sintan (problem 3.4.5)

Percentage Accurate: 1.6% → 99.9%

Time: 10.6s

Alternatives: 4

Speedup: 9.0×

Specification

\[-0.4 \leq \varepsilon \land \varepsilon \leq 0.4\]

Math

\[\begin{array}{l} \\ \frac{\varepsilon - \sin \varepsilon}{\varepsilon - \tan \varepsilon} \end{array} \]

FPCore

(FPCore (eps) :precision binary64 (/ (- eps (sin eps)) (- eps (tan eps))))

C

double code(double eps) {
	return (eps - sin(eps)) / (eps - tan(eps));
}

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(eps)
use fmin_fmax_functions
    real(8), intent (in) :: eps
    code = (eps - sin(eps)) / (eps - tan(eps))
end function

Java

public static double code(double eps) {
	return (eps - Math.sin(eps)) / (eps - Math.tan(eps));
}

Python

def code(eps):
	return (eps - math.sin(eps)) / (eps - math.tan(eps))

Julia

function code(eps)
	return Float64(Float64(eps - sin(eps)) / Float64(eps - tan(eps)))
end

MATLAB

function tmp = code(eps)
	tmp = (eps - sin(eps)) / (eps - tan(eps));
end

Wolfram

code[eps_] := N[(N[(eps - N[Sin[eps], $MachinePrecision]), $MachinePrecision] / N[(eps - N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 1.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\varepsilon - \sin \varepsilon}{\varepsilon - \tan \varepsilon} \end{array} \]

FPCore

(FPCore (eps) :precision binary64 (/ (- eps (sin eps)) (- eps (tan eps))))

C

double code(double eps) {
	return (eps - sin(eps)) / (eps - tan(eps));
}

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(eps)
use fmin_fmax_functions
    real(8), intent (in) :: eps
    code = (eps - sin(eps)) / (eps - tan(eps))
end function

Java

public static double code(double eps) {
	return (eps - Math.sin(eps)) / (eps - Math.tan(eps));
}

Python

def code(eps):
	return (eps - math.sin(eps)) / (eps - math.tan(eps))

Julia

function code(eps)
	return Float64(Float64(eps - sin(eps)) / Float64(eps - tan(eps)))
end

MATLAB

function tmp = code(eps)
	tmp = (eps - sin(eps)) / (eps - tan(eps));
end

Wolfram

code[eps_] := N[(N[(eps - N[Sin[eps], $MachinePrecision]), $MachinePrecision] / N[(eps - N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.00024107142857142857, -0.009642857142857142\right) \cdot \varepsilon, \varepsilon, 0.225\right), -0.5\right) \end{array} \]

FPCore

(FPCore (eps)
 :precision binary64
 (fma
  (* eps eps)
  (fma
   (* (fma (* eps eps) 0.00024107142857142857 -0.009642857142857142) eps)
   eps
   0.225)
  -0.5))

C

double code(double eps) {
	return fma((eps * eps), fma((fma((eps * eps), 0.00024107142857142857, -0.009642857142857142) * eps), eps, 0.225), -0.5);
}

Julia

function code(eps)
	return fma(Float64(eps * eps), fma(Float64(fma(Float64(eps * eps), 0.00024107142857142857, -0.009642857142857142) * eps), eps, 0.225), -0.5)
end

Wolfram

code[eps_] := N[(N[(eps * eps), $MachinePrecision] * N[(N[(N[(N[(eps * eps), $MachinePrecision] * 0.00024107142857142857 + -0.009642857142857142), $MachinePrecision] * eps), $MachinePrecision] * eps + 0.225), $MachinePrecision] + -0.5), $MachinePrecision]

Derivation

1. Initial program 1.6%

   \[\frac{\varepsilon - \sin \varepsilon}{\varepsilon - \tan \varepsilon} \]

2. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{{\varepsilon}^{2} \cdot \left(\frac{9}{40} + {\varepsilon}^{2} \cdot \left(\frac{27}{112000} \cdot {\varepsilon}^{2} - \frac{27}{2800}\right)\right) - \frac{1}{2}} \]
3. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto {\varepsilon}^{2} \cdot \left(\frac{9}{40} + {\varepsilon}^{2} \cdot \left(\frac{27}{112000} \cdot {\varepsilon}^{2} - \frac{27}{2800}\right)\right) - \color{blue}{\frac{1}{2}} \]

   2. *-commutative N/A

      \[\leadsto \left(\frac{9}{40} + {\varepsilon}^{2} \cdot \left(\frac{27}{112000} \cdot {\varepsilon}^{2} - \frac{27}{2800}\right)\right) \cdot {\varepsilon}^{2} - \frac{1}{2} \]

   3. lower-*.f64 N/A

      \[\leadsto \left(\frac{9}{40} + {\varepsilon}^{2} \cdot \left(\frac{27}{112000} \cdot {\varepsilon}^{2} - \frac{27}{2800}\right)\right) \cdot {\varepsilon}^{2} - \frac{1}{2} \]

   4. +-commutative N/A

      \[\leadsto \left({\varepsilon}^{2} \cdot \left(\frac{27}{112000} \cdot {\varepsilon}^{2} - \frac{27}{2800}\right) + \frac{9}{40}\right) \cdot {\varepsilon}^{2} - \frac{1}{2} \]

   5. *-commutative N/A

      \[\leadsto \left(\left(\frac{27}{112000} \cdot {\varepsilon}^{2} - \frac{27}{2800}\right) \cdot {\varepsilon}^{2} + \frac{9}{40}\right) \cdot {\varepsilon}^{2} - \frac{1}{2} \]

   6. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{27}{112000} \cdot {\varepsilon}^{2} - \frac{27}{2800}, {\varepsilon}^{2}, \frac{9}{40}\right) \cdot {\varepsilon}^{2} - \frac{1}{2} \]

   7. lower--.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{27}{112000} \cdot {\varepsilon}^{2} - \frac{27}{2800}, {\varepsilon}^{2}, \frac{9}{40}\right) \cdot {\varepsilon}^{2} - \frac{1}{2} \]

   8. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{27}{112000} \cdot {\varepsilon}^{2} - \frac{27}{2800}, {\varepsilon}^{2}, \frac{9}{40}\right) \cdot {\varepsilon}^{2} - \frac{1}{2} \]

   9. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\frac{27}{112000} \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{27}{2800}, {\varepsilon}^{2}, \frac{9}{40}\right) \cdot {\varepsilon}^{2} - \frac{1}{2} \]

   10. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{27}{112000} \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{27}{2800}, {\varepsilon}^{2}, \frac{9}{40}\right) \cdot {\varepsilon}^{2} - \frac{1}{2} \]

   11. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\frac{27}{112000} \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{27}{2800}, \varepsilon \cdot \varepsilon, \frac{9}{40}\right) \cdot {\varepsilon}^{2} - \frac{1}{2} \]

   12. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{27}{112000} \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{27}{2800}, \varepsilon \cdot \varepsilon, \frac{9}{40}\right) \cdot {\varepsilon}^{2} - \frac{1}{2} \]

   13. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\frac{27}{112000} \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{27}{2800}, \varepsilon \cdot \varepsilon, \frac{9}{40}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{2} \]

   14. lower-*.f64 99.9

       \[\leadsto \mathsf{fma}\left(0.00024107142857142857 \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.009642857142857142, \varepsilon \cdot \varepsilon, 0.225\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.5 \]

4. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.00024107142857142857 \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.009642857142857142, \varepsilon \cdot \varepsilon, 0.225\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.5} \]

5. Taylor expanded in eps around inf

   \[\leadsto {\varepsilon}^{6} \cdot \color{blue}{\left(\left(\frac{27}{112000} + \frac{\frac{9}{40}}{{\varepsilon}^{4}}\right) - \left(\frac{27}{2800} \cdot \frac{1}{{\varepsilon}^{2}} + \frac{1}{2} \cdot \frac{1}{{\varepsilon}^{6}}\right)\right)} \]
6. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto {\varepsilon}^{\left(3 + 3\right)} \cdot \left(\left(\frac{27}{112000} + \frac{\frac{9}{40}}{{\varepsilon}^{4}}\right) - \left(\frac{27}{2800} \cdot \color{blue}{\frac{1}{{\varepsilon}^{2}}} + \frac{1}{2} \cdot \frac{1}{{\varepsilon}^{6}}\right)\right) \]

   2. pow-prod-up N/A

      \[\leadsto \left({\varepsilon}^{3} \cdot {\varepsilon}^{3}\right) \cdot \left(\left(\frac{27}{112000} + \frac{\frac{9}{40}}{{\varepsilon}^{4}}\right) - \left(\color{blue}{\frac{27}{2800} \cdot \frac{1}{{\varepsilon}^{2}}} + \frac{1}{2} \cdot \frac{1}{{\varepsilon}^{6}}\right)\right) \]

   3. pow-prod-down N/A

      \[\leadsto {\left(\varepsilon \cdot \varepsilon\right)}^{3} \cdot \left(\left(\frac{27}{112000} + \frac{\frac{9}{40}}{{\varepsilon}^{4}}\right) - \left(\color{blue}{\frac{27}{2800} \cdot \frac{1}{{\varepsilon}^{2}}} + \frac{1}{2} \cdot \frac{1}{{\varepsilon}^{6}}\right)\right) \]

   4. pow2 N/A

      \[\leadsto {\left({\varepsilon}^{2}\right)}^{3} \cdot \left(\left(\frac{27}{112000} + \frac{\frac{9}{40}}{{\varepsilon}^{4}}\right) - \left(\color{blue}{\frac{27}{2800}} \cdot \frac{1}{{\varepsilon}^{2}} + \frac{1}{2} \cdot \frac{1}{{\varepsilon}^{6}}\right)\right) \]

   5. lower-*.f64 N/A

      \[\leadsto {\left({\varepsilon}^{2}\right)}^{3} \cdot \left(\left(\frac{27}{112000} + \frac{\frac{9}{40}}{{\varepsilon}^{4}}\right) - \color{blue}{\left(\frac{27}{2800} \cdot \frac{1}{{\varepsilon}^{2}} + \frac{1}{2} \cdot \frac{1}{{\varepsilon}^{6}}\right)}\right) \]

   6. pow2 N/A

      \[\leadsto {\left(\varepsilon \cdot \varepsilon\right)}^{3} \cdot \left(\left(\frac{27}{112000} + \frac{\frac{9}{40}}{{\varepsilon}^{4}}\right) - \left(\color{blue}{\frac{27}{2800}} \cdot \frac{1}{{\varepsilon}^{2}} + \frac{1}{2} \cdot \frac{1}{{\varepsilon}^{6}}\right)\right) \]

   7. pow-prod-down N/A

      \[\leadsto \left({\varepsilon}^{3} \cdot {\varepsilon}^{3}\right) \cdot \left(\left(\frac{27}{112000} + \frac{\frac{9}{40}}{{\varepsilon}^{4}}\right) - \left(\color{blue}{\frac{27}{2800} \cdot \frac{1}{{\varepsilon}^{2}}} + \frac{1}{2} \cdot \frac{1}{{\varepsilon}^{6}}\right)\right) \]

   8. pow-prod-up N/A

      \[\leadsto {\varepsilon}^{\left(3 + 3\right)} \cdot \left(\left(\frac{27}{112000} + \frac{\frac{9}{40}}{{\varepsilon}^{4}}\right) - \left(\color{blue}{\frac{27}{2800} \cdot \frac{1}{{\varepsilon}^{2}}} + \frac{1}{2} \cdot \frac{1}{{\varepsilon}^{6}}\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto {\varepsilon}^{6} \cdot \left(\left(\frac{27}{112000} + \frac{\frac{9}{40}}{{\varepsilon}^{4}}\right) - \left(\frac{27}{2800} \cdot \color{blue}{\frac{1}{{\varepsilon}^{2}}} + \frac{1}{2} \cdot \frac{1}{{\varepsilon}^{6}}\right)\right) \]

   10. lower-pow.f64 N/A

       \[\leadsto {\varepsilon}^{6} \cdot \left(\left(\frac{27}{112000} + \frac{\frac{9}{40}}{{\varepsilon}^{4}}\right) - \left(\color{blue}{\frac{27}{2800} \cdot \frac{1}{{\varepsilon}^{2}}} + \frac{1}{2} \cdot \frac{1}{{\varepsilon}^{6}}\right)\right) \]

   11. lower--.f64 N/A

       \[\leadsto {\varepsilon}^{6} \cdot \left(\left(\frac{27}{112000} + \frac{\frac{9}{40}}{{\varepsilon}^{4}}\right) - \left(\frac{27}{2800} \cdot \frac{1}{{\varepsilon}^{2}} + \color{blue}{\frac{1}{2} \cdot \frac{1}{{\varepsilon}^{6}}}\right)\right) \]

7. Applied rewrites 16.3%

   \[\leadsto {\varepsilon}^{6} \cdot \color{blue}{\left(\left(\frac{0.225}{{\varepsilon}^{4}} + 0.00024107142857142857\right) - \mathsf{fma}\left({\varepsilon}^{-6}, 0.5, {\varepsilon}^{-2} \cdot 0.009642857142857142\right)\right)} \]
8. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto {\varepsilon}^{6} \cdot \left(\left(\frac{\frac{9}{40}}{{\varepsilon}^{4}} + \frac{27}{112000}\right) - \mathsf{fma}\left({\varepsilon}^{-6}, \color{blue}{\frac{1}{2}}, {\varepsilon}^{-2} \cdot \frac{27}{2800}\right)\right) \]

   2. lift-+.f64 N/A

      \[\leadsto {\varepsilon}^{6} \cdot \left(\left(\frac{\frac{9}{40}}{{\varepsilon}^{4}} + \frac{27}{112000}\right) - \mathsf{fma}\left({\varepsilon}^{-6}, \frac{1}{2}, {\varepsilon}^{-2} \cdot \frac{27}{2800}\right)\right) \]

   3. lift-/.f64 N/A

      \[\leadsto {\varepsilon}^{6} \cdot \left(\left(\frac{\frac{9}{40}}{{\varepsilon}^{4}} + \frac{27}{112000}\right) - \mathsf{fma}\left({\varepsilon}^{-6}, \frac{1}{2}, {\varepsilon}^{-2} \cdot \frac{27}{2800}\right)\right) \]

   4. lift-pow.f64 N/A

      \[\leadsto {\varepsilon}^{6} \cdot \left(\left(\frac{\frac{9}{40}}{{\varepsilon}^{4}} + \frac{27}{112000}\right) - \mathsf{fma}\left({\varepsilon}^{-6}, \frac{1}{2}, {\varepsilon}^{-2} \cdot \frac{27}{2800}\right)\right) \]

   5. lift-pow.f64 N/A

      \[\leadsto {\varepsilon}^{6} \cdot \left(\left(\frac{\frac{9}{40}}{{\varepsilon}^{4}} + \frac{27}{112000}\right) - \mathsf{fma}\left({\varepsilon}^{-6}, \frac{1}{2}, {\varepsilon}^{-2} \cdot \frac{27}{2800}\right)\right) \]

   6. lift-fma.f64 N/A

      \[\leadsto {\varepsilon}^{6} \cdot \left(\left(\frac{\frac{9}{40}}{{\varepsilon}^{4}} + \frac{27}{112000}\right) - \left({\varepsilon}^{-6} \cdot \frac{1}{2} + {\varepsilon}^{-2} \cdot \color{blue}{\frac{27}{2800}}\right)\right) \]

   7. lift-*.f64 N/A

      \[\leadsto {\varepsilon}^{6} \cdot \left(\left(\frac{\frac{9}{40}}{{\varepsilon}^{4}} + \frac{27}{112000}\right) - \left({\varepsilon}^{-6} \cdot \frac{1}{2} + {\varepsilon}^{-2} \cdot \frac{27}{2800}\right)\right) \]

   8. lift-pow.f64 N/A

      \[\leadsto {\varepsilon}^{6} \cdot \left(\left(\frac{\frac{9}{40}}{{\varepsilon}^{4}} + \frac{27}{112000}\right) - \left({\varepsilon}^{-6} \cdot \frac{1}{2} + {\varepsilon}^{-2} \cdot \frac{27}{2800}\right)\right) \]

   9. associate--l+ N/A

      \[\leadsto {\varepsilon}^{6} \cdot \left(\frac{\frac{9}{40}}{{\varepsilon}^{4}} + \left(\frac{27}{112000} - \color{blue}{\left({\varepsilon}^{-6} \cdot \frac{1}{2} + {\varepsilon}^{-2} \cdot \frac{27}{2800}\right)}\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto {\varepsilon}^{6} \cdot \left(\frac{\frac{9}{40} \cdot 1}{{\varepsilon}^{4}} + \left(\frac{27}{112000} - \left(\color{blue}{{\varepsilon}^{-6}} \cdot \frac{1}{2} + {\varepsilon}^{-2} \cdot \frac{27}{2800}\right)\right)\right) \]

   11. associate-*r/ N/A

       \[\leadsto {\varepsilon}^{6} \cdot \left(\frac{9}{40} \cdot \frac{1}{{\varepsilon}^{4}} + \left(\frac{27}{112000} - \left(\color{blue}{{\varepsilon}^{-6} \cdot \frac{1}{2}} + {\varepsilon}^{-2} \cdot \frac{27}{2800}\right)\right)\right) \]

   12. *-commutative N/A

       \[\leadsto {\varepsilon}^{6} \cdot \left(\frac{1}{{\varepsilon}^{4}} \cdot \frac{9}{40} + \left(\frac{27}{112000} - \left(\color{blue}{{\varepsilon}^{-6} \cdot \frac{1}{2}} + {\varepsilon}^{-2} \cdot \frac{27}{2800}\right)\right)\right) \]

   13. lower-fma.f64 N/A

       \[\leadsto {\varepsilon}^{6} \cdot \mathsf{fma}\left(\frac{1}{{\varepsilon}^{4}}, \frac{9}{40}, \frac{27}{112000} - \left({\varepsilon}^{-6} \cdot \frac{1}{2} + {\varepsilon}^{-2} \cdot \frac{27}{2800}\right)\right) \]

   14. pow-flip N/A

       \[\leadsto {\varepsilon}^{6} \cdot \mathsf{fma}\left({\varepsilon}^{\left(\mathsf{neg}\left(4\right)\right)}, \frac{9}{40}, \frac{27}{112000} - \left({\varepsilon}^{-6} \cdot \frac{1}{2} + {\varepsilon}^{-2} \cdot \frac{27}{2800}\right)\right) \]

   15. lower-pow.f64 N/A

       \[\leadsto {\varepsilon}^{6} \cdot \mathsf{fma}\left({\varepsilon}^{\left(\mathsf{neg}\left(4\right)\right)}, \frac{9}{40}, \frac{27}{112000} - \left({\varepsilon}^{-6} \cdot \frac{1}{2} + {\varepsilon}^{-2} \cdot \frac{27}{2800}\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto {\varepsilon}^{6} \cdot \mathsf{fma}\left({\varepsilon}^{-4}, \frac{9}{40}, \frac{27}{112000} - \left({\varepsilon}^{-6} \cdot \frac{1}{2} + {\varepsilon}^{-2} \cdot \frac{27}{2800}\right)\right) \]

   17. lower--.f64 N/A

       \[\leadsto {\varepsilon}^{6} \cdot \mathsf{fma}\left({\varepsilon}^{-4}, \frac{9}{40}, \frac{27}{112000} - \left({\varepsilon}^{-6} \cdot \frac{1}{2} + {\varepsilon}^{-2} \cdot \frac{27}{2800}\right)\right) \]

   18. lift-pow.f64 N/A

       \[\leadsto {\varepsilon}^{6} \cdot \mathsf{fma}\left({\varepsilon}^{-4}, \frac{9}{40}, \frac{27}{112000} - \left({\varepsilon}^{-6} \cdot \frac{1}{2} + {\varepsilon}^{-2} \cdot \frac{27}{2800}\right)\right) \]

   19. lift-*.f64 N/A

       \[\leadsto {\varepsilon}^{6} \cdot \mathsf{fma}\left({\varepsilon}^{-4}, \frac{9}{40}, \frac{27}{112000} - \left({\varepsilon}^{-6} \cdot \frac{1}{2} + {\varepsilon}^{-2} \cdot \frac{27}{2800}\right)\right) \]

9. Applied rewrites 16.3%

   \[\leadsto {\varepsilon}^{6} \cdot \mathsf{fma}\left({\varepsilon}^{-4}, 0.225, 0.00024107142857142857 - \mathsf{fma}\left({\varepsilon}^{-6}, 0.5, {\varepsilon}^{-2} \cdot 0.009642857142857142\right)\right) \]

10. Taylor expanded in eps around 0

    \[\leadsto \color{blue}{{\varepsilon}^{2} \cdot \left(\frac{9}{40} + {\varepsilon}^{2} \cdot \left(\frac{27}{112000} \cdot {\varepsilon}^{2} - \frac{27}{2800}\right)\right) - \frac{1}{2}} \]
11. Step-by-step derivation

    1. metadata-eval N/A

       \[\leadsto {\varepsilon}^{2} \cdot \left(\frac{9}{40} + {\varepsilon}^{2} \cdot \left(\frac{27}{112000} \cdot {\varepsilon}^{2} - \frac{27}{2800}\right)\right) - \frac{1}{2} \cdot \color{blue}{1} \]

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

       \[\leadsto {\varepsilon}^{2} \cdot \left(\frac{9}{40} + {\varepsilon}^{2} \cdot \left(\frac{27}{112000} \cdot {\varepsilon}^{2} - \frac{27}{2800}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot 1} \]

    3. metadata-eval N/A

       \[\leadsto {\varepsilon}^{2} \cdot \left(\frac{9}{40} + {\varepsilon}^{2} \cdot \left(\frac{27}{112000} \cdot {\varepsilon}^{2} - \frac{27}{2800}\right)\right) + \frac{-1}{2} \cdot 1 \]

    4. metadata-eval N/A

       \[\leadsto {\varepsilon}^{2} \cdot \left(\frac{9}{40} + {\varepsilon}^{2} \cdot \left(\frac{27}{112000} \cdot {\varepsilon}^{2} - \frac{27}{2800}\right)\right) + \frac{-1}{2} \]

    5. pow2 N/A

       \[\leadsto {\varepsilon}^{2} \cdot \left(\frac{9}{40} + {\varepsilon}^{2} \cdot \left(\frac{27}{112000} \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{27}{2800}\right)\right) + \frac{-1}{2} \]

    6. *-commutative N/A

       \[\leadsto {\varepsilon}^{2} \cdot \left(\frac{9}{40} + \left(\frac{27}{112000} \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{27}{2800}\right) \cdot {\varepsilon}^{2}\right) + \frac{-1}{2} \]

    7. pow2 N/A

       \[\leadsto {\varepsilon}^{2} \cdot \left(\frac{9}{40} + \left(\frac{27}{112000} \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{27}{2800}\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + \frac{-1}{2} \]

    8. +-commutative N/A

       \[\leadsto {\varepsilon}^{2} \cdot \left(\left(\frac{27}{112000} \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{27}{2800}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) + \frac{9}{40}\right) + \frac{-1}{2} \]

    9. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left({\varepsilon}^{2}, \color{blue}{\left(\frac{27}{112000} \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{27}{2800}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) + \frac{9}{40}}, \frac{-1}{2}\right) \]

12. Applied rewrites 99.9%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.00024107142857142857, -0.009642857142857142\right) \cdot \varepsilon, \varepsilon, 0.225\right), -0.5\right)} \]
13. Add Preprocessing

Alternative 2: 99.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(-0.009642857142857142, \varepsilon \cdot \varepsilon, 0.225\right), \varepsilon \cdot \varepsilon, -0.5\right) \end{array} \]

FPCore

(FPCore (eps)
 :precision binary64
 (fma (fma -0.009642857142857142 (* eps eps) 0.225) (* eps eps) -0.5))

C

double code(double eps) {
	return fma(fma(-0.009642857142857142, (eps * eps), 0.225), (eps * eps), -0.5);
}

Julia

function code(eps)
	return fma(fma(-0.009642857142857142, Float64(eps * eps), 0.225), Float64(eps * eps), -0.5)
end

Wolfram

code[eps_] := N[(N[(-0.009642857142857142 * N[(eps * eps), $MachinePrecision] + 0.225), $MachinePrecision] * N[(eps * eps), $MachinePrecision] + -0.5), $MachinePrecision]

Derivation

1. Initial program 1.6%

   \[\frac{\varepsilon - \sin \varepsilon}{\varepsilon - \tan \varepsilon} \]

2. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{{\varepsilon}^{2} \cdot \left(\frac{9}{40} + {\varepsilon}^{2} \cdot \left(\frac{27}{112000} \cdot {\varepsilon}^{2} - \frac{27}{2800}\right)\right) - \frac{1}{2}} \]
3. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto {\varepsilon}^{2} \cdot \left(\frac{9}{40} + {\varepsilon}^{2} \cdot \left(\frac{27}{112000} \cdot {\varepsilon}^{2} - \frac{27}{2800}\right)\right) - \color{blue}{\frac{1}{2}} \]

   2. *-commutative N/A

      \[\leadsto \left(\frac{9}{40} + {\varepsilon}^{2} \cdot \left(\frac{27}{112000} \cdot {\varepsilon}^{2} - \frac{27}{2800}\right)\right) \cdot {\varepsilon}^{2} - \frac{1}{2} \]

   3. lower-*.f64 N/A

      \[\leadsto \left(\frac{9}{40} + {\varepsilon}^{2} \cdot \left(\frac{27}{112000} \cdot {\varepsilon}^{2} - \frac{27}{2800}\right)\right) \cdot {\varepsilon}^{2} - \frac{1}{2} \]

   4. +-commutative N/A

      \[\leadsto \left({\varepsilon}^{2} \cdot \left(\frac{27}{112000} \cdot {\varepsilon}^{2} - \frac{27}{2800}\right) + \frac{9}{40}\right) \cdot {\varepsilon}^{2} - \frac{1}{2} \]

   5. *-commutative N/A

      \[\leadsto \left(\left(\frac{27}{112000} \cdot {\varepsilon}^{2} - \frac{27}{2800}\right) \cdot {\varepsilon}^{2} + \frac{9}{40}\right) \cdot {\varepsilon}^{2} - \frac{1}{2} \]

   6. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{27}{112000} \cdot {\varepsilon}^{2} - \frac{27}{2800}, {\varepsilon}^{2}, \frac{9}{40}\right) \cdot {\varepsilon}^{2} - \frac{1}{2} \]

   7. lower--.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{27}{112000} \cdot {\varepsilon}^{2} - \frac{27}{2800}, {\varepsilon}^{2}, \frac{9}{40}\right) \cdot {\varepsilon}^{2} - \frac{1}{2} \]

   8. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{27}{112000} \cdot {\varepsilon}^{2} - \frac{27}{2800}, {\varepsilon}^{2}, \frac{9}{40}\right) \cdot {\varepsilon}^{2} - \frac{1}{2} \]

   9. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\frac{27}{112000} \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{27}{2800}, {\varepsilon}^{2}, \frac{9}{40}\right) \cdot {\varepsilon}^{2} - \frac{1}{2} \]

   10. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{27}{112000} \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{27}{2800}, {\varepsilon}^{2}, \frac{9}{40}\right) \cdot {\varepsilon}^{2} - \frac{1}{2} \]

   11. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\frac{27}{112000} \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{27}{2800}, \varepsilon \cdot \varepsilon, \frac{9}{40}\right) \cdot {\varepsilon}^{2} - \frac{1}{2} \]

   12. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{27}{112000} \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{27}{2800}, \varepsilon \cdot \varepsilon, \frac{9}{40}\right) \cdot {\varepsilon}^{2} - \frac{1}{2} \]

   13. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\frac{27}{112000} \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{27}{2800}, \varepsilon \cdot \varepsilon, \frac{9}{40}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{2} \]

   14. lower-*.f64 99.9

       \[\leadsto \mathsf{fma}\left(0.00024107142857142857 \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.009642857142857142, \varepsilon \cdot \varepsilon, 0.225\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.5 \]

4. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.00024107142857142857 \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.009642857142857142, \varepsilon \cdot \varepsilon, 0.225\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.5} \]

5. Taylor expanded in eps around inf

   \[\leadsto {\varepsilon}^{6} \cdot \color{blue}{\left(\left(\frac{27}{112000} + \frac{\frac{9}{40}}{{\varepsilon}^{4}}\right) - \left(\frac{27}{2800} \cdot \frac{1}{{\varepsilon}^{2}} + \frac{1}{2} \cdot \frac{1}{{\varepsilon}^{6}}\right)\right)} \]
6. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto {\varepsilon}^{\left(3 + 3\right)} \cdot \left(\left(\frac{27}{112000} + \frac{\frac{9}{40}}{{\varepsilon}^{4}}\right) - \left(\frac{27}{2800} \cdot \color{blue}{\frac{1}{{\varepsilon}^{2}}} + \frac{1}{2} \cdot \frac{1}{{\varepsilon}^{6}}\right)\right) \]

   2. pow-prod-up N/A

      \[\leadsto \left({\varepsilon}^{3} \cdot {\varepsilon}^{3}\right) \cdot \left(\left(\frac{27}{112000} + \frac{\frac{9}{40}}{{\varepsilon}^{4}}\right) - \left(\color{blue}{\frac{27}{2800} \cdot \frac{1}{{\varepsilon}^{2}}} + \frac{1}{2} \cdot \frac{1}{{\varepsilon}^{6}}\right)\right) \]

   3. pow-prod-down N/A

      \[\leadsto {\left(\varepsilon \cdot \varepsilon\right)}^{3} \cdot \left(\left(\frac{27}{112000} + \frac{\frac{9}{40}}{{\varepsilon}^{4}}\right) - \left(\color{blue}{\frac{27}{2800} \cdot \frac{1}{{\varepsilon}^{2}}} + \frac{1}{2} \cdot \frac{1}{{\varepsilon}^{6}}\right)\right) \]

   4. pow2 N/A

      \[\leadsto {\left({\varepsilon}^{2}\right)}^{3} \cdot \left(\left(\frac{27}{112000} + \frac{\frac{9}{40}}{{\varepsilon}^{4}}\right) - \left(\color{blue}{\frac{27}{2800}} \cdot \frac{1}{{\varepsilon}^{2}} + \frac{1}{2} \cdot \frac{1}{{\varepsilon}^{6}}\right)\right) \]

   5. lower-*.f64 N/A

      \[\leadsto {\left({\varepsilon}^{2}\right)}^{3} \cdot \left(\left(\frac{27}{112000} + \frac{\frac{9}{40}}{{\varepsilon}^{4}}\right) - \color{blue}{\left(\frac{27}{2800} \cdot \frac{1}{{\varepsilon}^{2}} + \frac{1}{2} \cdot \frac{1}{{\varepsilon}^{6}}\right)}\right) \]

   6. pow2 N/A

      \[\leadsto {\left(\varepsilon \cdot \varepsilon\right)}^{3} \cdot \left(\left(\frac{27}{112000} + \frac{\frac{9}{40}}{{\varepsilon}^{4}}\right) - \left(\color{blue}{\frac{27}{2800}} \cdot \frac{1}{{\varepsilon}^{2}} + \frac{1}{2} \cdot \frac{1}{{\varepsilon}^{6}}\right)\right) \]

   7. pow-prod-down N/A

      \[\leadsto \left({\varepsilon}^{3} \cdot {\varepsilon}^{3}\right) \cdot \left(\left(\frac{27}{112000} + \frac{\frac{9}{40}}{{\varepsilon}^{4}}\right) - \left(\color{blue}{\frac{27}{2800} \cdot \frac{1}{{\varepsilon}^{2}}} + \frac{1}{2} \cdot \frac{1}{{\varepsilon}^{6}}\right)\right) \]

   8. pow-prod-up N/A

      \[\leadsto {\varepsilon}^{\left(3 + 3\right)} \cdot \left(\left(\frac{27}{112000} + \frac{\frac{9}{40}}{{\varepsilon}^{4}}\right) - \left(\color{blue}{\frac{27}{2800} \cdot \frac{1}{{\varepsilon}^{2}}} + \frac{1}{2} \cdot \frac{1}{{\varepsilon}^{6}}\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto {\varepsilon}^{6} \cdot \left(\left(\frac{27}{112000} + \frac{\frac{9}{40}}{{\varepsilon}^{4}}\right) - \left(\frac{27}{2800} \cdot \color{blue}{\frac{1}{{\varepsilon}^{2}}} + \frac{1}{2} \cdot \frac{1}{{\varepsilon}^{6}}\right)\right) \]

   10. lower-pow.f64 N/A

       \[\leadsto {\varepsilon}^{6} \cdot \left(\left(\frac{27}{112000} + \frac{\frac{9}{40}}{{\varepsilon}^{4}}\right) - \left(\color{blue}{\frac{27}{2800} \cdot \frac{1}{{\varepsilon}^{2}}} + \frac{1}{2} \cdot \frac{1}{{\varepsilon}^{6}}\right)\right) \]

   11. lower--.f64 N/A

       \[\leadsto {\varepsilon}^{6} \cdot \left(\left(\frac{27}{112000} + \frac{\frac{9}{40}}{{\varepsilon}^{4}}\right) - \left(\frac{27}{2800} \cdot \frac{1}{{\varepsilon}^{2}} + \color{blue}{\frac{1}{2} \cdot \frac{1}{{\varepsilon}^{6}}}\right)\right) \]

7. Applied rewrites 16.3%

   \[\leadsto {\varepsilon}^{6} \cdot \color{blue}{\left(\left(\frac{0.225}{{\varepsilon}^{4}} + 0.00024107142857142857\right) - \mathsf{fma}\left({\varepsilon}^{-6}, 0.5, {\varepsilon}^{-2} \cdot 0.009642857142857142\right)\right)} \]
8. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto {\varepsilon}^{6} \cdot \left(\left(\frac{\frac{9}{40}}{{\varepsilon}^{4}} + \frac{27}{112000}\right) - \mathsf{fma}\left({\varepsilon}^{-6}, \color{blue}{\frac{1}{2}}, {\varepsilon}^{-2} \cdot \frac{27}{2800}\right)\right) \]

   2. lift-+.f64 N/A

      \[\leadsto {\varepsilon}^{6} \cdot \left(\left(\frac{\frac{9}{40}}{{\varepsilon}^{4}} + \frac{27}{112000}\right) - \mathsf{fma}\left({\varepsilon}^{-6}, \frac{1}{2}, {\varepsilon}^{-2} \cdot \frac{27}{2800}\right)\right) \]

   3. lift-/.f64 N/A

      \[\leadsto {\varepsilon}^{6} \cdot \left(\left(\frac{\frac{9}{40}}{{\varepsilon}^{4}} + \frac{27}{112000}\right) - \mathsf{fma}\left({\varepsilon}^{-6}, \frac{1}{2}, {\varepsilon}^{-2} \cdot \frac{27}{2800}\right)\right) \]

   4. lift-pow.f64 N/A

      \[\leadsto {\varepsilon}^{6} \cdot \left(\left(\frac{\frac{9}{40}}{{\varepsilon}^{4}} + \frac{27}{112000}\right) - \mathsf{fma}\left({\varepsilon}^{-6}, \frac{1}{2}, {\varepsilon}^{-2} \cdot \frac{27}{2800}\right)\right) \]

   5. lift-pow.f64 N/A

      \[\leadsto {\varepsilon}^{6} \cdot \left(\left(\frac{\frac{9}{40}}{{\varepsilon}^{4}} + \frac{27}{112000}\right) - \mathsf{fma}\left({\varepsilon}^{-6}, \frac{1}{2}, {\varepsilon}^{-2} \cdot \frac{27}{2800}\right)\right) \]

   6. lift-fma.f64 N/A

      \[\leadsto {\varepsilon}^{6} \cdot \left(\left(\frac{\frac{9}{40}}{{\varepsilon}^{4}} + \frac{27}{112000}\right) - \left({\varepsilon}^{-6} \cdot \frac{1}{2} + {\varepsilon}^{-2} \cdot \color{blue}{\frac{27}{2800}}\right)\right) \]

   7. lift-*.f64 N/A

      \[\leadsto {\varepsilon}^{6} \cdot \left(\left(\frac{\frac{9}{40}}{{\varepsilon}^{4}} + \frac{27}{112000}\right) - \left({\varepsilon}^{-6} \cdot \frac{1}{2} + {\varepsilon}^{-2} \cdot \frac{27}{2800}\right)\right) \]

   8. lift-pow.f64 N/A

      \[\leadsto {\varepsilon}^{6} \cdot \left(\left(\frac{\frac{9}{40}}{{\varepsilon}^{4}} + \frac{27}{112000}\right) - \left({\varepsilon}^{-6} \cdot \frac{1}{2} + {\varepsilon}^{-2} \cdot \frac{27}{2800}\right)\right) \]

   9. associate--l+ N/A

      \[\leadsto {\varepsilon}^{6} \cdot \left(\frac{\frac{9}{40}}{{\varepsilon}^{4}} + \left(\frac{27}{112000} - \color{blue}{\left({\varepsilon}^{-6} \cdot \frac{1}{2} + {\varepsilon}^{-2} \cdot \frac{27}{2800}\right)}\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto {\varepsilon}^{6} \cdot \left(\frac{\frac{9}{40} \cdot 1}{{\varepsilon}^{4}} + \left(\frac{27}{112000} - \left(\color{blue}{{\varepsilon}^{-6}} \cdot \frac{1}{2} + {\varepsilon}^{-2} \cdot \frac{27}{2800}\right)\right)\right) \]

   11. associate-*r/ N/A

       \[\leadsto {\varepsilon}^{6} \cdot \left(\frac{9}{40} \cdot \frac{1}{{\varepsilon}^{4}} + \left(\frac{27}{112000} - \left(\color{blue}{{\varepsilon}^{-6} \cdot \frac{1}{2}} + {\varepsilon}^{-2} \cdot \frac{27}{2800}\right)\right)\right) \]

   12. *-commutative N/A

       \[\leadsto {\varepsilon}^{6} \cdot \left(\frac{1}{{\varepsilon}^{4}} \cdot \frac{9}{40} + \left(\frac{27}{112000} - \left(\color{blue}{{\varepsilon}^{-6} \cdot \frac{1}{2}} + {\varepsilon}^{-2} \cdot \frac{27}{2800}\right)\right)\right) \]

   13. lower-fma.f64 N/A

       \[\leadsto {\varepsilon}^{6} \cdot \mathsf{fma}\left(\frac{1}{{\varepsilon}^{4}}, \frac{9}{40}, \frac{27}{112000} - \left({\varepsilon}^{-6} \cdot \frac{1}{2} + {\varepsilon}^{-2} \cdot \frac{27}{2800}\right)\right) \]

   14. pow-flip N/A

       \[\leadsto {\varepsilon}^{6} \cdot \mathsf{fma}\left({\varepsilon}^{\left(\mathsf{neg}\left(4\right)\right)}, \frac{9}{40}, \frac{27}{112000} - \left({\varepsilon}^{-6} \cdot \frac{1}{2} + {\varepsilon}^{-2} \cdot \frac{27}{2800}\right)\right) \]

   15. lower-pow.f64 N/A

       \[\leadsto {\varepsilon}^{6} \cdot \mathsf{fma}\left({\varepsilon}^{\left(\mathsf{neg}\left(4\right)\right)}, \frac{9}{40}, \frac{27}{112000} - \left({\varepsilon}^{-6} \cdot \frac{1}{2} + {\varepsilon}^{-2} \cdot \frac{27}{2800}\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto {\varepsilon}^{6} \cdot \mathsf{fma}\left({\varepsilon}^{-4}, \frac{9}{40}, \frac{27}{112000} - \left({\varepsilon}^{-6} \cdot \frac{1}{2} + {\varepsilon}^{-2} \cdot \frac{27}{2800}\right)\right) \]

   17. lower--.f64 N/A

       \[\leadsto {\varepsilon}^{6} \cdot \mathsf{fma}\left({\varepsilon}^{-4}, \frac{9}{40}, \frac{27}{112000} - \left({\varepsilon}^{-6} \cdot \frac{1}{2} + {\varepsilon}^{-2} \cdot \frac{27}{2800}\right)\right) \]

   18. lift-pow.f64 N/A

       \[\leadsto {\varepsilon}^{6} \cdot \mathsf{fma}\left({\varepsilon}^{-4}, \frac{9}{40}, \frac{27}{112000} - \left({\varepsilon}^{-6} \cdot \frac{1}{2} + {\varepsilon}^{-2} \cdot \frac{27}{2800}\right)\right) \]

   19. lift-*.f64 N/A

       \[\leadsto {\varepsilon}^{6} \cdot \mathsf{fma}\left({\varepsilon}^{-4}, \frac{9}{40}, \frac{27}{112000} - \left({\varepsilon}^{-6} \cdot \frac{1}{2} + {\varepsilon}^{-2} \cdot \frac{27}{2800}\right)\right) \]

9. Applied rewrites 16.3%

   \[\leadsto {\varepsilon}^{6} \cdot \mathsf{fma}\left({\varepsilon}^{-4}, 0.225, 0.00024107142857142857 - \mathsf{fma}\left({\varepsilon}^{-6}, 0.5, {\varepsilon}^{-2} \cdot 0.009642857142857142\right)\right) \]

10. Taylor expanded in eps around 0

    \[\leadsto \color{blue}{{\varepsilon}^{2} \cdot \left(\frac{9}{40} + \frac{-27}{2800} \cdot {\varepsilon}^{2}\right) - \frac{1}{2}} \]
11. Step-by-step derivation

    1. metadata-eval N/A

       \[\leadsto {\varepsilon}^{2} \cdot \left(\frac{9}{40} + \frac{-27}{2800} \cdot {\varepsilon}^{2}\right) - \frac{1}{2} \cdot \color{blue}{1} \]

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

       \[\leadsto {\varepsilon}^{2} \cdot \left(\frac{9}{40} + \frac{-27}{2800} \cdot {\varepsilon}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot 1} \]

    3. *-commutative N/A

       \[\leadsto \left(\frac{9}{40} + \frac{-27}{2800} \cdot {\varepsilon}^{2}\right) \cdot {\varepsilon}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot 1 \]

    4. metadata-eval N/A

       \[\leadsto \left(\frac{9}{40} + \frac{-27}{2800} \cdot {\varepsilon}^{2}\right) \cdot {\varepsilon}^{2} + \frac{-1}{2} \cdot 1 \]

    5. metadata-eval N/A

       \[\leadsto \left(\frac{9}{40} + \frac{-27}{2800} \cdot {\varepsilon}^{2}\right) \cdot {\varepsilon}^{2} + \frac{-1}{2} \]

    6. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{9}{40} + \frac{-27}{2800} \cdot {\varepsilon}^{2}, \color{blue}{{\varepsilon}^{2}}, \frac{-1}{2}\right) \]

    7. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(\frac{-27}{2800} \cdot {\varepsilon}^{2} + \frac{9}{40}, {\color{blue}{\varepsilon}}^{2}, \frac{-1}{2}\right) \]

    8. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-27}{2800}, {\varepsilon}^{2}, \frac{9}{40}\right), {\color{blue}{\varepsilon}}^{2}, \frac{-1}{2}\right) \]

    9. pow2 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-27}{2800}, \varepsilon \cdot \varepsilon, \frac{9}{40}\right), {\varepsilon}^{2}, \frac{-1}{2}\right) \]

    10. lower-*.f64 N/A

        \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-27}{2800}, \varepsilon \cdot \varepsilon, \frac{9}{40}\right), {\varepsilon}^{2}, \frac{-1}{2}\right) \]

    11. pow2 N/A

        \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-27}{2800}, \varepsilon \cdot \varepsilon, \frac{9}{40}\right), \varepsilon \cdot \color{blue}{\varepsilon}, \frac{-1}{2}\right) \]

    12. lower-*.f64 99.9

        \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.009642857142857142, \varepsilon \cdot \varepsilon, 0.225\right), \varepsilon \cdot \color{blue}{\varepsilon}, -0.5\right) \]

12. Applied rewrites 99.9%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.009642857142857142, \varepsilon \cdot \varepsilon, 0.225\right), \varepsilon \cdot \varepsilon, -0.5\right)} \]
13. Add Preprocessing

Alternative 3: 99.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.225, -0.5\right) \end{array} \]

FPCore

(FPCore (eps) :precision binary64 (fma (* eps eps) 0.225 -0.5))

C

double code(double eps) {
	return fma((eps * eps), 0.225, -0.5);
}

Julia

function code(eps)
	return fma(Float64(eps * eps), 0.225, -0.5)
end

Wolfram

code[eps_] := N[(N[(eps * eps), $MachinePrecision] * 0.225 + -0.5), $MachinePrecision]

Derivation

1. Initial program 1.6%

   \[\frac{\varepsilon - \sin \varepsilon}{\varepsilon - \tan \varepsilon} \]

2. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{{\varepsilon}^{2} \cdot \left(\frac{9}{40} + {\varepsilon}^{2} \cdot \left(\frac{27}{112000} \cdot {\varepsilon}^{2} - \frac{27}{2800}\right)\right) - \frac{1}{2}} \]
3. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto {\varepsilon}^{2} \cdot \left(\frac{9}{40} + {\varepsilon}^{2} \cdot \left(\frac{27}{112000} \cdot {\varepsilon}^{2} - \frac{27}{2800}\right)\right) - \color{blue}{\frac{1}{2}} \]

   2. *-commutative N/A

      \[\leadsto \left(\frac{9}{40} + {\varepsilon}^{2} \cdot \left(\frac{27}{112000} \cdot {\varepsilon}^{2} - \frac{27}{2800}\right)\right) \cdot {\varepsilon}^{2} - \frac{1}{2} \]

   3. lower-*.f64 N/A

      \[\leadsto \left(\frac{9}{40} + {\varepsilon}^{2} \cdot \left(\frac{27}{112000} \cdot {\varepsilon}^{2} - \frac{27}{2800}\right)\right) \cdot {\varepsilon}^{2} - \frac{1}{2} \]

   4. +-commutative N/A

      \[\leadsto \left({\varepsilon}^{2} \cdot \left(\frac{27}{112000} \cdot {\varepsilon}^{2} - \frac{27}{2800}\right) + \frac{9}{40}\right) \cdot {\varepsilon}^{2} - \frac{1}{2} \]

   5. *-commutative N/A

      \[\leadsto \left(\left(\frac{27}{112000} \cdot {\varepsilon}^{2} - \frac{27}{2800}\right) \cdot {\varepsilon}^{2} + \frac{9}{40}\right) \cdot {\varepsilon}^{2} - \frac{1}{2} \]

   6. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{27}{112000} \cdot {\varepsilon}^{2} - \frac{27}{2800}, {\varepsilon}^{2}, \frac{9}{40}\right) \cdot {\varepsilon}^{2} - \frac{1}{2} \]

   7. lower--.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{27}{112000} \cdot {\varepsilon}^{2} - \frac{27}{2800}, {\varepsilon}^{2}, \frac{9}{40}\right) \cdot {\varepsilon}^{2} - \frac{1}{2} \]

   8. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{27}{112000} \cdot {\varepsilon}^{2} - \frac{27}{2800}, {\varepsilon}^{2}, \frac{9}{40}\right) \cdot {\varepsilon}^{2} - \frac{1}{2} \]

   9. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\frac{27}{112000} \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{27}{2800}, {\varepsilon}^{2}, \frac{9}{40}\right) \cdot {\varepsilon}^{2} - \frac{1}{2} \]

   10. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{27}{112000} \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{27}{2800}, {\varepsilon}^{2}, \frac{9}{40}\right) \cdot {\varepsilon}^{2} - \frac{1}{2} \]

   11. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\frac{27}{112000} \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{27}{2800}, \varepsilon \cdot \varepsilon, \frac{9}{40}\right) \cdot {\varepsilon}^{2} - \frac{1}{2} \]

   12. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{27}{112000} \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{27}{2800}, \varepsilon \cdot \varepsilon, \frac{9}{40}\right) \cdot {\varepsilon}^{2} - \frac{1}{2} \]

   13. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\frac{27}{112000} \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{27}{2800}, \varepsilon \cdot \varepsilon, \frac{9}{40}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{2} \]

   14. lower-*.f64 99.9

       \[\leadsto \mathsf{fma}\left(0.00024107142857142857 \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.009642857142857142, \varepsilon \cdot \varepsilon, 0.225\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.5 \]

4. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.00024107142857142857 \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.009642857142857142, \varepsilon \cdot \varepsilon, 0.225\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.5} \]

5. Taylor expanded in eps around inf

   \[\leadsto {\varepsilon}^{6} \cdot \color{blue}{\left(\left(\frac{27}{112000} + \frac{\frac{9}{40}}{{\varepsilon}^{4}}\right) - \left(\frac{27}{2800} \cdot \frac{1}{{\varepsilon}^{2}} + \frac{1}{2} \cdot \frac{1}{{\varepsilon}^{6}}\right)\right)} \]
6. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto {\varepsilon}^{\left(3 + 3\right)} \cdot \left(\left(\frac{27}{112000} + \frac{\frac{9}{40}}{{\varepsilon}^{4}}\right) - \left(\frac{27}{2800} \cdot \color{blue}{\frac{1}{{\varepsilon}^{2}}} + \frac{1}{2} \cdot \frac{1}{{\varepsilon}^{6}}\right)\right) \]

   2. pow-prod-up N/A

      \[\leadsto \left({\varepsilon}^{3} \cdot {\varepsilon}^{3}\right) \cdot \left(\left(\frac{27}{112000} + \frac{\frac{9}{40}}{{\varepsilon}^{4}}\right) - \left(\color{blue}{\frac{27}{2800} \cdot \frac{1}{{\varepsilon}^{2}}} + \frac{1}{2} \cdot \frac{1}{{\varepsilon}^{6}}\right)\right) \]

   3. pow-prod-down N/A

      \[\leadsto {\left(\varepsilon \cdot \varepsilon\right)}^{3} \cdot \left(\left(\frac{27}{112000} + \frac{\frac{9}{40}}{{\varepsilon}^{4}}\right) - \left(\color{blue}{\frac{27}{2800} \cdot \frac{1}{{\varepsilon}^{2}}} + \frac{1}{2} \cdot \frac{1}{{\varepsilon}^{6}}\right)\right) \]

   4. pow2 N/A

      \[\leadsto {\left({\varepsilon}^{2}\right)}^{3} \cdot \left(\left(\frac{27}{112000} + \frac{\frac{9}{40}}{{\varepsilon}^{4}}\right) - \left(\color{blue}{\frac{27}{2800}} \cdot \frac{1}{{\varepsilon}^{2}} + \frac{1}{2} \cdot \frac{1}{{\varepsilon}^{6}}\right)\right) \]

   5. lower-*.f64 N/A

      \[\leadsto {\left({\varepsilon}^{2}\right)}^{3} \cdot \left(\left(\frac{27}{112000} + \frac{\frac{9}{40}}{{\varepsilon}^{4}}\right) - \color{blue}{\left(\frac{27}{2800} \cdot \frac{1}{{\varepsilon}^{2}} + \frac{1}{2} \cdot \frac{1}{{\varepsilon}^{6}}\right)}\right) \]

   6. pow2 N/A

      \[\leadsto {\left(\varepsilon \cdot \varepsilon\right)}^{3} \cdot \left(\left(\frac{27}{112000} + \frac{\frac{9}{40}}{{\varepsilon}^{4}}\right) - \left(\color{blue}{\frac{27}{2800}} \cdot \frac{1}{{\varepsilon}^{2}} + \frac{1}{2} \cdot \frac{1}{{\varepsilon}^{6}}\right)\right) \]

   7. pow-prod-down N/A

      \[\leadsto \left({\varepsilon}^{3} \cdot {\varepsilon}^{3}\right) \cdot \left(\left(\frac{27}{112000} + \frac{\frac{9}{40}}{{\varepsilon}^{4}}\right) - \left(\color{blue}{\frac{27}{2800} \cdot \frac{1}{{\varepsilon}^{2}}} + \frac{1}{2} \cdot \frac{1}{{\varepsilon}^{6}}\right)\right) \]

   8. pow-prod-up N/A

      \[\leadsto {\varepsilon}^{\left(3 + 3\right)} \cdot \left(\left(\frac{27}{112000} + \frac{\frac{9}{40}}{{\varepsilon}^{4}}\right) - \left(\color{blue}{\frac{27}{2800} \cdot \frac{1}{{\varepsilon}^{2}}} + \frac{1}{2} \cdot \frac{1}{{\varepsilon}^{6}}\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto {\varepsilon}^{6} \cdot \left(\left(\frac{27}{112000} + \frac{\frac{9}{40}}{{\varepsilon}^{4}}\right) - \left(\frac{27}{2800} \cdot \color{blue}{\frac{1}{{\varepsilon}^{2}}} + \frac{1}{2} \cdot \frac{1}{{\varepsilon}^{6}}\right)\right) \]

   10. lower-pow.f64 N/A

       \[\leadsto {\varepsilon}^{6} \cdot \left(\left(\frac{27}{112000} + \frac{\frac{9}{40}}{{\varepsilon}^{4}}\right) - \left(\color{blue}{\frac{27}{2800} \cdot \frac{1}{{\varepsilon}^{2}}} + \frac{1}{2} \cdot \frac{1}{{\varepsilon}^{6}}\right)\right) \]

   11. lower--.f64 N/A

       \[\leadsto {\varepsilon}^{6} \cdot \left(\left(\frac{27}{112000} + \frac{\frac{9}{40}}{{\varepsilon}^{4}}\right) - \left(\frac{27}{2800} \cdot \frac{1}{{\varepsilon}^{2}} + \color{blue}{\frac{1}{2} \cdot \frac{1}{{\varepsilon}^{6}}}\right)\right) \]

7. Applied rewrites 16.3%

   \[\leadsto {\varepsilon}^{6} \cdot \color{blue}{\left(\left(\frac{0.225}{{\varepsilon}^{4}} + 0.00024107142857142857\right) - \mathsf{fma}\left({\varepsilon}^{-6}, 0.5, {\varepsilon}^{-2} \cdot 0.009642857142857142\right)\right)} \]
8. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto {\varepsilon}^{6} \cdot \left(\left(\frac{\frac{9}{40}}{{\varepsilon}^{4}} + \frac{27}{112000}\right) - \mathsf{fma}\left({\varepsilon}^{-6}, \color{blue}{\frac{1}{2}}, {\varepsilon}^{-2} \cdot \frac{27}{2800}\right)\right) \]

   2. lift-+.f64 N/A

      \[\leadsto {\varepsilon}^{6} \cdot \left(\left(\frac{\frac{9}{40}}{{\varepsilon}^{4}} + \frac{27}{112000}\right) - \mathsf{fma}\left({\varepsilon}^{-6}, \frac{1}{2}, {\varepsilon}^{-2} \cdot \frac{27}{2800}\right)\right) \]

   3. lift-/.f64 N/A

      \[\leadsto {\varepsilon}^{6} \cdot \left(\left(\frac{\frac{9}{40}}{{\varepsilon}^{4}} + \frac{27}{112000}\right) - \mathsf{fma}\left({\varepsilon}^{-6}, \frac{1}{2}, {\varepsilon}^{-2} \cdot \frac{27}{2800}\right)\right) \]

   4. lift-pow.f64 N/A

      \[\leadsto {\varepsilon}^{6} \cdot \left(\left(\frac{\frac{9}{40}}{{\varepsilon}^{4}} + \frac{27}{112000}\right) - \mathsf{fma}\left({\varepsilon}^{-6}, \frac{1}{2}, {\varepsilon}^{-2} \cdot \frac{27}{2800}\right)\right) \]

   5. lift-pow.f64 N/A

      \[\leadsto {\varepsilon}^{6} \cdot \left(\left(\frac{\frac{9}{40}}{{\varepsilon}^{4}} + \frac{27}{112000}\right) - \mathsf{fma}\left({\varepsilon}^{-6}, \frac{1}{2}, {\varepsilon}^{-2} \cdot \frac{27}{2800}\right)\right) \]

   6. lift-fma.f64 N/A

      \[\leadsto {\varepsilon}^{6} \cdot \left(\left(\frac{\frac{9}{40}}{{\varepsilon}^{4}} + \frac{27}{112000}\right) - \left({\varepsilon}^{-6} \cdot \frac{1}{2} + {\varepsilon}^{-2} \cdot \color{blue}{\frac{27}{2800}}\right)\right) \]

   7. lift-*.f64 N/A

      \[\leadsto {\varepsilon}^{6} \cdot \left(\left(\frac{\frac{9}{40}}{{\varepsilon}^{4}} + \frac{27}{112000}\right) - \left({\varepsilon}^{-6} \cdot \frac{1}{2} + {\varepsilon}^{-2} \cdot \frac{27}{2800}\right)\right) \]

   8. lift-pow.f64 N/A

      \[\leadsto {\varepsilon}^{6} \cdot \left(\left(\frac{\frac{9}{40}}{{\varepsilon}^{4}} + \frac{27}{112000}\right) - \left({\varepsilon}^{-6} \cdot \frac{1}{2} + {\varepsilon}^{-2} \cdot \frac{27}{2800}\right)\right) \]

   9. associate--l+ N/A

      \[\leadsto {\varepsilon}^{6} \cdot \left(\frac{\frac{9}{40}}{{\varepsilon}^{4}} + \left(\frac{27}{112000} - \color{blue}{\left({\varepsilon}^{-6} \cdot \frac{1}{2} + {\varepsilon}^{-2} \cdot \frac{27}{2800}\right)}\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto {\varepsilon}^{6} \cdot \left(\frac{\frac{9}{40} \cdot 1}{{\varepsilon}^{4}} + \left(\frac{27}{112000} - \left(\color{blue}{{\varepsilon}^{-6}} \cdot \frac{1}{2} + {\varepsilon}^{-2} \cdot \frac{27}{2800}\right)\right)\right) \]

   11. associate-*r/ N/A

       \[\leadsto {\varepsilon}^{6} \cdot \left(\frac{9}{40} \cdot \frac{1}{{\varepsilon}^{4}} + \left(\frac{27}{112000} - \left(\color{blue}{{\varepsilon}^{-6} \cdot \frac{1}{2}} + {\varepsilon}^{-2} \cdot \frac{27}{2800}\right)\right)\right) \]

   12. *-commutative N/A

       \[\leadsto {\varepsilon}^{6} \cdot \left(\frac{1}{{\varepsilon}^{4}} \cdot \frac{9}{40} + \left(\frac{27}{112000} - \left(\color{blue}{{\varepsilon}^{-6} \cdot \frac{1}{2}} + {\varepsilon}^{-2} \cdot \frac{27}{2800}\right)\right)\right) \]

   13. lower-fma.f64 N/A

       \[\leadsto {\varepsilon}^{6} \cdot \mathsf{fma}\left(\frac{1}{{\varepsilon}^{4}}, \frac{9}{40}, \frac{27}{112000} - \left({\varepsilon}^{-6} \cdot \frac{1}{2} + {\varepsilon}^{-2} \cdot \frac{27}{2800}\right)\right) \]

   14. pow-flip N/A

       \[\leadsto {\varepsilon}^{6} \cdot \mathsf{fma}\left({\varepsilon}^{\left(\mathsf{neg}\left(4\right)\right)}, \frac{9}{40}, \frac{27}{112000} - \left({\varepsilon}^{-6} \cdot \frac{1}{2} + {\varepsilon}^{-2} \cdot \frac{27}{2800}\right)\right) \]

   15. lower-pow.f64 N/A

       \[\leadsto {\varepsilon}^{6} \cdot \mathsf{fma}\left({\varepsilon}^{\left(\mathsf{neg}\left(4\right)\right)}, \frac{9}{40}, \frac{27}{112000} - \left({\varepsilon}^{-6} \cdot \frac{1}{2} + {\varepsilon}^{-2} \cdot \frac{27}{2800}\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto {\varepsilon}^{6} \cdot \mathsf{fma}\left({\varepsilon}^{-4}, \frac{9}{40}, \frac{27}{112000} - \left({\varepsilon}^{-6} \cdot \frac{1}{2} + {\varepsilon}^{-2} \cdot \frac{27}{2800}\right)\right) \]

   17. lower--.f64 N/A

       \[\leadsto {\varepsilon}^{6} \cdot \mathsf{fma}\left({\varepsilon}^{-4}, \frac{9}{40}, \frac{27}{112000} - \left({\varepsilon}^{-6} \cdot \frac{1}{2} + {\varepsilon}^{-2} \cdot \frac{27}{2800}\right)\right) \]

   18. lift-pow.f64 N/A

       \[\leadsto {\varepsilon}^{6} \cdot \mathsf{fma}\left({\varepsilon}^{-4}, \frac{9}{40}, \frac{27}{112000} - \left({\varepsilon}^{-6} \cdot \frac{1}{2} + {\varepsilon}^{-2} \cdot \frac{27}{2800}\right)\right) \]

   19. lift-*.f64 N/A

       \[\leadsto {\varepsilon}^{6} \cdot \mathsf{fma}\left({\varepsilon}^{-4}, \frac{9}{40}, \frac{27}{112000} - \left({\varepsilon}^{-6} \cdot \frac{1}{2} + {\varepsilon}^{-2} \cdot \frac{27}{2800}\right)\right) \]

9. Applied rewrites 16.3%

   \[\leadsto {\varepsilon}^{6} \cdot \mathsf{fma}\left({\varepsilon}^{-4}, 0.225, 0.00024107142857142857 - \mathsf{fma}\left({\varepsilon}^{-6}, 0.5, {\varepsilon}^{-2} \cdot 0.009642857142857142\right)\right) \]

10. Taylor expanded in eps around 0

    \[\leadsto \color{blue}{\frac{9}{40} \cdot {\varepsilon}^{2} - \frac{1}{2}} \]
11. Step-by-step derivation

    1. metadata-eval N/A

       \[\leadsto \frac{9}{40} \cdot {\varepsilon}^{2} - \frac{1}{2} \cdot \color{blue}{1} \]

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

       \[\leadsto \frac{9}{40} \cdot {\varepsilon}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot 1} \]

    3. *-commutative N/A

       \[\leadsto {\varepsilon}^{2} \cdot \frac{9}{40} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot 1 \]

    4. metadata-eval N/A

       \[\leadsto {\varepsilon}^{2} \cdot \frac{9}{40} + \frac{-1}{2} \cdot 1 \]

    5. metadata-eval N/A

       \[\leadsto {\varepsilon}^{2} \cdot \frac{9}{40} + \frac{-1}{2} \]

    6. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left({\varepsilon}^{2}, \color{blue}{\frac{9}{40}}, \frac{-1}{2}\right) \]

    7. pow2 N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{9}{40}, \frac{-1}{2}\right) \]

    8. lower-*.f64 99.7

       \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.225, -0.5\right) \]

12. Applied rewrites 99.7%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.225, -0.5\right)} \]
13. Add Preprocessing

Alternative 4: 99.1% accurate, 9.0× speedup

Math

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

FPCore

(FPCore (eps) :precision binary64 -0.5)

C

double code(double eps) {
	return -0.5;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

public static double code(double eps) {
	return -0.5;
}

Python

def code(eps):
	return -0.5

Julia

function code(eps)
	return -0.5
end

MATLAB

function tmp = code(eps)
	tmp = -0.5;
end

Wolfram

code[eps_] := -0.5

Derivation

1. Initial program 1.6%

   \[\frac{\varepsilon - \sin \varepsilon}{\varepsilon - \tan \varepsilon} \]

2. Taylor expanded in eps around 0

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

4. Applied rewrites 99.1%

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

Reproduce

herbie shell --seed 2025101 
(FPCore (eps)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  :pre (and (<= -0.4 eps) (<= eps 0.4))
  (/ (- eps (sin eps)) (- eps (tan eps))))