bug500 (missed optimization)

Percentage Accurate: 69.6% → 98.6%

Time: 6.7s

Alternatives: 6

Speedup: 6.5×

Specification

\[-1000 < x \land x < 1000\]

Math

\[\begin{array}{l} \\ \sin x - x \end{array} \]

FPCore

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

C

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

Fortran

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

Java

public static double code(double x) {
	return Math.sin(x) - x;
}

Python

def code(x):
	return math.sin(x) - x

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sin x - x \end{array} \]

FPCore

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

C

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

Fortran

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

Java

public static double code(double x) {
	return Math.sin(x) - x;
}

Python

def code(x):
	return math.sin(x) - x

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left({x}^{4}, -0.0001984126984126984, \left(x \cdot x\right) \cdot 0.008333333333333333 - 0.16666666666666666\right) \cdot {x}^{3} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (fma
   (pow x 4.0)
   -0.0001984126984126984
   (- (* (* x x) 0.008333333333333333) 0.16666666666666666))
  (pow x 3.0)))

C

double code(double x) {
	return fma(pow(x, 4.0), -0.0001984126984126984, (((x * x) * 0.008333333333333333) - 0.16666666666666666)) * pow(x, 3.0);
}

Julia

function code(x)
	return Float64(fma((x ^ 4.0), -0.0001984126984126984, Float64(Float64(Float64(x * x) * 0.008333333333333333) - 0.16666666666666666)) * (x ^ 3.0))
end

Wolfram

code[x_] := N[(N[(N[Power[x, 4.0], $MachinePrecision] * -0.0001984126984126984 + N[(N[(N[(x * x), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.4%

   \[\sin x - x \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{3} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right) \cdot {x}^{3}} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right) \cdot {x}^{3}} \]

   3. +-commutative N/A

      \[\leadsto \left({x}^{2} \cdot \color{blue}{\left(\frac{-1}{5040} \cdot {x}^{2} + \frac{1}{120}\right)} - \frac{1}{6}\right) \cdot {x}^{3} \]

   4. distribute-lft-in N/A

      \[\leadsto \left(\color{blue}{\left({x}^{2} \cdot \left(\frac{-1}{5040} \cdot {x}^{2}\right) + {x}^{2} \cdot \frac{1}{120}\right)} - \frac{1}{6}\right) \cdot {x}^{3} \]

   5. *-commutative N/A

      \[\leadsto \left(\left({x}^{2} \cdot \left(\frac{-1}{5040} \cdot {x}^{2}\right) + \color{blue}{\frac{1}{120} \cdot {x}^{2}}\right) - \frac{1}{6}\right) \cdot {x}^{3} \]

   6. associate--l+ N/A

      \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{-1}{5040} \cdot {x}^{2}\right) + \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)} \cdot {x}^{3} \]

   7. *-commutative N/A

      \[\leadsto \left({x}^{2} \cdot \color{blue}{\left({x}^{2} \cdot \frac{-1}{5040}\right)} + \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right) \cdot {x}^{3} \]

   8. associate-*r* N/A

      \[\leadsto \left(\color{blue}{\left({x}^{2} \cdot {x}^{2}\right) \cdot \frac{-1}{5040}} + \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right) \cdot {x}^{3} \]

   9. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2} \cdot {x}^{2}, \frac{-1}{5040}, \frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)} \cdot {x}^{3} \]

   10. pow-sqr N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{\left(2 \cdot 2\right)}}, \frac{-1}{5040}, \frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot {x}^{3} \]

   11. lower-pow.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{\left(2 \cdot 2\right)}}, \frac{-1}{5040}, \frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot {x}^{3} \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left({x}^{\color{blue}{4}}, \frac{-1}{5040}, \frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot {x}^{3} \]

   13. lower--.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{4}, \frac{-1}{5040}, \color{blue}{\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}}\right) \cdot {x}^{3} \]

   14. *-commutative N/A

       \[\leadsto \mathsf{fma}\left({x}^{4}, \frac{-1}{5040}, \color{blue}{{x}^{2} \cdot \frac{1}{120}} - \frac{1}{6}\right) \cdot {x}^{3} \]

   15. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{4}, \frac{-1}{5040}, \color{blue}{{x}^{2} \cdot \frac{1}{120}} - \frac{1}{6}\right) \cdot {x}^{3} \]

   16. unpow2 N/A

       \[\leadsto \mathsf{fma}\left({x}^{4}, \frac{-1}{5040}, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{120} - \frac{1}{6}\right) \cdot {x}^{3} \]

   17. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{4}, \frac{-1}{5040}, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{120} - \frac{1}{6}\right) \cdot {x}^{3} \]

   18. lower-pow.f64 99.4

       \[\leadsto \mathsf{fma}\left({x}^{4}, -0.0001984126984126984, \left(x \cdot x\right) \cdot 0.008333333333333333 - 0.16666666666666666\right) \cdot \color{blue}{{x}^{3}} \]

5. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{4}, -0.0001984126984126984, \left(x \cdot x\right) \cdot 0.008333333333333333 - 0.16666666666666666\right) \cdot {x}^{3}} \]

6. Final simplification 99.4%

   \[\leadsto \mathsf{fma}\left({x}^{4}, -0.0001984126984126984, \left(x \cdot x\right) \cdot 0.008333333333333333 - 0.16666666666666666\right) \cdot {x}^{3} \]
7. Add Preprocessing

Alternative 2: 98.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(-0.0001984126984126984 \cdot x, x, 0.008333333333333333\right) - 0.16666666666666666\right) \cdot \left(\left(x \cdot x\right) \cdot x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (-
   (* (* x x) (fma (* -0.0001984126984126984 x) x 0.008333333333333333))
   0.16666666666666666)
  (* (* x x) x)))

C

double code(double x) {
	return (((x * x) * fma((-0.0001984126984126984 * x), x, 0.008333333333333333)) - 0.16666666666666666) * ((x * x) * x);
}

Julia

function code(x)
	return Float64(Float64(Float64(Float64(x * x) * fma(Float64(-0.0001984126984126984 * x), x, 0.008333333333333333)) - 0.16666666666666666) * Float64(Float64(x * x) * x))
end

Wolfram

code[x_] := N[(N[(N[(N[(x * x), $MachinePrecision] * N[(N[(-0.0001984126984126984 * x), $MachinePrecision] * x + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.4%

   \[\sin x - x \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{3} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right) \cdot {x}^{3}} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right) \cdot {x}^{3}} \]

   3. +-commutative N/A

      \[\leadsto \left({x}^{2} \cdot \color{blue}{\left(\frac{-1}{5040} \cdot {x}^{2} + \frac{1}{120}\right)} - \frac{1}{6}\right) \cdot {x}^{3} \]

   4. distribute-lft-in N/A

      \[\leadsto \left(\color{blue}{\left({x}^{2} \cdot \left(\frac{-1}{5040} \cdot {x}^{2}\right) + {x}^{2} \cdot \frac{1}{120}\right)} - \frac{1}{6}\right) \cdot {x}^{3} \]

   5. *-commutative N/A

      \[\leadsto \left(\left({x}^{2} \cdot \left(\frac{-1}{5040} \cdot {x}^{2}\right) + \color{blue}{\frac{1}{120} \cdot {x}^{2}}\right) - \frac{1}{6}\right) \cdot {x}^{3} \]

   6. associate--l+ N/A

      \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{-1}{5040} \cdot {x}^{2}\right) + \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)} \cdot {x}^{3} \]

   7. *-commutative N/A

      \[\leadsto \left({x}^{2} \cdot \color{blue}{\left({x}^{2} \cdot \frac{-1}{5040}\right)} + \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right) \cdot {x}^{3} \]

   8. associate-*r* N/A

      \[\leadsto \left(\color{blue}{\left({x}^{2} \cdot {x}^{2}\right) \cdot \frac{-1}{5040}} + \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right) \cdot {x}^{3} \]

   9. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2} \cdot {x}^{2}, \frac{-1}{5040}, \frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)} \cdot {x}^{3} \]

   10. pow-sqr N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{\left(2 \cdot 2\right)}}, \frac{-1}{5040}, \frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot {x}^{3} \]

   11. lower-pow.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{\left(2 \cdot 2\right)}}, \frac{-1}{5040}, \frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot {x}^{3} \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left({x}^{\color{blue}{4}}, \frac{-1}{5040}, \frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot {x}^{3} \]

   13. lower--.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{4}, \frac{-1}{5040}, \color{blue}{\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}}\right) \cdot {x}^{3} \]

   14. *-commutative N/A

       \[\leadsto \mathsf{fma}\left({x}^{4}, \frac{-1}{5040}, \color{blue}{{x}^{2} \cdot \frac{1}{120}} - \frac{1}{6}\right) \cdot {x}^{3} \]

   15. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{4}, \frac{-1}{5040}, \color{blue}{{x}^{2} \cdot \frac{1}{120}} - \frac{1}{6}\right) \cdot {x}^{3} \]

   16. unpow2 N/A

       \[\leadsto \mathsf{fma}\left({x}^{4}, \frac{-1}{5040}, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{120} - \frac{1}{6}\right) \cdot {x}^{3} \]

   17. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{4}, \frac{-1}{5040}, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{120} - \frac{1}{6}\right) \cdot {x}^{3} \]

   18. lower-pow.f64 99.4

       \[\leadsto \mathsf{fma}\left({x}^{4}, -0.0001984126984126984, \left(x \cdot x\right) \cdot 0.008333333333333333 - 0.16666666666666666\right) \cdot \color{blue}{{x}^{3}} \]

5. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{4}, -0.0001984126984126984, \left(x \cdot x\right) \cdot 0.008333333333333333 - 0.16666666666666666\right) \cdot {x}^{3}} \]
6. Step-by-step derivation

7. Applied rewrites 99.4%

   \[\leadsto \mathsf{fma}\left({x}^{4}, -0.0001984126984126984, \left(x \cdot x\right) \cdot 0.008333333333333333 - 0.16666666666666666\right) \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{x}\right) \]
8. Step-by-step derivation

9. Applied rewrites 99.4%

   \[\leadsto \left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.0001984126984126984 \cdot {x}^{4}\right) - 0.16666666666666666\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot x\right) \]
10. Step-by-step derivation

11. Applied rewrites 99.4%

    \[\leadsto \left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(-0.0001984126984126984 \cdot x, x, 0.008333333333333333\right) - 0.16666666666666666\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot x\right) \]

12. Final simplification 99.4%

    \[\leadsto \left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(-0.0001984126984126984 \cdot x, x, 0.008333333333333333\right) - 0.16666666666666666\right) \cdot \left(\left(x \cdot x\right) \cdot x\right) \]
13. Add Preprocessing

Alternative 3: 98.3% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right) \cdot \left(x \cdot x\right)\right) \cdot x \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (* (fma 0.008333333333333333 (* x x) -0.16666666666666666) (* x x)) x))

C

double code(double x) {
	return (fma(0.008333333333333333, (x * x), -0.16666666666666666) * (x * x)) * x;
}

Julia

function code(x)
	return Float64(Float64(fma(0.008333333333333333, Float64(x * x), -0.16666666666666666) * Float64(x * x)) * x)
end

Wolfram

code[x_] := N[(N[(N[(0.008333333333333333 * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 70.4%

   \[\sin x - x \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{3} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right) \cdot {x}^{3}} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right) \cdot {x}^{3}} \]

   3. +-commutative N/A

      \[\leadsto \left({x}^{2} \cdot \color{blue}{\left(\frac{-1}{5040} \cdot {x}^{2} + \frac{1}{120}\right)} - \frac{1}{6}\right) \cdot {x}^{3} \]

   4. distribute-lft-in N/A

      \[\leadsto \left(\color{blue}{\left({x}^{2} \cdot \left(\frac{-1}{5040} \cdot {x}^{2}\right) + {x}^{2} \cdot \frac{1}{120}\right)} - \frac{1}{6}\right) \cdot {x}^{3} \]

   5. *-commutative N/A

      \[\leadsto \left(\left({x}^{2} \cdot \left(\frac{-1}{5040} \cdot {x}^{2}\right) + \color{blue}{\frac{1}{120} \cdot {x}^{2}}\right) - \frac{1}{6}\right) \cdot {x}^{3} \]

   6. associate--l+ N/A

      \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{-1}{5040} \cdot {x}^{2}\right) + \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)} \cdot {x}^{3} \]

   7. *-commutative N/A

      \[\leadsto \left({x}^{2} \cdot \color{blue}{\left({x}^{2} \cdot \frac{-1}{5040}\right)} + \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right) \cdot {x}^{3} \]

   8. associate-*r* N/A

      \[\leadsto \left(\color{blue}{\left({x}^{2} \cdot {x}^{2}\right) \cdot \frac{-1}{5040}} + \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right) \cdot {x}^{3} \]

   9. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2} \cdot {x}^{2}, \frac{-1}{5040}, \frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)} \cdot {x}^{3} \]

   10. pow-sqr N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{\left(2 \cdot 2\right)}}, \frac{-1}{5040}, \frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot {x}^{3} \]

   11. lower-pow.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{\left(2 \cdot 2\right)}}, \frac{-1}{5040}, \frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot {x}^{3} \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left({x}^{\color{blue}{4}}, \frac{-1}{5040}, \frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot {x}^{3} \]

   13. lower--.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{4}, \frac{-1}{5040}, \color{blue}{\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}}\right) \cdot {x}^{3} \]

   14. *-commutative N/A

       \[\leadsto \mathsf{fma}\left({x}^{4}, \frac{-1}{5040}, \color{blue}{{x}^{2} \cdot \frac{1}{120}} - \frac{1}{6}\right) \cdot {x}^{3} \]

   15. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{4}, \frac{-1}{5040}, \color{blue}{{x}^{2} \cdot \frac{1}{120}} - \frac{1}{6}\right) \cdot {x}^{3} \]

   16. unpow2 N/A

       \[\leadsto \mathsf{fma}\left({x}^{4}, \frac{-1}{5040}, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{120} - \frac{1}{6}\right) \cdot {x}^{3} \]

   17. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{4}, \frac{-1}{5040}, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{120} - \frac{1}{6}\right) \cdot {x}^{3} \]

   18. lower-pow.f64 99.4

       \[\leadsto \mathsf{fma}\left({x}^{4}, -0.0001984126984126984, \left(x \cdot x\right) \cdot 0.008333333333333333 - 0.16666666666666666\right) \cdot \color{blue}{{x}^{3}} \]

5. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{4}, -0.0001984126984126984, \left(x \cdot x\right) \cdot 0.008333333333333333 - 0.16666666666666666\right) \cdot {x}^{3}} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{3} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot {x}^{3}} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot {x}^{3}} \]

   3. metadata-eval N/A

      \[\leadsto \left(\frac{1}{120} \cdot {x}^{2} - \color{blue}{\frac{1}{6} \cdot 1}\right) \cdot {x}^{3} \]

   4. metadata-eval N/A

      \[\leadsto \left(\frac{1}{120} \cdot {x}^{2} - \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{6}\right)\right)} \cdot 1\right) \cdot {x}^{3} \]

   5. lft-mult-inverse N/A

      \[\leadsto \left(\frac{1}{120} \cdot {x}^{2} - \left(\mathsf{neg}\left(\frac{-1}{6}\right)\right) \cdot \color{blue}{\left(\frac{1}{{x}^{2}} \cdot {x}^{2}\right)}\right) \cdot {x}^{3} \]

   6. fp-cancel-sign-sub-inv N/A

      \[\leadsto \color{blue}{\left(\frac{1}{120} \cdot {x}^{2} + \frac{-1}{6} \cdot \left(\frac{1}{{x}^{2}} \cdot {x}^{2}\right)\right)} \cdot {x}^{3} \]

   7. associate-*l* N/A

      \[\leadsto \left(\frac{1}{120} \cdot {x}^{2} + \color{blue}{\left(\frac{-1}{6} \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{2}}\right) \cdot {x}^{3} \]

   8. metadata-eval N/A

      \[\leadsto \left(\frac{1}{120} \cdot {x}^{2} + \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)} \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{2}\right) \cdot {x}^{3} \]

   9. metadata-eval N/A

      \[\leadsto \left(\frac{1}{120} \cdot {x}^{2} + \left(\color{blue}{\frac{-1}{6}} \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{2}\right) \cdot {x}^{3} \]

   10. associate-*l* N/A

       \[\leadsto \left(\frac{1}{120} \cdot {x}^{2} + \color{blue}{\frac{-1}{6} \cdot \left(\frac{1}{{x}^{2}} \cdot {x}^{2}\right)}\right) \cdot {x}^{3} \]

   11. lft-mult-inverse N/A

       \[\leadsto \left(\frac{1}{120} \cdot {x}^{2} + \frac{-1}{6} \cdot \color{blue}{1}\right) \cdot {x}^{3} \]

   12. metadata-eval N/A

       \[\leadsto \left(\frac{1}{120} \cdot {x}^{2} + \color{blue}{\frac{-1}{6}}\right) \cdot {x}^{3} \]

   13. *-commutative N/A

       \[\leadsto \left(\color{blue}{{x}^{2} \cdot \frac{1}{120}} + \frac{-1}{6}\right) \cdot {x}^{3} \]

   14. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{120}, \frac{-1}{6}\right)} \cdot {x}^{3} \]

   15. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{120}, \frac{-1}{6}\right) \cdot {x}^{3} \]

   16. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{120}, \frac{-1}{6}\right) \cdot {x}^{3} \]

   17. lower-pow.f64 99.3

       \[\leadsto \mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right) \cdot \color{blue}{{x}^{3}} \]

8. Applied rewrites 99.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right) \cdot {x}^{3}} \]
9. Step-by-step derivation

10. Applied rewrites 99.3%

    \[\leadsto \left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right) \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{x} \]

11. Final simplification 99.3%

    \[\leadsto \left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right) \cdot \left(x \cdot x\right)\right) \cdot x \]
12. Add Preprocessing

Alternative 4: 98.3% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right) \cdot \left(\left(x \cdot x\right) \cdot x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (fma (* x x) 0.008333333333333333 -0.16666666666666666) (* (* x x) x)))

C

double code(double x) {
	return fma((x * x), 0.008333333333333333, -0.16666666666666666) * ((x * x) * x);
}

Julia

function code(x)
	return Float64(fma(Float64(x * x), 0.008333333333333333, -0.16666666666666666) * Float64(Float64(x * x) * x))
end

Wolfram

code[x_] := N[(N[(N[(x * x), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.4%

   \[\sin x - x \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{3} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right) \cdot {x}^{3}} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right) \cdot {x}^{3}} \]

   3. +-commutative N/A

      \[\leadsto \left({x}^{2} \cdot \color{blue}{\left(\frac{-1}{5040} \cdot {x}^{2} + \frac{1}{120}\right)} - \frac{1}{6}\right) \cdot {x}^{3} \]

   4. distribute-lft-in N/A

      \[\leadsto \left(\color{blue}{\left({x}^{2} \cdot \left(\frac{-1}{5040} \cdot {x}^{2}\right) + {x}^{2} \cdot \frac{1}{120}\right)} - \frac{1}{6}\right) \cdot {x}^{3} \]

   5. *-commutative N/A

      \[\leadsto \left(\left({x}^{2} \cdot \left(\frac{-1}{5040} \cdot {x}^{2}\right) + \color{blue}{\frac{1}{120} \cdot {x}^{2}}\right) - \frac{1}{6}\right) \cdot {x}^{3} \]

   6. associate--l+ N/A

      \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{-1}{5040} \cdot {x}^{2}\right) + \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)} \cdot {x}^{3} \]

   7. *-commutative N/A

      \[\leadsto \left({x}^{2} \cdot \color{blue}{\left({x}^{2} \cdot \frac{-1}{5040}\right)} + \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right) \cdot {x}^{3} \]

   8. associate-*r* N/A

      \[\leadsto \left(\color{blue}{\left({x}^{2} \cdot {x}^{2}\right) \cdot \frac{-1}{5040}} + \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right) \cdot {x}^{3} \]

   9. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2} \cdot {x}^{2}, \frac{-1}{5040}, \frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)} \cdot {x}^{3} \]

   10. pow-sqr N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{\left(2 \cdot 2\right)}}, \frac{-1}{5040}, \frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot {x}^{3} \]

   11. lower-pow.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{\left(2 \cdot 2\right)}}, \frac{-1}{5040}, \frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot {x}^{3} \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left({x}^{\color{blue}{4}}, \frac{-1}{5040}, \frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot {x}^{3} \]

   13. lower--.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{4}, \frac{-1}{5040}, \color{blue}{\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}}\right) \cdot {x}^{3} \]

   14. *-commutative N/A

       \[\leadsto \mathsf{fma}\left({x}^{4}, \frac{-1}{5040}, \color{blue}{{x}^{2} \cdot \frac{1}{120}} - \frac{1}{6}\right) \cdot {x}^{3} \]

   15. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{4}, \frac{-1}{5040}, \color{blue}{{x}^{2} \cdot \frac{1}{120}} - \frac{1}{6}\right) \cdot {x}^{3} \]

   16. unpow2 N/A

       \[\leadsto \mathsf{fma}\left({x}^{4}, \frac{-1}{5040}, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{120} - \frac{1}{6}\right) \cdot {x}^{3} \]

   17. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{4}, \frac{-1}{5040}, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{120} - \frac{1}{6}\right) \cdot {x}^{3} \]

   18. lower-pow.f64 99.4

       \[\leadsto \mathsf{fma}\left({x}^{4}, -0.0001984126984126984, \left(x \cdot x\right) \cdot 0.008333333333333333 - 0.16666666666666666\right) \cdot \color{blue}{{x}^{3}} \]

5. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{4}, -0.0001984126984126984, \left(x \cdot x\right) \cdot 0.008333333333333333 - 0.16666666666666666\right) \cdot {x}^{3}} \]
6. Step-by-step derivation

7. Applied rewrites 99.4%

   \[\leadsto \mathsf{fma}\left({x}^{4}, -0.0001984126984126984, \left(x \cdot x\right) \cdot 0.008333333333333333 - 0.16666666666666666\right) \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{x}\right) \]

8. Taylor expanded in x around 0

   \[\leadsto \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot x\right) \]
9. Step-by-step derivation

10. Applied rewrites 99.2%

    \[\leadsto \mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot x\right) \]

11. Final simplification 99.2%

    \[\leadsto \mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right) \cdot \left(\left(x \cdot x\right) \cdot x\right) \]
12. Add Preprocessing

Alternative 5: 98.0% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(\left(x \cdot x\right) \cdot -0.16666666666666666\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x (* (* x x) -0.16666666666666666)))

C

double code(double x) {
	return x * ((x * x) * -0.16666666666666666);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * ((x * x) * (-0.16666666666666666d0))
end function

Java

public static double code(double x) {
	return x * ((x * x) * -0.16666666666666666);
}

Python

def code(x):
	return x * ((x * x) * -0.16666666666666666)

Julia

function code(x)
	return Float64(x * Float64(Float64(x * x) * -0.16666666666666666))
end

MATLAB

function tmp = code(x)
	tmp = x * ((x * x) * -0.16666666666666666);
end

Wolfram

code[x_] := N[(x * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.4%

   \[\sin x - x \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{-1}{6} \cdot {x}^{3}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{{x}^{3} \cdot \frac{-1}{6}} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{{x}^{3} \cdot \frac{-1}{6}} \]

   3. lower-pow.f64 98.8

      \[\leadsto \color{blue}{{x}^{3}} \cdot -0.16666666666666666 \]

5. Applied rewrites 98.8%

   \[\leadsto \color{blue}{{x}^{3} \cdot -0.16666666666666666} \]
6. Step-by-step derivation

7. Applied rewrites 98.8%

   \[\leadsto x \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot -0.16666666666666666\right)} \]

8. Final simplification 98.8%

   \[\leadsto x \cdot \left(\left(x \cdot x\right) \cdot -0.16666666666666666\right) \]
9. Add Preprocessing

Alternative 6: 67.1% accurate, 104.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 70.4%

   \[\sin x - x \]
2. Add Preprocessing

4. Applied rewrites 4.7%

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

5. Taylor expanded in x around 0

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

   1. unpow2 N/A

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

   2. rem-square-sqrt N/A

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

   3. metadata-eval N/A

      \[\leadsto x \cdot \color{blue}{0} \]

   4. mul0-rgt 68.8

      \[\leadsto \color{blue}{0} \]

7. Applied rewrites 68.8%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Developer Target 1: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| < 0.07:\\ \;\;\;\;-\left(\left(\frac{{x}^{3}}{6} - \frac{{x}^{5}}{120}\right) + \frac{{x}^{7}}{5040}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (< (fabs x) 0.07)
   (- (+ (- (/ (pow x 3.0) 6.0) (/ (pow x 5.0) 120.0)) (/ (pow x 7.0) 5040.0)))
   (- (sin x) x)))

C

double code(double x) {
	double tmp;
	if (fabs(x) < 0.07) {
		tmp = -(((pow(x, 3.0) / 6.0) - (pow(x, 5.0) / 120.0)) + (pow(x, 7.0) / 5040.0));
	} else {
		tmp = sin(x) - x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (abs(x) < 0.07d0) then
        tmp = -((((x ** 3.0d0) / 6.0d0) - ((x ** 5.0d0) / 120.0d0)) + ((x ** 7.0d0) / 5040.0d0))
    else
        tmp = sin(x) - x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) < 0.07) {
		tmp = -(((Math.pow(x, 3.0) / 6.0) - (Math.pow(x, 5.0) / 120.0)) + (Math.pow(x, 7.0) / 5040.0));
	} else {
		tmp = Math.sin(x) - x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) < 0.07:
		tmp = -(((math.pow(x, 3.0) / 6.0) - (math.pow(x, 5.0) / 120.0)) + (math.pow(x, 7.0) / 5040.0))
	else:
		tmp = math.sin(x) - x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) < 0.07)
		tmp = Float64(-Float64(Float64(Float64((x ^ 3.0) / 6.0) - Float64((x ^ 5.0) / 120.0)) + Float64((x ^ 7.0) / 5040.0)));
	else
		tmp = Float64(sin(x) - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) < 0.07)
		tmp = -((((x ^ 3.0) / 6.0) - ((x ^ 5.0) / 120.0)) + ((x ^ 7.0) / 5040.0));
	else
		tmp = sin(x) - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Less[N[Abs[x], $MachinePrecision], 0.07], (-N[(N[(N[(N[Power[x, 3.0], $MachinePrecision] / 6.0), $MachinePrecision] - N[(N[Power[x, 5.0], $MachinePrecision] / 120.0), $MachinePrecision]), $MachinePrecision] + N[(N[Power[x, 7.0], $MachinePrecision] / 5040.0), $MachinePrecision]), $MachinePrecision]), N[(N[Sin[x], $MachinePrecision] - x), $MachinePrecision]]

Reproduce

herbie shell --seed 2024332 
(FPCore (x)
  :name "bug500 (missed optimization)"
  :precision binary64
  :pre (and (< -1000.0 x) (< x 1000.0))

  :alt
  (! :herbie-platform default (if (< (fabs x) 7/100) (- (+ (- (/ (pow x 3) 6) (/ (pow x 5) 120)) (/ (pow x 7) 5040))) (- (sin x) x)))

  (- (sin x) x))