bug333 (missed optimization)

Percentage Accurate: 8.2% → 100.0%

Time: 7.5s

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: 8.2% 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

\[\begin{array}{l} \\ 0.5 \cdot \frac{x \cdot 4}{\sqrt{x + 1} + \sqrt{1 - x}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* 0.5 (/ (* x 4.0) (+ (sqrt (+ x 1.0)) (sqrt (- 1.0 x))))))

C

double code(double x) {
	return 0.5 * ((x * 4.0) / (sqrt((x + 1.0)) + sqrt((1.0 - x))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.5d0 * ((x * 4.0d0) / (sqrt((x + 1.0d0)) + sqrt((1.0d0 - x))))
end function

Java

public static double code(double x) {
	return 0.5 * ((x * 4.0) / (Math.sqrt((x + 1.0)) + Math.sqrt((1.0 - x))));
}

Python

def code(x):
	return 0.5 * ((x * 4.0) / (math.sqrt((x + 1.0)) + math.sqrt((1.0 - x))))

Julia

function code(x)
	return Float64(0.5 * Float64(Float64(x * 4.0) / Float64(sqrt(Float64(x + 1.0)) + sqrt(Float64(1.0 - x)))))
end

MATLAB

function tmp = code(x)
	tmp = 0.5 * ((x * 4.0) / (sqrt((x + 1.0)) + sqrt((1.0 - x))));
end

Wolfram

code[x_] := N[(0.5 * N[(N[(x * 4.0), $MachinePrecision] / N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 9.0%

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

   1. flip-- 9.0%

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

   2. flip-- 9.0%

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

   3. associate-/l/ 9.0%

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

3. Applied egg-rr 9.1%

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

   1. *-rgt-identity 9.1%

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

   2. *-commutative 9.1%

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

   3. *-commutative 9.1%

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

   4. times-frac 9.1%

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

   5. +-commutative 9.1%

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

   6. associate-+l- 9.1%

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

   7. +-inverses 9.1%

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

   8. metadata-eval 9.1%

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

   9. metadata-eval 9.1%

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

   10. metadata-eval 9.1%

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

5. Simplified 100.0%

   \[\leadsto \color{blue}{0.5 \cdot \frac{2 \cdot \left(x + x\right)}{\sqrt{x + 1} + \sqrt{1 - x}}} \]
6. Step-by-step derivation

   1. add-sqr-sqrt_binary64 49.2%

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

7. Applied rewrite-once 49.2%

   \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{\frac{2 \cdot \left(x + x\right)}{\sqrt{x + 1} + \sqrt{1 - x}}} \cdot \sqrt{\frac{2 \cdot \left(x + x\right)}{\sqrt{x + 1} + \sqrt{1 - x}}}\right)} \]
8. Step-by-step derivation

   1. rem-square-sqrt 100.0%

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

   2. distribute-rgt-in 100.0%

      \[\leadsto 0.5 \cdot \frac{\color{blue}{x \cdot 2 + x \cdot 2}}{\sqrt{x + 1} + \sqrt{1 - x}} \]

   3. distribute-lft-out 100.0%

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

   4. metadata-eval 100.0%

      \[\leadsto 0.5 \cdot \frac{x \cdot \color{blue}{4}}{\sqrt{x + 1} + \sqrt{1 - x}} \]

9. Simplified 100.0%

   \[\leadsto 0.5 \cdot \color{blue}{\frac{x \cdot 4}{\sqrt{x + 1} + \sqrt{1 - x}}} \]

10. Final simplification 100.0%

    \[\leadsto 0.5 \cdot \frac{x \cdot 4}{\sqrt{x + 1} + \sqrt{1 - x}} \]

Alternative 2: 99.6% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \frac{x \cdot 4}{2 + -0.25 \cdot \left(x \cdot x\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* 0.5 (/ (* x 4.0) (+ 2.0 (* -0.25 (* x x))))))

C

double code(double x) {
	return 0.5 * ((x * 4.0) / (2.0 + (-0.25 * (x * x))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.5d0 * ((x * 4.0d0) / (2.0d0 + ((-0.25d0) * (x * x))))
end function

Java

public static double code(double x) {
	return 0.5 * ((x * 4.0) / (2.0 + (-0.25 * (x * x))));
}

Python

def code(x):
	return 0.5 * ((x * 4.0) / (2.0 + (-0.25 * (x * x))))

Julia

function code(x)
	return Float64(0.5 * Float64(Float64(x * 4.0) / Float64(2.0 + Float64(-0.25 * Float64(x * x)))))
end

MATLAB

function tmp = code(x)
	tmp = 0.5 * ((x * 4.0) / (2.0 + (-0.25 * (x * x))));
end

Wolfram

code[x_] := N[(0.5 * N[(N[(x * 4.0), $MachinePrecision] / N[(2.0 + N[(-0.25 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 9.0%

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

   1. flip-- 9.0%

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

   2. flip-- 9.0%

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

   3. associate-/l/ 9.0%

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

3. Applied egg-rr 9.1%

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

   1. *-rgt-identity 9.1%

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

   2. *-commutative 9.1%

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

   3. *-commutative 9.1%

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

   4. times-frac 9.1%

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

   5. +-commutative 9.1%

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

   6. associate-+l- 9.1%

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

   7. +-inverses 9.1%

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

   8. metadata-eval 9.1%

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

   9. metadata-eval 9.1%

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

   10. metadata-eval 9.1%

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

5. Simplified 100.0%

   \[\leadsto \color{blue}{0.5 \cdot \frac{2 \cdot \left(x + x\right)}{\sqrt{x + 1} + \sqrt{1 - x}}} \]
6. Step-by-step derivation

   1. add-sqr-sqrt_binary64 49.2%

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

7. Applied rewrite-once 49.2%

   \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{\frac{2 \cdot \left(x + x\right)}{\sqrt{x + 1} + \sqrt{1 - x}}} \cdot \sqrt{\frac{2 \cdot \left(x + x\right)}{\sqrt{x + 1} + \sqrt{1 - x}}}\right)} \]
8. Step-by-step derivation

   1. rem-square-sqrt 100.0%

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

   2. distribute-rgt-in 100.0%

      \[\leadsto 0.5 \cdot \frac{\color{blue}{x \cdot 2 + x \cdot 2}}{\sqrt{x + 1} + \sqrt{1 - x}} \]

   3. distribute-lft-out 100.0%

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

   4. metadata-eval 100.0%

      \[\leadsto 0.5 \cdot \frac{x \cdot \color{blue}{4}}{\sqrt{x + 1} + \sqrt{1 - x}} \]

9. Simplified 100.0%

   \[\leadsto 0.5 \cdot \color{blue}{\frac{x \cdot 4}{\sqrt{x + 1} + \sqrt{1 - x}}} \]

10. Taylor expanded in x around 0 99.1%

    \[\leadsto 0.5 \cdot \frac{x \cdot 4}{\color{blue}{2 + -0.25 \cdot {x}^{2}}} \]
11. Step-by-step derivation

    1. unpow2 99.1%

       \[\leadsto 0.5 \cdot \frac{x \cdot 4}{2 + -0.25 \cdot \color{blue}{\left(x \cdot x\right)}} \]

12. Simplified 99.1%

    \[\leadsto 0.5 \cdot \frac{x \cdot 4}{\color{blue}{2 + -0.25 \cdot \left(x \cdot x\right)}} \]

13. Final simplification 99.1%

    \[\leadsto 0.5 \cdot \frac{x \cdot 4}{2 + -0.25 \cdot \left(x \cdot x\right)} \]

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 9.0%

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

2. Taylor expanded in x around 0 98.7%

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

3. Final simplification 98.7%

   \[\leadsto x \]

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 2023297 
(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))))