bug333 (missed optimization)

Percentage Accurate: 8.4% → 99.8%

Time: 11.3s

Alternatives: 8

Speedup: 2.1×

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, 0.8× speedup

Math

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

FPCore

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

C

double code(double x) {
	return fma((x * x), (x * fma(x, (x * fma((x * x), 0.0322265625, 0.0546875)), 0.125)), x);
}

Julia

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

Wolfram

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

Derivation

1. Initial program 9.1%

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

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

   2. distribute-rgt-in N/A

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

   3. associate-*l* N/A

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

   4. *-lft-identity N/A

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

   5. lower-fma.f64 N/A

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

5. Applied rewrites 99.5%

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

Alternative 2: 99.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	return fma(Float64(x * x), Float64(x * fma(x, Float64(x * 0.0546875), 0.125)), x)
end

Wolfram

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

Derivation

1. Initial program 9.1%

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

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

   2. distribute-rgt-in N/A

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

   3. associate-*l* N/A

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

   4. *-lft-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. +-commutative N/A

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

   11. unpow2 N/A

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

   12. associate-*r* N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 99.4

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

5. Applied rewrites 99.4%

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

Alternative 3: 99.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

double code(double x) {
	return fma(x, (x * (x * 0.125)), x);
}

Julia

function code(x)
	return fma(x, Float64(x * Float64(x * 0.125)), x)
end

Wolfram

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

Derivation

1. Initial program 9.1%

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

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

   4. lower-fma.f64 N/A

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

   5. unpow2 N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 99.2

       \[\leadsto \mathsf{fma}\left(x, x \cdot \color{blue}{\left(x \cdot 0.125\right)}, x\right) \]

5. Applied rewrites 99.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \left(x \cdot 0.125\right), x\right)} \]
6. Add Preprocessing

Alternative 4: 99.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ x \cdot 0.5 - x \cdot -0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (* x 0.5) (* x -0.5)))

C

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

Fortran

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

Java

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

Python

def code(x):
	return (x * 0.5) - (x * -0.5)

Julia

function code(x)
	return Float64(Float64(x * 0.5) - Float64(x * -0.5))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 9.1%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 7.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.5, 1\right)} - \sqrt{1 - x} \]

5. Applied rewrites 7.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.5, 1\right)} - \sqrt{1 - x} \]

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(x, \frac{1}{2}, 1\right) - \color{blue}{\left(1 + \frac{-1}{2} \cdot x\right)} \]
7. Step-by-step derivation

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 7.9

      \[\leadsto \mathsf{fma}\left(x, 0.5, 1\right) - \color{blue}{\mathsf{fma}\left(x, -0.5, 1\right)} \]

8. Applied rewrites 7.9%

   \[\leadsto \mathsf{fma}\left(x, 0.5, 1\right) - \color{blue}{\mathsf{fma}\left(x, -0.5, 1\right)} \]

9. Taylor expanded in x around inf

   \[\leadsto \mathsf{fma}\left(x, \frac{1}{2}, 1\right) - \frac{-1}{2} \cdot \color{blue}{x} \]
10. Step-by-step derivation

11. Applied rewrites 4.0%

    \[\leadsto \mathsf{fma}\left(x, 0.5, 1\right) - x \cdot \color{blue}{-0.5} \]

12. Taylor expanded in x around inf

    \[\leadsto \frac{1}{2} \cdot \color{blue}{x} - x \cdot \frac{-1}{2} \]
13. Step-by-step derivation

14. Applied rewrites 98.5%

    \[\leadsto x \cdot \color{blue}{0.5} - x \cdot -0.5 \]
15. Add Preprocessing

Alternative 5: 6.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, 0.5, 1\right) - 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (fma x 0.5 1.0) 1.0))

C

double code(double x) {
	return fma(x, 0.5, 1.0) - 1.0;
}

Julia

function code(x)
	return Float64(fma(x, 0.5, 1.0) - 1.0)
end

Wolfram

code[x_] := N[(N[(x * 0.5 + 1.0), $MachinePrecision] - 1.0), $MachinePrecision]

Derivation

1. Initial program 9.1%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 7.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.5, 1\right)} - \sqrt{1 - x} \]

5. Applied rewrites 7.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.5, 1\right)} - \sqrt{1 - x} \]

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(x, \frac{1}{2}, 1\right) - \color{blue}{1} \]
7. Step-by-step derivation

8. Applied rewrites 6.2%

   \[\leadsto \mathsf{fma}\left(x, 0.5, 1\right) - \color{blue}{1} \]
9. Add Preprocessing

Alternative 6: 6.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ 1 - \mathsf{fma}\left(x, -0.5, 1\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- 1.0 (fma x -0.5 1.0)))

C

double code(double x) {
	return 1.0 - fma(x, -0.5, 1.0);
}

Julia

function code(x)
	return Float64(1.0 - fma(x, -0.5, 1.0))
end

Wolfram

code[x_] := N[(1.0 - N[(x * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 9.1%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 7.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.5, 1\right)} - \sqrt{1 - x} \]

5. Applied rewrites 7.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.5, 1\right)} - \sqrt{1 - x} \]

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(x, \frac{1}{2}, 1\right) - \color{blue}{\left(1 + \frac{-1}{2} \cdot x\right)} \]
7. Step-by-step derivation

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 7.9

      \[\leadsto \mathsf{fma}\left(x, 0.5, 1\right) - \color{blue}{\mathsf{fma}\left(x, -0.5, 1\right)} \]

8. Applied rewrites 7.9%

   \[\leadsto \mathsf{fma}\left(x, 0.5, 1\right) - \color{blue}{\mathsf{fma}\left(x, -0.5, 1\right)} \]

9. Taylor expanded in x around 0

   \[\leadsto 1 - \mathsf{fma}\left(x, \frac{-1}{2}, 1\right) \]
10. Step-by-step derivation

11. Applied rewrites 6.2%

    \[\leadsto 1 - \mathsf{fma}\left(x, -0.5, 1\right) \]
12. Add Preprocessing

Alternative 7: 3.8% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ 1 - x \cdot -0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- 1.0 (* x -0.5)))

C

double code(double x) {
	return 1.0 - (x * -0.5);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 - (x * (-0.5d0))
end function

Java

public static double code(double x) {
	return 1.0 - (x * -0.5);
}

Python

def code(x):
	return 1.0 - (x * -0.5)

Julia

function code(x)
	return Float64(1.0 - Float64(x * -0.5))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 - (x * -0.5);
end

Wolfram

code[x_] := N[(1.0 - N[(x * -0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 9.1%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 7.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.5, 1\right)} - \sqrt{1 - x} \]

5. Applied rewrites 7.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.5, 1\right)} - \sqrt{1 - x} \]

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(x, \frac{1}{2}, 1\right) - \color{blue}{\left(1 + \frac{-1}{2} \cdot x\right)} \]
7. Step-by-step derivation

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 7.9

      \[\leadsto \mathsf{fma}\left(x, 0.5, 1\right) - \color{blue}{\mathsf{fma}\left(x, -0.5, 1\right)} \]

8. Applied rewrites 7.9%

   \[\leadsto \mathsf{fma}\left(x, 0.5, 1\right) - \color{blue}{\mathsf{fma}\left(x, -0.5, 1\right)} \]

9. Taylor expanded in x around inf

   \[\leadsto \mathsf{fma}\left(x, \frac{1}{2}, 1\right) - \frac{-1}{2} \cdot \color{blue}{x} \]
10. Step-by-step derivation

11. Applied rewrites 4.0%

    \[\leadsto \mathsf{fma}\left(x, 0.5, 1\right) - x \cdot \color{blue}{-0.5} \]

12. Taylor expanded in x around 0

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

14. Applied rewrites 4.0%

    \[\leadsto 1 - x \cdot -0.5 \]
15. Add Preprocessing

Alternative 8: 5.4% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ 1 - 1 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

public static double code(double x) {
	return 1.0 - 1.0;
}

Python

def code(x):
	return 1.0 - 1.0

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 9.1%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 7.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.5, 1\right)} - \sqrt{1 - x} \]

5. Applied rewrites 7.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.5, 1\right)} - \sqrt{1 - x} \]

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(x, \frac{1}{2}, 1\right) - \color{blue}{1} \]
7. Step-by-step derivation

8. Applied rewrites 6.2%

   \[\leadsto \mathsf{fma}\left(x, 0.5, 1\right) - \color{blue}{1} \]

9. Taylor expanded in x around 0

   \[\leadsto 1 - 1 \]
10. Step-by-step derivation

11. Applied rewrites 5.2%

    \[\leadsto 1 - 1 \]
12. Add Preprocessing

Developer Target 1: 100.0% accurate, 0.7× 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 2024220 
(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))))