bug333 (missed optimization)

Percentage Accurate: 8.2% → 99.8%

Time: 9.7s

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: 8.2% 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: 99.8% accurate, 9.9× speedup

Math

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

FPCore

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

C

double code(double x) {
	return x + (x * (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 + (x * (x * (x * (0.125d0 + ((x * x) * (0.0546875d0 + (x * (x * 0.0322265625d0))))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.6%

   \[\sqrt{1 + x} - \sqrt{1 - x} \]
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. unpow2 N/A

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

   4. associate-*l* N/A

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

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

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

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

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

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

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

   8. unpow2 N/A

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

   9. associate-*l* N/A

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

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

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

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

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

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

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

   13. *-commutative N/A

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

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

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

   15. unpow2 N/A

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

   16. *-lowering-*.f64 99.8%

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

5. Simplified 99.8%

   \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(x \cdot \left(0.125 + x \cdot \left(x \cdot \left(0.0546875 + \left(x \cdot x\right) \cdot 0.0322265625\right)\right)\right)\right)\right)} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto x \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{8} + x \cdot \left(x \cdot \left(\frac{7}{128} + \left(x \cdot x\right) \cdot \frac{33}{1024}\right)\right)\right)\right) + \color{blue}{1}\right) \]

   2. distribute-lft-in N/A

      \[\leadsto x \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{8} + x \cdot \left(x \cdot \left(\frac{7}{128} + \left(x \cdot x\right) \cdot \frac{33}{1024}\right)\right)\right)\right)\right) + \color{blue}{x \cdot 1} \]

   3. *-rgt-identity N/A

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

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

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

7. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(x \cdot \left(0.125 + \left(x \cdot x\right) \cdot \left(0.0546875 + x \cdot \left(x \cdot 0.0322265625\right)\right)\right)\right)\right) + x} \]

8. Final simplification 99.8%

   \[\leadsto x + x \cdot \left(x \cdot \left(x \cdot \left(0.125 + \left(x \cdot x\right) \cdot \left(0.0546875 + x \cdot \left(x \cdot 0.0322265625\right)\right)\right)\right)\right) \]
9. Add Preprocessing

Alternative 2: 99.8% accurate, 9.9× speedup

Math

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

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(x * Float64(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[(x * N[(x * 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]), $MachinePrecision]

Derivation

1. Initial program 8.6%

   \[\sqrt{1 + x} - \sqrt{1 - x} \]
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. unpow2 N/A

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

   4. associate-*l* N/A

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

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

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

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

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

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

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

   8. unpow2 N/A

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

   9. associate-*l* N/A

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

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

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

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

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

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

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

   13. *-commutative N/A

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

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

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

   15. unpow2 N/A

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

   16. *-lowering-*.f64 99.8%

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

5. Simplified 99.8%

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

Alternative 3: 99.8% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot 4}{4 + \left(x \cdot x\right) \cdot \left(-0.5 + x \cdot \left(x \cdot -0.15625\right)\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (* x 4.0) (+ 4.0 (* (* x x) (+ -0.5 (* x (* x -0.15625)))))))

C

double code(double x) {
	return (x * 4.0) / (4.0 + ((x * x) * (-0.5 + (x * (x * -0.15625)))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * 4.0d0) / (4.0d0 + ((x * x) * ((-0.5d0) + (x * (x * (-0.15625d0))))))
end function

Java

public static double code(double x) {
	return (x * 4.0) / (4.0 + ((x * x) * (-0.5 + (x * (x * -0.15625)))));
}

Python

def code(x):
	return (x * 4.0) / (4.0 + ((x * x) * (-0.5 + (x * (x * -0.15625)))))

Julia

function code(x)
	return Float64(Float64(x * 4.0) / Float64(4.0 + Float64(Float64(x * x) * Float64(-0.5 + Float64(x * Float64(x * -0.15625))))))
end

MATLAB

function tmp = code(x)
	tmp = (x * 4.0) / (4.0 + ((x * x) * (-0.5 + (x * (x * -0.15625)))));
end

Wolfram

code[x_] := N[(N[(x * 4.0), $MachinePrecision] / N[(4.0 + N[(N[(x * x), $MachinePrecision] * N[(-0.5 + N[(x * N[(x * -0.15625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.6%

   \[\sqrt{1 + x} - \sqrt{1 - x} \]
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. rem-square-sqrt N/A

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

   3. rem-square-sqrt N/A

      \[\leadsto \frac{\left(1 + x\right) - \left(1 - x\right)}{\sqrt{1 + x} + \sqrt{1 - x}} \]

   4. flip-- N/A

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

   5. associate-/l/ N/A

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

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

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

4. Applied egg-rr 8.7%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, x\right)\right)\right), \color{blue}{\left(4 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-21}{256} \cdot {x}^{2} - \frac{5}{32}\right) - \frac{1}{2}\right)\right)}\right) \]
6. Step-by-step derivation

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, x\right)\right)\right), \mathsf{+.f64}\left(4, \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-21}{256} \cdot {x}^{2} - \frac{5}{32}\right) - \frac{1}{2}\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, x\right)\right)\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left({x}^{2} \cdot \left(\frac{-21}{256} \cdot {x}^{2} - \frac{5}{32}\right) - \frac{1}{2}\right)}\right)\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, x\right)\right)\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{{x}^{2} \cdot \left(\frac{-21}{256} \cdot {x}^{2} - \frac{5}{32}\right)} - \frac{1}{2}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, x\right)\right)\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{{x}^{2} \cdot \left(\frac{-21}{256} \cdot {x}^{2} - \frac{5}{32}\right)} - \frac{1}{2}\right)\right)\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, x\right)\right)\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{-21}{256} \cdot {x}^{2} - \frac{5}{32}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, x\right)\right)\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{-21}{256} \cdot {x}^{2} - \frac{5}{32}\right) + \frac{-1}{2}\right)\right)\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, x\right)\right)\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{-1}{2} + \color{blue}{{x}^{2} \cdot \left(\frac{-21}{256} \cdot {x}^{2} - \frac{5}{32}\right)}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, x\right)\right)\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left({x}^{2} \cdot \left(\frac{-21}{256} \cdot {x}^{2} - \frac{5}{32}\right)\right)}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, x\right)\right)\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{-21}{256} \cdot {x}^{2} - \frac{5}{32}\right)}\right)\right)\right)\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, x\right)\right)\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{-21}{256} \cdot {x}^{2}} - \frac{5}{32}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, x\right)\right)\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{-21}{256} \cdot {x}^{2}} - \frac{5}{32}\right)\right)\right)\right)\right)\right) \]

   12. sub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, x\right)\right)\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{-21}{256} \cdot {x}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{5}{32}\right)\right)}\right)\right)\right)\right)\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, x\right)\right)\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{-21}{256} \cdot {x}^{2} + \frac{-5}{32}\right)\right)\right)\right)\right)\right) \]

   14. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, x\right)\right)\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{-5}{32} + \color{blue}{\frac{-21}{256} \cdot {x}^{2}}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, x\right)\right)\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-5}{32}, \color{blue}{\left(\frac{-21}{256} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

   16. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, x\right)\right)\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-5}{32}, \left({x}^{2} \cdot \color{blue}{\frac{-21}{256}}\right)\right)\right)\right)\right)\right)\right) \]

   17. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, x\right)\right)\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-5}{32}, \left(\left(x \cdot x\right) \cdot \frac{-21}{256}\right)\right)\right)\right)\right)\right)\right) \]

   18. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, x\right)\right)\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-5}{32}, \left(x \cdot \color{blue}{\left(x \cdot \frac{-21}{256}\right)}\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, x\right)\right)\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-5}{32}, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \frac{-21}{256}\right)}\right)\right)\right)\right)\right)\right)\right) \]

   20. *-lowering-*.f64 8.5%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, x\right)\right)\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-5}{32}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{-21}{256}}\right)\right)\right)\right)\right)\right)\right)\right) \]

7. Simplified 8.5%

   \[\leadsto \frac{\left(1 + x\right) \cdot \left(1 + x\right) - \left(1 - x\right) \cdot \left(1 - x\right)}{\color{blue}{4 + \left(x \cdot x\right) \cdot \left(-0.5 + \left(x \cdot x\right) \cdot \left(-0.15625 + x \cdot \left(x \cdot -0.08203125\right)\right)\right)}} \]

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(4 \cdot x\right)}, \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-5}{32}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{-21}{256}\right)\right)\right)\right)\right)\right)\right)\right) \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(x \cdot 4\right), \mathsf{+.f64}\left(\color{blue}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-5}{32}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{-21}{256}\right)\right)\right)\right)\right)\right)\right)\right) \]

   2. *-lowering-*.f64 99.8%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 4\right), \mathsf{+.f64}\left(\color{blue}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-5}{32}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{-21}{256}\right)\right)\right)\right)\right)\right)\right)\right) \]

10. Simplified 99.8%

    \[\leadsto \frac{\color{blue}{x \cdot 4}}{4 + \left(x \cdot x\right) \cdot \left(-0.5 + \left(x \cdot x\right) \cdot \left(-0.15625 + x \cdot \left(x \cdot -0.08203125\right)\right)\right)} \]

11. Taylor expanded in x around 0

    \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 4\right), \color{blue}{\left(4 + {x}^{2} \cdot \left(\frac{-5}{32} \cdot {x}^{2} - \frac{1}{2}\right)\right)}\right) \]
12. Step-by-step derivation

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 4\right), \mathsf{+.f64}\left(4, \color{blue}{\left({x}^{2} \cdot \left(\frac{-5}{32} \cdot {x}^{2} - \frac{1}{2}\right)\right)}\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 4\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{-5}{32} \cdot {x}^{2} - \frac{1}{2}\right)}\right)\right)\right) \]

    3. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 4\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{-5}{32} \cdot {x}^{2}} - \frac{1}{2}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 4\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{-5}{32} \cdot {x}^{2}} - \frac{1}{2}\right)\right)\right)\right) \]

    5. sub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 4\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{-5}{32} \cdot {x}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right) \]

    6. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 4\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{-5}{32} \cdot {x}^{2} + \frac{-1}{2}\right)\right)\right)\right) \]

    7. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 4\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{-1}{2} + \color{blue}{\frac{-5}{32} \cdot {x}^{2}}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 4\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{-5}{32} \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

    9. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 4\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left({x}^{2} \cdot \color{blue}{\frac{-5}{32}}\right)\right)\right)\right)\right) \]

    10. unpow2 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 4\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(\left(x \cdot x\right) \cdot \frac{-5}{32}\right)\right)\right)\right)\right) \]

    11. associate-*l* N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 4\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(x \cdot \color{blue}{\left(x \cdot \frac{-5}{32}\right)}\right)\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 4\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \frac{-5}{32}\right)}\right)\right)\right)\right)\right) \]

    13. *-lowering-*.f64 99.7%

        \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 4\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{-5}{32}}\right)\right)\right)\right)\right)\right) \]

13. Simplified 99.7%

    \[\leadsto \frac{x \cdot 4}{\color{blue}{4 + \left(x \cdot x\right) \cdot \left(-0.5 + x \cdot \left(x \cdot -0.15625\right)\right)}} \]
14. Add Preprocessing

Alternative 4: 99.8% accurate, 13.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.6%

   \[\sqrt{1 + x} - \sqrt{1 - x} \]
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. unpow2 N/A

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

   9. associate-*l* N/A

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

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

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

   11. *-lowering-*.f64 99.7%

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

5. Simplified 99.7%

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. *-lft-identity N/A

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

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

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

   5. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

   11. *-lowering-*.f64 99.7%

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

7. Applied egg-rr 99.7%

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

8. Final simplification 99.7%

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

Alternative 5: 99.8% accurate, 13.8× speedup

Math

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

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(x * Float64(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[(x * N[(x * 0.0546875), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.6%

   \[\sqrt{1 + x} - \sqrt{1 - x} \]
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. unpow2 N/A

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

   9. associate-*l* N/A

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

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

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

   11. *-lowering-*.f64 99.7%

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

5. Simplified 99.7%

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

Alternative 6: 99.6% accurate, 18.8× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot 4}{4 + x \cdot \left(x \cdot -0.5\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (* x 4.0) (+ 4.0 (* x (* x -0.5)))))

C

double code(double x) {
	return (x * 4.0) / (4.0 + (x * (x * -0.5)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * 4.0d0) / (4.0d0 + (x * (x * (-0.5d0))))
end function

Java

public static double code(double x) {
	return (x * 4.0) / (4.0 + (x * (x * -0.5)));
}

Python

def code(x):
	return (x * 4.0) / (4.0 + (x * (x * -0.5)))

Julia

function code(x)
	return Float64(Float64(x * 4.0) / Float64(4.0 + Float64(x * Float64(x * -0.5))))
end

MATLAB

function tmp = code(x)
	tmp = (x * 4.0) / (4.0 + (x * (x * -0.5)));
end

Wolfram

code[x_] := N[(N[(x * 4.0), $MachinePrecision] / N[(4.0 + N[(x * N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.6%

   \[\sqrt{1 + x} - \sqrt{1 - x} \]
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. rem-square-sqrt N/A

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

   3. rem-square-sqrt N/A

      \[\leadsto \frac{\left(1 + x\right) - \left(1 - x\right)}{\sqrt{1 + x} + \sqrt{1 - x}} \]

   4. flip-- N/A

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

   5. associate-/l/ N/A

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

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

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

4. Applied egg-rr 8.7%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, x\right)\right)\right), \color{blue}{\left(4 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-21}{256} \cdot {x}^{2} - \frac{5}{32}\right) - \frac{1}{2}\right)\right)}\right) \]
6. Step-by-step derivation

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, x\right)\right)\right), \mathsf{+.f64}\left(4, \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-21}{256} \cdot {x}^{2} - \frac{5}{32}\right) - \frac{1}{2}\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, x\right)\right)\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left({x}^{2} \cdot \left(\frac{-21}{256} \cdot {x}^{2} - \frac{5}{32}\right) - \frac{1}{2}\right)}\right)\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, x\right)\right)\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{{x}^{2} \cdot \left(\frac{-21}{256} \cdot {x}^{2} - \frac{5}{32}\right)} - \frac{1}{2}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, x\right)\right)\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{{x}^{2} \cdot \left(\frac{-21}{256} \cdot {x}^{2} - \frac{5}{32}\right)} - \frac{1}{2}\right)\right)\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, x\right)\right)\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{-21}{256} \cdot {x}^{2} - \frac{5}{32}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, x\right)\right)\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{-21}{256} \cdot {x}^{2} - \frac{5}{32}\right) + \frac{-1}{2}\right)\right)\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, x\right)\right)\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{-1}{2} + \color{blue}{{x}^{2} \cdot \left(\frac{-21}{256} \cdot {x}^{2} - \frac{5}{32}\right)}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, x\right)\right)\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left({x}^{2} \cdot \left(\frac{-21}{256} \cdot {x}^{2} - \frac{5}{32}\right)\right)}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, x\right)\right)\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{-21}{256} \cdot {x}^{2} - \frac{5}{32}\right)}\right)\right)\right)\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, x\right)\right)\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{-21}{256} \cdot {x}^{2}} - \frac{5}{32}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, x\right)\right)\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{-21}{256} \cdot {x}^{2}} - \frac{5}{32}\right)\right)\right)\right)\right)\right) \]

   12. sub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, x\right)\right)\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{-21}{256} \cdot {x}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{5}{32}\right)\right)}\right)\right)\right)\right)\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, x\right)\right)\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{-21}{256} \cdot {x}^{2} + \frac{-5}{32}\right)\right)\right)\right)\right)\right) \]

   14. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, x\right)\right)\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{-5}{32} + \color{blue}{\frac{-21}{256} \cdot {x}^{2}}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, x\right)\right)\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-5}{32}, \color{blue}{\left(\frac{-21}{256} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

   16. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, x\right)\right)\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-5}{32}, \left({x}^{2} \cdot \color{blue}{\frac{-21}{256}}\right)\right)\right)\right)\right)\right)\right) \]

   17. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, x\right)\right)\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-5}{32}, \left(\left(x \cdot x\right) \cdot \frac{-21}{256}\right)\right)\right)\right)\right)\right)\right) \]

   18. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, x\right)\right)\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-5}{32}, \left(x \cdot \color{blue}{\left(x \cdot \frac{-21}{256}\right)}\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, x\right)\right)\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-5}{32}, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \frac{-21}{256}\right)}\right)\right)\right)\right)\right)\right)\right) \]

   20. *-lowering-*.f64 8.5%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, x\right)\right)\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-5}{32}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{-21}{256}}\right)\right)\right)\right)\right)\right)\right)\right) \]

7. Simplified 8.5%

   \[\leadsto \frac{\left(1 + x\right) \cdot \left(1 + x\right) - \left(1 - x\right) \cdot \left(1 - x\right)}{\color{blue}{4 + \left(x \cdot x\right) \cdot \left(-0.5 + \left(x \cdot x\right) \cdot \left(-0.15625 + x \cdot \left(x \cdot -0.08203125\right)\right)\right)}} \]

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(4 \cdot x\right)}, \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-5}{32}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{-21}{256}\right)\right)\right)\right)\right)\right)\right)\right) \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(x \cdot 4\right), \mathsf{+.f64}\left(\color{blue}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-5}{32}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{-21}{256}\right)\right)\right)\right)\right)\right)\right)\right) \]

   2. *-lowering-*.f64 99.8%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 4\right), \mathsf{+.f64}\left(\color{blue}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-5}{32}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{-21}{256}\right)\right)\right)\right)\right)\right)\right)\right) \]

10. Simplified 99.8%

    \[\leadsto \frac{\color{blue}{x \cdot 4}}{4 + \left(x \cdot x\right) \cdot \left(-0.5 + \left(x \cdot x\right) \cdot \left(-0.15625 + x \cdot \left(x \cdot -0.08203125\right)\right)\right)} \]

11. Taylor expanded in x around 0

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

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

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

    2. unpow2 N/A

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

    3. associate-*r* N/A

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

    4. *-commutative N/A

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

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

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

    6. *-commutative N/A

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

    7. *-lowering-*.f64 99.4%

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

13. Simplified 99.4%

    \[\leadsto \frac{x \cdot 4}{\color{blue}{4 + x \cdot \left(x \cdot -0.5\right)}} \]
14. Add Preprocessing

Alternative 7: 99.6% accurate, 23.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.6%

   \[\sqrt{1 + x} - \sqrt{1 - x} \]
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. *-commutative N/A

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

   4. unpow2 N/A

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

   5. associate-*l* N/A

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

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

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

   7. *-lowering-*.f64 99.4%

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

5. Simplified 99.4%

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

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

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

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

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

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

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

   9. *-lowering-*.f64 99.4%

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

7. Applied egg-rr 99.4%

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

8. Final simplification 99.4%

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

Alternative 8: 99.6% accurate, 23.0× speedup

Math

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

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(x * Float64(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[(x * N[(x * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.6%

   \[\sqrt{1 + x} - \sqrt{1 - x} \]
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. *-commutative N/A

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

   4. unpow2 N/A

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

   5. associate-*l* N/A

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

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

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

   7. *-lowering-*.f64 99.4%

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

5. Simplified 99.4%

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

Alternative 9: 99.2% 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 8.6%

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

3. Taylor expanded in x around 0

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

5. Simplified 99.0%

   \[\leadsto \color{blue}{x} \]
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 2024148 
(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))))