ENA, Section 1.4, Mentioned, A

Percentage Accurate: 52.9% → 100.0%

Time: 10.5s

Alternatives: 9

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: 52.9% 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, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.2%

   \[1 - \cos x \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. frac-2neg N/A

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

   4. metadata-eval N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. 1-sub-cos N/A

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

   8. distribute-rgt-neg-in N/A

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

   9. neg-mul-1 N/A

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

   10. times-frac N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-neg.f64 N/A

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

   16. lower-sin.f64 N/A

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

   17. lower-+.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Final simplification 100.0%

   \[\leadsto \frac{\sin x}{-1} \cdot \frac{\sin x}{-1 - \cos x} \]
6. Add Preprocessing

Alternative 2: 100.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -2.48015873015873 \cdot 10^{-5}, 0.001388888888888889\right), -0.041666666666666664\right), 0.5\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  x
  (*
   x
   (fma
    (* x x)
    (fma
     (* x x)
     (fma x (* x -2.48015873015873e-5) 0.001388888888888889)
     -0.041666666666666664)
    0.5))))

C

double code(double x) {
	return x * (x * fma((x * x), fma((x * x), fma(x, (x * -2.48015873015873e-5), 0.001388888888888889), -0.041666666666666664), 0.5));
}

Julia

function code(x)
	return Float64(x * Float64(x * fma(Float64(x * x), fma(Float64(x * x), fma(x, Float64(x * -2.48015873015873e-5), 0.001388888888888889), -0.041666666666666664), 0.5)))
end

Wolfram

code[x_] := N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * -2.48015873015873e-5), $MachinePrecision] + 0.001388888888888889), $MachinePrecision] + -0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.2%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}\right)\right)} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}\right)\right) \]

   2. associate-*l* N/A

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}\right)\right)\right)} \]

   3. lower-*.f64 N/A

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}\right)\right)\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}\right)\right)\right)} \]

   5. +-commutative N/A

      \[\leadsto x \cdot \left(x \cdot \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}\right) + \frac{1}{2}\right)}\right) \]

   6. lower-fma.f64 N/A

      \[\leadsto x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, {x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}, \frac{1}{2}\right)}\right) \]

5. Applied rewrites 100.0%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -2.48015873015873 \cdot 10^{-5}, 0.001388888888888889\right), -0.041666666666666664\right), 0.5\right)\right)} \]
6. Add Preprocessing

Alternative 3: 100.0% accurate, 3.2× speedup

Math

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

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

Wolfram

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

Derivation

1. Initial program 52.2%

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

3. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. sub-neg N/A

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

   9. metadata-eval N/A

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

   10. *-commutative N/A

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

   11. unpow2 N/A

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

   12. associate-*l* N/A

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

   13. lower-fma.f64 N/A

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

   14. lower-*.f64 100.0

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

5. Applied rewrites 100.0%

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

Alternative 4: 99.9% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 52.2%

   \[1 - \cos x \]
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. lower-*.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. lower-*.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. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 99.9

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

5. Applied rewrites 99.9%

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

7. Applied rewrites 99.9%

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

Alternative 5: 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 52.2%

   \[1 - \cos x \]
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. lower-*.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. lower-*.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. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 99.9

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

5. Applied rewrites 99.9%

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

7. Applied rewrites 99.9%

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

8. Final simplification 99.9%

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

Alternative 6: 99.9% accurate, 4.7× speedup

Math

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

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(Float64(x * x), -0.041666666666666664, 0.5))
end

Wolfram

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

Derivation

1. Initial program 52.2%

   \[1 - \cos x \]
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. lower-*.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. lower-*.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. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 99.9

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

5. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, -0.041666666666666664, 0.5\right)} \]
6. 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 52.2%

   \[1 - \cos x \]
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. lower-*.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 99.5

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

5. Applied rewrites 99.5%

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

7. Applied rewrites 99.5%

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

8. Final simplification 99.5%

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

Alternative 8: 99.5% accurate, 9.5× speedup

Math

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

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(Float64(x * x) * 0.5)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.2%

   \[1 - \cos x \]
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. lower-*.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 99.5

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

5. Applied rewrites 99.5%

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

6. Final simplification 99.5%

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

Alternative 9: 52.0% accurate, 26.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return 1.0 - 1.0

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.2%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 51.1%

   \[\leadsto 1 - \color{blue}{1} \]
6. 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 2024233 
(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)))