bug333 (missed optimization)

Percentage Accurate: 8.2% → 100.0%

Time: 5.8s

Alternatives: 4

Speedup: 207.0×

Specification

\[-1 \leq x \land x \leq 1\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 8.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 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 9.3%

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

   1. +-commutative 9.3%

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

   2. metadata-eval 9.3%

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

   3. sub-neg 9.3%

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

3. Simplified 9.3%

   \[\leadsto \color{blue}{\sqrt{x - -1} - \sqrt{1 - x}} \]
4. Step-by-step derivation

   1. flip-- 9.3%

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

   2. add-sqr-sqrt 9.4%

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

   3. add-sqr-sqrt 9.5%

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

   4. associate--r- 21.7%

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

   5. sub-neg 21.7%

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

   6. metadata-eval 21.7%

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

   7. +-commutative 21.7%

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

   8. add-exp-log 21.7%

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

   9. log1p-udef 21.7%

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

   10. expm1-udef 100.0%

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

   11. expm1-log1p-u 100.0%

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

   12. sub-neg 100.0%

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

   13. metadata-eval 100.0%

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

5. Applied egg-rr 100.0%

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

6. Final simplification 100.0%

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

Alternative 2: 99.3% 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 9.3%

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

   1. +-commutative 9.3%

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

   2. metadata-eval 9.3%

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

   3. sub-neg 9.3%

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

3. Simplified 9.3%

   \[\leadsto \color{blue}{\sqrt{x - -1} - \sqrt{1 - x}} \]
4. Step-by-step derivation

   1. flip-- 9.3%

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

   2. div-inv 9.3%

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

   3. add-sqr-sqrt 9.4%

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

   4. add-sqr-sqrt 9.5%

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

   5. associate--r- 21.6%

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

   6. sub-neg 21.6%

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

   7. metadata-eval 21.6%

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

   8. +-commutative 21.6%

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

   9. add-exp-log 21.6%

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

   10. log1p-udef 21.6%

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

   11. expm1-udef 100.0%

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

   12. expm1-log1p-u 100.0%

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

   13. sub-neg 100.0%

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

   14. metadata-eval 100.0%

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

5. Applied egg-rr 100.0%

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

6. Taylor expanded in x around 0 99.0%

   \[\leadsto \left(x + x\right) \cdot \frac{1}{\color{blue}{2 + -0.25 \cdot {x}^{2}}} \]
7. Step-by-step derivation

   1. add-log-exp 8.5%

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

   2. div-inv 8.5%

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

   3. *-un-lft-identity 8.5%

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

   4. log-prod 8.5%

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

   5. metadata-eval 8.5%

      \[\leadsto \color{blue}{0} + \log \left(e^{\frac{x + x}{2 + -0.25 \cdot {x}^{2}}}\right) \]

   6. add-log-exp 99.0%

      \[\leadsto 0 + \color{blue}{\frac{x + x}{2 + -0.25 \cdot {x}^{2}}} \]

   7. count-2 99.0%

      \[\leadsto 0 + \frac{\color{blue}{2 \cdot x}}{2 + -0.25 \cdot {x}^{2}} \]

   8. *-un-lft-identity 99.0%

      \[\leadsto 0 + \frac{2 \cdot x}{\color{blue}{1 \cdot \left(2 + -0.25 \cdot {x}^{2}\right)}} \]

   9. times-frac 99.0%

      \[\leadsto 0 + \color{blue}{\frac{2}{1} \cdot \frac{x}{2 + -0.25 \cdot {x}^{2}}} \]

   10. metadata-eval 99.0%

       \[\leadsto 0 + \color{blue}{2} \cdot \frac{x}{2 + -0.25 \cdot {x}^{2}} \]

   11. +-commutative 99.0%

       \[\leadsto 0 + 2 \cdot \frac{x}{\color{blue}{-0.25 \cdot {x}^{2} + 2}} \]

   12. *-commutative 99.0%

       \[\leadsto 0 + 2 \cdot \frac{x}{\color{blue}{{x}^{2} \cdot -0.25} + 2} \]

   13. unpow2 99.0%

       \[\leadsto 0 + 2 \cdot \frac{x}{\color{blue}{\left(x \cdot x\right)} \cdot -0.25 + 2} \]

   14. associate-*l* 99.0%

       \[\leadsto 0 + 2 \cdot \frac{x}{\color{blue}{x \cdot \left(x \cdot -0.25\right)} + 2} \]

   15. fma-def 99.0%

       \[\leadsto 0 + 2 \cdot \frac{x}{\color{blue}{\mathsf{fma}\left(x, x \cdot -0.25, 2\right)}} \]

8. Applied egg-rr 99.0%

   \[\leadsto \color{blue}{0 + 2 \cdot \frac{x}{\mathsf{fma}\left(x, x \cdot -0.25, 2\right)}} \]
9. Step-by-step derivation

   1. +-lft-identity 99.0%

      \[\leadsto \color{blue}{2 \cdot \frac{x}{\mathsf{fma}\left(x, x \cdot -0.25, 2\right)}} \]

   2. associate-*r/ 99.0%

      \[\leadsto \color{blue}{\frac{2 \cdot x}{\mathsf{fma}\left(x, x \cdot -0.25, 2\right)}} \]

   3. associate-/l* 98.7%

      \[\leadsto \color{blue}{\frac{2}{\frac{\mathsf{fma}\left(x, x \cdot -0.25, 2\right)}{x}}} \]

10. Simplified 98.7%

    \[\leadsto \color{blue}{\frac{2}{\frac{\mathsf{fma}\left(x, x \cdot -0.25, 2\right)}{x}}} \]

11. Taylor expanded in x around 0 98.7%

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

12. Final simplification 98.7%

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

Alternative 3: 99.6% accurate, 18.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 9.3%

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

   1. +-commutative 9.3%

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

   2. metadata-eval 9.3%

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

   3. sub-neg 9.3%

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

3. Simplified 9.3%

   \[\leadsto \color{blue}{\sqrt{x - -1} - \sqrt{1 - x}} \]
4. Step-by-step derivation

   1. flip-- 9.3%

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

   2. add-sqr-sqrt 9.4%

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

   3. add-sqr-sqrt 9.5%

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

   4. associate--r- 21.7%

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

   5. sub-neg 21.7%

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

   6. metadata-eval 21.7%

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

   7. +-commutative 21.7%

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

   8. add-exp-log 21.7%

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

   9. log1p-udef 21.7%

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

   10. expm1-udef 100.0%

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

   11. expm1-log1p-u 100.0%

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

   12. sub-neg 100.0%

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

   13. metadata-eval 100.0%

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

5. Applied egg-rr 100.0%

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

6. Taylor expanded in x around 0 99.0%

   \[\leadsto \frac{x + x}{\color{blue}{2 + -0.25 \cdot {x}^{2}}} \]
7. Step-by-step derivation

   1. add-log-exp 99.0%

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

   2. *-un-lft-identity 99.0%

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

   3. log-prod 99.0%

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

   4. metadata-eval 99.0%

      \[\leadsto \frac{x + x}{2 + \left(\color{blue}{0} + \log \left(e^{-0.25 \cdot {x}^{2}}\right)\right)} \]

   5. add-log-exp 99.0%

      \[\leadsto \frac{x + x}{2 + \left(0 + \color{blue}{-0.25 \cdot {x}^{2}}\right)} \]

   6. *-commutative 99.0%

      \[\leadsto \frac{x + x}{2 + \left(0 + \color{blue}{{x}^{2} \cdot -0.25}\right)} \]

   7. unpow2 99.0%

      \[\leadsto \frac{x + x}{2 + \left(0 + \color{blue}{\left(x \cdot x\right)} \cdot -0.25\right)} \]

   8. associate-*l* 99.0%

      \[\leadsto \frac{x + x}{2 + \left(0 + \color{blue}{x \cdot \left(x \cdot -0.25\right)}\right)} \]

8. Applied egg-rr 99.0%

   \[\leadsto \frac{x + x}{2 + \color{blue}{\left(0 + x \cdot \left(x \cdot -0.25\right)\right)}} \]
9. Step-by-step derivation

   1. +-lft-identity 99.0%

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

10. Simplified 99.0%

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

11. Final simplification 99.0%

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

Alternative 4: 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 9.3%

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

   1. +-commutative 9.3%

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

   2. metadata-eval 9.3%

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

   3. sub-neg 9.3%

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

3. Simplified 9.3%

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

4. Taylor expanded in x around 0 98.6%

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

5. Final simplification 98.6%

   \[\leadsto x \]

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 2023227 
(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))))