cos2 (problem 3.4.1)

Percentage Accurate: 50.5% → 99.6%

Time: 3.8s

Alternatives: 6

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: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.02:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\mathsf{fma}\left(-2.48015873015873 \cdot 10^{-5}, x\_m \cdot x\_m, 0.001388888888888889\right) \cdot x\_m\right) \cdot x\_m - 0.041666666666666664\right) \cdot x\_m, x\_m, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \left(\frac{x\_m}{2}\right) \cdot \frac{\sin x\_m}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.02)
   (fma
    (*
     (-
      (*
       (* (fma -2.48015873015873e-5 (* x_m x_m) 0.001388888888888889) x_m)
       x_m)
      0.041666666666666664)
     x_m)
    x_m
    0.5)
   (* (tan (/ x_m 2.0)) (/ (sin x_m) (* x_m x_m)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.02) {
		tmp = 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 = tan((x_m / 2.0)) * (sin(x_m) / (x_m * x_m));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.02)
		tmp = fma(Float64(Float64(Float64(Float64(fma(-2.48015873015873e-5, Float64(x_m * x_m), 0.001388888888888889) * x_m) * x_m) - 0.041666666666666664) * x_m), x_m, 0.5);
	else
		tmp = Float64(tan(Float64(x_m / 2.0)) * Float64(sin(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, 0.02], N[(N[(N[(N[(N[(N[(-2.48015873015873e-5 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.001388888888888889), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] - 0.041666666666666664), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m + 0.5), $MachinePrecision], N[(N[Tan[N[(x$95$m / 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Sin[x$95$m], $MachinePrecision] / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0200000000000000004

   1. Initial program 50.5%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}\right)} \]
   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 51.0%

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

   if 0.0200000000000000004 < x

   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. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log.f64 48.5

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

   3. Applied rewrites 48.5%

      \[\leadsto \frac{1 - \cos x}{\color{blue}{e^{\log x \cdot 2}}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lower--.f64 N/A

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

      4. flip-- N/A

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

      5. metadata-eval N/A

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

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

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

      7. +-commutative N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. pow-to-exp N/A

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

      12. associate-/l/ N/A

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

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

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

      14. 1-sub-cos N/A

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

      15. times-frac N/A

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

      16. lower-*.f64 N/A

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

   5. Applied rewrites 75.7%

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

   6. Taylor expanded in x around inf

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

      1. hang-0p-tan N/A

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

      2. lower-tan.f64 N/A

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

      3. lower-/.f64 76.0

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

   8. Applied rewrites 76.0%

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

Alternative 2: 99.5% accurate, 0.4× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (/ (* (sin x_m) (/ (sin x_m) x_m)) (fma (cos x_m) x_m x_m)))

C

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

Julia

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[(N[Sin[x$95$m], $MachinePrecision] * N[(N[Sin[x$95$m], $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[x$95$m], $MachinePrecision] * x$95$m + x$95$m), $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. pow2 N/A

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

   3. pow-to-exp N/A

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

   4. lower-exp.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-log.f64 48.5

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

3. Applied rewrites 48.5%

   \[\leadsto \frac{1 - \cos x}{\color{blue}{e^{\log x \cdot 2}}} \]
4. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lower--.f64 N/A

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

   4. flip-- N/A

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

   5. metadata-eval N/A

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

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

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

   7. +-commutative N/A

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

   8. lift-exp.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-log.f64 N/A

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

   11. pow-to-exp N/A

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

   12. associate-/l/ N/A

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

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

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

   14. 1-sub-cos N/A

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

   15. times-frac N/A

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

   16. lower-*.f64 N/A

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

5. Applied rewrites 75.7%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-sin.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-/r* N/A

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

   10. frac-times N/A

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

   11. lower-/.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lift-sin.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lift-sin.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. lift-cos.f64 N/A

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

   18. lift--.f64 99.6

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

7. Applied rewrites 99.6%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. metadata-eval N/A

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

   5. fp-cancel-sub-sign N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. +-commutative N/A

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

   9. *-commutative N/A

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

   10. +-commutative N/A

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

   11. distribute-rgt-in N/A

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

   12. *-lft-identity N/A

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

   13. lower-fma.f64 N/A

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

   14. lift-cos.f64 99.6

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

9. Applied rewrites 99.6%

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

Alternative 3: 99.3% accurate, 0.9× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.105)
   (fma
    (*
     (-
      (*
       (* (fma -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(-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(Float64(Float64(Float64(fma(-2.48015873015873e-5, Float64(x_m * x_m), 0.001388888888888889) * 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[(N[(N[(-2.48015873015873e-5 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.001388888888888889), $MachinePrecision] * x$95$m), $MachinePrecision] * 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 50.5%

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

   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 51.0%

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

   if 0.104999999999999996 < x

   1. Initial program 50.5%

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

Alternative 4: 75.7% accurate, 2.0× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 4.3e+38)
   (fma
    (- (* 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(((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(Float64(Float64(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[(N[(0.001388888888888889 * N[(x$95$m * x$95$m), $MachinePrecision]), $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 50.5%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 51.5

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

   4. Applied rewrites 51.5%

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

   if 4.2999999999999997e38 < x

   1. Initial program 50.5%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 26.1%

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

Alternative 5: 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 50.5%

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

   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 51.0

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

   4. Applied rewrites 51.0%

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

   if 3.39999999999999991 < x

   1. Initial program 50.5%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 26.1%

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

Alternative 6: 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)))