bug333 (missed optimization)

Percentage Accurate: 8.3% → 100.0%

Time: 7.2s

Alternatives: 6

Speedup: 1.8×

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, 0.7× 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]

Derivation

1. Initial program 8.4%

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

4. Applied rewrites 8.3%

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

6. Applied rewrites 8.6%

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

7. Taylor expanded in x around 0

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

   1. lower-*.f64 100.0

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

9. Applied rewrites 100.0%

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

10. Final simplification 100.0%

    \[\leadsto \frac{2 \cdot x}{\sqrt{1 - x} + \sqrt{1 + x}} \]
11. Add Preprocessing

Alternative 2: 99.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.0546875, 0.125\right) \cdot \left(x \cdot x\right), x, x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma (* (fma (* x x) 0.0546875 0.125) (* x x)) x x))

C

double code(double x) {
	return fma((fma((x * x), 0.0546875, 0.125) * (x * x)), x, x);
}

Julia

function code(x)
	return fma(Float64(fma(Float64(x * x), 0.0546875, 0.125) * Float64(x * x)), x, x)
end

Wolfram

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

Derivation

1. Initial program 8.4%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{8} + \frac{7}{128} \cdot {x}^{2}\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{8} + \frac{7}{128} \cdot {x}^{2}\right) + 1\right)} \]

   2. distribute-lft-in N/A

      \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{1}{8} + \frac{7}{128} \cdot {x}^{2}\right)\right) + x \cdot 1} \]

   3. associate-*r* N/A

      \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left(\frac{1}{8} + \frac{7}{128} \cdot {x}^{2}\right)} + x \cdot 1 \]

   4. unpow2 N/A

      \[\leadsto \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(\frac{1}{8} + \frac{7}{128} \cdot {x}^{2}\right) + x \cdot 1 \]

   5. cube-mult N/A

      \[\leadsto \color{blue}{{x}^{3}} \cdot \left(\frac{1}{8} + \frac{7}{128} \cdot {x}^{2}\right) + x \cdot 1 \]

   6. *-rgt-identity N/A

      \[\leadsto {x}^{3} \cdot \left(\frac{1}{8} + \frac{7}{128} \cdot {x}^{2}\right) + \color{blue}{x} \]

   7. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \frac{1}{8} + \frac{7}{128} \cdot {x}^{2}, x\right)} \]

   8. lower-pow.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{3}}, \frac{1}{8} + \frac{7}{128} \cdot {x}^{2}, x\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\frac{7}{128} \cdot {x}^{2} + \frac{1}{8}}, x\right) \]

   10. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\mathsf{fma}\left(\frac{7}{128}, {x}^{2}, \frac{1}{8}\right)}, x\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\frac{7}{128}, \color{blue}{x \cdot x}, \frac{1}{8}\right), x\right) \]

   12. lower-*.f64 99.6

       \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.0546875, \color{blue}{x \cdot x}, 0.125\right), x\right) \]

5. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.0546875, x \cdot x, 0.125\right), x\right)} \]
6. Step-by-step derivation

7. Applied rewrites 99.6%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.0546875, 0.125\right) \cdot \left(x \cdot x\right), \color{blue}{x}, x\right) \]
8. Add Preprocessing

Alternative 3: 99.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

double code(double x) {
	return fma((0.125 * (x * x)), x, x);
}

Julia

function code(x)
	return fma(Float64(0.125 * Float64(x * x)), x, x)
end

Wolfram

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

Derivation

1. Initial program 8.4%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{8} + \frac{7}{128} \cdot {x}^{2}\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{8} + \frac{7}{128} \cdot {x}^{2}\right) + 1\right)} \]

   2. distribute-lft-in N/A

      \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{1}{8} + \frac{7}{128} \cdot {x}^{2}\right)\right) + x \cdot 1} \]

   3. associate-*r* N/A

      \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left(\frac{1}{8} + \frac{7}{128} \cdot {x}^{2}\right)} + x \cdot 1 \]

   4. unpow2 N/A

      \[\leadsto \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(\frac{1}{8} + \frac{7}{128} \cdot {x}^{2}\right) + x \cdot 1 \]

   5. cube-mult N/A

      \[\leadsto \color{blue}{{x}^{3}} \cdot \left(\frac{1}{8} + \frac{7}{128} \cdot {x}^{2}\right) + x \cdot 1 \]

   6. *-rgt-identity N/A

      \[\leadsto {x}^{3} \cdot \left(\frac{1}{8} + \frac{7}{128} \cdot {x}^{2}\right) + \color{blue}{x} \]

   7. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \frac{1}{8} + \frac{7}{128} \cdot {x}^{2}, x\right)} \]

   8. lower-pow.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{3}}, \frac{1}{8} + \frac{7}{128} \cdot {x}^{2}, x\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\frac{7}{128} \cdot {x}^{2} + \frac{1}{8}}, x\right) \]

   10. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\mathsf{fma}\left(\frac{7}{128}, {x}^{2}, \frac{1}{8}\right)}, x\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\frac{7}{128}, \color{blue}{x \cdot x}, \frac{1}{8}\right), x\right) \]

   12. lower-*.f64 99.6

       \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.0546875, \color{blue}{x \cdot x}, 0.125\right), x\right) \]

5. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.0546875, x \cdot x, 0.125\right), x\right)} \]
6. Step-by-step derivation

7. Applied rewrites 99.6%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.0546875, 0.125\right) \cdot \left(x \cdot x\right), \color{blue}{x}, x\right) \]

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(\frac{1}{8} \cdot \left(x \cdot x\right), x, x\right) \]
9. Step-by-step derivation

10. Applied rewrites 99.4%

    \[\leadsto \mathsf{fma}\left(0.125 \cdot \left(x \cdot x\right), x, x\right) \]
11. Add Preprocessing

Alternative 4: 7.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.5, x, 1\right) - \mathsf{fma}\left(-0.5, x, 1\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (fma 0.5 x 1.0) (fma -0.5 x 1.0)))

C

double code(double x) {
	return fma(0.5, x, 1.0) - fma(-0.5, x, 1.0);
}

Julia

function code(x)
	return Float64(fma(0.5, x, 1.0) - fma(-0.5, x, 1.0))
end

Wolfram

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

Derivation

1. Initial program 8.4%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 7.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, x, 1\right)} - \sqrt{1 - x} \]

5. Applied rewrites 7.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, x, 1\right)} - \sqrt{1 - x} \]

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x, 1\right) - \color{blue}{\left(\frac{-1}{2} \cdot x + 1\right)} \]

   2. lower-fma.f64 7.7

      \[\leadsto \mathsf{fma}\left(0.5, x, 1\right) - \color{blue}{\mathsf{fma}\left(-0.5, x, 1\right)} \]

8. Applied rewrites 7.7%

   \[\leadsto \mathsf{fma}\left(0.5, x, 1\right) - \color{blue}{\mathsf{fma}\left(-0.5, x, 1\right)} \]
9. Add Preprocessing

Alternative 5: 6.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ 1 - \mathsf{fma}\left(-0.5, x, 1\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- 1.0 (fma -0.5 x 1.0)))

C

double code(double x) {
	return 1.0 - fma(-0.5, x, 1.0);
}

Julia

function code(x)
	return Float64(1.0 - fma(-0.5, x, 1.0))
end

Wolfram

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

Derivation

1. Initial program 8.4%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} - \sqrt{1 - x} \]
4. Step-by-step derivation

5. Applied rewrites 6.3%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 6.4

      \[\leadsto 1 - \color{blue}{\mathsf{fma}\left(-0.5, x, 1\right)} \]

8. Applied rewrites 6.4%

   \[\leadsto 1 - \color{blue}{\mathsf{fma}\left(-0.5, x, 1\right)} \]
9. Add Preprocessing

Alternative 6: 3.8% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ 1 - -0.5 \cdot x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- 1.0 (* -0.5 x)))

C

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

Fortran

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

Java

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

Python

def code(x):
	return 1.0 - (-0.5 * x)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.4%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} - \sqrt{1 - x} \]
4. Step-by-step derivation

5. Applied rewrites 6.3%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 6.4

      \[\leadsto 1 - \color{blue}{\mathsf{fma}\left(-0.5, x, 1\right)} \]

8. Applied rewrites 6.4%

   \[\leadsto 1 - \color{blue}{\mathsf{fma}\left(-0.5, x, 1\right)} \]

9. Taylor expanded in x around inf

   \[\leadsto 1 - \frac{-1}{2} \cdot \color{blue}{x} \]
10. Step-by-step derivation

11. Applied rewrites 3.6%

    \[\leadsto 1 - -0.5 \cdot \color{blue}{x} \]
12. Add Preprocessing

Developer Target 1: 100.0% accurate, 0.7× 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 2024331 
(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))))