bug333 (missed optimization)

Percentage Accurate: 8.4% → 99.8%

Time: 6.4s

Alternatives: 6

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: 8.4% 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

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

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 7.6%

   \[\sqrt{1 + x} - \sqrt{1 - x} \]
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. *-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 100.0%

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

5. Simplified 100.0%

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

Alternative 2: 99.7% accurate, 13.8× speedup

Math

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

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

Derivation

1. Initial program 7.6%

   \[\sqrt{1 + x} - \sqrt{1 - x} \]
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. *-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 100.0%

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

5. Simplified 100.0%

   \[\leadsto \color{blue}{x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.125 + \left(x \cdot x\right) \cdot 0.0546875\right)\right)} \]
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. pow3 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. associate-*l* N/A

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

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

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

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

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

   13. cube-unmult N/A

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

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

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

   15. *-lowering-*.f64 100.0%

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

7. Applied egg-rr 100.0%

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

8. Final simplification 100.0%

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

Alternative 3: 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 7.6%

   \[\sqrt{1 + x} - \sqrt{1 - x} \]
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. *-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 100.0%

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

5. Simplified 100.0%

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

Alternative 4: 99.6% accurate, 23.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.6%

   \[\sqrt{1 + x} - \sqrt{1 - x} \]
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. *-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. Step-by-step derivation

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

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

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. pow3 N/A

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

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

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

   9. cube-unmult N/A

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

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

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

   11. *-lowering-*.f64 99.8%

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

7. Applied egg-rr 99.8%

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

8. Final simplification 99.8%

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

Alternative 5: 99.6% 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 7.6%

   \[\sqrt{1 + x} - \sqrt{1 - x} \]
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. *-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 6: 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 7.6%

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

3. Taylor expanded in x around 0

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

5. Simplified 99.6%

   \[\leadsto \color{blue}{x} \]
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 2024158 
(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))))