bug333 (missed optimization)

Percentage Accurate: 8.5% → 100.0%

Time: 9.9s

Alternatives: 9

Speedup: 207.0×

Specification

\[-1 \leq x \land x \leq 1\]

Math

\[\begin{array}{l} \\ \sqrt{1 + x} - \sqrt{1 - x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (sqrt (+ 1.0 x)) (sqrt (- 1.0 x))))

C

double code(double x) {
	return sqrt((1.0 + x)) - sqrt((1.0 - x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt((1.0d0 + x)) - sqrt((1.0d0 - x))
end function

Java

public static double code(double x) {
	return Math.sqrt((1.0 + x)) - Math.sqrt((1.0 - x));
}

Python

def code(x):
	return math.sqrt((1.0 + x)) - math.sqrt((1.0 - x))

Julia

function code(x)
	return Float64(sqrt(Float64(1.0 + x)) - sqrt(Float64(1.0 - x)))
end

MATLAB

function tmp = code(x)
	tmp = sqrt((1.0 + x)) - sqrt((1.0 - x));
end

Wolfram

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

Initial Program: 8.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{1 + x} - \sqrt{1 - x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (sqrt (+ 1.0 x)) (sqrt (- 1.0 x))))

C

double code(double x) {
	return sqrt((1.0 + x)) - sqrt((1.0 - x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt((1.0d0 + x)) - sqrt((1.0d0 - x))
end function

Java

public static double code(double x) {
	return Math.sqrt((1.0 + x)) - Math.sqrt((1.0 - x));
}

Python

def code(x):
	return math.sqrt((1.0 + x)) - math.sqrt((1.0 - x))

Julia

function code(x)
	return Float64(sqrt(Float64(1.0 + x)) - sqrt(Float64(1.0 - x)))
end

MATLAB

function tmp = code(x)
	tmp = sqrt((1.0 + x)) - sqrt((1.0 - x));
end

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot 2}{\sqrt{x + 1} + \sqrt{1 - x}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (* x 2.0) (+ (sqrt (+ x 1.0)) (sqrt (- 1.0 x)))))

C

double code(double x) {
	return (x * 2.0) / (sqrt((x + 1.0)) + sqrt((1.0 - x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * 2.0d0) / (sqrt((x + 1.0d0)) + sqrt((1.0d0 - x)))
end function

Java

public static double code(double x) {
	return (x * 2.0) / (Math.sqrt((x + 1.0)) + Math.sqrt((1.0 - x)));
}

Python

def code(x):
	return (x * 2.0) / (math.sqrt((x + 1.0)) + math.sqrt((1.0 - x)))

Julia

function code(x)
	return Float64(Float64(x * 2.0) / Float64(sqrt(Float64(x + 1.0)) + sqrt(Float64(1.0 - x))))
end

MATLAB

function tmp = code(x)
	tmp = (x * 2.0) / (sqrt((x + 1.0)) + sqrt((1.0 - x)));
end

Wolfram

code[x_] := N[(N[(x * 2.0), $MachinePrecision] / N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.7%

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

   1. flip-- N/A

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

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{1 - x} \cdot \sqrt{1 - x}\right), \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 - x}\right)}\right) \]

   3. rem-square-sqrt N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + x\right) - \sqrt{1 - x} \cdot \sqrt{1 - x}\right), \left(\sqrt{\color{blue}{1 + x}} + \sqrt{1 - x}\right)\right) \]

   4. rem-square-sqrt N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + x\right) - \left(1 - x\right)\right), \left(\sqrt{1 + x} + \sqrt{1 - x}\right)\right) \]

   5. associate--l+ N/A

      \[\leadsto \mathsf{/.f64}\left(\left(1 + \left(x - \left(1 - x\right)\right)\right), \left(\color{blue}{\sqrt{1 + x}} + \sqrt{1 - x}\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(x - \left(1 - x\right)\right)\right), \left(\color{blue}{\sqrt{1 + x}} + \sqrt{1 - x}\right)\right) \]

   7. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{\_.f64}\left(x, \left(1 - x\right)\right)\right), \left(\sqrt{1 + x} + \sqrt{1 - x}\right)\right) \]

   8. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right)\right), \left(\sqrt{1 + x} + \sqrt{1 - x}\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right)\right), \mathsf{+.f64}\left(\left(\sqrt{1 + x}\right), \color{blue}{\left(\sqrt{1 - x}\right)}\right)\right) \]

   10. pow1/2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right)\right), \mathsf{+.f64}\left(\left({\left(1 + x\right)}^{\frac{1}{2}}\right), \left(\sqrt{\color{blue}{1 - x}}\right)\right)\right) \]

   11. pow-lowering-pow.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\left(1 + x\right), \frac{1}{2}\right), \left(\sqrt{\color{blue}{1 - x}}\right)\right)\right) \]

   12. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{1}{2}\right), \left(\sqrt{\color{blue}{1} - x}\right)\right)\right) \]

   13. pow1/2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{1}{2}\right), \left({\left(1 - x\right)}^{\color{blue}{\frac{1}{2}}}\right)\right)\right) \]

   14. pow-lowering-pow.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{1}{2}\right), \mathsf{pow.f64}\left(\left(1 - x\right), \color{blue}{\frac{1}{2}}\right)\right)\right) \]

   15. --lowering--.f64 7.8%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{1}{2}\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(1, x\right), \frac{1}{2}\right)\right)\right) \]

4. Applied egg-rr 7.8%

   \[\leadsto \color{blue}{\frac{1 + \left(x - \left(1 - x\right)\right)}{{\left(1 + x\right)}^{0.5} + {\left(1 - x\right)}^{0.5}}} \]

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(2 \cdot x\right)}, \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{1}{2}\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(1, x\right), \frac{1}{2}\right)\right)\right) \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(x \cdot 2\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{1}{2}\right)}, \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(1, x\right), \frac{1}{2}\right)\right)\right) \]

   2. *-lowering-*.f64 100.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{1}{2}\right)}, \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(1, x\right), \frac{1}{2}\right)\right)\right) \]

7. Simplified 100.0%

   \[\leadsto \frac{\color{blue}{x \cdot 2}}{{\left(1 + x\right)}^{0.5} + {\left(1 - x\right)}^{0.5}} \]
8. Step-by-step derivation

   1. unpow1/2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{1}{2}\right), \left(\sqrt{1 - x}\right)\right)\right) \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{1}{2}\right), \mathsf{sqrt.f64}\left(\left(1 - x\right)\right)\right)\right) \]

   3. --lowering--.f64 100.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{1}{2}\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, x\right)\right)\right)\right) \]

9. Applied egg-rr 100.0%

   \[\leadsto \frac{x \cdot 2}{{\left(1 + x\right)}^{0.5} + \color{blue}{\sqrt{1 - x}}} \]
10. Step-by-step derivation

    1. unpow1/2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{+.f64}\left(\left(\sqrt{1 + x}\right), \mathsf{sqrt.f64}\left(\color{blue}{\mathsf{\_.f64}\left(1, x\right)}\right)\right)\right) \]

    2. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{+.f64}\left(\mathsf{sqrt.f64}\left(\left(1 + x\right)\right), \mathsf{sqrt.f64}\left(\color{blue}{\mathsf{\_.f64}\left(1, x\right)}\right)\right)\right) \]

    3. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{+.f64}\left(\mathsf{sqrt.f64}\left(\left(x + 1\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\color{blue}{1}, x\right)\right)\right)\right) \]

    4. +-lowering-+.f64 100.0%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{+.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(x, 1\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\color{blue}{1}, x\right)\right)\right)\right) \]

11. Applied egg-rr 100.0%

    \[\leadsto \frac{x \cdot 2}{\color{blue}{\sqrt{x + 1}} + \sqrt{1 - x}} \]
12. Add Preprocessing

Alternative 2: 99.8% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot 2}{2 + \left(x \cdot x\right) \cdot \left(-0.25 + x \cdot \left(x \cdot \left(-0.078125 + \left(x \cdot x\right) \cdot -0.041015625\right)\right)\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (* x 2.0)
  (+
   2.0
   (* (* x x) (+ -0.25 (* x (* x (+ -0.078125 (* (* x x) -0.041015625)))))))))

C

double code(double x) {
	return (x * 2.0) / (2.0 + ((x * x) * (-0.25 + (x * (x * (-0.078125 + ((x * x) * -0.041015625)))))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * 2.0d0) / (2.0d0 + ((x * x) * ((-0.25d0) + (x * (x * ((-0.078125d0) + ((x * x) * (-0.041015625d0))))))))
end function

Java

public static double code(double x) {
	return (x * 2.0) / (2.0 + ((x * x) * (-0.25 + (x * (x * (-0.078125 + ((x * x) * -0.041015625)))))));
}

Python

def code(x):
	return (x * 2.0) / (2.0 + ((x * x) * (-0.25 + (x * (x * (-0.078125 + ((x * x) * -0.041015625)))))))

Julia

function code(x)
	return Float64(Float64(x * 2.0) / Float64(2.0 + Float64(Float64(x * x) * Float64(-0.25 + Float64(x * Float64(x * Float64(-0.078125 + Float64(Float64(x * x) * -0.041015625))))))))
end

MATLAB

function tmp = code(x)
	tmp = (x * 2.0) / (2.0 + ((x * x) * (-0.25 + (x * (x * (-0.078125 + ((x * x) * -0.041015625)))))));
end

Wolfram

code[x_] := N[(N[(x * 2.0), $MachinePrecision] / N[(2.0 + N[(N[(x * x), $MachinePrecision] * N[(-0.25 + N[(x * N[(x * N[(-0.078125 + N[(N[(x * x), $MachinePrecision] * -0.041015625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.7%

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

   1. flip-- N/A

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

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{1 - x} \cdot \sqrt{1 - x}\right), \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 - x}\right)}\right) \]

   3. rem-square-sqrt N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + x\right) - \sqrt{1 - x} \cdot \sqrt{1 - x}\right), \left(\sqrt{\color{blue}{1 + x}} + \sqrt{1 - x}\right)\right) \]

   4. rem-square-sqrt N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + x\right) - \left(1 - x\right)\right), \left(\sqrt{1 + x} + \sqrt{1 - x}\right)\right) \]

   5. associate--l+ N/A

      \[\leadsto \mathsf{/.f64}\left(\left(1 + \left(x - \left(1 - x\right)\right)\right), \left(\color{blue}{\sqrt{1 + x}} + \sqrt{1 - x}\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(x - \left(1 - x\right)\right)\right), \left(\color{blue}{\sqrt{1 + x}} + \sqrt{1 - x}\right)\right) \]

   7. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{\_.f64}\left(x, \left(1 - x\right)\right)\right), \left(\sqrt{1 + x} + \sqrt{1 - x}\right)\right) \]

   8. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right)\right), \left(\sqrt{1 + x} + \sqrt{1 - x}\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right)\right), \mathsf{+.f64}\left(\left(\sqrt{1 + x}\right), \color{blue}{\left(\sqrt{1 - x}\right)}\right)\right) \]

   10. pow1/2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right)\right), \mathsf{+.f64}\left(\left({\left(1 + x\right)}^{\frac{1}{2}}\right), \left(\sqrt{\color{blue}{1 - x}}\right)\right)\right) \]

   11. pow-lowering-pow.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\left(1 + x\right), \frac{1}{2}\right), \left(\sqrt{\color{blue}{1 - x}}\right)\right)\right) \]

   12. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{1}{2}\right), \left(\sqrt{\color{blue}{1} - x}\right)\right)\right) \]

   13. pow1/2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{1}{2}\right), \left({\left(1 - x\right)}^{\color{blue}{\frac{1}{2}}}\right)\right)\right) \]

   14. pow-lowering-pow.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{1}{2}\right), \mathsf{pow.f64}\left(\left(1 - x\right), \color{blue}{\frac{1}{2}}\right)\right)\right) \]

   15. --lowering--.f64 7.8%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{1}{2}\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(1, x\right), \frac{1}{2}\right)\right)\right) \]

4. Applied egg-rr 7.8%

   \[\leadsto \color{blue}{\frac{1 + \left(x - \left(1 - x\right)\right)}{{\left(1 + x\right)}^{0.5} + {\left(1 - x\right)}^{0.5}}} \]

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(2 \cdot x\right)}, \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{1}{2}\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(1, x\right), \frac{1}{2}\right)\right)\right) \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(x \cdot 2\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{1}{2}\right)}, \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(1, x\right), \frac{1}{2}\right)\right)\right) \]

   2. *-lowering-*.f64 100.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{1}{2}\right)}, \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(1, x\right), \frac{1}{2}\right)\right)\right) \]

7. Simplified 100.0%

   \[\leadsto \frac{\color{blue}{x \cdot 2}}{{\left(1 + x\right)}^{0.5} + {\left(1 - x\right)}^{0.5}} \]

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \color{blue}{\left(2 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-21}{512} \cdot {x}^{2} - \frac{5}{64}\right) - \frac{1}{4}\right)\right)}\right) \]
9. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{+.f64}\left(2, \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-21}{512} \cdot {x}^{2} - \frac{5}{64}\right) - \frac{1}{4}\right)\right)}\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left({x}^{2} \cdot \left(\frac{-21}{512} \cdot {x}^{2} - \frac{5}{64}\right) - \frac{1}{4}\right)}\right)\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{{x}^{2} \cdot \left(\frac{-21}{512} \cdot {x}^{2} - \frac{5}{64}\right)} - \frac{1}{4}\right)\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{{x}^{2} \cdot \left(\frac{-21}{512} \cdot {x}^{2} - \frac{5}{64}\right)} - \frac{1}{4}\right)\right)\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{-21}{512} \cdot {x}^{2} - \frac{5}{64}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}\right)\right)\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{-21}{512} \cdot {x}^{2} - \frac{5}{64}\right) + \frac{-1}{4}\right)\right)\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{-1}{4} + \color{blue}{{x}^{2} \cdot \left(\frac{-21}{512} \cdot {x}^{2} - \frac{5}{64}\right)}\right)\right)\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{4}, \color{blue}{\left({x}^{2} \cdot \left(\frac{-21}{512} \cdot {x}^{2} - \frac{5}{64}\right)\right)}\right)\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{4}, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{-21}{512} \cdot {x}^{2}} - \frac{5}{64}\right)\right)\right)\right)\right)\right) \]

   10. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{4}, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{-21}{512} \cdot {x}^{2} - \frac{5}{64}\right)\right)}\right)\right)\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{-21}{512} \cdot {x}^{2} - \frac{5}{64}\right)\right)}\right)\right)\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{-21}{512} \cdot {x}^{2} - \frac{5}{64}\right)}\right)\right)\right)\right)\right)\right) \]

   13. sub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{-21}{512} \cdot {x}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{5}{64}\right)\right)}\right)\right)\right)\right)\right)\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{-21}{512} \cdot {x}^{2} + \frac{-5}{64}\right)\right)\right)\right)\right)\right)\right) \]

   15. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{-5}{64} + \color{blue}{\frac{-21}{512} \cdot {x}^{2}}\right)\right)\right)\right)\right)\right)\right) \]

   16. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-5}{64}, \color{blue}{\left(\frac{-21}{512} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-5}{64}, \left({x}^{2} \cdot \color{blue}{\frac{-21}{512}}\right)\right)\right)\right)\right)\right)\right)\right) \]

   18. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-5}{64}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-21}{512}}\right)\right)\right)\right)\right)\right)\right)\right) \]

   19. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-5}{64}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-21}{512}\right)\right)\right)\right)\right)\right)\right)\right) \]

   20. *-lowering-*.f64 99.9%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-5}{64}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-21}{512}\right)\right)\right)\right)\right)\right)\right)\right) \]

10. Simplified 99.9%

    \[\leadsto \frac{x \cdot 2}{\color{blue}{2 + \left(x \cdot x\right) \cdot \left(-0.25 + x \cdot \left(x \cdot \left(-0.078125 + \left(x \cdot x\right) \cdot -0.041015625\right)\right)\right)}} \]
11. Add Preprocessing

Alternative 3: 99.8% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.125 + x \cdot \left(x \cdot \left(0.0546875 + \left(x \cdot x\right) \cdot 0.0322265625\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  x
  (+
   1.0
   (* (* x x) (+ 0.125 (* x (* x (+ 0.0546875 (* (* x x) 0.0322265625)))))))))

C

double code(double x) {
	return x * (1.0 + ((x * x) * (0.125 + (x * (x * (0.0546875 + ((x * x) * 0.0322265625)))))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (1.0d0 + ((x * x) * (0.125d0 + (x * (x * (0.0546875d0 + ((x * x) * 0.0322265625d0)))))))
end function

Java

public static double code(double x) {
	return x * (1.0 + ((x * x) * (0.125 + (x * (x * (0.0546875 + ((x * x) * 0.0322265625)))))));
}

Python

def code(x):
	return x * (1.0 + ((x * x) * (0.125 + (x * (x * (0.0546875 + ((x * x) * 0.0322265625)))))))

Julia

function code(x)
	return Float64(x * Float64(1.0 + Float64(Float64(x * x) * Float64(0.125 + Float64(x * Float64(x * Float64(0.0546875 + Float64(Float64(x * x) * 0.0322265625))))))))
end

MATLAB

function tmp = code(x)
	tmp = x * (1.0 + ((x * x) * (0.125 + (x * (x * (0.0546875 + ((x * x) * 0.0322265625)))))));
end

Wolfram

code[x_] := N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(0.125 + N[(x * N[(x * N[(0.0546875 + N[(N[(x * x), $MachinePrecision] * 0.0322265625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.7%

   \[\sqrt{1 + x} - \sqrt{1 - x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{7}{128} + \frac{33}{1024} \cdot {x}^{2}\right)\right)\right)} \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{7}{128} + \frac{33}{1024} \cdot {x}^{2}\right)\right)\right)}\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{7}{128} + \frac{33}{1024} \cdot {x}^{2}\right)\right)\right)}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{7}{128} + \frac{33}{1024} \cdot {x}^{2}\right)\right)}\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{8}} + {x}^{2} \cdot \left(\frac{7}{128} + \frac{33}{1024} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{8}} + {x}^{2} \cdot \left(\frac{7}{128} + \frac{33}{1024} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{8}, \color{blue}{\left({x}^{2} \cdot \left(\frac{7}{128} + \frac{33}{1024} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{8}, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{7}{128}} + \frac{33}{1024} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

   8. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{8}, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{7}{128} + \frac{33}{1024} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{7}{128} + \frac{33}{1024} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{7}{128} + \frac{33}{1024} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

   11. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{7}{128}, \color{blue}{\left(\frac{33}{1024} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{7}{128}, \left({x}^{2} \cdot \color{blue}{\frac{33}{1024}}\right)\right)\right)\right)\right)\right)\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{7}{128}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{33}{1024}}\right)\right)\right)\right)\right)\right)\right)\right) \]

   14. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{7}{128}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{33}{1024}\right)\right)\right)\right)\right)\right)\right)\right) \]

   15. *-lowering-*.f64 99.9%

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{7}{128}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{33}{1024}\right)\right)\right)\right)\right)\right)\right)\right) \]

5. Simplified 99.9%

   \[\leadsto \color{blue}{x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.125 + x \cdot \left(x \cdot \left(0.0546875 + \left(x \cdot x\right) \cdot 0.0322265625\right)\right)\right)\right)} \]
6. Add Preprocessing

Alternative 4: 99.7% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot 2}{2 + \left(x \cdot x\right) \cdot \left(-0.25 + \left(x \cdot x\right) \cdot -0.078125\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (* x 2.0) (+ 2.0 (* (* x x) (+ -0.25 (* (* x x) -0.078125))))))

C

double code(double x) {
	return (x * 2.0) / (2.0 + ((x * x) * (-0.25 + ((x * x) * -0.078125))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * 2.0d0) / (2.0d0 + ((x * x) * ((-0.25d0) + ((x * x) * (-0.078125d0)))))
end function

Java

public static double code(double x) {
	return (x * 2.0) / (2.0 + ((x * x) * (-0.25 + ((x * x) * -0.078125))));
}

Python

def code(x):
	return (x * 2.0) / (2.0 + ((x * x) * (-0.25 + ((x * x) * -0.078125))))

Julia

function code(x)
	return Float64(Float64(x * 2.0) / Float64(2.0 + Float64(Float64(x * x) * Float64(-0.25 + Float64(Float64(x * x) * -0.078125)))))
end

MATLAB

function tmp = code(x)
	tmp = (x * 2.0) / (2.0 + ((x * x) * (-0.25 + ((x * x) * -0.078125))));
end

Wolfram

code[x_] := N[(N[(x * 2.0), $MachinePrecision] / N[(2.0 + N[(N[(x * x), $MachinePrecision] * N[(-0.25 + N[(N[(x * x), $MachinePrecision] * -0.078125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.7%

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

   1. flip-- N/A

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

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{1 - x} \cdot \sqrt{1 - x}\right), \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 - x}\right)}\right) \]

   3. rem-square-sqrt N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + x\right) - \sqrt{1 - x} \cdot \sqrt{1 - x}\right), \left(\sqrt{\color{blue}{1 + x}} + \sqrt{1 - x}\right)\right) \]

   4. rem-square-sqrt N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + x\right) - \left(1 - x\right)\right), \left(\sqrt{1 + x} + \sqrt{1 - x}\right)\right) \]

   5. associate--l+ N/A

      \[\leadsto \mathsf{/.f64}\left(\left(1 + \left(x - \left(1 - x\right)\right)\right), \left(\color{blue}{\sqrt{1 + x}} + \sqrt{1 - x}\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(x - \left(1 - x\right)\right)\right), \left(\color{blue}{\sqrt{1 + x}} + \sqrt{1 - x}\right)\right) \]

   7. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{\_.f64}\left(x, \left(1 - x\right)\right)\right), \left(\sqrt{1 + x} + \sqrt{1 - x}\right)\right) \]

   8. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right)\right), \left(\sqrt{1 + x} + \sqrt{1 - x}\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right)\right), \mathsf{+.f64}\left(\left(\sqrt{1 + x}\right), \color{blue}{\left(\sqrt{1 - x}\right)}\right)\right) \]

   10. pow1/2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right)\right), \mathsf{+.f64}\left(\left({\left(1 + x\right)}^{\frac{1}{2}}\right), \left(\sqrt{\color{blue}{1 - x}}\right)\right)\right) \]

   11. pow-lowering-pow.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\left(1 + x\right), \frac{1}{2}\right), \left(\sqrt{\color{blue}{1 - x}}\right)\right)\right) \]

   12. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{1}{2}\right), \left(\sqrt{\color{blue}{1} - x}\right)\right)\right) \]

   13. pow1/2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{1}{2}\right), \left({\left(1 - x\right)}^{\color{blue}{\frac{1}{2}}}\right)\right)\right) \]

   14. pow-lowering-pow.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{1}{2}\right), \mathsf{pow.f64}\left(\left(1 - x\right), \color{blue}{\frac{1}{2}}\right)\right)\right) \]

   15. --lowering--.f64 7.8%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{1}{2}\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(1, x\right), \frac{1}{2}\right)\right)\right) \]

4. Applied egg-rr 7.8%

   \[\leadsto \color{blue}{\frac{1 + \left(x - \left(1 - x\right)\right)}{{\left(1 + x\right)}^{0.5} + {\left(1 - x\right)}^{0.5}}} \]

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(2 \cdot x\right)}, \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{1}{2}\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(1, x\right), \frac{1}{2}\right)\right)\right) \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(x \cdot 2\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{1}{2}\right)}, \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(1, x\right), \frac{1}{2}\right)\right)\right) \]

   2. *-lowering-*.f64 100.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{1}{2}\right)}, \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(1, x\right), \frac{1}{2}\right)\right)\right) \]

7. Simplified 100.0%

   \[\leadsto \frac{\color{blue}{x \cdot 2}}{{\left(1 + x\right)}^{0.5} + {\left(1 - x\right)}^{0.5}} \]

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \color{blue}{\left(2 + {x}^{2} \cdot \left(\frac{-5}{64} \cdot {x}^{2} - \frac{1}{4}\right)\right)}\right) \]
9. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{+.f64}\left(2, \color{blue}{\left({x}^{2} \cdot \left(\frac{-5}{64} \cdot {x}^{2} - \frac{1}{4}\right)\right)}\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{-5}{64} \cdot {x}^{2} - \frac{1}{4}\right)}\right)\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{-5}{64} \cdot {x}^{2}} - \frac{1}{4}\right)\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{-5}{64} \cdot {x}^{2}} - \frac{1}{4}\right)\right)\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{-5}{64} \cdot {x}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}\right)\right)\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{-5}{64} \cdot {x}^{2} + \frac{-1}{4}\right)\right)\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{-1}{4} + \color{blue}{\frac{-5}{64} \cdot {x}^{2}}\right)\right)\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{4}, \color{blue}{\left(\frac{-5}{64} \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{4}, \left({x}^{2} \cdot \color{blue}{\frac{-5}{64}}\right)\right)\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-5}{64}}\right)\right)\right)\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-5}{64}\right)\right)\right)\right)\right) \]

   12. *-lowering-*.f64 99.9%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-5}{64}\right)\right)\right)\right)\right) \]

10. Simplified 99.9%

    \[\leadsto \frac{x \cdot 2}{\color{blue}{2 + \left(x \cdot x\right) \cdot \left(-0.25 + \left(x \cdot x\right) \cdot -0.078125\right)}} \]
11. Add Preprocessing

Alternative 5: 99.7% accurate, 13.8× speedup

Math

\[\begin{array}{l} \\ x + \left(0.125 + \left(x \cdot x\right) \cdot 0.0546875\right) \cdot \left(x \cdot \left(x \cdot x\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ x (* (+ 0.125 (* (* x x) 0.0546875)) (* x (* x x)))))

C

double code(double x) {
	return x + ((0.125 + ((x * x) * 0.0546875)) * (x * (x * x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x + ((0.125d0 + ((x * x) * 0.0546875d0)) * (x * (x * x)))
end function

Java

public static double code(double x) {
	return x + ((0.125 + ((x * x) * 0.0546875)) * (x * (x * x)));
}

Python

def code(x):
	return x + ((0.125 + ((x * x) * 0.0546875)) * (x * (x * x)))

Julia

function code(x)
	return Float64(x + Float64(Float64(0.125 + Float64(Float64(x * x) * 0.0546875)) * Float64(x * Float64(x * x))))
end

MATLAB

function tmp = code(x)
	tmp = x + ((0.125 + ((x * x) * 0.0546875)) * (x * (x * x)));
end

Wolfram

code[x_] := N[(x + N[(N[(0.125 + N[(N[(x * x), $MachinePrecision] * 0.0546875), $MachinePrecision]), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.7%

   \[\sqrt{1 + x} - \sqrt{1 - x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{8} + \frac{7}{128} \cdot {x}^{2}\right)\right)} \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{8} + \frac{7}{128} \cdot {x}^{2}\right)\right)}\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{8} + \frac{7}{128} \cdot {x}^{2}\right)\right)}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{8} + \frac{7}{128} \cdot {x}^{2}\right)}\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{8}} + \frac{7}{128} \cdot {x}^{2}\right)\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{8}} + \frac{7}{128} \cdot {x}^{2}\right)\right)\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{8}, \color{blue}{\left(\frac{7}{128} \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{8}, \left({x}^{2} \cdot \color{blue}{\frac{7}{128}}\right)\right)\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{7}{128}}\right)\right)\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{7}{128}\right)\right)\right)\right)\right) \]

   10. *-lowering-*.f64 99.9%

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{7}{128}\right)\right)\right)\right)\right) \]

5. Simplified 99.9%

   \[\leadsto \color{blue}{x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.125 + \left(x \cdot x\right) \cdot 0.0546875\right)\right)} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto x \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{8} + \left(x \cdot x\right) \cdot \frac{7}{128}\right) + \color{blue}{1}\right) \]

   2. distribute-rgt-in N/A

      \[\leadsto \left(\left(x \cdot x\right) \cdot \left(\frac{1}{8} + \left(x \cdot x\right) \cdot \frac{7}{128}\right)\right) \cdot x + \color{blue}{1 \cdot x} \]

   3. *-lft-identity N/A

      \[\leadsto \left(\left(x \cdot x\right) \cdot \left(\frac{1}{8} + \left(x \cdot x\right) \cdot \frac{7}{128}\right)\right) \cdot x + x \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(\left(x \cdot x\right) \cdot \left(\frac{1}{8} + \left(x \cdot x\right) \cdot \frac{7}{128}\right)\right) \cdot x\right), \color{blue}{x}\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(\left(\frac{1}{8} + \left(x \cdot x\right) \cdot \frac{7}{128}\right) \cdot \left(x \cdot x\right)\right) \cdot x\right), x\right) \]

   6. associate-*l* N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(\frac{1}{8} + \left(x \cdot x\right) \cdot \frac{7}{128}\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), x\right) \]

   7. unpow3 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(\frac{1}{8} + \left(x \cdot x\right) \cdot \frac{7}{128}\right) \cdot {x}^{3}\right), x\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{8} + \left(x \cdot x\right) \cdot \frac{7}{128}\right), \left({x}^{3}\right)\right), x\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{8}, \left(\left(x \cdot x\right) \cdot \frac{7}{128}\right)\right), \left({x}^{3}\right)\right), x\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{7}{128}\right)\right), \left({x}^{3}\right)\right), x\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{7}{128}\right)\right), \left({x}^{3}\right)\right), x\right) \]

   12. cube-mult N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{7}{128}\right)\right), \left(x \cdot \left(x \cdot x\right)\right)\right), x\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{7}{128}\right)\right), \mathsf{*.f64}\left(x, \left(x \cdot x\right)\right)\right), x\right) \]

   14. *-lowering-*.f64 99.9%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{7}{128}\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), x\right) \]

7. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\left(0.125 + \left(x \cdot x\right) \cdot 0.0546875\right) \cdot \left(x \cdot \left(x \cdot x\right)\right) + x} \]

8. Final simplification 99.9%

   \[\leadsto x + \left(0.125 + \left(x \cdot x\right) \cdot 0.0546875\right) \cdot \left(x \cdot \left(x \cdot x\right)\right) \]
9. Add Preprocessing

Alternative 6: 99.7% accurate, 13.8× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.125 + \left(x \cdot x\right) \cdot 0.0546875\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* x (+ 1.0 (* (* x x) (+ 0.125 (* (* x x) 0.0546875))))))

C

double code(double x) {
	return x * (1.0 + ((x * x) * (0.125 + ((x * x) * 0.0546875))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (1.0d0 + ((x * x) * (0.125d0 + ((x * x) * 0.0546875d0))))
end function

Java

public static double code(double x) {
	return x * (1.0 + ((x * x) * (0.125 + ((x * x) * 0.0546875))));
}

Python

def code(x):
	return x * (1.0 + ((x * x) * (0.125 + ((x * x) * 0.0546875))))

Julia

function code(x)
	return Float64(x * Float64(1.0 + Float64(Float64(x * x) * Float64(0.125 + Float64(Float64(x * x) * 0.0546875)))))
end

MATLAB

function tmp = code(x)
	tmp = x * (1.0 + ((x * x) * (0.125 + ((x * x) * 0.0546875))));
end

Wolfram

code[x_] := N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(0.125 + N[(N[(x * x), $MachinePrecision] * 0.0546875), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.7%

   \[\sqrt{1 + x} - \sqrt{1 - x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{8} + \frac{7}{128} \cdot {x}^{2}\right)\right)} \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{8} + \frac{7}{128} \cdot {x}^{2}\right)\right)}\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{8} + \frac{7}{128} \cdot {x}^{2}\right)\right)}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{8} + \frac{7}{128} \cdot {x}^{2}\right)}\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{8}} + \frac{7}{128} \cdot {x}^{2}\right)\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{8}} + \frac{7}{128} \cdot {x}^{2}\right)\right)\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{8}, \color{blue}{\left(\frac{7}{128} \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{8}, \left({x}^{2} \cdot \color{blue}{\frac{7}{128}}\right)\right)\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{7}{128}}\right)\right)\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{7}{128}\right)\right)\right)\right)\right) \]

   10. *-lowering-*.f64 99.9%

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{7}{128}\right)\right)\right)\right)\right) \]

5. Simplified 99.9%

   \[\leadsto \color{blue}{x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.125 + \left(x \cdot x\right) \cdot 0.0546875\right)\right)} \]
6. Add Preprocessing

Alternative 7: 99.5% accurate, 18.8× speedup

Math

\[\begin{array}{l} \\ x \cdot \frac{2}{2 + x \cdot \left(x \cdot -0.25\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x (/ 2.0 (+ 2.0 (* x (* x -0.25))))))

C

double code(double x) {
	return x * (2.0 / (2.0 + (x * (x * -0.25))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (2.0d0 / (2.0d0 + (x * (x * (-0.25d0)))))
end function

Java

public static double code(double x) {
	return x * (2.0 / (2.0 + (x * (x * -0.25))));
}

Python

def code(x):
	return x * (2.0 / (2.0 + (x * (x * -0.25))))

Julia

function code(x)
	return Float64(x * Float64(2.0 / Float64(2.0 + Float64(x * Float64(x * -0.25)))))
end

MATLAB

function tmp = code(x)
	tmp = x * (2.0 / (2.0 + (x * (x * -0.25))));
end

Wolfram

code[x_] := N[(x * N[(2.0 / N[(2.0 + N[(x * N[(x * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.7%

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

   1. flip-- N/A

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

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{1 - x} \cdot \sqrt{1 - x}\right), \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 - x}\right)}\right) \]

   3. rem-square-sqrt N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + x\right) - \sqrt{1 - x} \cdot \sqrt{1 - x}\right), \left(\sqrt{\color{blue}{1 + x}} + \sqrt{1 - x}\right)\right) \]

   4. rem-square-sqrt N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + x\right) - \left(1 - x\right)\right), \left(\sqrt{1 + x} + \sqrt{1 - x}\right)\right) \]

   5. associate--l+ N/A

      \[\leadsto \mathsf{/.f64}\left(\left(1 + \left(x - \left(1 - x\right)\right)\right), \left(\color{blue}{\sqrt{1 + x}} + \sqrt{1 - x}\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(x - \left(1 - x\right)\right)\right), \left(\color{blue}{\sqrt{1 + x}} + \sqrt{1 - x}\right)\right) \]

   7. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{\_.f64}\left(x, \left(1 - x\right)\right)\right), \left(\sqrt{1 + x} + \sqrt{1 - x}\right)\right) \]

   8. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right)\right), \left(\sqrt{1 + x} + \sqrt{1 - x}\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right)\right), \mathsf{+.f64}\left(\left(\sqrt{1 + x}\right), \color{blue}{\left(\sqrt{1 - x}\right)}\right)\right) \]

   10. pow1/2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right)\right), \mathsf{+.f64}\left(\left({\left(1 + x\right)}^{\frac{1}{2}}\right), \left(\sqrt{\color{blue}{1 - x}}\right)\right)\right) \]

   11. pow-lowering-pow.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\left(1 + x\right), \frac{1}{2}\right), \left(\sqrt{\color{blue}{1 - x}}\right)\right)\right) \]

   12. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{1}{2}\right), \left(\sqrt{\color{blue}{1} - x}\right)\right)\right) \]

   13. pow1/2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{1}{2}\right), \left({\left(1 - x\right)}^{\color{blue}{\frac{1}{2}}}\right)\right)\right) \]

   14. pow-lowering-pow.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{1}{2}\right), \mathsf{pow.f64}\left(\left(1 - x\right), \color{blue}{\frac{1}{2}}\right)\right)\right) \]

   15. --lowering--.f64 7.8%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right)\right), \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{1}{2}\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(1, x\right), \frac{1}{2}\right)\right)\right) \]

4. Applied egg-rr 7.8%

   \[\leadsto \color{blue}{\frac{1 + \left(x - \left(1 - x\right)\right)}{{\left(1 + x\right)}^{0.5} + {\left(1 - x\right)}^{0.5}}} \]

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(2 \cdot x\right)}, \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{1}{2}\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(1, x\right), \frac{1}{2}\right)\right)\right) \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(x \cdot 2\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{1}{2}\right)}, \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(1, x\right), \frac{1}{2}\right)\right)\right) \]

   2. *-lowering-*.f64 100.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{1}{2}\right)}, \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(1, x\right), \frac{1}{2}\right)\right)\right) \]

7. Simplified 100.0%

   \[\leadsto \frac{\color{blue}{x \cdot 2}}{{\left(1 + x\right)}^{0.5} + {\left(1 - x\right)}^{0.5}} \]

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \color{blue}{\left(2 + \frac{-1}{4} \cdot {x}^{2}\right)}\right) \]
9. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{+.f64}\left(2, \color{blue}{\left(\frac{-1}{4} \cdot {x}^{2}\right)}\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{+.f64}\left(2, \left({x}^{2} \cdot \color{blue}{\frac{-1}{4}}\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-1}{4}}\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{4}\right)\right)\right) \]

   5. *-lowering-*.f64 99.8%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{4}\right)\right)\right) \]

10. Simplified 99.8%

    \[\leadsto \frac{x \cdot 2}{\color{blue}{2 + \left(x \cdot x\right) \cdot -0.25}} \]
11. Step-by-step derivation

    1. associate-/l* N/A

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

    2. *-commutative N/A

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

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{2 + \left(x \cdot x\right) \cdot \frac{-1}{4}}\right), \color{blue}{x}\right) \]

    4. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(2 + \left(x \cdot x\right) \cdot \frac{-1}{4}\right)\right), x\right) \]

    5. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \left(\left(x \cdot x\right) \cdot \frac{-1}{4}\right)\right)\right), x\right) \]

    6. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \left(x \cdot \left(x \cdot \frac{-1}{4}\right)\right)\right)\right), x\right) \]

    7. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \left(x \cdot \frac{-1}{4}\right)\right)\right)\right), x\right) \]

    8. *-lowering-*.f64 99.8%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{-1}{4}\right)\right)\right)\right), x\right) \]

12. Applied egg-rr 99.8%

    \[\leadsto \color{blue}{\frac{2}{2 + x \cdot \left(x \cdot -0.25\right)} \cdot x} \]

13. Final simplification 99.8%

    \[\leadsto x \cdot \frac{2}{2 + x \cdot \left(x \cdot -0.25\right)} \]
14. Add Preprocessing

Alternative 8: 99.5% accurate, 23.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 + \left(x \cdot x\right) \cdot 0.125\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x (+ 1.0 (* (* x x) 0.125))))

C

double code(double x) {
	return x * (1.0 + ((x * x) * 0.125));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (1.0d0 + ((x * x) * 0.125d0))
end function

Java

public static double code(double x) {
	return x * (1.0 + ((x * x) * 0.125));
}

Python

def code(x):
	return x * (1.0 + ((x * x) * 0.125))

Julia

function code(x)
	return Float64(x * Float64(1.0 + Float64(Float64(x * x) * 0.125)))
end

MATLAB

function tmp = code(x)
	tmp = x * (1.0 + ((x * x) * 0.125));
end

Wolfram

code[x_] := N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.7%

   \[\sqrt{1 + x} - \sqrt{1 - x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + \frac{1}{8} \cdot {x}^{2}\right)} \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + \frac{1}{8} \cdot {x}^{2}\right)}\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{8} \cdot {x}^{2}\right)}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{8}, \color{blue}{\left({x}^{2}\right)}\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{8}, \left(x \cdot \color{blue}{x}\right)\right)\right)\right) \]

   5. *-lowering-*.f64 99.8%

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right) \]

5. Simplified 99.8%

   \[\leadsto \color{blue}{x \cdot \left(1 + 0.125 \cdot \left(x \cdot x\right)\right)} \]

6. Final simplification 99.8%

   \[\leadsto x \cdot \left(1 + \left(x \cdot x\right) \cdot 0.125\right) \]
7. Add Preprocessing

Alternative 9: 99.0% accurate, 207.0× 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 7.7%

   \[\sqrt{1 + x} - \sqrt{1 - x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

5. Simplified 99.5%

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

Developer Target 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{2 \cdot x}{\sqrt{1 + x} + \sqrt{1 - x}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (* 2.0 x) (+ (sqrt (+ 1.0 x)) (sqrt (- 1.0 x)))))

C

double code(double x) {
	return (2.0 * x) / (sqrt((1.0 + x)) + sqrt((1.0 - x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (2.0d0 * x) / (sqrt((1.0d0 + x)) + sqrt((1.0d0 - x)))
end function

Java

public static double code(double x) {
	return (2.0 * x) / (Math.sqrt((1.0 + x)) + Math.sqrt((1.0 - x)));
}

Python

def code(x):
	return (2.0 * x) / (math.sqrt((1.0 + x)) + math.sqrt((1.0 - x)))

Julia

function code(x)
	return Float64(Float64(2.0 * x) / Float64(sqrt(Float64(1.0 + x)) + sqrt(Float64(1.0 - x))))
end

MATLAB

function tmp = code(x)
	tmp = (2.0 * x) / (sqrt((1.0 + x)) + sqrt((1.0 - x)));
end

Wolfram

code[x_] := N[(N[(2.0 * x), $MachinePrecision] / N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024154 
(FPCore (x)
  :name "bug333 (missed optimization)"
  :precision binary64
  :pre (and (<= -1.0 x) (<= x 1.0))

  :alt
  (! :herbie-platform default (/ (* 2 x) (+ (sqrt (+ 1 x)) (sqrt (- 1 x)))))

  (- (sqrt (+ 1.0 x)) (sqrt (- 1.0 x))))