ENA, Section 1.4, Mentioned, A

Percentage Accurate: 52.9% → 100.0%

Time: 9.9s

Alternatives: 7

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, 0.5× speedup

Math

\[\begin{array}{l} \\ \tan \left(\frac{x}{2}\right) \cdot \sin x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (tan (/ x 2.0)) (sin x)))

C

double code(double x) {
	return tan((x / 2.0)) * sin(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = tan((x / 2.0d0)) * sin(x)
end function

Java

public static double code(double x) {
	return Math.tan((x / 2.0)) * Math.sin(x);
}

Python

def code(x):
	return math.tan((x / 2.0)) * math.sin(x)

Julia

function code(x)
	return Float64(tan(Float64(x / 2.0)) * sin(x))
end

MATLAB

function tmp = code(x)
	tmp = tan((x / 2.0)) * sin(x);
end

Wolfram

code[x_] := N[(N[Tan[N[(x / 2.0), $MachinePrecision]], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.5%

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

4. Applied rewrites 100.0%

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

5. Final simplification 100.0%

   \[\leadsto \tan \left(\frac{x}{2}\right) \cdot \sin x \]
6. Add Preprocessing

Alternative 2: 100.0% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 50.5%

   \[1 - \cos x \]
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. 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

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

   12. lower-fma.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 100.0

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

5. Applied rewrites 100.0%

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

Alternative 3: 99.9% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 50.5%

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

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 99.9

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

5. Applied rewrites 99.9%

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

7. Applied rewrites 99.9%

   \[\leadsto \mathsf{fma}\left(0.5 \cdot x, \color{blue}{x}, {x}^{4} \cdot -0.041666666666666664\right) \]
8. Step-by-step derivation

9. Applied rewrites 99.9%

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

Alternative 4: 99.9% accurate, 4.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 50.5%

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

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 99.9

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

5. Applied rewrites 99.9%

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

Alternative 5: 99.5% accurate, 9.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.5%

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

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

   2. lower-*.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 99.5

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

5. Applied rewrites 99.5%

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

6. Final simplification 99.5%

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

Alternative 6: 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 50.5%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 49.6%

   \[\leadsto 1 - \color{blue}{1} \]
6. Add Preprocessing

Alternative 7: 4.2% accurate, 104.0× speedup

Math

\[\begin{array}{l} \\ 2 \end{array} \]

FPCore

(FPCore (x) :precision binary64 2.0)

C

double code(double x) {
	return 2.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0
end function

Java

public static double code(double x) {
	return 2.0;
}

Python

def code(x):
	return 2.0

Julia

function code(x)
	return 2.0
end

MATLAB

function tmp = code(x)
	tmp = 2.0;
end

Wolfram

code[x_] := 2.0

Derivation

1. Initial program 50.5%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 49.6%

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

7. Applied rewrites 4.2%

   \[\leadsto \color{blue}{1 + 1} \]

8. Taylor expanded in x around 0

   \[\leadsto \color{blue}{2} \]
9. Step-by-step derivation

10. Applied rewrites 4.2%

    \[\leadsto \color{blue}{2} \]
11. 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 2024331 
(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)))