bug333 (missed optimization)

Percentage Accurate: 8.3% → 100.0%

Time: 4.5s

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: 8.3% 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, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.8%

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

   1. flip-- 8.8%

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

   2. add-sqr-sqrt 8.8%

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

   3. add-sqr-sqrt 8.9%

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

   4. associate--r- 21.4%

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

   5. add-exp-log 21.4%

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

   6. log1p-udef 21.4%

      \[\leadsto \frac{\left(e^{\color{blue}{\mathsf{log1p}\left(x\right)}} - 1\right) + x}{\sqrt{1 + x} + \sqrt{1 - x}} \]

   7. expm1-udef 100.0%

      \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x\right)\right)} + x}{\sqrt{1 + x} + \sqrt{1 - x}} \]

   8. expm1-log1p-u 100.0%

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

3. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.8%

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

   1. flip-- 8.8%

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

   2. div-inv 8.8%

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

   3. add-sqr-sqrt 8.8%

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

   4. add-sqr-sqrt 8.9%

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

   5. associate--r- 21.4%

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

   6. add-exp-log 21.4%

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

   7. log1p-udef 21.4%

      \[\leadsto \left(\left(e^{\color{blue}{\mathsf{log1p}\left(x\right)}} - 1\right) + x\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{1 - x}} \]

   8. expm1-udef 100.0%

      \[\leadsto \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x\right)\right)} + x\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{1 - x}} \]

   9. expm1-log1p-u 100.0%

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

3. Applied egg-rr 100.0%

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

   1. associate-*r/ 100.0%

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

   2. *-rgt-identity 100.0%

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

   3. count-2 100.0%

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

   4. associate-/l* 99.7%

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

   5. remove-double-neg 99.7%

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

   6. sub-neg 99.7%

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

   7. sub-neg 99.7%

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

   8. +-commutative 99.7%

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

   9. remove-double-neg 99.7%

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

5. Simplified 99.7%

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

6. Final simplification 99.7%

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

Alternative 3: 99.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{2}{x \cdot -0.25 + \left(-0.078125 \cdot {x}^{3} + 2 \cdot \frac{1}{x}\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ 2.0 (+ (* x -0.25) (+ (* -0.078125 (pow x 3.0)) (* 2.0 (/ 1.0 x))))))

C

double code(double x) {
	return 2.0 / ((x * -0.25) + ((-0.078125 * pow(x, 3.0)) + (2.0 * (1.0 / x))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 / ((x * (-0.25d0)) + (((-0.078125d0) * (x ** 3.0d0)) + (2.0d0 * (1.0d0 / x))))
end function

Java

public static double code(double x) {
	return 2.0 / ((x * -0.25) + ((-0.078125 * Math.pow(x, 3.0)) + (2.0 * (1.0 / x))));
}

Python

def code(x):
	return 2.0 / ((x * -0.25) + ((-0.078125 * math.pow(x, 3.0)) + (2.0 * (1.0 / x))))

Julia

function code(x)
	return Float64(2.0 / Float64(Float64(x * -0.25) + Float64(Float64(-0.078125 * (x ^ 3.0)) + Float64(2.0 * Float64(1.0 / x)))))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 / ((x * -0.25) + ((-0.078125 * (x ^ 3.0)) + (2.0 * (1.0 / x))));
end

Wolfram

code[x_] := N[(2.0 / N[(N[(x * -0.25), $MachinePrecision] + N[(N[(-0.078125 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.8%

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

   1. flip-- 8.8%

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

   2. div-inv 8.8%

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

   3. add-sqr-sqrt 8.8%

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

   4. add-sqr-sqrt 8.9%

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

   5. associate--r- 21.4%

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

   6. add-exp-log 21.4%

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

   7. log1p-udef 21.4%

      \[\leadsto \left(\left(e^{\color{blue}{\mathsf{log1p}\left(x\right)}} - 1\right) + x\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{1 - x}} \]

   8. expm1-udef 100.0%

      \[\leadsto \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x\right)\right)} + x\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{1 - x}} \]

   9. expm1-log1p-u 100.0%

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

3. Applied egg-rr 100.0%

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

   1. associate-*r/ 100.0%

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

   2. *-rgt-identity 100.0%

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

   3. count-2 100.0%

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

   4. associate-/l* 99.7%

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

   5. remove-double-neg 99.7%

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

   6. sub-neg 99.7%

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

   7. sub-neg 99.7%

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

   8. +-commutative 99.7%

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

   9. remove-double-neg 99.7%

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

5. Simplified 99.7%

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

6. Taylor expanded in x around 0 98.6%

   \[\leadsto \frac{2}{\color{blue}{-0.25 \cdot x + \left(-0.078125 \cdot {x}^{3} + 2 \cdot \frac{1}{x}\right)}} \]

7. Final simplification 98.6%

   \[\leadsto \frac{2}{x \cdot -0.25 + \left(-0.078125 \cdot {x}^{3} + 2 \cdot \frac{1}{x}\right)} \]

Alternative 4: 99.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ x + {x}^{3} \cdot 0.125 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ x (* (pow x 3.0) 0.125)))

C

double code(double x) {
	return x + (pow(x, 3.0) * 0.125);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x + ((x ** 3.0d0) * 0.125d0)
end function

Java

public static double code(double x) {
	return x + (Math.pow(x, 3.0) * 0.125);
}

Python

def code(x):
	return x + (math.pow(x, 3.0) * 0.125)

Julia

function code(x)
	return Float64(x + Float64((x ^ 3.0) * 0.125))
end

MATLAB

function tmp = code(x)
	tmp = x + ((x ^ 3.0) * 0.125);
end

Wolfram

code[x_] := N[(x + N[(N[Power[x, 3.0], $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.8%

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

2. Taylor expanded in x around 0 98.6%

   \[\leadsto \color{blue}{x + 0.125 \cdot {x}^{3}} \]

3. Final simplification 98.6%

   \[\leadsto x + {x}^{3} \cdot 0.125 \]

Alternative 5: 99.2% accurate, 18.8× speedup

Math

\[\begin{array}{l} \\ \frac{2}{x \cdot -0.25 + 2 \cdot \frac{1}{x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 2.0 (+ (* x -0.25) (* 2.0 (/ 1.0 x)))))

C

double code(double x) {
	return 2.0 / ((x * -0.25) + (2.0 * (1.0 / x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 / ((x * (-0.25d0)) + (2.0d0 * (1.0d0 / x)))
end function

Java

public static double code(double x) {
	return 2.0 / ((x * -0.25) + (2.0 * (1.0 / x)));
}

Python

def code(x):
	return 2.0 / ((x * -0.25) + (2.0 * (1.0 / x)))

Julia

function code(x)
	return Float64(2.0 / Float64(Float64(x * -0.25) + Float64(2.0 * Float64(1.0 / x))))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 / ((x * -0.25) + (2.0 * (1.0 / x)));
end

Wolfram

code[x_] := N[(2.0 / N[(N[(x * -0.25), $MachinePrecision] + N[(2.0 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.8%

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

   1. flip-- 8.8%

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

   2. div-inv 8.8%

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

   3. add-sqr-sqrt 8.8%

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

   4. add-sqr-sqrt 8.9%

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

   5. associate--r- 21.4%

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

   6. add-exp-log 21.4%

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

   7. log1p-udef 21.4%

      \[\leadsto \left(\left(e^{\color{blue}{\mathsf{log1p}\left(x\right)}} - 1\right) + x\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{1 - x}} \]

   8. expm1-udef 100.0%

      \[\leadsto \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x\right)\right)} + x\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{1 - x}} \]

   9. expm1-log1p-u 100.0%

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

3. Applied egg-rr 100.0%

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

   1. associate-*r/ 100.0%

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

   2. *-rgt-identity 100.0%

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

   3. count-2 100.0%

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

   4. associate-/l* 99.7%

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

   5. remove-double-neg 99.7%

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

   6. sub-neg 99.7%

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

   7. sub-neg 99.7%

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

   8. +-commutative 99.7%

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

   9. remove-double-neg 99.7%

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

5. Simplified 99.7%

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

6. Taylor expanded in x around 0 98.3%

   \[\leadsto \frac{2}{\color{blue}{-0.25 \cdot x + 2 \cdot \frac{1}{x}}} \]

7. Final simplification 98.3%

   \[\leadsto \frac{2}{x \cdot -0.25 + 2 \cdot \frac{1}{x}} \]

Alternative 6: 99.1% 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 8.8%

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

2. Taylor expanded in x around 0 98.1%

   \[\leadsto \color{blue}{x} \]

3. Final simplification 98.1%

   \[\leadsto x \]

Developer target: 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 2023315 
(FPCore (x)
  :name "bug333 (missed optimization)"
  :precision binary64
  :pre (and (<= -1.0 x) (<= x 1.0))

  :herbie-target
  (/ (* 2.0 x) (+ (sqrt (+ 1.0 x)) (sqrt (- 1.0 x))))

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