pow x_m 5.0) (pow x_m 3.0))) (+ 1.0 (pow x_m 6.0)))
    (- (/ 1.0 x_m) (pow x_m -3.0)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 8000.0) {
		tmp = (x_m + (pow(x_m, 5.0) - pow(x_m, 3.0))) / (1.0 + pow(x_m, 6.0));
	} else {
		tmp = (1.0 / x_m) - pow(x_m, -3.0);
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 8000.0d0) then
        tmp = (x_m + ((x_m ** 5.0d0) - (x_m ** 3.0d0))) / (1.0d0 + (x_m ** 6.0d0))
    else
        tmp = (1.0d0 / x_m) - (x_m ** (-3.0d0))
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 8000.0) {
		tmp = (x_m + (Math.pow(x_m, 5.0) - Math.pow(x_m, 3.0))) / (1.0 + Math.pow(x_m, 6.0));
	} else {
		tmp = (1.0 / x_m) - Math.pow(x_m, -3.0);
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 8000.0:
		tmp = (x_m + (math.pow(x_m, 5.0) - math.pow(x_m, 3.0))) / (1.0 + math.pow(x_m, 6.0))
	else:
		tmp = (1.0 / x_m) - math.pow(x_m, -3.0)
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 8000.0)
		tmp = Float64(Float64(x_m + Float64((x_m ^ 5.0) - (x_m ^ 3.0))) / Float64(1.0 + (x_m ^ 6.0)));
	else
		tmp = Float64(Float64(1.0 / x_m) - (x_m ^ -3.0));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 8000.0)
		tmp = (x_m + ((x_m ^ 5.0) - (x_m ^ 3.0))) / (1.0 + (x_m ^ 6.0));
	else
		tmp = (1.0 / x_m) - (x_m ^ -3.0);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 8000.0], N[(N[(x$95$m + N[(N[Power[x$95$m, 5.0], $MachinePrecision] - N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Power[x$95$m, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x$95$m), $MachinePrecision] - N[Power[x$95$m, -3.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 8e3

   1. Initial program 84.3%

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

      1. flip3-+ 75.7%

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

      2. associate-/r/ 75.7%

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

      3. pow3 75.7%

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

      4. metadata-eval 75.7%

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

      5. +-commutative 75.7%

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

      6. pow3 75.7%

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

      7. pow2 75.7%

         \[\leadsto \frac{x}{1 + {\color{blue}{\left({x}^{2}\right)}}^{3}} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(1 \cdot 1 - \left(x \cdot x\right) \cdot 1\right)\right) \]

      8. pow-pow 75.7%

         \[\leadsto \frac{x}{1 + \color{blue}{{x}^{\left(2 \cdot 3\right)}}} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(1 \cdot 1 - \left(x \cdot x\right) \cdot 1\right)\right) \]

      9. metadata-eval 75.7%

         \[\leadsto \frac{x}{1 + {x}^{\color{blue}{6}}} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(1 \cdot 1 - \left(x \cdot x\right) \cdot 1\right)\right) \]

      10. pow2 75.7%

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

      11. pow2 75.7%

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

      12. pow-prod-up 75.7%

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

      13. metadata-eval 75.7%

          \[\leadsto \frac{x}{1 + {x}^{6}} \cdot \left({x}^{\color{blue}{4}} + \left(1 \cdot 1 - \left(x \cdot x\right) \cdot 1\right)\right) \]

      14. metadata-eval 75.7%

          \[\leadsto \frac{x}{1 + {x}^{6}} \cdot \left({x}^{4} + \left(\color{blue}{1} - \left(x \cdot x\right) \cdot 1\right)\right) \]

      15. *-rgt-identity 75.7%

          \[\leadsto \frac{x}{1 + {x}^{6}} \cdot \left({x}^{4} + \left(1 - \color{blue}{x \cdot x}\right)\right) \]

      16. pow2 75.7%

          \[\leadsto \frac{x}{1 + {x}^{6}} \cdot \left({x}^{4} + \left(1 - \color{blue}{{x}^{2}}\right)\right) \]

   4. Applied egg-rr 75.7%

      \[\leadsto \color{blue}{\frac{x}{1 + {x}^{6}} \cdot \left({x}^{4} + \left(1 - {x}^{2}\right)\right)} \]
   5. Step-by-step derivation

      1. associate-*l/ 75.7%

         \[\leadsto \color{blue}{\frac{x \cdot \left({x}^{4} + \left(1 - {x}^{2}\right)\right)}{1 + {x}^{6}}} \]

      2. +-commutative 75.7%

         \[\leadsto \frac{x \cdot \color{blue}{\left(\left(1 - {x}^{2}\right) + {x}^{4}\right)}}{1 + {x}^{6}} \]

      3. distribute-lft-out 75.7%

         \[\leadsto \frac{\color{blue}{x \cdot \left(1 - {x}^{2}\right) + x \cdot {x}^{4}}}{1 + {x}^{6}} \]

      4. distribute-lft-out-- 75.7%

         \[\leadsto \frac{\color{blue}{\left(x \cdot 1 - x \cdot {x}^{2}\right)} + x \cdot {x}^{4}}{1 + {x}^{6}} \]

      5. *-rgt-identity 75.7%

         \[\leadsto \frac{\left(\color{blue}{x} - x \cdot {x}^{2}\right) + x \cdot {x}^{4}}{1 + {x}^{6}} \]

      6. unpow2 75.7%

         \[\leadsto \frac{\left(x - x \cdot \color{blue}{\left(x \cdot x\right)}\right) + x \cdot {x}^{4}}{1 + {x}^{6}} \]

      7. cube-mult 75.7%

         \[\leadsto \frac{\left(x - \color{blue}{{x}^{3}}\right) + x \cdot {x}^{4}}{1 + {x}^{6}} \]

      8. unsub-neg 75.7%

         \[\leadsto \frac{\color{blue}{\left(x + \left(-{x}^{3}\right)\right)} + x \cdot {x}^{4}}{1 + {x}^{6}} \]

      9. mul-1-neg 75.7%

         \[\leadsto \frac{\left(x + \color{blue}{-1 \cdot {x}^{3}}\right) + x \cdot {x}^{4}}{1 + {x}^{6}} \]

      10. *-commutative 75.7%

          \[\leadsto \frac{\left(x + -1 \cdot {x}^{3}\right) + \color{blue}{{x}^{4} \cdot x}}{1 + {x}^{6}} \]

      11. pow-plus 75.7%

          \[\leadsto \frac{\left(x + -1 \cdot {x}^{3}\right) + \color{blue}{{x}^{\left(4 + 1\right)}}}{1 + {x}^{6}} \]

      12. metadata-eval 75.7%

          \[\leadsto \frac{\left(x + -1 \cdot {x}^{3}\right) + {x}^{\color{blue}{5}}}{1 + {x}^{6}} \]

      13. associate-+r+ 75.7%

          \[\leadsto \frac{\color{blue}{x + \left(-1 \cdot {x}^{3} + {x}^{5}\right)}}{1 + {x}^{6}} \]

      14. +-commutative 75.7%

          \[\leadsto \frac{x + \color{blue}{\left({x}^{5} + -1 \cdot {x}^{3}\right)}}{1 + {x}^{6}} \]

      15. mul-1-neg 75.7%

          \[\leadsto \frac{x + \left({x}^{5} + \color{blue}{\left(-{x}^{3}\right)}\right)}{1 + {x}^{6}} \]

      16. unsub-neg 75.7%

          \[\leadsto \frac{x + \color{blue}{\left({x}^{5} - {x}^{3}\right)}}{1 + {x}^{6}} \]

   6. Simplified 75.7%

      \[\leadsto \color{blue}{\frac{x + \left({x}^{5} - {x}^{3}\right)}{1 + {x}^{6}}} \]

   if 8e3 < x

   1. Initial program 49.7%

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

      1. flip-+ 23.4%

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

      2. associate-/r/ 23.4%

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

      3. metadata-eval 23.4%

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

      4. sub-neg 23.4%

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

      5. pow2 23.4%

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

      6. pow2 23.4%

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

      7. pow-prod-up 23.3%

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

      8. metadata-eval 23.3%

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

      9. metadata-eval 23.3%

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

      10. fma-neg 23.3%

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

      11. metadata-eval 23.3%

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

   4. Applied egg-rr 23.3%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. exp-to-pow 100.0%

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

      2. exp-neg 100.0%

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

      3. distribute-rgt-neg-in 100.0%

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

      4. metadata-eval 100.0%

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

      5. exp-to-pow 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8000:\\ \;\;\;\;\frac{x + \left({x}^{5} - {x}^{3}\right)}{1 + {x}^{6}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - {x}^{-3}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 100.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 10000:\\ \;\;\;\;\frac{{x\_m}^{3} - x\_m}{{x\_m}^{4} + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x\_m} - {x\_m}^{-3}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 10000.0)
    (/ (- (pow x_m 3.0) x_m) (+ (pow x_m 4.0) -1.0))
    (- (/ 1.0 x_m) (pow x_m -3.0)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 10000.0) {
		tmp = (pow(x_m, 3.0) - x_m) / (pow(x_m, 4.0) + -1.0);
	} else {
		tmp = (1.0 / x_m) - pow(x_m, -3.0);
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 10000.0d0) then
        tmp = ((x_m ** 3.0d0) - x_m) / ((x_m ** 4.0d0) + (-1.0d0))
    else
        tmp = (1.0d0 / x_m) - (x_m ** (-3.0d0))
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 10000.0) {
		tmp = (Math.pow(x_m, 3.0) - x_m) / (Math.pow(x_m, 4.0) + -1.0);
	} else {
		tmp = (1.0 / x_m) - Math.pow(x_m, -3.0);
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 10000.0:
		tmp = (math.pow(x_m, 3.0) - x_m) / (math.pow(x_m, 4.0) + -1.0)
	else:
		tmp = (1.0 / x_m) - math.pow(x_m, -3.0)
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 10000.0)
		tmp = Float64(Float64((x_m ^ 3.0) - x_m) / Float64((x_m ^ 4.0) + -1.0));
	else
		tmp = Float64(Float64(1.0 / x_m) - (x_m ^ -3.0));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 10000.0)
		tmp = ((x_m ^ 3.0) - x_m) / ((x_m ^ 4.0) + -1.0);
	else
		tmp = (1.0 / x_m) - (x_m ^ -3.0);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 10000.0], N[(N[(N[Power[x$95$m, 3.0], $MachinePrecision] - x$95$m), $MachinePrecision] / N[(N[Power[x$95$m, 4.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x$95$m), $MachinePrecision] - N[Power[x$95$m, -3.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1e4

   1. Initial program 84.3%

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

      1. flip-+ 77.5%

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

      2. associate-/r/ 77.5%

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

      3. metadata-eval 77.5%

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

      4. sub-neg 77.5%

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

      5. pow2 77.5%

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

      6. pow2 77.5%

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

      7. pow-prod-up 77.5%

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

      8. metadata-eval 77.5%

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

      9. metadata-eval 77.5%

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

      10. fma-neg 77.5%

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

      11. metadata-eval 77.5%

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

   4. Applied egg-rr 77.5%

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

      1. associate-*l/ 77.3%

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

      2. fma-udef 77.3%

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

      3. distribute-rgt-in 77.3%

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

      4. unpow3 77.3%

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

      5. neg-mul-1 77.3%

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

      6. *-rgt-identity 77.3%

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

      7. metadata-eval 77.3%

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

      8. div-inv 77.3%

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

      9. sub-neg 77.3%

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

      10. /-rgt-identity 77.3%

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

   6. Applied egg-rr 77.3%

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

   if 1e4 < x

   1. Initial program 49.7%

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

      1. flip-+ 23.4%

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

      2. associate-/r/ 23.4%

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

      3. metadata-eval 23.4%

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

      4. sub-neg 23.4%

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

      5. pow2 23.4%

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

      6. pow2 23.4%

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

      7. pow-prod-up 23.3%

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

      8. metadata-eval 23.3%

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

      9. metadata-eval 23.3%

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

      10. fma-neg 23.3%

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

      11. metadata-eval 23.3%

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

   4. Applied egg-rr 23.3%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. exp-to-pow 100.0%

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

      2. exp-neg 100.0%

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

      3. distribute-rgt-neg-in 100.0%

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

      4. metadata-eval 100.0%

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

      5. exp-to-pow 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10000:\\ \;\;\;\;\frac{{x}^{3} - x}{{x}^{4} + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - {x}^{-3}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 100.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1000000:\\ \;\;\;\;\frac{x\_m}{1 + x\_m \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x\_m} - {x\_m}^{-3}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 1000000.0)
    (/ x_m (+ 1.0 (* x_m x_m)))
    (- (/ 1.0 x_m) (pow x_m -3.0)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 1000000.0) {
		tmp = x_m / (1.0 + (x_m * x_m));
	} else {
		tmp = (1.0 / x_m) - pow(x_m, -3.0);
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 1000000.0d0) then
        tmp = x_m / (1.0d0 + (x_m * x_m))
    else
        tmp = (1.0d0 / x_m) - (x_m ** (-3.0d0))
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 1000000.0) {
		tmp = x_m / (1.0 + (x_m * x_m));
	} else {
		tmp = (1.0 / x_m) - Math.pow(x_m, -3.0);
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 1000000.0:
		tmp = x_m / (1.0 + (x_m * x_m))
	else:
		tmp = (1.0 / x_m) - math.pow(x_m, -3.0)
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 1000000.0)
		tmp = Float64(x_m / Float64(1.0 + Float64(x_m * x_m)));
	else
		tmp = Float64(Float64(1.0 / x_m) - (x_m ^ -3.0));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 1000000.0)
		tmp = x_m / (1.0 + (x_m * x_m));
	else
		tmp = (1.0 / x_m) - (x_m ^ -3.0);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 1000000.0], N[(x$95$m / N[(1.0 + N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x$95$m), $MachinePrecision] - N[Power[x$95$m, -3.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1e6

   1. Initial program 84.5%

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

   if 1e6 < x

   1. Initial program 48.2%

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

      1. flip-+ 21.2%

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

      2. associate-/r/ 21.2%

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

      3. metadata-eval 21.2%

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

      4. sub-neg 21.2%

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

      5. pow2 21.2%

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

      6. pow2 21.2%

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

      7. pow-prod-up 21.2%

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

      8. metadata-eval 21.2%

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

      9. metadata-eval 21.2%

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

      10. fma-neg 21.2%

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

      11. metadata-eval 21.2%

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

   4. Applied egg-rr 21.2%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. exp-to-pow 100.0%

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

      2. exp-neg 100.0%

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

      3. distribute-rgt-neg-in 100.0%

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

      4. metadata-eval 100.0%

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

      5. exp-to-pow 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1000000:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - {x}^{-3}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 100.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 20000000:\\ \;\;\;\;\frac{x\_m}{1 + x\_m \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (if (<= x_m 20000000.0) (/ x_m (+ 1.0 (* x_m x_m))) (/ 1.0 x_m))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 20000000.0) {
		tmp = x_m / (1.0 + (x_m * x_m));
	} else {
		tmp = 1.0 / x_m;
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 20000000.0d0) then
        tmp = x_m / (1.0d0 + (x_m * x_m))
    else
        tmp = 1.0d0 / x_m
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 20000000.0) {
		tmp = x_m / (1.0 + (x_m * x_m));
	} else {
		tmp = 1.0 / x_m;
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 20000000.0:
		tmp = x_m / (1.0 + (x_m * x_m))
	else:
		tmp = 1.0 / x_m
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 20000000.0)
		tmp = Float64(x_m / Float64(1.0 + Float64(x_m * x_m)));
	else
		tmp = Float64(1.0 / x_m);
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 20000000.0)
		tmp = x_m / (1.0 + (x_m * x_m));
	else
		tmp = 1.0 / x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 20000000.0], N[(x$95$m / N[(1.0 + N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2e7

   1. Initial program 84.5%

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

   if 2e7 < x

   1. Initial program 48.2%

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

   3. Taylor expanded in x around inf 100.0%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 20000000:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1:\\ \;\;\;\;x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (if (<= x_m 1.0) x_m (/ 1.0 x_m))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 1.0) {
		tmp = x_m;
	} else {
		tmp = 1.0 / x_m;
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 1.0d0) then
        tmp = x_m
    else
        tmp = 1.0d0 / x_m
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 1.0) {
		tmp = x_m;
	} else {
		tmp = 1.0 / x_m;
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 1.0:
		tmp = x_m
	else:
		tmp = 1.0 / x_m
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 1.0)
		tmp = x_m;
	else
		tmp = Float64(1.0 / x_m);
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 1.0)
		tmp = x_m;
	else
		tmp = 1.0 / x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 1.0], x$95$m, N[(1.0 / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 84.3%

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

   3. Taylor expanded in x around 0 68.3%

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

   if 1 < x

   1. Initial program 49.7%

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

   3. Taylor expanded in x around inf 99.2%

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

4. Final simplification 76.8%

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

Alternative 6: 51.3% accurate, 7.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot x\_m \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m) :precision binary64 (* x_s x_m))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * x_m;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    code = x_s * x_m
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	return x_s * x_m;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * x_m

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * x_m)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * x_m;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * x$95$m), $MachinePrecision]

Derivation

1. Initial program 74.7%

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

3. Taylor expanded in x around 0 50.4%

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

4. Final simplification 50.4%

   \[\leadsto x \]
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 2024040 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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