bug333 (missed optimization)

Percentage Accurate: 8.3% → 100.0%

Time: 9.5s

Alternatives: 5

Speedup: 2.7×

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: 8.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.0%

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

   1. lift-sqrt.f64 N/A

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

   2. lift--.f64 N/A

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

   3. flip-- N/A

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

   4. lift-+.f64 N/A

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

   5. sqrt-div N/A

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

   6. lift-sqrt.f64 N/A

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

   7. div-inv N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. metadata-eval N/A

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

   11. lower--.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-/.f64 7.9

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

4. Applied rewrites 7.9%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. div-inv N/A

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

   4. lower-*.f64 N/A

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

6. Applied rewrites 8.2%

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

7. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 100.0

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

9. Applied rewrites 100.0%

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

Alternative 2: 100.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.0%

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

   1. lift-sqrt.f64 N/A

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

   2. lift--.f64 N/A

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

   3. flip-- N/A

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

   4. lift-+.f64 N/A

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

   5. sqrt-div N/A

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

   6. lift-sqrt.f64 N/A

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

   7. div-inv N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. metadata-eval N/A

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

   11. lower--.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-/.f64 7.9

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

4. Applied rewrites 7.9%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. lower-/.f64 N/A

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

6. Applied rewrites 8.2%

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

7. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 100.0

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

9. Applied rewrites 100.0%

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

Alternative 3: 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 8.0%

   \[\sqrt{1 + x} - \sqrt{1 - x} \]
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 99.8

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

5. Applied rewrites 99.8%

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

Alternative 4: 99.6% 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 8.0%

   \[\sqrt{1 + x} - \sqrt{1 - x} \]
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 99.7

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

5. Applied rewrites 99.7%

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

Alternative 5: 99.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot 2\right) \cdot 0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (* x 2.0) 0.5))

C

double code(double x) {
	return (x * 2.0) * 0.5;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * 2.0d0) * 0.5d0
end function

Java

public static double code(double x) {
	return (x * 2.0) * 0.5;
}

Python

def code(x):
	return (x * 2.0) * 0.5

Julia

function code(x)
	return Float64(Float64(x * 2.0) * 0.5)
end

MATLAB

function tmp = code(x)
	tmp = (x * 2.0) * 0.5;
end

Wolfram

code[x_] := N[(N[(x * 2.0), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 8.0%

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

   1. lift-sqrt.f64 N/A

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

   2. lift--.f64 N/A

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

   3. flip-- N/A

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

   4. lift-+.f64 N/A

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

   5. sqrt-div N/A

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

   6. lift-sqrt.f64 N/A

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

   7. div-inv N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. metadata-eval N/A

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

   11. lower--.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-/.f64 7.9

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

4. Applied rewrites 7.9%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. div-inv N/A

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

   4. lower-*.f64 N/A

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

6. Applied rewrites 8.2%

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

7. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 100.0

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

9. Applied rewrites 100.0%

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

10. Taylor expanded in x around 0

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

12. Applied rewrites 99.2%

    \[\leadsto \left(x \cdot 2\right) \cdot \color{blue}{0.5} \]
13. 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 2024233 
(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))))