bug333 (missed optimization)

Percentage Accurate: 8.4% → 99.8%

Time: 5.6s

Alternatives: 3

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: 8.4% 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: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \left(0.0546875 \cdot {x}^{5} + 0.125 \cdot {x}^{3}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ x (+ (* 0.0546875 (pow x 5.0)) (* 0.125 (pow x 3.0)))))

C

double code(double x) {
	return x + ((0.0546875 * pow(x, 5.0)) + (0.125 * pow(x, 3.0)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x + ((0.0546875d0 * (x ** 5.0d0)) + (0.125d0 * (x ** 3.0d0)))
end function

Java

public static double code(double x) {
	return x + ((0.0546875 * Math.pow(x, 5.0)) + (0.125 * Math.pow(x, 3.0)));
}

Python

def code(x):
	return x + ((0.0546875 * math.pow(x, 5.0)) + (0.125 * math.pow(x, 3.0)))

Julia

function code(x)
	return Float64(x + Float64(Float64(0.0546875 * (x ^ 5.0)) + Float64(0.125 * (x ^ 3.0))))
end

MATLAB

function tmp = code(x)
	tmp = x + ((0.0546875 * (x ^ 5.0)) + (0.125 * (x ^ 3.0)));
end

Wolfram

code[x_] := N[(x + N[(N[(0.0546875 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(0.125 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.0%

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

2. Taylor expanded in x around 0 100.0%

   \[\leadsto \color{blue}{x + \left(0.0546875 \cdot {x}^{5} + 0.125 \cdot {x}^{3}\right)} \]

3. Final simplification 100.0%

   \[\leadsto x + \left(0.0546875 \cdot {x}^{5} + 0.125 \cdot {x}^{3}\right) \]

Alternative 2: 99.6% accurate, 23.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.0%

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

2. Taylor expanded in x around 0 99.9%

   \[\leadsto \color{blue}{x + 0.125 \cdot {x}^{3}} \]
3. Step-by-step derivation

   1. expm1-log1p-u 99.9%

      \[\leadsto x + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.125 \cdot {x}^{3}\right)\right)} \]

   2. expm1-udef 99.8%

      \[\leadsto x + \color{blue}{\left(e^{\mathsf{log1p}\left(0.125 \cdot {x}^{3}\right)} - 1\right)} \]

   3. log1p-udef 99.8%

      \[\leadsto x + \left(e^{\color{blue}{\log \left(1 + 0.125 \cdot {x}^{3}\right)}} - 1\right) \]

   4. add-exp-log 99.8%

      \[\leadsto x + \left(\color{blue}{\left(1 + 0.125 \cdot {x}^{3}\right)} - 1\right) \]

4. Applied egg-rr 99.8%

   \[\leadsto x + \color{blue}{\left(\left(1 + 0.125 \cdot {x}^{3}\right) - 1\right)} \]
5. Step-by-step derivation

   1. add-exp-log 99.8%

      \[\leadsto x + \left(\color{blue}{e^{\log \left(1 + 0.125 \cdot {x}^{3}\right)}} - 1\right) \]

   2. log1p-udef 99.8%

      \[\leadsto x + \left(e^{\color{blue}{\mathsf{log1p}\left(0.125 \cdot {x}^{3}\right)}} - 1\right) \]

   3. expm1-udef 99.9%

      \[\leadsto x + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.125 \cdot {x}^{3}\right)\right)} \]

   4. expm1-log1p-u 99.9%

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

   5. *-commutative 99.9%

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

   6. unpow3 99.9%

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

   7. associate-*l* 99.9%

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

   \[\leadsto x + \left(x \cdot x\right) \cdot \left(x \cdot 0.125\right) \]

Alternative 3: 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 8.0%

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

2. Taylor expanded in x around 0 99.6%

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

3. Final simplification 99.6%

   \[\leadsto x \]

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 2023285 
(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))))