x / (x^2 + 1)

Percentage Accurate: 77.1% → 99.9%

Time: 743.0ms

Alternatives: 2

Speedup: 1.0×

Specification

Math

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

FPCore

(FPCore (x)
  :precision binary64
  (/ x (+ (* x x) 1)))

C

double code(double x) {
	return x / ((x * x) + 1.0);
}

Fortran

real(8) function code(x)
use fmin_fmax_functions
    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), $MachinePrecision]), $MachinePrecision]

Initial Program: 77.1% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
  :precision binary64
  (/ x (+ (* x x) 1)))

C

double code(double x) {
	return x / ((x * x) + 1.0);
}

Fortran

real(8) function code(x)
use fmin_fmax_functions
    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), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 0.8× speedup

Math

\[\frac{1}{x - \frac{-1}{x}} \]

FPCore

(FPCore (x)
  :precision binary64
  (/ 1 (- x (/ -1 x))))

C

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

Fortran

real(8) function code(x)
use fmin_fmax_functions
    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 / N[(x - N[(-1 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.1%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. frac-2neg N/A

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

   6. add-sound-/ N/A

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

   7. lower-/.f64 N/A

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

   8. metadata-eval N/A

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

   9. lift-+.f64 N/A

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

   10. add-flip N/A

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

   11. sub-negate N/A

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

   12. lower--.f64 N/A

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

   13. metadata-eval 77.1%

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

3. Applied rewrites 77.1%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. frac-2neg N/A

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

   4. metadata-eval N/A

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

   5. lift--.f64 N/A

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

   6. sub-negate-rev N/A

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

   7. lift--.f64 N/A

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

   8. associate-/r/ N/A

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

   9. lift-/.f64 N/A

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

   10. lift-/.f64 77.1%

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

   11. lift-/.f64 N/A

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

   12. lift--.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. sub-to-fraction-rev N/A

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

   15. lower--.f64 N/A

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

   16. add-sound-/ N/A

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

   17. lower-/.f64 99.9%

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

5. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\frac{1}{x - \frac{-1}{x}}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2025326 -o generate:taylor -o generate:evaluate
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64
  (/ x (+ (* x x) 1)))