bug333 (missed optimization)

Percentage Accurate: 8.2% → 99.8%

Time: 6.5s

Alternatives: 4

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: 99.8% accurate, 13.8× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.125 + \left(x \cdot x\right) \cdot 0.0546875\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* x (+ 1.0 (* (* x x) (+ 0.125 (* (* x x) 0.0546875))))))

C

double code(double x) {
	return x * (1.0 + ((x * x) * (0.125 + ((x * x) * 0.0546875))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (1.0d0 + ((x * x) * (0.125d0 + ((x * x) * 0.0546875d0))))
end function

Java

public static double code(double x) {
	return x * (1.0 + ((x * x) * (0.125 + ((x * x) * 0.0546875))));
}

Python

def code(x):
	return x * (1.0 + ((x * x) * (0.125 + ((x * x) * 0.0546875))))

Julia

function code(x)
	return Float64(x * Float64(1.0 + Float64(Float64(x * x) * Float64(0.125 + Float64(Float64(x * x) * 0.0546875)))))
end

MATLAB

function tmp = code(x)
	tmp = x * (1.0 + ((x * x) * (0.125 + ((x * x) * 0.0546875))));
end

Wolfram

code[x_] := N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(0.125 + N[(N[(x * x), $MachinePrecision] * 0.0546875), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 9.1%

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

3. Taylor expanded in x around 0 100.0%

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(0.125 + 0.0546875 \cdot {x}^{2}\right)\right)} \]
4. Step-by-step derivation

   1. *-commutative 100.0%

      \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(0.125 + \color{blue}{{x}^{2} \cdot 0.0546875}\right)\right) \]

5. Simplified 100.0%

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(0.125 + {x}^{2} \cdot 0.0546875\right)\right)} \]
6. Step-by-step derivation

   1. unpow2 100.0%

      \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(0.125 + \color{blue}{\left(x \cdot x\right)} \cdot 0.0546875\right)\right) \]

7. Applied egg-rr 100.0%

   \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(0.125 + \color{blue}{\left(x \cdot x\right)} \cdot 0.0546875\right)\right) \]
8. Step-by-step derivation

   1. unpow2 100.0%

      \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(0.125 + \color{blue}{\left(x \cdot x\right)} \cdot 0.0546875\right)\right) \]

9. Applied egg-rr 100.0%

   \[\leadsto x \cdot \left(1 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.125 + \left(x \cdot x\right) \cdot 0.0546875\right)\right) \]

10. Final simplification 100.0%

    \[\leadsto x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.125 + \left(x \cdot x\right) \cdot 0.0546875\right)\right) \]
11. Add Preprocessing

Alternative 2: 99.7% accurate, 18.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(x + x), $MachinePrecision] / N[(2.0 + N[(N[(x * x), $MachinePrecision] * -0.25), $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. add-sqr-sqrt 9.2%

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

   3. add-sqr-sqrt 9.2%

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

   4. associate--r- 21.9%

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

   5. add-exp-log 21.9%

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

   6. expm1-undefine 21.9%

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

   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 99.7%

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

   1. unpow2 100.0%

      \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(0.125 + \color{blue}{\left(x \cdot x\right)} \cdot 0.0546875\right)\right) \]

7. Applied egg-rr 99.7%

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

8. Final simplification 99.7%

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

Alternative 3: 99.7% accurate, 23.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * 0.125), $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 99.7%

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

   1. unpow2 100.0%

      \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(0.125 + \color{blue}{\left(x \cdot x\right)} \cdot 0.0546875\right)\right) \]

5. Applied egg-rr 99.7%

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

6. Final simplification 99.7%

   \[\leadsto x \cdot \left(1 + \left(x \cdot x\right) \cdot 0.125\right) \]
7. Add Preprocessing

Alternative 4: 99.2% 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 98.3%

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

4. Final simplification 98.3%

   \[\leadsto x \]
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 2024055 
(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))))