bug333 (missed optimization)

Percentage Accurate: 8.5% → 100.0%

Time: 7.8s

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 8.5%

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

   1. flip-- 8.5%

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

   2. add-sqr-sqrt 8.4%

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

   3. add-sqr-sqrt 8.6%

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

   4. associate--r- 20.9%

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

   5. add-exp-log 20.9%

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

   6. expm1-undefine 20.9%

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

   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: 99.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{x + x}{\left(1 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.0625 - 0.125\right)\right)\right) + \sqrt{\left(2 - x\right) + -1}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (+ x x)
  (+
   (+ 1.0 (* x (+ 0.5 (* x (- (* x 0.0625) 0.125)))))
   (sqrt (+ (- 2.0 x) -1.0)))))

C

double code(double x) {
	return (x + x) / ((1.0 + (x * (0.5 + (x * ((x * 0.0625) - 0.125))))) + sqrt(((2.0 - x) + -1.0)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x + x) / ((1.0d0 + (x * (0.5d0 + (x * ((x * 0.0625d0) - 0.125d0))))) + sqrt(((2.0d0 - x) + (-1.0d0))))
end function

Java

public static double code(double x) {
	return (x + x) / ((1.0 + (x * (0.5 + (x * ((x * 0.0625) - 0.125))))) + Math.sqrt(((2.0 - x) + -1.0)));
}

Python

def code(x):
	return (x + x) / ((1.0 + (x * (0.5 + (x * ((x * 0.0625) - 0.125))))) + math.sqrt(((2.0 - x) + -1.0)))

Julia

function code(x)
	return Float64(Float64(x + x) / Float64(Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * Float64(Float64(x * 0.0625) - 0.125))))) + sqrt(Float64(Float64(2.0 - x) + -1.0))))
end

MATLAB

function tmp = code(x)
	tmp = (x + x) / ((1.0 + (x * (0.5 + (x * ((x * 0.0625) - 0.125))))) + sqrt(((2.0 - x) + -1.0)));
end

Wolfram

code[x_] := N[(N[(x + x), $MachinePrecision] / N[(N[(1.0 + N[(x * N[(0.5 + N[(x * N[(N[(x * 0.0625), $MachinePrecision] - 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[(N[(2.0 - x), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.5%

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

   1. flip-- 8.5%

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

   2. add-sqr-sqrt 8.4%

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

   3. add-sqr-sqrt 8.6%

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

   4. associate--r- 20.9%

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

   5. add-exp-log 20.9%

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

   6. expm1-undefine 20.9%

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

   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 99.8%

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

   1. expm1-log1p-u 99.8%

      \[\leadsto \frac{x + x}{\left(1 + x \cdot \left(0.5 + x \cdot \left(0.0625 \cdot x - 0.125\right)\right)\right) + \sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - x\right)\right)}}} \]

7. Applied egg-rr 99.8%

   \[\leadsto \frac{x + x}{\left(1 + x \cdot \left(0.5 + x \cdot \left(0.0625 \cdot x - 0.125\right)\right)\right) + \sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - x\right)\right)}}} \]
8. Step-by-step derivation

   1. expm1-undefine 99.8%

      \[\leadsto \frac{x + x}{\left(1 + x \cdot \left(0.5 + x \cdot \left(0.0625 \cdot x - 0.125\right)\right)\right) + \sqrt{\color{blue}{e^{\mathsf{log1p}\left(1 - x\right)} - 1}}} \]

   2. sub-neg 99.8%

      \[\leadsto \frac{x + x}{\left(1 + x \cdot \left(0.5 + x \cdot \left(0.0625 \cdot x - 0.125\right)\right)\right) + \sqrt{\color{blue}{e^{\mathsf{log1p}\left(1 - x\right)} + \left(-1\right)}}} \]

   3. log1p-undefine 99.8%

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

   4. rem-exp-log 99.8%

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

   5. associate-+r- 99.8%

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

   6. metadata-eval 99.8%

      \[\leadsto \frac{x + x}{\left(1 + x \cdot \left(0.5 + x \cdot \left(0.0625 \cdot x - 0.125\right)\right)\right) + \sqrt{\left(\color{blue}{2} - x\right) + \left(-1\right)}} \]

   7. metadata-eval 99.8%

      \[\leadsto \frac{x + x}{\left(1 + x \cdot \left(0.5 + x \cdot \left(0.0625 \cdot x - 0.125\right)\right)\right) + \sqrt{\left(2 - x\right) + \color{blue}{-1}}} \]

9. Simplified 99.8%

   \[\leadsto \frac{x + x}{\left(1 + x \cdot \left(0.5 + x \cdot \left(0.0625 \cdot x - 0.125\right)\right)\right) + \sqrt{\color{blue}{\left(2 - x\right) + -1}}} \]

10. Final simplification 99.8%

    \[\leadsto \frac{x + x}{\left(1 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.0625 - 0.125\right)\right)\right) + \sqrt{\left(2 - x\right) + -1}} \]
11. 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 8.5%

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

   1. flip-- 8.5%

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

   2. add-sqr-sqrt 8.4%

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

   3. add-sqr-sqrt 8.6%

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

   4. associate--r- 20.9%

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

   5. add-exp-log 20.9%

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

   6. expm1-undefine 20.9%

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

   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 99.8%

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

6. Final simplification 99.8%

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

Alternative 4: 99.6% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Python

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

Wolfram

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

Derivation

1. Initial program 8.5%

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

   1. flip-- 8.5%

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

   2. add-sqr-sqrt 8.4%

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

   3. add-sqr-sqrt 8.6%

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

   4. associate--r- 20.9%

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

   5. add-exp-log 20.9%

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

   6. expm1-undefine 20.9%

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

   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 99.8%

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

6. Taylor expanded in x around 0 99.6%

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

7. Taylor expanded in x around 0 99.8%

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

   1. +-commutative 99.8%

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

   2. *-commutative 99.8%

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

9. Simplified 99.8%

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

10. Final simplification 99.8%

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

Alternative 5: 99.3% accurate, 18.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.5%

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

   1. flip-- 8.5%

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

   2. clear-num 8.5%

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

   3. add-sqr-sqrt 8.4%

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

   4. add-sqr-sqrt 8.6%

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

   5. associate--r- 20.9%

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

   6. add-exp-log 20.9%

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

   7. expm1-undefine 20.9%

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

   8. log1p-define 99.8%

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

   9. expm1-log1p-u 99.8%

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in x around 0 99.6%

   \[\leadsto \frac{1}{\color{blue}{\frac{1 + -0.125 \cdot {x}^{2}}{x}}} \]
6. Step-by-step derivation

   1. unpow2 99.6%

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

7. Applied egg-rr 99.6%

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

Alternative 6: 99.3% accurate, 23.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x \cdot -0.125 + \frac{1}{x}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.5%

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

   1. flip-- 8.5%

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

   2. clear-num 8.5%

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

   3. add-sqr-sqrt 8.4%

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

   4. add-sqr-sqrt 8.6%

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

   5. associate--r- 20.9%

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

   6. add-exp-log 20.9%

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

   7. expm1-undefine 20.9%

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

   8. log1p-define 99.8%

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

   9. expm1-log1p-u 99.8%

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in x around 0 99.6%

   \[\leadsto \frac{1}{\color{blue}{\frac{1 + -0.125 \cdot {x}^{2}}{x}}} \]

6. Taylor expanded in x around inf 55.9%

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

   1. sub-neg 55.9%

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

   2. unpow2 55.9%

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

   3. associate-/r* 56.0%

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

   4. *-lft-identity 56.0%

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

   5. associate-*l/ 55.8%

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

   6. unpow-1 55.8%

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

   7. unpow-1 55.8%

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

   8. pow-sqr 56.0%

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

   9. metadata-eval 56.0%

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

   10. metadata-eval 56.0%

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

8. Simplified 56.0%

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

   1. *-un-lft-identity 56.0%

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

   2. distribute-lft-in 56.0%

      \[\leadsto 1 \cdot \frac{1}{\color{blue}{x \cdot {x}^{-2} + x \cdot -0.125}} \]

   3. pow1 56.0%

      \[\leadsto 1 \cdot \frac{1}{\color{blue}{{x}^{1}} \cdot {x}^{-2} + x \cdot -0.125} \]

   4. pow-prod-up 99.6%

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

   5. metadata-eval 99.6%

      \[\leadsto 1 \cdot \frac{1}{{x}^{\color{blue}{-1}} + x \cdot -0.125} \]

   6. inv-pow 99.6%

      \[\leadsto 1 \cdot \frac{1}{\color{blue}{\frac{1}{x}} + x \cdot -0.125} \]

10. Applied egg-rr 99.6%

    \[\leadsto \color{blue}{1 \cdot \frac{1}{\frac{1}{x} + x \cdot -0.125}} \]
11. Step-by-step derivation

    1. *-lft-identity 99.6%

       \[\leadsto \color{blue}{\frac{1}{\frac{1}{x} + x \cdot -0.125}} \]

12. Simplified 99.6%

    \[\leadsto \color{blue}{\frac{1}{\frac{1}{x} + x \cdot -0.125}} \]

13. Final simplification 99.6%

    \[\leadsto \frac{1}{x \cdot -0.125 + \frac{1}{x}} \]
14. Add Preprocessing

Alternative 7: 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 8.5%

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

3. Taylor expanded in x around 0 99.1%

   \[\leadsto \color{blue}{x} \]
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 2024150 
(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))))