bug333 (missed optimization)

Percentage Accurate: 8.4% → 100.0%

Time: 6.2s

Alternatives: 6

Speedup: 5.0×

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: 100.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{2 \cdot x}{\sqrt{1 - x} + \sqrt{x + 1}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 9.5%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. lower-/.f64 N/A

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around 0

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

   1. lower-*.f64 100.0

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

7. Applied rewrites 100.0%

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

Alternative 2: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 9.5%

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

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 99.3

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

5. Applied rewrites 99.3%

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

7. Applied rewrites 99.3%

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

Alternative 3: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 9.5%

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

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 99.3

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

5. Applied rewrites 99.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.0546875, x \cdot x, 0.125\right), x \cdot x, 1\right) \cdot x} \]
6. Add Preprocessing

Alternative 4: 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 9.5%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 99.1

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

5. Applied rewrites 99.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, 0.125, 1\right) \cdot x} \]
6. Step-by-step derivation

7. Applied rewrites 99.2%

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

Alternative 5: 99.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	return Float64(fma(Float64(x * x), 0.125, 1.0) * x)
end

Wolfram

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

Derivation

1. Initial program 9.5%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 99.1

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

5. Applied rewrites 99.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, 0.125, 1\right) \cdot x} \]
6. Add Preprocessing

Alternative 6: 99.1% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return 1.0 * x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 9.5%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 99.1

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

5. Applied rewrites 99.1%

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

6. Taylor expanded in x around 0

   \[\leadsto 1 \cdot x \]
7. Step-by-step derivation

8. Applied rewrites 98.9%

   \[\leadsto 1 \cdot x \]
9. 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 2024305 
(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))))