cos2 (problem 3.4.1)

Percentage Accurate: 51.3% → 99.7%

Time: 3.4s

Alternatives: 9

Speedup: 3.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 51.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.3%

   \[\frac{1 - \cos x}{x \cdot x} \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. pow2 N/A

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

   6. div-sub N/A

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

   7. lower--.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. pow2 N/A

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

   10. lift-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lift-cos.f64 N/A

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

   13. pow2 N/A

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

   14. lift-*.f64 51.5

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

3. Applied rewrites 51.5%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-cos.f64 N/A

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

   8. pow2 N/A

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

   9. sub-div N/A

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

   10. flip-- N/A

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

   11. metadata-eval N/A

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

   12. 1-sub-cos N/A

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

   13. sqr-sin-a-rev N/A

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

   14. frac-2neg N/A

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

   15. +-commutative N/A

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

   16. associate-/l/ N/A

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

   17. distribute-lft-neg-in N/A

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

5. Applied rewrites 75.8%

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

   1. lift-/.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. pow2 N/A

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

   8. associate-/r* N/A

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

   9. pow2 N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

7. Applied rewrites 99.4%

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

8. Taylor expanded in x around inf

   \[\leadsto \sin x \cdot \frac{\frac{\color{blue}{\frac{\sin x}{1 + \cos x}}}{x}}{x} \]
9. Step-by-step derivation

   1. hang-0p-tan N/A

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

   2. lower-tan.f64 N/A

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

   3. lower-/.f64 99.7

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

10. Applied rewrites 99.7%

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

Alternative 2: 75.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, -0.041666666666666664, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin x}{x \cdot x} \cdot \tan \left(\frac{x}{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 4e-5)
   (fma (* x x) -0.041666666666666664 0.5)
   (* (/ (sin x) (* x x)) (tan (/ x 2.0)))))

C

double code(double x) {
	double tmp;
	if (x <= 4e-5) {
		tmp = fma((x * x), -0.041666666666666664, 0.5);
	} else {
		tmp = (sin(x) / (x * x)) * tan((x / 2.0));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 4e-5)
		tmp = fma(Float64(x * x), -0.041666666666666664, 0.5);
	else
		tmp = Float64(Float64(sin(x) / Float64(x * x)) * tan(Float64(x / 2.0)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 4e-5], N[(N[(x * x), $MachinePrecision] * -0.041666666666666664 + 0.5), $MachinePrecision], N[(N[(N[Sin[x], $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision] * N[Tan[N[(x / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.00000000000000033e-5

   1. Initial program 51.3%

      \[\frac{1 - \cos x}{x \cdot x} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + \frac{-1}{24} \cdot {x}^{2}} \]
   3. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

         \[\leadsto {x}^{2} \cdot \frac{-1}{24} + \frac{1}{2} \]

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 50.0

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

   4. Applied rewrites 50.0%

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

   if 4.00000000000000033e-5 < x

   1. Initial program 51.3%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. pow2 N/A

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

      6. div-sub N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 51.5

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

   3. Applied rewrites 51.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. pow2 N/A

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

      9. sub-div N/A

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

      10. flip-- N/A

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

      11. metadata-eval N/A

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

      12. 1-sub-cos N/A

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

      13. sqr-sin-a-rev N/A

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

      14. frac-2neg N/A

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

      15. +-commutative N/A

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

      16. associate-/l/ N/A

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

      17. distribute-lft-neg-in N/A

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

   5. Applied rewrites 75.8%

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

      1. lift-/.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. associate-/r* N/A

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

      9. pow2 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   7. Applied rewrites 99.4%

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

   8. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. hang-0p-tan N/A

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

      9. lower-tan.f64 N/A

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

      10. lower-/.f64 76.1

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

   10. Applied rewrites 76.1%

       \[\leadsto \color{blue}{\frac{\sin x}{x \cdot x} \cdot \tan \left(\frac{x}{2}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 75.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.028:\\ \;\;\;\;\mathsf{fma}\left(0.001388888888888889 \cdot \left(x \cdot x\right) - 0.041666666666666664, x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.028)
   (fma (- (* 0.001388888888888889 (* x x)) 0.041666666666666664) (* x x) 0.5)
   (/ (/ (- 1.0 (cos x)) x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 0.028) {
		tmp = fma(((0.001388888888888889 * (x * x)) - 0.041666666666666664), (x * x), 0.5);
	} else {
		tmp = ((1.0 - cos(x)) / x) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.028)
		tmp = fma(Float64(Float64(0.001388888888888889 * Float64(x * x)) - 0.041666666666666664), Float64(x * x), 0.5);
	else
		tmp = Float64(Float64(Float64(1.0 - cos(x)) / x) / x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.028], N[(N[(N[(0.001388888888888889 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0280000000000000006

   1. Initial program 51.3%

      \[\frac{1 - \cos x}{x \cdot x} \]

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 50.5

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

   4. Applied rewrites 50.5%

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

   if 0.0280000000000000006 < x

   1. Initial program 51.3%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift--.f64 52.6

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

   3. Applied rewrites 52.6%

      \[\leadsto \color{blue}{\frac{\frac{1 - \cos x}{x}}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 75.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.028:\\ \;\;\;\;\mathsf{fma}\left(0.001388888888888889 \cdot \left(x \cdot x\right) - 0.041666666666666664, x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.028)
   (fma (- (* 0.001388888888888889 (* x x)) 0.041666666666666664) (* x x) 0.5)
   (/ (- 1.0 (cos x)) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.028) {
		tmp = fma(((0.001388888888888889 * (x * x)) - 0.041666666666666664), (x * x), 0.5);
	} else {
		tmp = (1.0 - cos(x)) / (x * x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.028)
		tmp = fma(Float64(Float64(0.001388888888888889 * Float64(x * x)) - 0.041666666666666664), Float64(x * x), 0.5);
	else
		tmp = Float64(Float64(1.0 - cos(x)) / Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.028], N[(N[(N[(0.001388888888888889 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0280000000000000006

   1. Initial program 51.3%

      \[\frac{1 - \cos x}{x \cdot x} \]

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 50.5

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

   4. Applied rewrites 50.5%

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

   if 0.0280000000000000006 < x

   1. Initial program 51.3%

      \[\frac{1 - \cos x}{x \cdot x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 63.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+23}:\\ \;\;\;\;\mathsf{fma}\left(0.001388888888888889 \cdot \left(x \cdot x\right) - 0.041666666666666664, x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{x \cdot x}, \frac{1}{x}, \frac{-x}{\left(x \cdot x\right) \cdot x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2e+23)
   (fma (- (* 0.001388888888888889 (* x x)) 0.041666666666666664) (* x x) 0.5)
   (fma (/ x (* x x)) (/ 1.0 x) (/ (- x) (* (* x x) x)))))

C

double code(double x) {
	double tmp;
	if (x <= 2e+23) {
		tmp = fma(((0.001388888888888889 * (x * x)) - 0.041666666666666664), (x * x), 0.5);
	} else {
		tmp = fma((x / (x * x)), (1.0 / x), (-x / ((x * x) * x)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 2e+23)
		tmp = fma(Float64(Float64(0.001388888888888889 * Float64(x * x)) - 0.041666666666666664), Float64(x * x), 0.5);
	else
		tmp = fma(Float64(x / Float64(x * x)), Float64(1.0 / x), Float64(Float64(-x) / Float64(Float64(x * x) * x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 2e+23], N[(N[(N[(0.001388888888888889 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(x / N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(1.0 / x), $MachinePrecision] + N[((-x) / N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.9999999999999998e23

   1. Initial program 51.3%

      \[\frac{1 - \cos x}{x \cdot x} \]

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 50.5

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

   4. Applied rewrites 50.5%

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

   if 1.9999999999999998e23 < x

   1. Initial program 51.3%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. pow2 N/A

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

      6. div-sub N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 51.5

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

   3. Applied rewrites 51.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. associate-/r* N/A

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

      9. frac-sub N/A

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

      10. pow2 N/A

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

      11. unpow3 N/A

          \[\leadsto \frac{1 \cdot x - {x}^{2} \cdot \frac{\cos x}{x}}{\color{blue}{{x}^{3}}} \]

      12. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1 \cdot x - {x}^{2} \cdot \frac{\cos x}{x}}{{x}^{3}}} \]

      13. lower--.f64 N/A

          \[\leadsto \frac{\color{blue}{1 \cdot x - {x}^{2} \cdot \frac{\cos x}{x}}}{{x}^{3}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{1 \cdot x} - {x}^{2} \cdot \frac{\cos x}{x}}{{x}^{3}} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{1 \cdot x - \color{blue}{{x}^{2} \cdot \frac{\cos x}{x}}}{{x}^{3}} \]

      16. pow2 N/A

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

      17. lift-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lift-cos.f64 N/A

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

      20. unpow3 N/A

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

      21. pow2 N/A

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

      22. lower-*.f64 N/A

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

   5. Applied rewrites 19.3%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. pow2 N/A

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

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

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

      9. lift-*.f64 N/A

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

      10. *-lft-identity N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. pow3 N/A

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

      14. div-add N/A

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

   7. Applied rewrites 19.5%

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

   8. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{x \cdot x}, \frac{1}{x}, \frac{\mathsf{neg}\left(x\right)}{\left(x \cdot x\right) \cdot x}\right) \]

      2. lift-neg.f64 28.3

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

   10. Applied rewrites 28.3%

       \[\leadsto \mathsf{fma}\left(\frac{x}{x \cdot x}, \frac{1}{x}, \frac{\color{blue}{-x}}{\left(x \cdot x\right) \cdot x}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 63.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(0.001388888888888889 \cdot \left(x \cdot x\right) - 0.041666666666666664, x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{x \cdot x}, \frac{1}{x}, \frac{-1}{x \cdot x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 9e+36)
   (fma (- (* 0.001388888888888889 (* x x)) 0.041666666666666664) (* x x) 0.5)
   (fma (/ x (* x x)) (/ 1.0 x) (/ -1.0 (* x x)))))

C

double code(double x) {
	double tmp;
	if (x <= 9e+36) {
		tmp = fma(((0.001388888888888889 * (x * x)) - 0.041666666666666664), (x * x), 0.5);
	} else {
		tmp = fma((x / (x * x)), (1.0 / x), (-1.0 / (x * x)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 9e+36)
		tmp = fma(Float64(Float64(0.001388888888888889 * Float64(x * x)) - 0.041666666666666664), Float64(x * x), 0.5);
	else
		tmp = fma(Float64(x / Float64(x * x)), Float64(1.0 / x), Float64(-1.0 / Float64(x * x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 9e+36], N[(N[(N[(0.001388888888888889 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(x / N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(1.0 / x), $MachinePrecision] + N[(-1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 8.99999999999999994e36

   1. Initial program 51.3%

      \[\frac{1 - \cos x}{x \cdot x} \]

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 50.5

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

   4. Applied rewrites 50.5%

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

   if 8.99999999999999994e36 < x

   1. Initial program 51.3%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. pow2 N/A

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

      6. div-sub N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 51.5

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

   3. Applied rewrites 51.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. associate-/r* N/A

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

      9. frac-sub N/A

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

      10. pow2 N/A

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

      11. unpow3 N/A

          \[\leadsto \frac{1 \cdot x - {x}^{2} \cdot \frac{\cos x}{x}}{\color{blue}{{x}^{3}}} \]

      12. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1 \cdot x - {x}^{2} \cdot \frac{\cos x}{x}}{{x}^{3}}} \]

      13. lower--.f64 N/A

          \[\leadsto \frac{\color{blue}{1 \cdot x - {x}^{2} \cdot \frac{\cos x}{x}}}{{x}^{3}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{1 \cdot x} - {x}^{2} \cdot \frac{\cos x}{x}}{{x}^{3}} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{1 \cdot x - \color{blue}{{x}^{2} \cdot \frac{\cos x}{x}}}{{x}^{3}} \]

      16. pow2 N/A

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

      17. lift-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lift-cos.f64 N/A

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

      20. unpow3 N/A

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

      21. pow2 N/A

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

      22. lower-*.f64 N/A

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

   5. Applied rewrites 19.3%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. pow2 N/A

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

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

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

      9. lift-*.f64 N/A

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

      10. *-lft-identity N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. pow3 N/A

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

      14. div-add N/A

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

   7. Applied rewrites 19.5%

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

   8. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 27.5

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

   10. Applied rewrites 27.5%

       \[\leadsto \mathsf{fma}\left(\frac{x}{x \cdot x}, \frac{1}{x}, \color{blue}{\frac{-1}{x \cdot x}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 63.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.5 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(0.001388888888888889 \cdot \left(x \cdot x\right) - 0.041666666666666664, x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - 1}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 7.5e+38)
   (fma (- (* 0.001388888888888889 (* x x)) 0.041666666666666664) (* x x) 0.5)
   (/ (- 1.0 1.0) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 7.5e+38) {
		tmp = fma(((0.001388888888888889 * (x * x)) - 0.041666666666666664), (x * x), 0.5);
	} else {
		tmp = (1.0 - 1.0) / (x * x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 7.5e+38)
		tmp = fma(Float64(Float64(0.001388888888888889 * Float64(x * x)) - 0.041666666666666664), Float64(x * x), 0.5);
	else
		tmp = Float64(Float64(1.0 - 1.0) / Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 7.5e+38], N[(N[(N[(0.001388888888888889 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(1.0 - 1.0), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.4999999999999999e38

   1. Initial program 51.3%

      \[\frac{1 - \cos x}{x \cdot x} \]

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 50.5

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

   4. Applied rewrites 50.5%

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

   if 7.4999999999999999e38 < x

   1. Initial program 51.3%

      \[\frac{1 - \cos x}{x \cdot x} \]

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 27.0%

      \[\leadsto \frac{1 - \color{blue}{1}}{x \cdot x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 62.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.45:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, -0.041666666666666664, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - 1}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3.45)
   (fma (* x x) -0.041666666666666664 0.5)
   (/ (- 1.0 1.0) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 3.45) {
		tmp = fma((x * x), -0.041666666666666664, 0.5);
	} else {
		tmp = (1.0 - 1.0) / (x * x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.45)
		tmp = fma(Float64(x * x), -0.041666666666666664, 0.5);
	else
		tmp = Float64(Float64(1.0 - 1.0) / Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 3.45], N[(N[(x * x), $MachinePrecision] * -0.041666666666666664 + 0.5), $MachinePrecision], N[(N[(1.0 - 1.0), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.4500000000000002

   1. Initial program 51.3%

      \[\frac{1 - \cos x}{x \cdot x} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + \frac{-1}{24} \cdot {x}^{2}} \]
   3. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

         \[\leadsto {x}^{2} \cdot \frac{-1}{24} + \frac{1}{2} \]

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 50.0

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

   4. Applied rewrites 50.0%

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

   if 3.4500000000000002 < x

   1. Initial program 51.3%

      \[\frac{1 - \cos x}{x \cdot x} \]

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 27.0%

      \[\leadsto \frac{1 - \color{blue}{1}}{x \cdot x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 51.0% accurate, 41.8× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.5)

C

double code(double x) {
	return 0.5;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

def code(x):
	return 0.5

Julia

function code(x)
	return 0.5
end

MATLAB

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

Wolfram

code[x_] := 0.5

Derivation

1. Initial program 51.3%

   \[\frac{1 - \cos x}{x \cdot x} \]

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{2}} \]
3. Step-by-step derivation

4. Applied rewrites 51.0%

   \[\leadsto \color{blue}{0.5} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025131 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))