ENA, Section 1.4, Mentioned, A

Percentage Accurate: 53.1% → 100.0%

Time: 15.9s

Alternatives: 8

Speedup: 9.5×

Specification

\[-0.01 \leq x \land x \leq 0.01\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return 1.0 - math.cos(x)

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return 1.0 - math.cos(x)

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \sin x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.004166666666666667, 0.041666666666666664\right), 0.5\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (sin x)
  (*
   x
   (fma x (* x (fma x (* x 0.004166666666666667) 0.041666666666666664)) 0.5))))

C

double code(double x) {
	return sin(x) * (x * fma(x, (x * fma(x, (x * 0.004166666666666667), 0.041666666666666664)), 0.5));
}

Julia

function code(x)
	return Float64(sin(x) * Float64(x * fma(x, Float64(x * fma(x, Float64(x * 0.004166666666666667), 0.041666666666666664)), 0.5)))
end

Wolfram

code[x_] := N[(N[Sin[x], $MachinePrecision] * N[(x * N[(x * N[(x * N[(x * N[(x * 0.004166666666666667), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.0%

   \[1 - \cos x \]
2. Add Preprocessing

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0

   \[\leadsto \sin x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{240} \cdot {x}^{2}\right)\right)\right)} \]
6. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \sin x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{240} \cdot {x}^{2}\right)\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto \sin x \cdot \left(x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{240} \cdot {x}^{2}\right) + \frac{1}{2}\right)}\right) \]

   3. unpow2 N/A

      \[\leadsto \sin x \cdot \left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{24} + \frac{1}{240} \cdot {x}^{2}\right) + \frac{1}{2}\right)\right) \]

   4. associate-*l* N/A

      \[\leadsto \sin x \cdot \left(x \cdot \left(\color{blue}{x \cdot \left(x \cdot \left(\frac{1}{24} + \frac{1}{240} \cdot {x}^{2}\right)\right)} + \frac{1}{2}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \sin x \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(\left(\frac{1}{24} + \frac{1}{240} \cdot {x}^{2}\right) \cdot x\right)} + \frac{1}{2}\right)\right) \]

   6. accelerator-lowering-fma.f64 N/A

      \[\leadsto \sin x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left(x, \left(\frac{1}{24} + \frac{1}{240} \cdot {x}^{2}\right) \cdot x, \frac{1}{2}\right)}\right) \]

   7. *-commutative N/A

      \[\leadsto \sin x \cdot \left(x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{24} + \frac{1}{240} \cdot {x}^{2}\right)}, \frac{1}{2}\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \sin x \cdot \left(x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{24} + \frac{1}{240} \cdot {x}^{2}\right)}, \frac{1}{2}\right)\right) \]

   9. +-commutative N/A

      \[\leadsto \sin x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{1}{240} \cdot {x}^{2} + \frac{1}{24}\right)}, \frac{1}{2}\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \sin x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{1}{240}} + \frac{1}{24}\right), \frac{1}{2}\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \sin x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{240} + \frac{1}{24}\right), \frac{1}{2}\right)\right) \]

   12. associate-*l* N/A

       \[\leadsto \sin x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{1}{240}\right)} + \frac{1}{24}\right), \frac{1}{2}\right)\right) \]

   13. accelerator-lowering-fma.f64 N/A

       \[\leadsto \sin x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{240}, \frac{1}{24}\right)}, \frac{1}{2}\right)\right) \]

   14. *-lowering-*.f64 100.0

       \[\leadsto \sin x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot 0.004166666666666667}, 0.041666666666666664\right), 0.5\right)\right) \]

7. Simplified 100.0%

   \[\leadsto \sin x \cdot \color{blue}{\left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.004166666666666667, 0.041666666666666664\right), 0.5\right)\right)} \]
8. Add Preprocessing

Alternative 2: 100.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.004166666666666667, 0.041666666666666664\right), 0.5\right)\right) \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.008333333333333333, -0.16666666666666666\right), 1\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (*
   x
   (fma x (* x (fma x (* x 0.004166666666666667) 0.041666666666666664)) 0.5))
  (*
   x
   (fma (* x x) (fma x (* x 0.008333333333333333) -0.16666666666666666) 1.0))))

C

double code(double x) {
	return (x * fma(x, (x * fma(x, (x * 0.004166666666666667), 0.041666666666666664)), 0.5)) * (x * fma((x * x), fma(x, (x * 0.008333333333333333), -0.16666666666666666), 1.0));
}

Julia

function code(x)
	return Float64(Float64(x * fma(x, Float64(x * fma(x, Float64(x * 0.004166666666666667), 0.041666666666666664)), 0.5)) * Float64(x * fma(Float64(x * x), fma(x, Float64(x * 0.008333333333333333), -0.16666666666666666), 1.0)))
end

Wolfram

code[x_] := N[(N[(x * N[(x * N[(x * N[(x * N[(x * 0.004166666666666667), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] * N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.0%

   \[1 - \cos x \]
2. Add Preprocessing

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0

   \[\leadsto \sin x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{240} \cdot {x}^{2}\right)\right)\right)} \]
6. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \sin x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{240} \cdot {x}^{2}\right)\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto \sin x \cdot \left(x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{240} \cdot {x}^{2}\right) + \frac{1}{2}\right)}\right) \]

   3. unpow2 N/A

      \[\leadsto \sin x \cdot \left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{24} + \frac{1}{240} \cdot {x}^{2}\right) + \frac{1}{2}\right)\right) \]

   4. associate-*l* N/A

      \[\leadsto \sin x \cdot \left(x \cdot \left(\color{blue}{x \cdot \left(x \cdot \left(\frac{1}{24} + \frac{1}{240} \cdot {x}^{2}\right)\right)} + \frac{1}{2}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \sin x \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(\left(\frac{1}{24} + \frac{1}{240} \cdot {x}^{2}\right) \cdot x\right)} + \frac{1}{2}\right)\right) \]

   6. accelerator-lowering-fma.f64 N/A

      \[\leadsto \sin x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left(x, \left(\frac{1}{24} + \frac{1}{240} \cdot {x}^{2}\right) \cdot x, \frac{1}{2}\right)}\right) \]

   7. *-commutative N/A

      \[\leadsto \sin x \cdot \left(x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{24} + \frac{1}{240} \cdot {x}^{2}\right)}, \frac{1}{2}\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \sin x \cdot \left(x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{24} + \frac{1}{240} \cdot {x}^{2}\right)}, \frac{1}{2}\right)\right) \]

   9. +-commutative N/A

      \[\leadsto \sin x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{1}{240} \cdot {x}^{2} + \frac{1}{24}\right)}, \frac{1}{2}\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \sin x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{1}{240}} + \frac{1}{24}\right), \frac{1}{2}\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \sin x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{240} + \frac{1}{24}\right), \frac{1}{2}\right)\right) \]

   12. associate-*l* N/A

       \[\leadsto \sin x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{1}{240}\right)} + \frac{1}{24}\right), \frac{1}{2}\right)\right) \]

   13. accelerator-lowering-fma.f64 N/A

       \[\leadsto \sin x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{240}, \frac{1}{24}\right)}, \frac{1}{2}\right)\right) \]

   14. *-lowering-*.f64 100.0

       \[\leadsto \sin x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot 0.004166666666666667}, 0.041666666666666664\right), 0.5\right)\right) \]

7. Simplified 100.0%

   \[\leadsto \sin x \cdot \color{blue}{\left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.004166666666666667, 0.041666666666666664\right), 0.5\right)\right)} \]

8. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\left(x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)\right)} \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{240}, \frac{1}{24}\right), \frac{1}{2}\right)\right) \]
9. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\left(x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)\right)} \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{240}, \frac{1}{24}\right), \frac{1}{2}\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \left(x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) + 1\right)}\right) \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{240}, \frac{1}{24}\right), \frac{1}{2}\right)\right) \]

   3. accelerator-lowering-fma.f64 N/A

      \[\leadsto \left(x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{120} \cdot {x}^{2} - \frac{1}{6}, 1\right)}\right) \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{240}, \frac{1}{24}\right), \frac{1}{2}\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{120} \cdot {x}^{2} - \frac{1}{6}, 1\right)\right) \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{240}, \frac{1}{24}\right), \frac{1}{2}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{120} \cdot {x}^{2} - \frac{1}{6}, 1\right)\right) \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{240}, \frac{1}{24}\right), \frac{1}{2}\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{120} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, 1\right)\right) \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{240}, \frac{1}{24}\right), \frac{1}{2}\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{120}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), 1\right)\right) \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{240}, \frac{1}{24}\right), \frac{1}{2}\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{120} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), 1\right)\right) \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{240}, \frac{1}{24}\right), \frac{1}{2}\right)\right) \]

   9. associate-*l* N/A

      \[\leadsto \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \frac{1}{120}\right)} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), 1\right)\right) \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{240}, \frac{1}{24}\right), \frac{1}{2}\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \left(x \cdot \mathsf{fma}\left(x \cdot x, x \cdot \left(x \cdot \frac{1}{120}\right) + \color{blue}{\frac{-1}{6}}, 1\right)\right) \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{240}, \frac{1}{24}\right), \frac{1}{2}\right)\right) \]

   11. accelerator-lowering-fma.f64 N/A

       \[\leadsto \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{120}, \frac{-1}{6}\right)}, 1\right)\right) \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{240}, \frac{1}{24}\right), \frac{1}{2}\right)\right) \]

   12. *-lowering-*.f64 100.0

       \[\leadsto \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot 0.008333333333333333}, -0.16666666666666666\right), 1\right)\right) \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.004166666666666667, 0.041666666666666664\right), 0.5\right)\right) \]

10. Simplified 100.0%

    \[\leadsto \color{blue}{\left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.008333333333333333, -0.16666666666666666\right), 1\right)\right)} \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.004166666666666667, 0.041666666666666664\right), 0.5\right)\right) \]

11. Final simplification 100.0%

    \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.004166666666666667, 0.041666666666666664\right), 0.5\right)\right) \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.008333333333333333, -0.16666666666666666\right), 1\right)\right) \]
12. Add Preprocessing

Alternative 3: 100.0% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, -0.041666666666666664\right), 0.5\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  x
  (*
   x
   (fma
    (* x x)
    (fma (* x x) 0.001388888888888889 -0.041666666666666664)
    0.5))))

C

double code(double x) {
	return x * (x * fma((x * x), fma((x * x), 0.001388888888888889, -0.041666666666666664), 0.5));
}

Julia

function code(x)
	return Float64(x * Float64(x * fma(Float64(x * x), fma(Float64(x * x), 0.001388888888888889, -0.041666666666666664), 0.5)))
end

Wolfram

code[x_] := N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + -0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.0%

   \[1 - \cos x \]
2. Add Preprocessing

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}\right)\right)} \]
6. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}\right)\right) \]

   2. associate-*l* N/A

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}\right)\right)\right)} \]

   3. *-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left(\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}\right)\right) \cdot x\right)} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{x \cdot \left(\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}\right)\right) \cdot x\right)} \]

   5. *-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}\right)\right)\right)} \]

   6. *-lowering-*.f64 N/A

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}\right)\right)\right)} \]

   7. +-commutative N/A

      \[\leadsto x \cdot \left(x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}\right) + \frac{1}{2}\right)}\right) \]

   8. accelerator-lowering-fma.f64 N/A

      \[\leadsto x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{720} \cdot {x}^{2} - \frac{1}{24}, \frac{1}{2}\right)}\right) \]

   9. unpow2 N/A

      \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{720} \cdot {x}^{2} - \frac{1}{24}, \frac{1}{2}\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{720} \cdot {x}^{2} - \frac{1}{24}, \frac{1}{2}\right)\right) \]

   11. sub-neg N/A

       \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{720} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{24}\right)\right)}, \frac{1}{2}\right)\right) \]

   12. *-commutative N/A

       \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{720}} + \left(\mathsf{neg}\left(\frac{1}{24}\right)\right), \frac{1}{2}\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, {x}^{2} \cdot \frac{1}{720} + \color{blue}{\frac{-1}{24}}, \frac{1}{2}\right)\right) \]

   14. accelerator-lowering-fma.f64 N/A

       \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{720}, \frac{-1}{24}\right)}, \frac{1}{2}\right)\right) \]

   15. unpow2 N/A

       \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{720}, \frac{-1}{24}\right), \frac{1}{2}\right)\right) \]

   16. *-lowering-*.f64 100.0

       \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.001388888888888889, -0.041666666666666664\right), 0.5\right)\right) \]

7. Simplified 100.0%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, -0.041666666666666664\right), 0.5\right)\right)} \]
8. Add Preprocessing

Alternative 4: 99.9% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x \cdot 0.5, x, -0.041666666666666664 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma (* x 0.5) x (* -0.041666666666666664 (* x (* x (* x x))))))

C

double code(double x) {
	return fma((x * 0.5), x, (-0.041666666666666664 * (x * (x * (x * x)))));
}

Julia

function code(x)
	return fma(Float64(x * 0.5), x, Float64(-0.041666666666666664 * Float64(x * Float64(x * Float64(x * x)))))
end

Wolfram

code[x_] := N[(N[(x * 0.5), $MachinePrecision] * x + N[(-0.041666666666666664 * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.0%

   \[1 - \cos x \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{24} \cdot {x}^{2}\right)} \]
4. Step-by-step derivation

   1. +-rgt-identity N/A

      \[\leadsto {x}^{2} \cdot \color{blue}{\left(\left(\frac{1}{2} + \frac{-1}{24} \cdot {x}^{2}\right) + 0\right)} \]

   2. distribute-lft-in N/A

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{24} \cdot {x}^{2}\right) + {x}^{2} \cdot 0} \]

   3. mul0-rgt N/A

      \[\leadsto {x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{24} \cdot {x}^{2}\right) + \color{blue}{0} \]

   4. accelerator-lowering-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + \frac{-1}{24} \cdot {x}^{2}, 0\right)} \]

   5. +-rgt-identity N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{2} + 0}, \frac{1}{2} + \frac{-1}{24} \cdot {x}^{2}, 0\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x} + 0, \frac{1}{2} + \frac{-1}{24} \cdot {x}^{2}, 0\right) \]

   7. accelerator-lowering-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x, x, 0\right)}, \frac{1}{2} + \frac{-1}{24} \cdot {x}^{2}, 0\right) \]

   8. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\frac{-1}{24} \cdot {x}^{2} + \frac{1}{2}}, 0\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{{x}^{2} \cdot \frac{-1}{24}} + \frac{1}{2}, 0\right) \]

   10. accelerator-lowering-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{24}, \frac{1}{2}\right)}, 0\right) \]

   11. +-rgt-identity N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(\color{blue}{{x}^{2} + 0}, \frac{-1}{24}, \frac{1}{2}\right), 0\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(\color{blue}{x \cdot x} + 0, \frac{-1}{24}, \frac{1}{2}\right), 0\right) \]

   13. accelerator-lowering-fma.f64 99.9

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x, x, 0\right)}, -0.041666666666666664, 0.5\right), 0\right) \]

5. Simplified 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), -0.041666666666666664, 0.5\right), 0\right)} \]
6. Step-by-step derivation

   1. +-rgt-identity N/A

      \[\leadsto \color{blue}{\left(x \cdot x + 0\right) \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{-1}{24} + \frac{1}{2}\right)} \]

   2. +-rgt-identity N/A

      \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{-1}{24} + \frac{1}{2}\right) \]

   3. +-commutative N/A

      \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\left(\frac{1}{2} + \left(x \cdot x + 0\right) \cdot \frac{-1}{24}\right)} \]

   4. distribute-rgt-in N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(x \cdot x\right) + \left(\left(x \cdot x + 0\right) \cdot \frac{-1}{24}\right) \cdot \left(x \cdot x\right)} \]

   5. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot x\right) \cdot x} + \left(\left(x \cdot x + 0\right) \cdot \frac{-1}{24}\right) \cdot \left(x \cdot x\right) \]

   6. metadata-eval N/A

      \[\leadsto \left(\color{blue}{\frac{1}{2}} \cdot x\right) \cdot x + \left(\left(x \cdot x + 0\right) \cdot \frac{-1}{24}\right) \cdot \left(x \cdot x\right) \]

   7. associate-/r/ N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{2}{x}}} \cdot x + \left(\left(x \cdot x + 0\right) \cdot \frac{-1}{24}\right) \cdot \left(x \cdot x\right) \]

   8. clear-num N/A

      \[\leadsto \color{blue}{\frac{x}{2}} \cdot x + \left(\left(x \cdot x + 0\right) \cdot \frac{-1}{24}\right) \cdot \left(x \cdot x\right) \]

   9. accelerator-lowering-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{2}, x, \left(\left(x \cdot x + 0\right) \cdot \frac{-1}{24}\right) \cdot \left(x \cdot x\right)\right)} \]

   10. div-inv N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot \frac{1}{2}}, x, \left(\left(x \cdot x + 0\right) \cdot \frac{-1}{24}\right) \cdot \left(x \cdot x\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(x \cdot \color{blue}{\frac{1}{2}}, x, \left(\left(x \cdot x + 0\right) \cdot \frac{-1}{24}\right) \cdot \left(x \cdot x\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot \frac{1}{2}}, x, \left(\left(x \cdot x + 0\right) \cdot \frac{-1}{24}\right) \cdot \left(x \cdot x\right)\right) \]

   13. +-rgt-identity N/A

       \[\leadsto \mathsf{fma}\left(x \cdot \frac{1}{2}, x, \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{24}\right) \cdot \left(x \cdot x\right)\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(x \cdot \frac{1}{2}, x, \color{blue}{\left(\frac{-1}{24} \cdot \left(x \cdot x\right)\right)} \cdot \left(x \cdot x\right)\right) \]

   15. associate-*l* N/A

       \[\leadsto \mathsf{fma}\left(x \cdot \frac{1}{2}, x, \color{blue}{\frac{-1}{24} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}\right) \]

   16. --rgt-identity N/A

       \[\leadsto \mathsf{fma}\left(x \cdot \frac{1}{2}, x, \frac{-1}{24} \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 0\right)}\right) \]

   17. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(x \cdot \frac{1}{2}, x, \frac{-1}{24} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - \color{blue}{0 \cdot 0}\right)\right) \]

   18. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(x \cdot \frac{1}{2}, x, \frac{-1}{24} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - \color{blue}{\left(1 - 1\right)} \cdot 0\right)\right) \]

   19. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(x \cdot \frac{1}{2}, x, \frac{-1}{24} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - \left(1 - 1\right) \cdot \color{blue}{\left(1 - 1\right)}\right)\right) \]

   20. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(x \cdot \frac{1}{2}, x, \color{blue}{\frac{-1}{24} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - \left(1 - 1\right) \cdot \left(1 - 1\right)\right)}\right) \]

7. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot 0.5, x, -0.041666666666666664 \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, x, 0\right), 0\right)\right)} \]

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(x \cdot \frac{1}{2}, x, \color{blue}{\frac{-1}{24} \cdot {x}^{4}}\right) \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(x \cdot \frac{1}{2}, x, \color{blue}{{x}^{4} \cdot \frac{-1}{24}}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(x \cdot \frac{1}{2}, x, \color{blue}{{x}^{4} \cdot \frac{-1}{24}}\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(x \cdot \frac{1}{2}, x, {x}^{\color{blue}{\left(3 + 1\right)}} \cdot \frac{-1}{24}\right) \]

   4. pow-plus N/A

      \[\leadsto \mathsf{fma}\left(x \cdot \frac{1}{2}, x, \color{blue}{\left({x}^{3} \cdot x\right)} \cdot \frac{-1}{24}\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(x \cdot \frac{1}{2}, x, \color{blue}{\left(x \cdot {x}^{3}\right)} \cdot \frac{-1}{24}\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(x \cdot \frac{1}{2}, x, \color{blue}{\left(x \cdot {x}^{3}\right)} \cdot \frac{-1}{24}\right) \]

   7. cube-mult N/A

      \[\leadsto \mathsf{fma}\left(x \cdot \frac{1}{2}, x, \left(x \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)}\right) \cdot \frac{-1}{24}\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(x \cdot \frac{1}{2}, x, \left(x \cdot \left(x \cdot \color{blue}{{x}^{2}}\right)\right) \cdot \frac{-1}{24}\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(x \cdot \frac{1}{2}, x, \left(x \cdot \color{blue}{\left(x \cdot {x}^{2}\right)}\right) \cdot \frac{-1}{24}\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(x \cdot \frac{1}{2}, x, \left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \cdot \frac{-1}{24}\right) \]

   11. *-lowering-*.f64 99.9

       \[\leadsto \mathsf{fma}\left(x \cdot 0.5, x, \left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \cdot -0.041666666666666664\right) \]

10. Simplified 99.9%

    \[\leadsto \mathsf{fma}\left(x \cdot 0.5, x, \color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot -0.041666666666666664}\right) \]

11. Final simplification 99.9%

    \[\leadsto \mathsf{fma}\left(x \cdot 0.5, x, -0.041666666666666664 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \]
12. Add Preprocessing

Alternative 5: 99.9% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot -0.041666666666666664, 0.5\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (* x x) (fma x (* x -0.041666666666666664) 0.5)))

C

double code(double x) {
	return (x * x) * fma(x, (x * -0.041666666666666664), 0.5);
}

Julia

function code(x)
	return Float64(Float64(x * x) * fma(x, Float64(x * -0.041666666666666664), 0.5))
end

Wolfram

code[x_] := N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * -0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.0%

   \[1 - \cos x \]
2. Add Preprocessing

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0

   \[\leadsto \sin x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{240} \cdot {x}^{2}\right)\right)\right)} \]
6. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \sin x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{240} \cdot {x}^{2}\right)\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto \sin x \cdot \left(x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{240} \cdot {x}^{2}\right) + \frac{1}{2}\right)}\right) \]

   3. unpow2 N/A

      \[\leadsto \sin x \cdot \left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{24} + \frac{1}{240} \cdot {x}^{2}\right) + \frac{1}{2}\right)\right) \]

   4. associate-*l* N/A

      \[\leadsto \sin x \cdot \left(x \cdot \left(\color{blue}{x \cdot \left(x \cdot \left(\frac{1}{24} + \frac{1}{240} \cdot {x}^{2}\right)\right)} + \frac{1}{2}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \sin x \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(\left(\frac{1}{24} + \frac{1}{240} \cdot {x}^{2}\right) \cdot x\right)} + \frac{1}{2}\right)\right) \]

   6. accelerator-lowering-fma.f64 N/A

      \[\leadsto \sin x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left(x, \left(\frac{1}{24} + \frac{1}{240} \cdot {x}^{2}\right) \cdot x, \frac{1}{2}\right)}\right) \]

   7. *-commutative N/A

      \[\leadsto \sin x \cdot \left(x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{24} + \frac{1}{240} \cdot {x}^{2}\right)}, \frac{1}{2}\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \sin x \cdot \left(x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{24} + \frac{1}{240} \cdot {x}^{2}\right)}, \frac{1}{2}\right)\right) \]

   9. +-commutative N/A

      \[\leadsto \sin x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{1}{240} \cdot {x}^{2} + \frac{1}{24}\right)}, \frac{1}{2}\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \sin x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{1}{240}} + \frac{1}{24}\right), \frac{1}{2}\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \sin x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{240} + \frac{1}{24}\right), \frac{1}{2}\right)\right) \]

   12. associate-*l* N/A

       \[\leadsto \sin x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{1}{240}\right)} + \frac{1}{24}\right), \frac{1}{2}\right)\right) \]

   13. accelerator-lowering-fma.f64 N/A

       \[\leadsto \sin x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{240}, \frac{1}{24}\right)}, \frac{1}{2}\right)\right) \]

   14. *-lowering-*.f64 100.0

       \[\leadsto \sin x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot 0.004166666666666667}, 0.041666666666666664\right), 0.5\right)\right) \]

7. Simplified 100.0%

   \[\leadsto \sin x \cdot \color{blue}{\left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.004166666666666667, 0.041666666666666664\right), 0.5\right)\right)} \]
8. Step-by-step derivation

   1. distribute-lft-in N/A

      \[\leadsto \sin x \cdot \color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{240}\right) + \frac{1}{24}\right)\right)\right) + x \cdot \frac{1}{2}\right)} \]

   2. associate-*r* N/A

      \[\leadsto \sin x \cdot \left(\color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{240}\right) + \frac{1}{24}\right)\right)} + x \cdot \frac{1}{2}\right) \]

   3. accelerator-lowering-fma.f64 N/A

      \[\leadsto \sin x \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, x \cdot \left(x \cdot \left(x \cdot \frac{1}{240}\right) + \frac{1}{24}\right), x \cdot \frac{1}{2}\right)} \]

   4. +-rgt-identity N/A

      \[\leadsto \sin x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x + 0}, x \cdot \left(x \cdot \left(x \cdot \frac{1}{240}\right) + \frac{1}{24}\right), x \cdot \frac{1}{2}\right) \]

   5. accelerator-lowering-fma.f64 N/A

      \[\leadsto \sin x \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x, x, 0\right)}, x \cdot \left(x \cdot \left(x \cdot \frac{1}{240}\right) + \frac{1}{24}\right), x \cdot \frac{1}{2}\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \sin x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{x \cdot \left(x \cdot \left(x \cdot \frac{1}{240}\right) + \frac{1}{24}\right)}, x \cdot \frac{1}{2}\right) \]

   7. associate-*r* N/A

      \[\leadsto \sin x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), x \cdot \left(\color{blue}{\left(x \cdot x\right) \cdot \frac{1}{240}} + \frac{1}{24}\right), x \cdot \frac{1}{2}\right) \]

   8. *-commutative N/A

      \[\leadsto \sin x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), x \cdot \left(\color{blue}{\frac{1}{240} \cdot \left(x \cdot x\right)} + \frac{1}{24}\right), x \cdot \frac{1}{2}\right) \]

   9. accelerator-lowering-fma.f64 N/A

      \[\leadsto \sin x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), x \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{240}, x \cdot x, \frac{1}{24}\right)}, x \cdot \frac{1}{2}\right) \]

   10. +-rgt-identity N/A

       \[\leadsto \sin x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), x \cdot \mathsf{fma}\left(\frac{1}{240}, \color{blue}{x \cdot x + 0}, \frac{1}{24}\right), x \cdot \frac{1}{2}\right) \]

   11. accelerator-lowering-fma.f64 N/A

       \[\leadsto \sin x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), x \cdot \mathsf{fma}\left(\frac{1}{240}, \color{blue}{\mathsf{fma}\left(x, x, 0\right)}, \frac{1}{24}\right), x \cdot \frac{1}{2}\right) \]

   12. *-lowering-*.f64 100.0

       \[\leadsto \sin x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), x \cdot \mathsf{fma}\left(0.004166666666666667, \mathsf{fma}\left(x, x, 0\right), 0.041666666666666664\right), \color{blue}{x \cdot 0.5}\right) \]

9. Applied egg-rr 100.0%

   \[\leadsto \sin x \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), x \cdot \mathsf{fma}\left(0.004166666666666667, \mathsf{fma}\left(x, x, 0\right), 0.041666666666666664\right), x \cdot 0.5\right)} \]

10. Taylor expanded in x around 0

    \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{24} \cdot {x}^{2}\right)} \]
11. Step-by-step derivation

    1. *-lowering-*.f64 N/A

       \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{24} \cdot {x}^{2}\right)} \]

    2. unpow2 N/A

       \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{2} + \frac{-1}{24} \cdot {x}^{2}\right) \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{2} + \frac{-1}{24} \cdot {x}^{2}\right) \]

    4. +-commutative N/A

       \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\left(\frac{-1}{24} \cdot {x}^{2} + \frac{1}{2}\right)} \]

    5. unpow2 N/A

       \[\leadsto \left(x \cdot x\right) \cdot \left(\frac{-1}{24} \cdot \color{blue}{\left(x \cdot x\right)} + \frac{1}{2}\right) \]

    6. associate-*r* N/A

       \[\leadsto \left(x \cdot x\right) \cdot \left(\color{blue}{\left(\frac{-1}{24} \cdot x\right) \cdot x} + \frac{1}{2}\right) \]

    7. *-commutative N/A

       \[\leadsto \left(x \cdot x\right) \cdot \left(\color{blue}{x \cdot \left(\frac{-1}{24} \cdot x\right)} + \frac{1}{2}\right) \]

    8. accelerator-lowering-fma.f64 N/A

       \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(x, \frac{-1}{24} \cdot x, \frac{1}{2}\right)} \]

    9. *-commutative N/A

       \[\leadsto \left(x \cdot x\right) \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-1}{24}}, \frac{1}{2}\right) \]

    10. *-lowering-*.f64 99.9

        \[\leadsto \left(x \cdot x\right) \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot -0.041666666666666664}, 0.5\right) \]

12. Simplified 99.9%

    \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot -0.041666666666666664, 0.5\right)} \]
13. Add Preprocessing

Alternative 6: 99.9% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot -0.041666666666666664, 0.5\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* x (* x (fma x (* x -0.041666666666666664) 0.5))))

C

double code(double x) {
	return x * (x * fma(x, (x * -0.041666666666666664), 0.5));
}

Julia

function code(x)
	return Float64(x * Float64(x * fma(x, Float64(x * -0.041666666666666664), 0.5)))
end

Wolfram

code[x_] := N[(x * N[(x * N[(x * N[(x * -0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.0%

   \[1 - \cos x \]
2. Add Preprocessing

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{24} \cdot {x}^{2}\right)} \]
6. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{2} + \frac{-1}{24} \cdot {x}^{2}\right) \]

   2. associate-*l* N/A

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{2} + \frac{-1}{24} \cdot {x}^{2}\right)\right)} \]

   3. *-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left(\left(\frac{1}{2} + \frac{-1}{24} \cdot {x}^{2}\right) \cdot x\right)} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{x \cdot \left(\left(\frac{1}{2} + \frac{-1}{24} \cdot {x}^{2}\right) \cdot x\right)} \]

   5. *-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{2} + \frac{-1}{24} \cdot {x}^{2}\right)\right)} \]

   6. *-lowering-*.f64 N/A

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{2} + \frac{-1}{24} \cdot {x}^{2}\right)\right)} \]

   7. +-commutative N/A

      \[\leadsto x \cdot \left(x \cdot \color{blue}{\left(\frac{-1}{24} \cdot {x}^{2} + \frac{1}{2}\right)}\right) \]

   8. *-commutative N/A

      \[\leadsto x \cdot \left(x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{24}} + \frac{1}{2}\right)\right) \]

   9. unpow2 N/A

      \[\leadsto x \cdot \left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{24} + \frac{1}{2}\right)\right) \]

   10. associate-*l* N/A

       \[\leadsto x \cdot \left(x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{-1}{24}\right)} + \frac{1}{2}\right)\right) \]

   11. accelerator-lowering-fma.f64 N/A

       \[\leadsto x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{-1}{24}, \frac{1}{2}\right)}\right) \]

   12. *-lowering-*.f64 99.9

       \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot -0.041666666666666664}, 0.5\right)\right) \]

7. Simplified 99.9%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot -0.041666666666666664, 0.5\right)\right)} \]
8. Add Preprocessing

Alternative 7: 99.5% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(x \cdot 0.5\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.0%

   \[1 - \cos x \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. +-rgt-identity N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot {x}^{2} + 0} \]

   2. accelerator-lowering-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {x}^{2}, 0\right)} \]

   3. +-rgt-identity N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{{x}^{2} + 0}, 0\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{x \cdot x} + 0, 0\right) \]

   5. accelerator-lowering-fma.f64 99.2

      \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{\mathsf{fma}\left(x, x, 0\right)}, 0\right) \]

5. Simplified 99.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(x, x, 0\right), 0\right)} \]
6. Step-by-step derivation

   1. +-rgt-identity N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(x \cdot x + 0\right)} \]

   2. +-rgt-identity N/A

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

   3. associate-*r* N/A

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

   4. metadata-eval N/A

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

   5. associate-/r/ N/A

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

   6. clear-num N/A

      \[\leadsto \color{blue}{\frac{x}{2}} \cdot x \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\frac{x}{2} \cdot x} \]

   8. div-inv N/A

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

   9. metadata-eval N/A

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

   10. *-lowering-*.f64 99.2

       \[\leadsto \color{blue}{\left(x \cdot 0.5\right)} \cdot x \]

7. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{\left(x \cdot 0.5\right) \cdot x} \]

8. Final simplification 99.2%

   \[\leadsto x \cdot \left(x \cdot 0.5\right) \]
9. Add Preprocessing

Alternative 8: 52.0% 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 52.0%

   \[1 - \cos x \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

5. Simplified 50.7%

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

   1. metadata-eval 50.7

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

7. Applied egg-rr 50.7%

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

Developer Target 1: 100.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{\sin x \cdot \sin x}{1 + \cos x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (* (sin x) (sin x)) (+ 1.0 (cos x))))

C

double code(double x) {
	return (sin(x) * sin(x)) / (1.0 + cos(x));
}

Fortran

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

Java

public static double code(double x) {
	return (Math.sin(x) * Math.sin(x)) / (1.0 + Math.cos(x));
}

Python

def code(x):
	return (math.sin(x) * math.sin(x)) / (1.0 + math.cos(x))

Julia

function code(x)
	return Float64(Float64(sin(x) * sin(x)) / Float64(1.0 + cos(x)))
end

MATLAB

function tmp = code(x)
	tmp = (sin(x) * sin(x)) / (1.0 + cos(x));
end

Wolfram

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

Reproduce

herbie shell --seed 2024195 
(FPCore (x)
  :name "ENA, Section 1.4, Mentioned, A"
  :precision binary64
  :pre (and (<= -0.01 x) (<= x 0.01))

  :alt
  (! :herbie-platform default (/ (* (sin x) (sin x)) (+ 1 (cos x))))

  (- 1.0 (cos x)))