bug333 (missed optimization)

Percentage Accurate: 8.5% → 100.0%

Time: 9.0s

Alternatives: 7

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: 8.5% 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 9.5%

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

   1. flip-- 9.5%

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

   2. add-sqr-sqrt 9.5%

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

   3. add-sqr-sqrt 9.6%

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

   4. associate--r- 22.1%

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

   5. add-exp-log 22.1%

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

   6. expm1-undefine 22.1%

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

   7. log1p-define 100.0%

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

   8. expm1-log1p-u 100.0%

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

4. Applied egg-rr 100.0%

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

5. Final simplification 100.0%

   \[\leadsto \frac{x + x}{\sqrt{x + 1} + \sqrt{1 - x}} \]
6. Add Preprocessing

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 9.5%

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

   1. flip-- 9.5%

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

   2. clear-num 9.5%

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

   3. add-sqr-sqrt 9.5%

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

   4. add-sqr-sqrt 9.6%

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

   5. associate--r- 22.1%

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

   6. add-exp-log 22.1%

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

   7. expm1-undefine 22.1%

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

   8. log1p-define 99.7%

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

   9. expm1-log1p-u 99.7%

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

4. Applied egg-rr 99.7%

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

   1. *-un-lft-identity 99.7%

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

   2. associate-/r/ 100.0%

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

   3. add-sqr-sqrt 51.2%

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

   4. hypot-1-def 51.2%

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

   5. count-2 51.2%

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

6. Applied egg-rr 51.2%

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

   1. *-lft-identity 51.2%

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

   2. associate-*r* 51.2%

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

   3. *-commutative 51.2%

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

   4. associate-*l/ 51.2%

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

   5. metadata-eval 51.2%

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

   6. hypot-undefine 51.2%

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

   7. metadata-eval 51.2%

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

   8. rem-square-sqrt 100.0%

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

   9. +-commutative 100.0%

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

8. Simplified 100.0%

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

Alternative 3: 99.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{x + x}{\sqrt{1 - x} + \left(1 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.0625 - 0.125\right)\right)\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (+ x x)
  (+ (sqrt (- 1.0 x)) (+ 1.0 (* x (+ 0.5 (* x (- (* x 0.0625) 0.125))))))))

C

double code(double x) {
	return (x + x) / (sqrt((1.0 - x)) + (1.0 + (x * (0.5 + (x * ((x * 0.0625) - 0.125))))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x + x) / (sqrt((1.0d0 - x)) + (1.0d0 + (x * (0.5d0 + (x * ((x * 0.0625d0) - 0.125d0))))))
end function

Java

public static double code(double x) {
	return (x + x) / (Math.sqrt((1.0 - x)) + (1.0 + (x * (0.5 + (x * ((x * 0.0625) - 0.125))))));
}

Python

def code(x):
	return (x + x) / (math.sqrt((1.0 - x)) + (1.0 + (x * (0.5 + (x * ((x * 0.0625) - 0.125))))))

Julia

function code(x)
	return Float64(Float64(x + x) / Float64(sqrt(Float64(1.0 - x)) + Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * Float64(Float64(x * 0.0625) - 0.125)))))))
end

MATLAB

function tmp = code(x)
	tmp = (x + x) / (sqrt((1.0 - x)) + (1.0 + (x * (0.5 + (x * ((x * 0.0625) - 0.125))))));
end

Wolfram

code[x_] := N[(N[(x + x), $MachinePrecision] / N[(N[Sqrt[N[(1.0 - x), $MachinePrecision]], $MachinePrecision] + N[(1.0 + N[(x * N[(0.5 + N[(x * N[(N[(x * 0.0625), $MachinePrecision] - 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 9.5%

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

   1. flip-- 9.5%

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

   2. add-sqr-sqrt 9.5%

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

   3. add-sqr-sqrt 9.6%

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

   4. associate--r- 22.1%

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

   5. add-exp-log 22.1%

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

   6. expm1-undefine 22.1%

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

   7. log1p-define 100.0%

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

   8. expm1-log1p-u 100.0%

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0 99.3%

   \[\leadsto \frac{x + x}{\color{blue}{\left(1 + x \cdot \left(0.5 + x \cdot \left(0.0625 \cdot x - 0.125\right)\right)\right)} + \sqrt{1 - x}} \]

6. Final simplification 99.3%

   \[\leadsto \frac{x + x}{\sqrt{1 - x} + \left(1 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.0625 - 0.125\right)\right)\right)} \]
7. Add Preprocessing

Alternative 4: 99.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 + {x}^{2} \cdot \left(0.125 + x \cdot \left(x \cdot 0.02734375\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* x (+ 1.0 (* (pow x 2.0) (+ 0.125 (* x (* x 0.02734375)))))))

C

double code(double x) {
	return x * (1.0 + (pow(x, 2.0) * (0.125 + (x * (x * 0.02734375)))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (1.0d0 + ((x ** 2.0d0) * (0.125d0 + (x * (x * 0.02734375d0)))))
end function

Java

public static double code(double x) {
	return x * (1.0 + (Math.pow(x, 2.0) * (0.125 + (x * (x * 0.02734375)))));
}

Python

def code(x):
	return x * (1.0 + (math.pow(x, 2.0) * (0.125 + (x * (x * 0.02734375)))))

Julia

function code(x)
	return Float64(x * Float64(1.0 + Float64((x ^ 2.0) * Float64(0.125 + Float64(x * Float64(x * 0.02734375))))))
end

MATLAB

function tmp = code(x)
	tmp = x * (1.0 + ((x ^ 2.0) * (0.125 + (x * (x * 0.02734375)))));
end

Wolfram

code[x_] := N[(x * N[(1.0 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(0.125 + N[(x * N[(x * 0.02734375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 9.5%

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

3. Taylor expanded in x around 0 8.7%

   \[\leadsto \color{blue}{\left(1 + x \cdot \left(0.5 + x \cdot \left(0.0625 \cdot x - 0.125\right)\right)\right)} - \sqrt{1 - x} \]

4. Taylor expanded in x around 0 99.1%

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(0.125 + x \cdot \left(0.0390625 + 0.02734375 \cdot x\right)\right)\right)} \]

5. Taylor expanded in x around inf 99.3%

   \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(0.125 + x \cdot \color{blue}{\left(0.02734375 \cdot x\right)}\right)\right) \]
6. Step-by-step derivation

   1. *-commutative 99.3%

      \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(0.125 + x \cdot \color{blue}{\left(x \cdot 0.02734375\right)}\right)\right) \]

7. Simplified 99.3%

   \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(0.125 + x \cdot \color{blue}{\left(x \cdot 0.02734375\right)}\right)\right) \]
8. Add Preprocessing

Alternative 5: 99.5% accurate, 18.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 9.5%

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

   1. flip-- 9.5%

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

   2. add-sqr-sqrt 9.5%

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

   3. add-sqr-sqrt 9.6%

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

   4. associate--r- 22.1%

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

   5. add-exp-log 22.1%

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

   6. expm1-undefine 22.1%

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

   7. log1p-define 100.0%

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

   8. expm1-log1p-u 100.0%

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0 99.3%

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

   1. unpow2 99.3%

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

7. Applied egg-rr 99.3%

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

Alternative 6: 99.5% accurate, 23.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (* x (+ 1.0 (* 0.125 (* x x)))))

C

double code(double x) {
	return x * (1.0 + (0.125 * (x * x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (1.0d0 + (0.125d0 * (x * x)))
end function

Java

public static double code(double x) {
	return x * (1.0 + (0.125 * (x * x)));
}

Python

def code(x):
	return x * (1.0 + (0.125 * (x * x)))

Julia

function code(x)
	return Float64(x * Float64(1.0 + Float64(0.125 * Float64(x * x))))
end

MATLAB

function tmp = code(x)
	tmp = x * (1.0 + (0.125 * (x * x)));
end

Wolfram

code[x_] := N[(x * N[(1.0 + N[(0.125 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 9.5%

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

3. Taylor expanded in x around 0 99.3%

   \[\leadsto \color{blue}{x \cdot \left(1 + 0.125 \cdot {x}^{2}\right)} \]
4. Step-by-step derivation

   1. unpow2 99.3%

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

5. Applied egg-rr 99.3%

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

Alternative 7: 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 9.5%

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

3. Taylor expanded in x around 0 98.8%

   \[\leadsto \color{blue}{x} \]
4. 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 2024144 
(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))))