bug333 (missed optimization)

Percentage Accurate: 8.3% → 100.0%

Time: 8.5s

Alternatives: 8

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 9.1%

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

   1. flip-- 9.1%

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

   2. div-inv 9.1%

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

   3. add-sqr-sqrt 9.1%

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

   4. add-sqr-sqrt 9.2%

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

   5. associate--r- 21.6%

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

   6. add-exp-log 21.6%

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

   7. expm1-undefine 21.6%

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

   8. log1p-define 100.0%

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

   9. expm1-log1p-u 100.0%

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

4. Applied egg-rr 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 99.6% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 9.1%

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

   1. flip-- 9.1%

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

   2. div-inv 9.1%

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

   3. add-sqr-sqrt 9.1%

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

   4. add-sqr-sqrt 9.2%

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

   5. associate--r- 21.6%

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

   6. add-exp-log 21.6%

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

   7. expm1-undefine 21.6%

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

   8. log1p-define 100.0%

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

   9. expm1-log1p-u 100.0%

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0 99.8%

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

6. Final simplification 99.8%

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

Alternative 3: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 9.1%

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

3. Taylor expanded in x around 0 99.8%

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

   1. distribute-rgt-in 99.8%

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

   2. *-lft-identity 99.8%

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

   3. associate-*l* 99.8%

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

   4. unpow2 99.8%

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

   5. unpow3 99.8%

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

5. Simplified 99.8%

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

6. Final simplification 99.8%

   \[\leadsto x + 0.125 \cdot {x}^{3} \]
7. Add Preprocessing

Alternative 4: 99.6% accurate, 6.3× speedup

Math

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

FPCore

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

C

double code(double x) {
	return (x + x) * (1.0 / ((1.0 + (x * ((x * ((x * -0.0625) - 0.125)) - 0.5))) + (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) * (1.0d0 / ((1.0d0 + (x * ((x * ((x * (-0.0625d0)) - 0.125d0)) - 0.5d0))) + (1.0d0 + (x * (0.5d0 + (x * ((x * 0.0625d0) - 0.125d0)))))))
end function

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 9.1%

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

   1. flip-- 9.1%

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

   2. div-inv 9.1%

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

   3. add-sqr-sqrt 9.1%

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

   4. add-sqr-sqrt 9.2%

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

   5. associate--r- 21.6%

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

   6. add-exp-log 21.6%

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

   7. expm1-undefine 21.6%

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

   8. log1p-define 100.0%

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

   9. expm1-log1p-u 100.0%

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0 99.8%

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

6. Taylor expanded in x around 0 99.8%

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

7. Final simplification 99.8%

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

Alternative 5: 99.1% accurate, 8.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 9.1%

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

   1. flip-- 9.1%

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

   2. div-inv 9.1%

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

   3. add-sqr-sqrt 9.1%

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

   4. add-sqr-sqrt 9.2%

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

   5. associate--r- 21.6%

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

   6. add-exp-log 21.6%

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

   7. expm1-undefine 21.6%

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

   8. log1p-define 100.0%

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

   9. expm1-log1p-u 100.0%

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0 99.8%

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

6. Taylor expanded in x around 0 99.3%

   \[\leadsto \left(x + x\right) \cdot \frac{1}{\color{blue}{\left(1 + 0.5 \cdot x\right)} + \left(1 + x \cdot \left(x \cdot \left(-0.0625 \cdot x - 0.125\right) - 0.5\right)\right)} \]
7. Step-by-step derivation

   1. *-commutative 99.3%

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

8. Simplified 99.3%

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

9. Final simplification 99.3%

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

Alternative 6: 99.6% accurate, 8.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return (x + x) * (1.0 / ((1.0 + (x * (0.5 + (x * -0.125)))) + (1.0 + (x * ((x * -0.125) - 0.5)))))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 9.1%

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

   1. flip-- 9.1%

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

   2. div-inv 9.1%

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

   3. add-sqr-sqrt 9.1%

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

   4. add-sqr-sqrt 9.2%

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

   5. associate--r- 21.6%

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

   6. add-exp-log 21.6%

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

   7. expm1-undefine 21.6%

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

   8. log1p-define 100.0%

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

   9. expm1-log1p-u 100.0%

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0 99.8%

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

6. Taylor expanded in x around 0 99.6%

   \[\leadsto \left(x + x\right) \cdot \frac{1}{\color{blue}{\left(1 + x \cdot \left(0.5 + -0.125 \cdot x\right)\right)} + \left(1 + x \cdot \left(x \cdot \left(-0.0625 \cdot x - 0.125\right) - 0.5\right)\right)} \]
7. Step-by-step derivation

   1. *-commutative 99.6%

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

8. Simplified 99.6%

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

9. Taylor expanded in x around 0 99.8%

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

10. Final simplification 99.8%

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

Alternative 7: 99.1% accurate, 9.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return (x + x) * (1.0 / ((1.0 + (x * (0.5 + (x * -0.125)))) + (1.0 + (x * -0.5))))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 9.1%

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

   1. flip-- 9.1%

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

   2. div-inv 9.1%

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

   3. add-sqr-sqrt 9.1%

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

   4. add-sqr-sqrt 9.2%

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

   5. associate--r- 21.6%

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

   6. add-exp-log 21.6%

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

   7. expm1-undefine 21.6%

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

   8. log1p-define 100.0%

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

   9. expm1-log1p-u 100.0%

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0 99.8%

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

6. Taylor expanded in x around 0 99.6%

   \[\leadsto \left(x + x\right) \cdot \frac{1}{\color{blue}{\left(1 + x \cdot \left(0.5 + -0.125 \cdot x\right)\right)} + \left(1 + x \cdot \left(x \cdot \left(-0.0625 \cdot x - 0.125\right) - 0.5\right)\right)} \]
7. Step-by-step derivation

   1. *-commutative 99.6%

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

8. Simplified 99.6%

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

9. Taylor expanded in x around 0 99.3%

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

10. Final simplification 99.3%

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

Alternative 8: 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 9.1%

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

3. Taylor expanded in x around 0 99.2%

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

4. Final simplification 99.2%

   \[\leadsto x \]
5. Add Preprocessing

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 2024084 
(FPCore (x)
  :name "bug333 (missed optimization)"
  :precision binary64
  :pre (and (<= -1.0 x) (<= x 1.0))

  :alt
  (/ (* 2.0 x) (+ (sqrt (+ 1.0 x)) (sqrt (- 1.0 x))))

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