x / (x^2 + 1)

Percentage Accurate: 75.4% → 100.0%

Time: 5.9s

Alternatives: 6

Speedup: 0.2×

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: 75.4% 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, 0.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 5000000:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(x\_m, x\_m, 1\right)} \cdot \left(-x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;{x\_m}^{-1}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 5000000.0)
    (* (/ -1.0 (fma x_m x_m 1.0)) (- x_m))
    (pow x_m -1.0))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 5000000.0) {
		tmp = (-1.0 / fma(x_m, x_m, 1.0)) * -x_m;
	} else {
		tmp = pow(x_m, -1.0);
	}
	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 <= 5000000.0)
		tmp = Float64(Float64(-1.0 / fma(x_m, x_m, 1.0)) * Float64(-x_m));
	else
		tmp = x_m ^ -1.0;
	end
	return Float64(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, 5000000.0], N[(N[(-1.0 / N[(x$95$m * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] * (-x$95$m)), $MachinePrecision], N[Power[x$95$m, -1.0], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 5e6

   1. Initial program 82.7%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 82.7

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

   4. Applied rewrites 82.7%

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

   if 5e6 < x

   1. Initial program 58.7%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5000000:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(x, x, 1\right)} \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.2× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m) :precision binary64 (* x_s (pow (- x_m (/ -1.0 x_m)) -1.0)))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * pow((x_m - (-1.0 / x_m)), -1.0);
}

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 - ((-1.0d0) / x_m)) ** (-1.0d0))
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 * Math.pow((x_m - (-1.0 / x_m)), -1.0);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * math.pow((x_m - (-1.0 / x_m)), -1.0)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * (Float64(x_m - Float64(-1.0 / x_m)) ^ -1.0))
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 - (-1.0 / x_m)) ^ -1.0);
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 * N[Power[N[(x$95$m - N[(-1.0 / x$95$m), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.0%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. frac-2neg N/A

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

   4. associate-/r/ N/A

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

   5. lower-*.f64 N/A

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

   6. metadata-eval N/A

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

   7. frac-2neg N/A

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

   8. lower-/.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lower-fma.f64 N/A

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

   12. lower-neg.f64 78.0

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

4. Applied rewrites 78.0%

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

   1. lift-neg.f64 N/A

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

   2. /-rgt-identity N/A

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

   3. clear-num N/A

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

   4. lift-/.f64 N/A

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

   5. distribute-neg-frac2 N/A

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

   6. neg-mul-1 N/A

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

   7. lower-/.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. div-inv N/A

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

   10. lower-/.f64 77.7

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

6. Applied rewrites 77.7%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. lift-/.f64 N/A

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

   5. un-div-inv N/A

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

   6. lift-/.f64 N/A

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

   7. associate-/r/ N/A

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

   8. metadata-eval N/A

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

   9. associate-*r/ N/A

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

   10. *-inverses N/A

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

   11. times-frac N/A

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

   12. *-commutative N/A

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

   13. lift-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. clear-num N/A

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

   16. lower-/.f64 N/A

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

   17. lift-*.f64 N/A

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

   18. associate-/l* N/A

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

   19. lift-*.f64 N/A

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

   20. associate-/r* N/A

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

   21. *-inverses N/A

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

   22. div-inv N/A

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

8. Applied rewrites 99.8%

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

9. Final simplification 99.8%

   \[\leadsto {\left(x - \frac{-1}{x}\right)}^{-1} \]
10. Add Preprocessing

Alternative 3: 100.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 5000000:\\ \;\;\;\;\frac{x\_m}{\mathsf{fma}\left(x\_m, x\_m, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;{x\_m}^{-1}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (if (<= x_m 5000000.0) (/ x_m (fma x_m x_m 1.0)) (pow x_m -1.0))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 5000000.0) {
		tmp = x_m / fma(x_m, x_m, 1.0);
	} else {
		tmp = pow(x_m, -1.0);
	}
	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 <= 5000000.0)
		tmp = Float64(x_m / fma(x_m, x_m, 1.0));
	else
		tmp = x_m ^ -1.0;
	end
	return Float64(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, 5000000.0], N[(x$95$m / N[(x$95$m * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision], N[Power[x$95$m, -1.0], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 5e6

   1. Initial program 82.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 82.7

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

   4. Applied rewrites 82.7%

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

   if 5e6 < x

   1. Initial program 58.7%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5000000:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.86:\\ \;\;\;\;\mathsf{fma}\left(x\_m, x\_m, -1\right) \cdot \left(-x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;{x\_m}^{-1}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (if (<= x_m 0.86) (* (fma x_m x_m -1.0) (- x_m)) (pow x_m -1.0))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.86) {
		tmp = fma(x_m, x_m, -1.0) * -x_m;
	} else {
		tmp = pow(x_m, -1.0);
	}
	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 <= 0.86)
		tmp = Float64(fma(x_m, x_m, -1.0) * Float64(-x_m));
	else
		tmp = x_m ^ -1.0;
	end
	return Float64(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, 0.86], N[(N[(x$95$m * x$95$m + -1.0), $MachinePrecision] * (-x$95$m)), $MachinePrecision], N[Power[x$95$m, -1.0], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.859999999999999987

   1. Initial program 82.5%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 82.5

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

   4. Applied rewrites 82.5%

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

   5. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. unpow2 N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 66.0

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

   7. Applied rewrites 66.0%

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

   if 0.859999999999999987 < x

   1. Initial program 60.2%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 98.2

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

   5. Applied rewrites 98.2%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.86:\\ \;\;\;\;\mathsf{fma}\left(x, x, -1\right) \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.9% accurate, 0.2× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (if (<= x_m 1.0) (* -1.0 (- x_m)) (pow x_m -1.0))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 1.0) {
		tmp = -1.0 * -x_m;
	} else {
		tmp = pow(x_m, -1.0);
	}
	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 <= 1.0d0) then
        tmp = (-1.0d0) * -x_m
    else
        tmp = x_m ** (-1.0d0)
    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 <= 1.0) {
		tmp = -1.0 * -x_m;
	} else {
		tmp = Math.pow(x_m, -1.0);
	}
	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 <= 1.0:
		tmp = -1.0 * -x_m
	else:
		tmp = math.pow(x_m, -1.0)
	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 <= 1.0)
		tmp = Float64(-1.0 * Float64(-x_m));
	else
		tmp = x_m ^ -1.0;
	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 <= 1.0)
		tmp = -1.0 * -x_m;
	else
		tmp = x_m ^ -1.0;
	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, 1.0], N[(-1.0 * (-x$95$m)), $MachinePrecision], N[Power[x$95$m, -1.0], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 82.5%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 82.5

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

   4. Applied rewrites 82.5%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 66.3%

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

   if 1 < x

   1. Initial program 60.2%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 98.2

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

   5. Applied rewrites 98.2%

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

4. Final simplification 72.8%

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

Alternative 6: 50.4% accurate, 2.5× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m) :precision binary64 (* x_s (* -1.0 (- x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * (-1.0 * -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 * ((-1.0d0) * -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 * (-1.0 * -x_m);
}

Python

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

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(-1.0 * Float64(-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 * (-1.0 * -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 * N[(-1.0 * (-x$95$m)), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.0%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. frac-2neg N/A

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

   4. associate-/r/ N/A

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

   5. lower-*.f64 N/A

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

   6. metadata-eval N/A

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

   7. frac-2neg N/A

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

   8. lower-/.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lower-fma.f64 N/A

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

   12. lower-neg.f64 78.0

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

4. Applied rewrites 78.0%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 53.7%

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

Developer Target 1: 99.9% accurate, 0.8× 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 2024323 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

  :alt
  (! :herbie-platform default (/ 1 (+ x (/ 1 x))))

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