cos2 (problem 3.4.1)

Percentage Accurate: 50.5% → 99.5%

Time: 4.1s

Alternatives: 10

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: 50.5% 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, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.105:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2.48015873015873 \cdot 10^{-5}, x\_m \cdot x\_m, 0.001388888888888889\right), x\_m \cdot x\_m, -0.041666666666666664\right) \cdot x\_m, x\_m, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x\_m}{\frac{1 - \cos x\_m}{x\_m}}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.105)
   (fma
    (*
     (fma
      (fma -2.48015873015873e-5 (* x_m x_m) 0.001388888888888889)
      (* x_m x_m)
      -0.041666666666666664)
     x_m)
    x_m
    0.5)
   (/ 1.0 (/ x_m (/ (- 1.0 (cos x_m)) x_m)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.105) {
		tmp = fma((fma(fma(-2.48015873015873e-5, (x_m * x_m), 0.001388888888888889), (x_m * x_m), -0.041666666666666664) * x_m), x_m, 0.5);
	} else {
		tmp = 1.0 / (x_m / ((1.0 - cos(x_m)) / x_m));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.105)
		tmp = fma(Float64(fma(fma(-2.48015873015873e-5, Float64(x_m * x_m), 0.001388888888888889), Float64(x_m * x_m), -0.041666666666666664) * x_m), x_m, 0.5);
	else
		tmp = Float64(1.0 / Float64(x_m / Float64(Float64(1.0 - cos(x_m)) / x_m)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.105], N[(N[(N[(N[(-2.48015873015873e-5 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.001388888888888889), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + -0.041666666666666664), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m + 0.5), $MachinePrecision], N[(1.0 / N[(x$95$m / N[(N[(1.0 - N[Cos[x$95$m], $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.104999999999999996

   1. Initial program 2.8%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. pow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

   4. Applied rewrites 100.0%

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

   if 0.104999999999999996 < x

   1. Initial program 98.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. associate-/r* N/A

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

      6. division-flip N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift--.f64 98.5

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

   3. Applied rewrites 98.5%

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

Alternative 2: 99.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.105:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2.48015873015873 \cdot 10^{-5}, x\_m \cdot x\_m, 0.001388888888888889\right), x\_m \cdot x\_m, -0.041666666666666664\right) \cdot x\_m, x\_m, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x\_m}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.105)
   (fma
    (*
     (fma
      (fma -2.48015873015873e-5 (* x_m x_m) 0.001388888888888889)
      (* x_m x_m)
      -0.041666666666666664)
     x_m)
    x_m
    0.5)
   (/ (- 1.0 (cos x_m)) (* x_m x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.105) {
		tmp = fma((fma(fma(-2.48015873015873e-5, (x_m * x_m), 0.001388888888888889), (x_m * x_m), -0.041666666666666664) * x_m), x_m, 0.5);
	} else {
		tmp = (1.0 - cos(x_m)) / (x_m * x_m);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.105)
		tmp = fma(Float64(fma(fma(-2.48015873015873e-5, Float64(x_m * x_m), 0.001388888888888889), Float64(x_m * x_m), -0.041666666666666664) * x_m), x_m, 0.5);
	else
		tmp = Float64(Float64(1.0 - cos(x_m)) / Float64(x_m * x_m));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.105], N[(N[(N[(N[(-2.48015873015873e-5 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.001388888888888889), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + -0.041666666666666664), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m + 0.5), $MachinePrecision], N[(N[(1.0 - N[Cos[x$95$m], $MachinePrecision]), $MachinePrecision] / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.104999999999999996

   1. Initial program 2.8%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. pow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

   4. Applied rewrites 100.0%

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

   if 0.104999999999999996 < x

   1. Initial program 98.6%

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

Alternative 3: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{{\left(\frac{\sin x\_m}{x\_m}\right)}^{2}}{\cos x\_m - -1} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (/ (pow (/ (sin x_m) x_m) 2.0) (- (cos x_m) -1.0)))

C

x_m = fabs(x);
double code(double x_m) {
	return pow((sin(x_m) / x_m), 2.0) / (cos(x_m) - -1.0);
}

Fortran

x_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    code = ((sin(x_m) / x_m) ** 2.0d0) / (cos(x_m) - (-1.0d0))
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return Math.pow((Math.sin(x_m) / x_m), 2.0) / (Math.cos(x_m) - -1.0);
}

Python

x_m = math.fabs(x)
def code(x_m):
	return math.pow((math.sin(x_m) / x_m), 2.0) / (math.cos(x_m) - -1.0)

Julia

x_m = abs(x)
function code(x_m)
	return Float64((Float64(sin(x_m) / x_m) ^ 2.0) / Float64(cos(x_m) - -1.0))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = ((sin(x_m) / x_m) ^ 2.0) / (cos(x_m) - -1.0);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[Power[N[(N[Sin[x$95$m], $MachinePrecision] / x$95$m), $MachinePrecision], 2.0], $MachinePrecision] / N[(N[Cos[x$95$m], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.5%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. pow2 N/A

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

   6. mult-flip N/A

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

   7. flip-- N/A

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

   8. associate-*l/ N/A

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

   9. lower-/.f64 N/A

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

3. Applied rewrites 50.2%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. associate-/r* N/A

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

   10. mult-flip-rev N/A

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

   11. frac-times N/A

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

   12. metadata-eval N/A

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

   13. pow2 N/A

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

   14. associate--r+ N/A

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

   15. metadata-eval N/A

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

   16. mult-flip N/A

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

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

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

   18. pow2 N/A

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

5. Applied rewrites 99.5%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-sin.f64 N/A

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

   6. pow2 N/A

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

   7. lower-pow.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. lift-/.f64 99.5

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

7. Applied rewrites 99.5%

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

   1. lift-+.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. metadata-eval N/A

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

   4. sub-flip-reverse N/A

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

   5. lower--.f64 N/A

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

   6. lift-cos.f64 99.5

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

9. Applied rewrites 99.5%

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

Alternative 4: 76.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 4.7:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2.48015873015873 \cdot 10^{-5}, x\_m \cdot x\_m, 0.001388888888888889\right), x\_m \cdot x\_m, -0.041666666666666664\right) \cdot x\_m, x\_m, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{x\_m}, \frac{1}{x\_m}, \frac{\left(-x\_m\right) \cdot 1}{\left(x\_m \cdot x\_m\right) \cdot x\_m}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 4.7)
   (fma
    (*
     (fma
      (fma -2.48015873015873e-5 (* x_m x_m) 0.001388888888888889)
      (* x_m x_m)
      -0.041666666666666664)
     x_m)
    x_m
    0.5)
   (fma (/ 1.0 x_m) (/ 1.0 x_m) (/ (* (- x_m) 1.0) (* (* x_m x_m) x_m)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 4.7) {
		tmp = fma((fma(fma(-2.48015873015873e-5, (x_m * x_m), 0.001388888888888889), (x_m * x_m), -0.041666666666666664) * x_m), x_m, 0.5);
	} else {
		tmp = fma((1.0 / x_m), (1.0 / x_m), ((-x_m * 1.0) / ((x_m * x_m) * x_m)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 4.7)
		tmp = fma(Float64(fma(fma(-2.48015873015873e-5, Float64(x_m * x_m), 0.001388888888888889), Float64(x_m * x_m), -0.041666666666666664) * x_m), x_m, 0.5);
	else
		tmp = fma(Float64(1.0 / x_m), Float64(1.0 / x_m), Float64(Float64(Float64(-x_m) * 1.0) / Float64(Float64(x_m * x_m) * x_m)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 4.7], N[(N[(N[(N[(-2.48015873015873e-5 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.001388888888888889), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + -0.041666666666666664), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m + 0.5), $MachinePrecision], N[(N[(1.0 / x$95$m), $MachinePrecision] * N[(1.0 / x$95$m), $MachinePrecision] + N[(N[((-x$95$m) * 1.0), $MachinePrecision] / N[(N[(x$95$m * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.70000000000000018

   1. Initial program 3.3%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. pow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

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

   if 4.70000000000000018 < x

   1. Initial program 98.6%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 51.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. pow2 N/A

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

      4. lift--.f64 N/A

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

      5. div-sub N/A

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

      6. pow2 N/A

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

      7. associate-/r* N/A

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

      8. frac-sub N/A

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

      9. pow2 N/A

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

      10. cube-mult N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. unpow3 N/A

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

      19. pow2 N/A

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

      20. lower-*.f64 N/A

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

   6. Applied rewrites 2.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. inv-pow N/A

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

      9. pow2 N/A

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

      10. pow-prod-up N/A

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

      11. metadata-eval N/A

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

      12. unpow1 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. pow3 N/A

          \[\leadsto \frac{x + \left(\mathsf{neg}\left(x\right)\right) \cdot 1}{\color{blue}{{x}^{3}}} \]

      16. div-add N/A

          \[\leadsto \color{blue}{\frac{x}{{x}^{3}} + \frac{\left(\mathsf{neg}\left(x\right)\right) \cdot 1}{{x}^{3}}} \]

   8. Applied rewrites 53.6%

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

Alternative 5: 76.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 5.2 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x\_m \cdot x\_m, -0.041666666666666664\right), x\_m \cdot x\_m, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x\_m \cdot x\_m} + \frac{\left(-x\_m\right) \cdot 1}{\left(x\_m \cdot x\_m\right) \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 5.2e+20)
   (fma
    (fma 0.001388888888888889 (* x_m x_m) -0.041666666666666664)
    (* x_m x_m)
    0.5)
   (+ (/ 1.0 (* x_m x_m)) (/ (* (- x_m) 1.0) (* (* x_m x_m) x_m)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 5.2e+20) {
		tmp = fma(fma(0.001388888888888889, (x_m * x_m), -0.041666666666666664), (x_m * x_m), 0.5);
	} else {
		tmp = (1.0 / (x_m * x_m)) + ((-x_m * 1.0) / ((x_m * x_m) * x_m));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 5.2e+20)
		tmp = fma(fma(0.001388888888888889, Float64(x_m * x_m), -0.041666666666666664), Float64(x_m * x_m), 0.5);
	else
		tmp = Float64(Float64(1.0 / Float64(x_m * x_m)) + Float64(Float64(Float64(-x_m) * 1.0) / Float64(Float64(x_m * x_m) * x_m)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 5.2e+20], N[(N[(0.001388888888888889 * N[(x$95$m * x$95$m), $MachinePrecision] + -0.041666666666666664), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(1.0 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[((-x$95$m) * 1.0), $MachinePrecision] / N[(N[(x$95$m * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.2e20

   1. Initial program 9.1%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. 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 94.0

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

   4. Applied rewrites 94.0%

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

   if 5.2e20 < x

   1. Initial program 98.6%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 54.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. pow2 N/A

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

      4. lift--.f64 N/A

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

      5. div-sub N/A

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

      6. pow2 N/A

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

      7. associate-/r* N/A

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

      8. frac-sub N/A

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

      9. pow2 N/A

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

      10. cube-mult N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. unpow3 N/A

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

      19. pow2 N/A

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

      20. lower-*.f64 N/A

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

   6. Applied rewrites 2.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. inv-pow N/A

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

      9. pow2 N/A

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

      10. pow-prod-up N/A

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

      11. metadata-eval N/A

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

      12. unpow1 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. pow3 N/A

          \[\leadsto \frac{x + \left(\mathsf{neg}\left(x\right)\right) \cdot 1}{\color{blue}{{x}^{3}}} \]

      16. div-add N/A

          \[\leadsto \color{blue}{\frac{x}{{x}^{3}} + \frac{\left(\mathsf{neg}\left(x\right)\right) \cdot 1}{{x}^{3}}} \]

   8. Applied rewrites 56.6%

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

Alternative 6: 76.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 90000000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x\_m \cdot x\_m, -0.041666666666666664\right), x\_m \cdot x\_m, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{x\_m}, \frac{1}{x\_m}, \frac{-1}{x\_m \cdot x\_m}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 90000000000.0)
   (fma
    (fma 0.001388888888888889 (* x_m x_m) -0.041666666666666664)
    (* x_m x_m)
    0.5)
   (fma (/ 1.0 x_m) (/ 1.0 x_m) (/ -1.0 (* x_m x_m)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 90000000000.0) {
		tmp = fma(fma(0.001388888888888889, (x_m * x_m), -0.041666666666666664), (x_m * x_m), 0.5);
	} else {
		tmp = fma((1.0 / x_m), (1.0 / x_m), (-1.0 / (x_m * x_m)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 90000000000.0)
		tmp = fma(fma(0.001388888888888889, Float64(x_m * x_m), -0.041666666666666664), Float64(x_m * x_m), 0.5);
	else
		tmp = fma(Float64(1.0 / x_m), Float64(1.0 / x_m), Float64(-1.0 / Float64(x_m * x_m)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 90000000000.0], N[(N[(0.001388888888888889 * N[(x$95$m * x$95$m), $MachinePrecision] + -0.041666666666666664), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(1.0 / x$95$m), $MachinePrecision] * N[(1.0 / x$95$m), $MachinePrecision] + N[(-1.0 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9e10

   1. Initial program 6.4%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. 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 96.7

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

   4. Applied rewrites 96.7%

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

   if 9e10 < x

   1. Initial program 98.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. sub-flip N/A

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

      6. pow2 N/A

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

      7. div-add N/A

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

      8. metadata-eval N/A

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

      9. pow2 N/A

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

      10. times-frac N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lift-cos.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 98.2

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

   3. Applied rewrites 98.2%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 53.5%

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

Alternative 7: 76.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 6 \cdot 10^{+26}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x\_m \cdot x\_m, -0.041666666666666664\right), x\_m \cdot x\_m, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{x\_m}, \frac{-1}{x\_m}, \frac{1}{x\_m \cdot x\_m}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 6e+26)
   (fma
    (fma 0.001388888888888889 (* x_m x_m) -0.041666666666666664)
    (* x_m x_m)
    0.5)
   (fma (/ 1.0 x_m) (/ -1.0 x_m) (/ 1.0 (* x_m x_m)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 6e+26) {
		tmp = fma(fma(0.001388888888888889, (x_m * x_m), -0.041666666666666664), (x_m * x_m), 0.5);
	} else {
		tmp = fma((1.0 / x_m), (-1.0 / x_m), (1.0 / (x_m * x_m)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 6e+26)
		tmp = fma(fma(0.001388888888888889, Float64(x_m * x_m), -0.041666666666666664), Float64(x_m * x_m), 0.5);
	else
		tmp = fma(Float64(1.0 / x_m), Float64(-1.0 / x_m), Float64(1.0 / Float64(x_m * x_m)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 6e+26], N[(N[(0.001388888888888889 * N[(x$95$m * x$95$m), $MachinePrecision] + -0.041666666666666664), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(1.0 / x$95$m), $MachinePrecision] * N[(-1.0 / x$95$m), $MachinePrecision] + N[(1.0 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.99999999999999994e26

   1. Initial program 11.0%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. 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 92.1

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

   4. Applied rewrites 92.1%

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

   if 5.99999999999999994e26 < x

   1. Initial program 98.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. pow2 N/A

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

      6. division-flip N/A

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

      7. associate-/r/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. lift--.f64 98.6

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

   3. Applied rewrites 98.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. mult-flip-rev N/A

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

      6. lift--.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. frac-times N/A

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

      10. metadata-eval N/A

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

      11. pow2 N/A

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

      12. mult-flip N/A

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

      13. pow2 N/A

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

      14. associate-/r* N/A

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

      15. sub-flip N/A

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

      16. +-commutative N/A

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

      17. div-add N/A

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

      18. div-add-rev N/A

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

      19. associate-/r* N/A

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

      20. mul-1-neg N/A

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

      21. *-commutative N/A

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

      22. times-frac N/A

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

      23. mult-flip-rev N/A

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in x around 0

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

      1. lift-/.f64 56.4

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

   8. Applied rewrites 56.4%

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

Alternative 8: 75.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 4.3 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x\_m \cdot x\_m, -0.041666666666666664\right), x\_m \cdot x\_m, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - 1}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 4.3e+38)
   (fma
    (fma 0.001388888888888889 (* x_m x_m) -0.041666666666666664)
    (* x_m x_m)
    0.5)
   (/ (- 1.0 1.0) (* x_m x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 4.3e+38) {
		tmp = fma(fma(0.001388888888888889, (x_m * x_m), -0.041666666666666664), (x_m * x_m), 0.5);
	} else {
		tmp = (1.0 - 1.0) / (x_m * x_m);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 4.3e+38)
		tmp = fma(fma(0.001388888888888889, Float64(x_m * x_m), -0.041666666666666664), Float64(x_m * x_m), 0.5);
	else
		tmp = Float64(Float64(1.0 - 1.0) / Float64(x_m * x_m));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 4.3e+38], N[(N[(0.001388888888888889 * N[(x$95$m * x$95$m), $MachinePrecision] + -0.041666666666666664), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(1.0 - 1.0), $MachinePrecision] / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.2999999999999997e38

   1. Initial program 13.8%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. 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 89.3

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

   4. Applied rewrites 89.3%

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

   if 4.2999999999999997e38 < x

   1. Initial program 98.6%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 57.9%

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

Alternative 9: 75.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 3.4:\\ \;\;\;\;\mathsf{fma}\left(x\_m \cdot x\_m, -0.041666666666666664, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - 1}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 3.4)
   (fma (* x_m x_m) -0.041666666666666664 0.5)
   (/ (- 1.0 1.0) (* x_m x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 3.4) {
		tmp = fma((x_m * x_m), -0.041666666666666664, 0.5);
	} else {
		tmp = (1.0 - 1.0) / (x_m * x_m);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 3.4)
		tmp = fma(Float64(x_m * x_m), -0.041666666666666664, 0.5);
	else
		tmp = Float64(Float64(1.0 - 1.0) / Float64(x_m * x_m));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 3.4], N[(N[(x$95$m * x$95$m), $MachinePrecision] * -0.041666666666666664 + 0.5), $MachinePrecision], N[(N[(1.0 - 1.0), $MachinePrecision] / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.39999999999999991

   1. Initial program 3.3%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 99.5

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

   4. Applied rewrites 99.5%

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

   if 3.39999999999999991 < x

   1. Initial program 98.6%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 51.0%

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

Alternative 10: 52.0% accurate, 41.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 0.5 \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 0.5)

C

x_m = fabs(x);
double code(double x_m) {
	return 0.5;
}

Fortran

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

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 0.5;
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 0.5

Julia

x_m = abs(x)
function code(x_m)
	return 0.5
end

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := 0.5

Derivation

1. Initial program 50.5%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 52.0%

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

Reproduce

herbie shell --seed 2025127 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))