cos2 (problem 3.4.1)

Percentage Accurate: 50.4% → 99.5%

Time: 3.2s

Alternatives: 8

Speedup: 41.8×

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: 50.4% 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: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{\sin x}{\mathsf{fma}\left(\cos x, x, x\right)} \cdot \frac{\sin x}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (/ (sin x) (fma (cos x) x x)) (/ (sin x) x)))

C

double code(double x) {
	return (sin(x) / fma(cos(x), x, x)) * (sin(x) / x);
}

Julia

function code(x)
	return Float64(Float64(sin(x) / fma(cos(x), x, x)) * Float64(sin(x) / x))
end

Wolfram

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

Derivation

1. Initial program 50.4%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. flip-- N/A

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

   4. associate-/l/ N/A

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

   5. lower-/.f64 N/A

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

   6. metadata-eval N/A

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

   7. lower--.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. lift-cos.f64 N/A

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

   10. sqr-cos-a N/A

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

   11. lower-+.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 50.1

       \[\leadsto \frac{1 - \left(0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)\right)}{\color{blue}{\left(\cos x + 1\right)} \cdot \left(x \cdot x\right)} \]

3. Applied rewrites 50.1%

   \[\leadsto \color{blue}{\frac{1 - \left(0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)\right)}{\left(\cos x + 1\right) \cdot \left(x \cdot x\right)}} \]
4. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. sqr-cos-a-rev N/A

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

   7. 1-sub-cos N/A

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

   8. pow2 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-sin.f64 75.3

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

5. Applied rewrites 75.3%

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

   1. lift-/.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow2 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. times-frac N/A

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

   8. lower-*.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. distribute-rgt1-in N/A

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

   11. *-lft-identity N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. *-lft-identity N/A

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

   18. +-commutative N/A

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

   19. lift-*.f64 N/A

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

   20. *-commutative N/A

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

   21. lower-fma.f64 N/A

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

   22. lower-/.f64 99.5

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

7. Applied rewrites 99.5%

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

Alternative 2: 75.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.034:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, -0.041666666666666664\right), x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.034)
   (fma (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 <= 0.034) {
		tmp = fma(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 <= 0.034)
		tmp = fma(fma(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, 0.034], N[(N[(0.001388888888888889 * N[(x * x), $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 < 0.034000000000000002

   1. Initial program 50.4%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. sqr-cos-a N/A

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

      11. lower-+.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 50.1

          \[\leadsto \frac{1 - \left(0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)\right)}{\color{blue}{\left(\cos x + 1\right)} \cdot \left(x \cdot x\right)} \]

   3. Applied rewrites 50.1%

      \[\leadsto \color{blue}{\frac{1 - \left(0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)\right)}{\left(\cos x + 1\right) \cdot \left(x \cdot x\right)}} \]
   4. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. sqr-cos-a-rev N/A

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

      7. 1-sub-cos N/A

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

      8. pow2 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 75.3

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

   5. Applied rewrites 75.3%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. distribute-rgt1-in N/A

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

      11. *-lft-identity N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-lft-identity N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 N/A

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

      22. lower-/.f64 99.5

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}\right)} \]
   9. 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. sub-flip N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 51.4

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

   10. Applied rewrites 51.4%

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

   if 0.034000000000000002 < x

   1. Initial program 50.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 51.5

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

   3. Applied rewrites 51.5%

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

Alternative 3: 75.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.034:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, -0.041666666666666664\right), x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.034)
   (fma (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 <= 0.034) {
		tmp = fma(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 <= 0.034)
		tmp = fma(fma(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, 0.034], N[(N[(0.001388888888888889 * N[(x * x), $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 < 0.034000000000000002

   1. Initial program 50.4%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. sqr-cos-a N/A

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

      11. lower-+.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 50.1

          \[\leadsto \frac{1 - \left(0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)\right)}{\color{blue}{\left(\cos x + 1\right)} \cdot \left(x \cdot x\right)} \]

   3. Applied rewrites 50.1%

      \[\leadsto \color{blue}{\frac{1 - \left(0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)\right)}{\left(\cos x + 1\right) \cdot \left(x \cdot x\right)}} \]
   4. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. sqr-cos-a-rev N/A

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

      7. 1-sub-cos N/A

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

      8. pow2 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 75.3

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

   5. Applied rewrites 75.3%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. distribute-rgt1-in N/A

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

      11. *-lft-identity N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-lft-identity N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 N/A

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

      22. lower-/.f64 99.5

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}\right)} \]
   9. 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. sub-flip N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 51.4

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

   10. Applied rewrites 51.4%

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

   if 0.034000000000000002 < x

   1. Initial program 50.4%

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

Alternative 4: 65.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{x}}\\ \mathbf{if}\;x \leq 3.9:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, -0.041666666666666664\right), x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 \cdot t\_0}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 x))))
   (if (<= x 3.9)
     (fma (fma 0.001388888888888889 (* x x) -0.041666666666666664) (* x x) 0.5)
     (/ (* t_0 t_0) x))))

C

double code(double x) {
	double t_0 = sqrt((1.0 / x));
	double tmp;
	if (x <= 3.9) {
		tmp = fma(fma(0.001388888888888889, (x * x), -0.041666666666666664), (x * x), 0.5);
	} else {
		tmp = (t_0 * t_0) / x;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 / x))
	tmp = 0.0
	if (x <= 3.9)
		tmp = fma(fma(0.001388888888888889, Float64(x * x), -0.041666666666666664), Float64(x * x), 0.5);
	else
		tmp = Float64(Float64(t_0 * t_0) / x);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 3.9], N[(N[(0.001388888888888889 * N[(x * x), $MachinePrecision] + -0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(t$95$0 * t$95$0), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 3.89999999999999991

   1. Initial program 50.4%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. sqr-cos-a N/A

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

      11. lower-+.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 50.1

          \[\leadsto \frac{1 - \left(0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)\right)}{\color{blue}{\left(\cos x + 1\right)} \cdot \left(x \cdot x\right)} \]

   3. Applied rewrites 50.1%

      \[\leadsto \color{blue}{\frac{1 - \left(0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)\right)}{\left(\cos x + 1\right) \cdot \left(x \cdot x\right)}} \]
   4. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. sqr-cos-a-rev N/A

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

      7. 1-sub-cos N/A

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

      8. pow2 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 75.3

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

   5. Applied rewrites 75.3%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. distribute-rgt1-in N/A

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

      11. *-lft-identity N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-lft-identity N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 N/A

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

      22. lower-/.f64 99.5

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}\right)} \]
   9. 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. sub-flip N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 51.4

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

   10. Applied rewrites 51.4%

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

   if 3.89999999999999991 < x

   1. Initial program 50.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 51.5

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

   3. Applied rewrites 51.5%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-flip N/A

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

      4. lift-neg.f64 N/A

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

      5. div-add N/A

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

      6. inv-pow N/A

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

      7. sqr-pow N/A

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

      8. lower-fma.f64 N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-pow.f64 N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-pow.f64 N/A

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

      16. metadata-eval N/A

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

      17. frac-2neg N/A

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

      18. lift-neg.f64 N/A

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

      19. remove-double-neg N/A

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

      20. lower-/.f64 N/A

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

      21. lower-neg.f64 25.4

          \[\leadsto \frac{\mathsf{fma}\left({x}^{-0.5}, {x}^{-0.5}, \frac{\cos x}{\color{blue}{-x}}\right)}{x} \]

   5. Applied rewrites 25.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{-0.5}, {x}^{-0.5}, \frac{\cos x}{-x}\right)}}{x} \]

   6. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 N/A

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

      3. inv-pow N/A

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

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. pow2 N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. pow2 N/A

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

      10. pow-prod-up N/A

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

      11. metadata-eval N/A

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

      12. inv-pow N/A

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

      13. lower-/.f64 N/A

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

      14. inv-pow N/A

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

      15. metadata-eval N/A

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

      16. pow-prod-up N/A

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

      17. pow2 N/A

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

      18. unpow1/2 N/A

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

      19. lower-sqrt.f64 N/A

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

      20. pow2 N/A

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

      21. pow-prod-up N/A

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

      22. metadata-eval N/A

          \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot \sqrt{{x}^{-1}}}{x} \]

      23. inv-pow N/A

          \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot \sqrt{\frac{1}{x}}}{x} \]

      24. lower-/.f64 15.6

          \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot \sqrt{\frac{1}{x}}}{x} \]

   8. Applied rewrites 15.6%

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

Alternative 5: 64.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 72000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, -0.041666666666666664\right), x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{x}, x, -1\right)}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 72000000.0)
   (fma (fma 0.001388888888888889 (* x x) -0.041666666666666664) (* x x) 0.5)
   (/ (fma (/ 1.0 x) x -1.0) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 72000000.0) {
		tmp = fma(fma(0.001388888888888889, (x * x), -0.041666666666666664), (x * x), 0.5);
	} else {
		tmp = fma((1.0 / x), x, -1.0) / (x * x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 72000000.0)
		tmp = fma(fma(0.001388888888888889, Float64(x * x), -0.041666666666666664), Float64(x * x), 0.5);
	else
		tmp = Float64(fma(Float64(1.0 / x), x, -1.0) / Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 72000000.0], N[(N[(0.001388888888888889 * N[(x * x), $MachinePrecision] + -0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(N[(1.0 / x), $MachinePrecision] * x + -1.0), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.2e7

   1. Initial program 50.4%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. sqr-cos-a N/A

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

      11. lower-+.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 50.1

          \[\leadsto \frac{1 - \left(0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)\right)}{\color{blue}{\left(\cos x + 1\right)} \cdot \left(x \cdot x\right)} \]

   3. Applied rewrites 50.1%

      \[\leadsto \color{blue}{\frac{1 - \left(0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)\right)}{\left(\cos x + 1\right) \cdot \left(x \cdot x\right)}} \]
   4. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. sqr-cos-a-rev N/A

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

      7. 1-sub-cos N/A

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

      8. pow2 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 75.3

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

   5. Applied rewrites 75.3%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. distribute-rgt1-in N/A

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

      11. *-lft-identity N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-lft-identity N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 N/A

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

      22. lower-/.f64 99.5

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}\right)} \]
   9. 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. sub-flip N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 51.4

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

   10. Applied rewrites 51.4%

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

   if 7.2e7 < x

   1. Initial program 50.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 51.5

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

   3. Applied rewrites 51.5%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-flip N/A

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

      4. lift-neg.f64 N/A

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

      5. div-add N/A

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

      6. inv-pow N/A

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

      7. sqr-pow N/A

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

      8. lower-fma.f64 N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-pow.f64 N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-pow.f64 N/A

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

      16. metadata-eval N/A

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

      17. frac-2neg N/A

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

      18. lift-neg.f64 N/A

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

      19. remove-double-neg N/A

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

      20. lower-/.f64 N/A

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

      21. lower-neg.f64 25.4

          \[\leadsto \frac{\mathsf{fma}\left({x}^{-0.5}, {x}^{-0.5}, \frac{\cos x}{\color{blue}{-x}}\right)}{x} \]

   5. Applied rewrites 25.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{-0.5}, {x}^{-0.5}, \frac{\cos x}{-x}\right)}}{x} \]

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 27.6%

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

Alternative 6: 64.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.65 \cdot 10^{+35}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{x}, x, -1\right)}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.65e+35) 0.5 (/ (fma (/ 1.0 x) x -1.0) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 1.65e+35) {
		tmp = 0.5;
	} else {
		tmp = fma((1.0 / x), x, -1.0) / (x * x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.65e+35)
		tmp = 0.5;
	else
		tmp = Float64(fma(Float64(1.0 / x), x, -1.0) / Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.65e+35], 0.5, N[(N[(N[(1.0 / x), $MachinePrecision] * x + -1.0), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.6500000000000001e35

   1. Initial program 50.4%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. sqr-cos-a N/A

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

      11. lower-+.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 50.1

          \[\leadsto \frac{1 - \left(0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)\right)}{\color{blue}{\left(\cos x + 1\right)} \cdot \left(x \cdot x\right)} \]

   3. Applied rewrites 50.1%

      \[\leadsto \color{blue}{\frac{1 - \left(0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)\right)}{\left(\cos x + 1\right) \cdot \left(x \cdot x\right)}} \]
   4. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. sqr-cos-a-rev N/A

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

      7. 1-sub-cos N/A

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

      8. pow2 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 75.3

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

   5. Applied rewrites 75.3%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. distribute-rgt1-in N/A

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

      11. *-lft-identity N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-lft-identity N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 N/A

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

      22. lower-/.f64 99.5

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 52.0%

       \[\leadsto \color{blue}{0.5} \]

   if 1.6500000000000001e35 < x

   1. Initial program 50.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 51.5

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

   3. Applied rewrites 51.5%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-flip N/A

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

      4. lift-neg.f64 N/A

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

      5. div-add N/A

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

      6. inv-pow N/A

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

      7. sqr-pow N/A

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

      8. lower-fma.f64 N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-pow.f64 N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-pow.f64 N/A

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

      16. metadata-eval N/A

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

      17. frac-2neg N/A

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

      18. lift-neg.f64 N/A

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

      19. remove-double-neg N/A

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

      20. lower-/.f64 N/A

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

      21. lower-neg.f64 25.4

          \[\leadsto \frac{\mathsf{fma}\left({x}^{-0.5}, {x}^{-0.5}, \frac{\cos x}{\color{blue}{-x}}\right)}{x} \]

   5. Applied rewrites 25.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{-0.5}, {x}^{-0.5}, \frac{\cos x}{-x}\right)}}{x} \]

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 27.6%

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

Alternative 7: 64.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9.5 \cdot 10^{+76}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 1\right) \cdot \frac{\frac{-1}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 9.5e+76) 0.5 (* (- 1.0 1.0) (/ (/ -1.0 x) x))))

C

double code(double x) {
	double tmp;
	if (x <= 9.5e+76) {
		tmp = 0.5;
	} else {
		tmp = (1.0 - 1.0) * ((-1.0 / x) / x);
	}
	return tmp;
}

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
    real(8) :: tmp
    if (x <= 9.5d+76) then
        tmp = 0.5d0
    else
        tmp = (1.0d0 - 1.0d0) * (((-1.0d0) / x) / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 9.5e+76) {
		tmp = 0.5;
	} else {
		tmp = (1.0 - 1.0) * ((-1.0 / x) / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 9.5e+76:
		tmp = 0.5
	else:
		tmp = (1.0 - 1.0) * ((-1.0 / x) / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 9.5e+76)
		tmp = 0.5;
	else
		tmp = Float64(Float64(1.0 - 1.0) * Float64(Float64(-1.0 / x) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 9.5e+76)
		tmp = 0.5;
	else
		tmp = (1.0 - 1.0) * ((-1.0 / x) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 9.5e+76], 0.5, N[(N[(1.0 - 1.0), $MachinePrecision] * N[(N[(-1.0 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.5000000000000003e76

   1. Initial program 50.4%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. sqr-cos-a N/A

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

      11. lower-+.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 50.1

          \[\leadsto \frac{1 - \left(0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)\right)}{\color{blue}{\left(\cos x + 1\right)} \cdot \left(x \cdot x\right)} \]

   3. Applied rewrites 50.1%

      \[\leadsto \color{blue}{\frac{1 - \left(0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)\right)}{\left(\cos x + 1\right) \cdot \left(x \cdot x\right)}} \]
   4. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. sqr-cos-a-rev N/A

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

      7. 1-sub-cos N/A

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

      8. pow2 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 75.3

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

   5. Applied rewrites 75.3%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. distribute-rgt1-in N/A

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

      11. *-lft-identity N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-lft-identity N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 N/A

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

      22. lower-/.f64 99.5

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 52.0%

       \[\leadsto \color{blue}{0.5} \]

   if 9.5000000000000003e76 < x

   1. Initial program 50.4%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. sqr-cos-a N/A

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

      11. lower-+.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 50.1

          \[\leadsto \frac{1 - \left(0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)\right)}{\color{blue}{\left(\cos x + 1\right)} \cdot \left(x \cdot x\right)} \]

   3. Applied rewrites 50.1%

      \[\leadsto \color{blue}{\frac{1 - \left(0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)\right)}{\left(\cos x + 1\right) \cdot \left(x \cdot x\right)}} \]
   4. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. sqr-cos-a-rev N/A

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

      7. 1-sub-cos N/A

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

      8. pow2 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 75.3

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

   5. Applied rewrites 75.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. 1-sub-cos N/A

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

      9. metadata-eval N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. flip-- N/A

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

      15. lift--.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. sqr-neg-rev N/A

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

      18. lift-neg.f64 N/A

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

      19. lift-neg.f64 N/A

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

      20. associate-/r* N/A

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

   7. Applied rewrites 50.7%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 26.7%

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

Alternative 8: 52.0% 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 50.4%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. flip-- N/A

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

   4. associate-/l/ N/A

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

   5. lower-/.f64 N/A

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

   6. metadata-eval N/A

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

   7. lower--.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. lift-cos.f64 N/A

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

   10. sqr-cos-a N/A

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

   11. lower-+.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 50.1

       \[\leadsto \frac{1 - \left(0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)\right)}{\color{blue}{\left(\cos x + 1\right)} \cdot \left(x \cdot x\right)} \]

3. Applied rewrites 50.1%

   \[\leadsto \color{blue}{\frac{1 - \left(0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)\right)}{\left(\cos x + 1\right) \cdot \left(x \cdot x\right)}} \]
4. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. sqr-cos-a-rev N/A

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

   7. 1-sub-cos N/A

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

   8. pow2 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-sin.f64 75.3

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

5. Applied rewrites 75.3%

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

   1. lift-/.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow2 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. times-frac N/A

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

   8. lower-*.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. distribute-rgt1-in N/A

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

   11. *-lft-identity N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. *-lft-identity N/A

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

   18. +-commutative N/A

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

   19. lift-*.f64 N/A

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

   20. *-commutative N/A

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

   21. lower-fma.f64 N/A

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

   22. lower-/.f64 99.5

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

7. Applied rewrites 99.5%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 52.0%

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

Reproduce

herbie shell --seed 2025136 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))