bug333 (missed optimization)

Percentage Accurate: 8.4% → 99.8%

Time: 9.5s

Alternatives: 9

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: 8.4% 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, 0.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x \cdot x, 0.125 \cdot x, \mathsf{fma}\left(\mathsf{fma}\left(0.0322265625, x \cdot x, 0.0546875\right), {x}^{5}, x\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma
  (* x x)
  (* 0.125 x)
  (fma (fma 0.0322265625 (* x x) 0.0546875) (pow x 5.0) x)))

C

double code(double x) {
	return fma((x * x), (0.125 * x), fma(fma(0.0322265625, (x * x), 0.0546875), pow(x, 5.0), x));
}

Julia

function code(x)
	return fma(Float64(x * x), Float64(0.125 * x), fma(fma(0.0322265625, Float64(x * x), 0.0546875), (x ^ 5.0), x))
end

Wolfram

code[x_] := N[(N[(x * x), $MachinePrecision] * N[(0.125 * x), $MachinePrecision] + N[(N[(0.0322265625 * N[(x * x), $MachinePrecision] + 0.0546875), $MachinePrecision] * N[Power[x, 5.0], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.9%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{7}{128} + \frac{33}{1024} \cdot {x}^{2}\right)\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{7}{128} + \frac{33}{1024} \cdot {x}^{2}\right)\right) + 1\right)} \]

   2. distribute-lft-in N/A

      \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{7}{128} + \frac{33}{1024} \cdot {x}^{2}\right)\right)\right) + x \cdot 1} \]

   3. associate-*r* N/A

      \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{7}{128} + \frac{33}{1024} \cdot {x}^{2}\right)\right)} + x \cdot 1 \]

   4. unpow2 N/A

      \[\leadsto \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{7}{128} + \frac{33}{1024} \cdot {x}^{2}\right)\right) + x \cdot 1 \]

   5. cube-mult N/A

      \[\leadsto \color{blue}{{x}^{3}} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{7}{128} + \frac{33}{1024} \cdot {x}^{2}\right)\right) + x \cdot 1 \]

   6. *-rgt-identity N/A

      \[\leadsto {x}^{3} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{7}{128} + \frac{33}{1024} \cdot {x}^{2}\right)\right) + \color{blue}{x} \]

   7. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \frac{1}{8} + {x}^{2} \cdot \left(\frac{7}{128} + \frac{33}{1024} \cdot {x}^{2}\right), x\right)} \]

   8. lower-pow.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{3}}, \frac{1}{8} + {x}^{2} \cdot \left(\frac{7}{128} + \frac{33}{1024} \cdot {x}^{2}\right), x\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{{x}^{2} \cdot \left(\frac{7}{128} + \frac{33}{1024} \cdot {x}^{2}\right) + \frac{1}{8}}, x\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\left(\frac{7}{128} + \frac{33}{1024} \cdot {x}^{2}\right) \cdot {x}^{2}} + \frac{1}{8}, x\right) \]

   11. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\mathsf{fma}\left(\frac{7}{128} + \frac{33}{1024} \cdot {x}^{2}, {x}^{2}, \frac{1}{8}\right)}, x\right) \]

   12. +-commutative N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\color{blue}{\frac{33}{1024} \cdot {x}^{2} + \frac{7}{128}}, {x}^{2}, \frac{1}{8}\right), x\right) \]

   13. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{33}{1024}, {x}^{2}, \frac{7}{128}\right)}, {x}^{2}, \frac{1}{8}\right), x\right) \]

   14. unpow2 N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{33}{1024}, \color{blue}{x \cdot x}, \frac{7}{128}\right), {x}^{2}, \frac{1}{8}\right), x\right) \]

   15. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{33}{1024}, \color{blue}{x \cdot x}, \frac{7}{128}\right), {x}^{2}, \frac{1}{8}\right), x\right) \]

   16. unpow2 N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{33}{1024}, x \cdot x, \frac{7}{128}\right), \color{blue}{x \cdot x}, \frac{1}{8}\right), x\right) \]

   17. lower-*.f64 99.4

       \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(0.0322265625, x \cdot x, 0.0546875\right), \color{blue}{x \cdot x}, 0.125\right), x\right) \]

5. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(0.0322265625, x \cdot x, 0.0546875\right), x \cdot x, 0.125\right), x\right)} \]
6. Step-by-step derivation

7. Applied rewrites 99.4%

   \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(x \cdot x, 0.0322265625, 0.0546875\right) \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right), \color{blue}{x}, \mathsf{fma}\left(0.125, {x}^{3}, x\right)\right) \]
8. Step-by-step derivation

9. Applied rewrites 99.4%

   \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{0.125 \cdot x}, x + \mathsf{fma}\left(0.0322265625, x \cdot x, 0.0546875\right) \cdot {x}^{5}\right) \]
10. Step-by-step derivation

11. Applied rewrites 99.4%

    \[\leadsto \mathsf{fma}\left(x \cdot x, 0.125 \cdot x, \mathsf{fma}\left(\mathsf{fma}\left(0.0322265625, x \cdot x, 0.0546875\right), {x}^{5}, x\right)\right) \]
12. Add Preprocessing

Alternative 2: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.0322265625, 0.0546875\right), x \cdot x, 0.125\right) \cdot \left(x \cdot x\right), x, x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma
  (* (fma (fma (* x x) 0.0322265625 0.0546875) (* x x) 0.125) (* x x))
  x
  x))

C

double code(double x) {
	return fma((fma(fma((x * x), 0.0322265625, 0.0546875), (x * x), 0.125) * (x * x)), x, x);
}

Julia

function code(x)
	return fma(Float64(fma(fma(Float64(x * x), 0.0322265625, 0.0546875), Float64(x * x), 0.125) * Float64(x * x)), x, x)
end

Wolfram

code[x_] := N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.0322265625 + 0.0546875), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.125), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]

Derivation

1. Initial program 8.9%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{7}{128} + \frac{33}{1024} \cdot {x}^{2}\right)\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{7}{128} + \frac{33}{1024} \cdot {x}^{2}\right)\right) + 1\right)} \]

   2. distribute-lft-in N/A

      \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{7}{128} + \frac{33}{1024} \cdot {x}^{2}\right)\right)\right) + x \cdot 1} \]

   3. associate-*r* N/A

      \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{7}{128} + \frac{33}{1024} \cdot {x}^{2}\right)\right)} + x \cdot 1 \]

   4. unpow2 N/A

      \[\leadsto \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{7}{128} + \frac{33}{1024} \cdot {x}^{2}\right)\right) + x \cdot 1 \]

   5. cube-mult N/A

      \[\leadsto \color{blue}{{x}^{3}} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{7}{128} + \frac{33}{1024} \cdot {x}^{2}\right)\right) + x \cdot 1 \]

   6. *-rgt-identity N/A

      \[\leadsto {x}^{3} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{7}{128} + \frac{33}{1024} \cdot {x}^{2}\right)\right) + \color{blue}{x} \]

   7. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \frac{1}{8} + {x}^{2} \cdot \left(\frac{7}{128} + \frac{33}{1024} \cdot {x}^{2}\right), x\right)} \]

   8. lower-pow.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{3}}, \frac{1}{8} + {x}^{2} \cdot \left(\frac{7}{128} + \frac{33}{1024} \cdot {x}^{2}\right), x\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{{x}^{2} \cdot \left(\frac{7}{128} + \frac{33}{1024} \cdot {x}^{2}\right) + \frac{1}{8}}, x\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\left(\frac{7}{128} + \frac{33}{1024} \cdot {x}^{2}\right) \cdot {x}^{2}} + \frac{1}{8}, x\right) \]

   11. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\mathsf{fma}\left(\frac{7}{128} + \frac{33}{1024} \cdot {x}^{2}, {x}^{2}, \frac{1}{8}\right)}, x\right) \]

   12. +-commutative N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\color{blue}{\frac{33}{1024} \cdot {x}^{2} + \frac{7}{128}}, {x}^{2}, \frac{1}{8}\right), x\right) \]

   13. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{33}{1024}, {x}^{2}, \frac{7}{128}\right)}, {x}^{2}, \frac{1}{8}\right), x\right) \]

   14. unpow2 N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{33}{1024}, \color{blue}{x \cdot x}, \frac{7}{128}\right), {x}^{2}, \frac{1}{8}\right), x\right) \]

   15. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{33}{1024}, \color{blue}{x \cdot x}, \frac{7}{128}\right), {x}^{2}, \frac{1}{8}\right), x\right) \]

   16. unpow2 N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{33}{1024}, x \cdot x, \frac{7}{128}\right), \color{blue}{x \cdot x}, \frac{1}{8}\right), x\right) \]

   17. lower-*.f64 99.4

       \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(0.0322265625, x \cdot x, 0.0546875\right), \color{blue}{x \cdot x}, 0.125\right), x\right) \]

5. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(0.0322265625, x \cdot x, 0.0546875\right), x \cdot x, 0.125\right), x\right)} \]
6. Step-by-step derivation

7. Applied rewrites 99.4%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.0322265625, 0.0546875\right), x \cdot x, 0.125\right) \cdot \left(x \cdot x\right), \color{blue}{x}, x\right) \]
8. Add Preprocessing

Alternative 3: 99.8% accurate, 1.1× speedup

Math

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

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(fma(0.0546875, Float64(x * x), 0.125) * Float64(x * x)), x, x)
end

Wolfram

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

Derivation

1. Initial program 8.9%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{7}{128} + \frac{33}{1024} \cdot {x}^{2}\right)\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{7}{128} + \frac{33}{1024} \cdot {x}^{2}\right)\right) + 1\right)} \]

   2. distribute-lft-in N/A

      \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{7}{128} + \frac{33}{1024} \cdot {x}^{2}\right)\right)\right) + x \cdot 1} \]

   3. associate-*r* N/A

      \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{7}{128} + \frac{33}{1024} \cdot {x}^{2}\right)\right)} + x \cdot 1 \]

   4. unpow2 N/A

      \[\leadsto \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{7}{128} + \frac{33}{1024} \cdot {x}^{2}\right)\right) + x \cdot 1 \]

   5. cube-mult N/A

      \[\leadsto \color{blue}{{x}^{3}} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{7}{128} + \frac{33}{1024} \cdot {x}^{2}\right)\right) + x \cdot 1 \]

   6. *-rgt-identity N/A

      \[\leadsto {x}^{3} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{7}{128} + \frac{33}{1024} \cdot {x}^{2}\right)\right) + \color{blue}{x} \]

   7. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \frac{1}{8} + {x}^{2} \cdot \left(\frac{7}{128} + \frac{33}{1024} \cdot {x}^{2}\right), x\right)} \]

   8. lower-pow.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{3}}, \frac{1}{8} + {x}^{2} \cdot \left(\frac{7}{128} + \frac{33}{1024} \cdot {x}^{2}\right), x\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{{x}^{2} \cdot \left(\frac{7}{128} + \frac{33}{1024} \cdot {x}^{2}\right) + \frac{1}{8}}, x\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\left(\frac{7}{128} + \frac{33}{1024} \cdot {x}^{2}\right) \cdot {x}^{2}} + \frac{1}{8}, x\right) \]

   11. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\mathsf{fma}\left(\frac{7}{128} + \frac{33}{1024} \cdot {x}^{2}, {x}^{2}, \frac{1}{8}\right)}, x\right) \]

   12. +-commutative N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\color{blue}{\frac{33}{1024} \cdot {x}^{2} + \frac{7}{128}}, {x}^{2}, \frac{1}{8}\right), x\right) \]

   13. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{33}{1024}, {x}^{2}, \frac{7}{128}\right)}, {x}^{2}, \frac{1}{8}\right), x\right) \]

   14. unpow2 N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{33}{1024}, \color{blue}{x \cdot x}, \frac{7}{128}\right), {x}^{2}, \frac{1}{8}\right), x\right) \]

   15. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{33}{1024}, \color{blue}{x \cdot x}, \frac{7}{128}\right), {x}^{2}, \frac{1}{8}\right), x\right) \]

   16. unpow2 N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{33}{1024}, x \cdot x, \frac{7}{128}\right), \color{blue}{x \cdot x}, \frac{1}{8}\right), x\right) \]

   17. lower-*.f64 99.4

       \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(0.0322265625, x \cdot x, 0.0546875\right), \color{blue}{x \cdot x}, 0.125\right), x\right) \]

5. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(0.0322265625, x \cdot x, 0.0546875\right), x \cdot x, 0.125\right), x\right)} \]
6. Step-by-step derivation

7. Applied rewrites 99.4%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.0322265625, 0.0546875\right), x \cdot x, 0.125\right) \cdot \left(x \cdot x\right), \color{blue}{x}, x\right) \]

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{7}{128}, x \cdot x, \frac{1}{8}\right) \cdot \left(x \cdot x\right), x, x\right) \]
9. Step-by-step derivation

10. Applied rewrites 99.3%

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

Alternative 4: 99.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.9%

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

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

   2. distribute-lft-in N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. unpow2 N/A

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

   6. cube-mult N/A

      \[\leadsto \color{blue}{{x}^{3}} \cdot \frac{1}{8} + x \cdot 1 \]

   7. *-rgt-identity N/A

      \[\leadsto {x}^{3} \cdot \frac{1}{8} + \color{blue}{x} \]

   8. lower-fma.f64 N/A

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

   9. lower-pow.f64 98.9

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

5. Applied rewrites 98.9%

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

7. Applied rewrites 98.9%

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

Alternative 5: 7.8% 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 8.9%

   \[\sqrt{1 + x} - \sqrt{1 - x} \]
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 6.2%

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

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 6.2

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

8. Applied rewrites 6.2%

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

9. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. lower-fma.f64 7.6

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

11. Applied rewrites 7.6%

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

Alternative 6: 6.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ 1 - \mathsf{fma}\left(\mathsf{fma}\left(-0.125, x, -0.5\right), x, 1\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- 1.0 (fma (fma -0.125 x -0.5) x 1.0)))

C

double code(double x) {
	return 1.0 - fma(fma(-0.125, x, -0.5), x, 1.0);
}

Julia

function code(x)
	return Float64(1.0 - fma(fma(-0.125, x, -0.5), x, 1.0))
end

Wolfram

code[x_] := N[(1.0 - N[(N[(-0.125 * x + -0.5), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.9%

   \[\sqrt{1 + x} - \sqrt{1 - x} \]
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 6.2%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. sub-neg N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 6.2

      \[\leadsto 1 - \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-0.125, x, -0.5\right)}, x, 1\right) \]

8. Applied rewrites 6.2%

   \[\leadsto 1 - \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.125, x, -0.5\right), x, 1\right)} \]
9. Add Preprocessing

Alternative 7: 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 8.9%

   \[\sqrt{1 + x} - \sqrt{1 - x} \]
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 6.2%

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

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 6.2

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

8. Applied rewrites 6.2%

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

Alternative 8: 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 8.9%

   \[\sqrt{1 + x} - \sqrt{1 - x} \]
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 6.2%

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

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 6.2

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

8. Applied rewrites 6.2%

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

9. Taylor expanded in x around inf

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

11. Applied rewrites 3.8%

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

Alternative 9: 5.4% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ 1 - 1 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return 1.0 - 1.0

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.9%

   \[\sqrt{1 + x} - \sqrt{1 - x} \]
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 6.2%

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

6. Taylor expanded in x around 0

   \[\leadsto 1 - \color{blue}{1} \]
7. Step-by-step derivation

8. Applied rewrites 5.2%

   \[\leadsto 1 - \color{blue}{1} \]
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 2024249 
(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))))