bug500 (missed optimization)

Percentage Accurate: 69.0% → 99.1%

Time: 6.5s

Alternatives: 9

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: 69.0% 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: 99.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-3.571428571428572 \cdot 10^{-5}, x \cdot x, -0.007857142857142858\right), x \cdot x, -0.3\right), x \cdot x, -6\right)} \cdot x \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/
   (* x x)
   (fma
    (fma
     (fma -3.571428571428572e-5 (* x x) -0.007857142857142858)
     (* x x)
     -0.3)
    (* x x)
    -6.0))
  x))

C

double code(double x) {
	return ((x * x) / fma(fma(fma(-3.571428571428572e-5, (x * x), -0.007857142857142858), (x * x), -0.3), (x * x), -6.0)) * x;
}

Julia

function code(x)
	return Float64(Float64(Float64(x * x) / fma(fma(fma(-3.571428571428572e-5, Float64(x * x), -0.007857142857142858), Float64(x * x), -0.3), Float64(x * x), -6.0)) * x)
end

Wolfram

code[x_] := N[(N[(N[(x * x), $MachinePrecision] / N[(N[(N[(-3.571428571428572e-5 * N[(x * x), $MachinePrecision] + -0.007857142857142858), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.3), $MachinePrecision] * N[(x * x), $MachinePrecision] + -6.0), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 68.5%

   \[\sin x - x \]
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. sub-neg N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-pow.f64 98.6

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

5. Applied rewrites 98.6%

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

7. Applied rewrites 98.6%

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

9. Applied rewrites 98.6%

   \[\leadsto \frac{x \cdot x}{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right), x \cdot x, -0.16666666666666666\right)}} \cdot x \]

10. Taylor expanded in x around 0

    \[\leadsto \frac{x \cdot x}{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-1}{28000} \cdot {x}^{2} - \frac{11}{1400}\right) - \frac{3}{10}\right) - 6} \cdot x \]
11. Step-by-step derivation

12. Applied rewrites 98.7%

    \[\leadsto \frac{x \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-3.571428571428572 \cdot 10^{-5}, x \cdot x, -0.007857142857142858\right), x \cdot x, -0.3\right), x \cdot x, -6\right)} \cdot x \]
13. Add Preprocessing

Alternative 2: 99.1% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 68.5%

   \[\sin x - x \]
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. sub-neg N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-pow.f64 98.6

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

5. Applied rewrites 98.6%

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

7. Applied rewrites 98.6%

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

Alternative 3: 98.9% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 68.5%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{3} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)} \]
4. 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. sub-neg N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-pow.f64 98.3

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

5. Applied rewrites 98.3%

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

7. Applied rewrites 98.3%

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

8. Final simplification 98.3%

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

Alternative 4: 98.9% accurate, 3.9× speedup

Math

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

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(Float64(fma(0.008333333333333333, Float64(x * x), -0.16666666666666666) * x) * x) * x)
end

Wolfram

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

Derivation

1. Initial program 68.5%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{3} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)} \]
4. 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. sub-neg N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-pow.f64 98.3

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

5. Applied rewrites 98.3%

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

7. Applied rewrites 98.2%

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

9. Applied rewrites 98.3%

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

Alternative 5: 98.5% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot x}{-6} \cdot x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (/ (* x x) -6.0) x))

C

double code(double x) {
	return ((x * x) / -6.0) * x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((x * x) / (-6.0d0)) * x
end function

Java

public static double code(double x) {
	return ((x * x) / -6.0) * x;
}

Python

def code(x):
	return ((x * x) / -6.0) * x

Julia

function code(x)
	return Float64(Float64(Float64(x * x) / -6.0) * x)
end

MATLAB

function tmp = code(x)
	tmp = ((x * x) / -6.0) * x;
end

Wolfram

code[x_] := N[(N[(N[(x * x), $MachinePrecision] / -6.0), $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 68.5%

   \[\sin x - x \]
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. sub-neg N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-pow.f64 98.6

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

5. Applied rewrites 98.6%

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

7. Applied rewrites 98.6%

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

9. Applied rewrites 98.6%

   \[\leadsto \frac{x \cdot x}{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right), x \cdot x, -0.16666666666666666\right)}} \cdot x \]

10. Taylor expanded in x around 0

    \[\leadsto \frac{x \cdot x}{-6} \cdot x \]
11. Step-by-step derivation

12. Applied rewrites 98.2%

    \[\leadsto \frac{x \cdot x}{-6} \cdot x \]
13. Add Preprocessing

Alternative 6: 98.5% accurate, 6.5× speedup

Math

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

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(Float64(Float64(x * x) * x) * -0.16666666666666666)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.5%

   \[\sin x - x \]
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 98.1

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

5. Applied rewrites 98.1%

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

7. Applied rewrites 98.1%

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

Alternative 7: 98.5% accurate, 6.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.5%

   \[\sin x - x \]
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 98.1

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

5. Applied rewrites 98.1%

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

7. Applied rewrites 98.1%

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

Alternative 8: 67.1% accurate, 11.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.5%

   \[\sin x - x \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. div-sub N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-sin.f64 N/A

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

   6. sin-mult N/A

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

   7. associate-/l/ N/A

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

   8. frac-sub N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 19.4%

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

5. Taylor expanded in x around inf

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

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \frac{-2 \cdot \sin x - -4 \cdot \sin x}{x} - 1\right) \cdot x} \]

   2. distribute-rgt-out-- N/A

      \[\leadsto \left(\frac{1}{2} \cdot \frac{\color{blue}{\sin x \cdot \left(-2 - -4\right)}}{x} - 1\right) \cdot x \]

   3. metadata-eval N/A

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

   4. *-commutative N/A

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

   5. associate-*r/ N/A

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

   6. associate-*r* N/A

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

   7. metadata-eval N/A

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

   8. *-lft-identity N/A

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

   9. lower-*.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-sin.f64 68.4

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

7. Applied rewrites 68.4%

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

8. Taylor expanded in x around 0

   \[\leadsto \left(1 - 1\right) \cdot x \]
9. Step-by-step derivation

10. Applied rewrites 66.1%

    \[\leadsto \left(1 - 1\right) \cdot x \]
11. Add Preprocessing

Alternative 9: 6.4% accurate, 34.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return -x

Julia

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

MATLAB

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

Wolfram

code[x_] := (-x)

Derivation

1. Initial program 68.5%

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

3. Taylor expanded in x around inf

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

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(x\right)} \]

   2. lower-neg.f64 6.6

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

5. Applied rewrites 6.6%

   \[\leadsto \color{blue}{-x} \]
6. 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 2024248 
(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))