2atan (example 3.5)

Percentage Accurate: 8.4% → 99.7%

Time: 2.5s

Alternatives: 8

Speedup: 1.9×

Specification

\[N > 1 \land N < 10^{+100}\]

Math

\[\begin{array}{l} \\ \tan^{-1} \left(N + 1\right) - \tan^{-1} N \end{array} \]

FPCore

(FPCore (N) :precision binary64 (- (atan (+ N 1.0)) (atan N)))

C

double code(double N) {
	return atan((N + 1.0)) - atan(N);
}

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(n)
use fmin_fmax_functions
    real(8), intent (in) :: n
    code = atan((n + 1.0d0)) - atan(n)
end function

Java

public static double code(double N) {
	return Math.atan((N + 1.0)) - Math.atan(N);
}

Python

def code(N):
	return math.atan((N + 1.0)) - math.atan(N)

Julia

function code(N)
	return Float64(atan(Float64(N + 1.0)) - atan(N))
end

MATLAB

function tmp = code(N)
	tmp = atan((N + 1.0)) - atan(N);
end

Wolfram

code[N_] := N[(N[ArcTan[N[(N + 1.0), $MachinePrecision]], $MachinePrecision] - N[ArcTan[N], $MachinePrecision]), $MachinePrecision]

Initial Program: 8.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \tan^{-1} \left(N + 1\right) - \tan^{-1} N \end{array} \]

FPCore

(FPCore (N) :precision binary64 (- (atan (+ N 1.0)) (atan N)))

C

double code(double N) {
	return atan((N + 1.0)) - atan(N);
}

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(n)
use fmin_fmax_functions
    real(8), intent (in) :: n
    code = atan((n + 1.0d0)) - atan(n)
end function

Java

public static double code(double N) {
	return Math.atan((N + 1.0)) - Math.atan(N);
}

Python

def code(N):
	return math.atan((N + 1.0)) - math.atan(N)

Julia

function code(N)
	return Float64(atan(Float64(N + 1.0)) - atan(N))
end

MATLAB

function tmp = code(N)
	tmp = atan((N + 1.0)) - atan(N);
end

Wolfram

code[N_] := N[(N[ArcTan[N[(N + 1.0), $MachinePrecision]], $MachinePrecision] - N[ArcTan[N], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \tan^{-1}_* \frac{1}{\left(N - -1\right) + \frac{1}{{N}^{-2}}} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (atan2 1.0 (+ (- N -1.0) (/ 1.0 (pow N -2.0)))))

C

double code(double N) {
	return atan2(1.0, ((N - -1.0) + (1.0 / pow(N, -2.0))));
}

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(n)
use fmin_fmax_functions
    real(8), intent (in) :: n
    code = atan2(1.0d0, ((n - (-1.0d0)) + (1.0d0 / (n ** (-2.0d0)))))
end function

Java

public static double code(double N) {
	return Math.atan2(1.0, ((N - -1.0) + (1.0 / Math.pow(N, -2.0))));
}

Python

def code(N):
	return math.atan2(1.0, ((N - -1.0) + (1.0 / math.pow(N, -2.0))))

Julia

function code(N)
	return atan(1.0, Float64(Float64(N - -1.0) + Float64(1.0 / (N ^ -2.0))))
end

MATLAB

function tmp = code(N)
	tmp = atan2(1.0, ((N - -1.0) + (1.0 / (N ^ -2.0))));
end

Wolfram

code[N_] := N[ArcTan[1.0 / N[(N[(N - -1.0), $MachinePrecision] + N[(1.0 / N[Power[N, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 6.9%

   \[\tan^{-1} \left(N + 1\right) - \tan^{-1} N \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\tan^{-1} \left(N + 1\right) - \tan^{-1} N} \]

   2. lift-+.f64 N/A

      \[\leadsto \tan^{-1} \color{blue}{\left(N + 1\right)} - \tan^{-1} N \]

   3. lift-atan.f64 N/A

      \[\leadsto \color{blue}{\tan^{-1} \left(N + 1\right)} - \tan^{-1} N \]

   4. lift-atan.f64 N/A

      \[\leadsto \tan^{-1} \left(N + 1\right) - \color{blue}{\tan^{-1} N} \]

   5. diff-atan N/A

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\left(N + 1\right) - N}{1 + \left(N + 1\right) \cdot N}} \]

   6. lower-atan2.f64 N/A

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\left(N + 1\right) - N}{1 + \left(N + 1\right) \cdot N}} \]

   7. lower--.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(N + 1\right) - N}}{1 + \left(N + 1\right) \cdot N} \]

   8. metadata-eval N/A

      \[\leadsto \tan^{-1}_* \frac{\left(N + \color{blue}{1 \cdot 1}\right) - N}{1 + \left(N + 1\right) \cdot N} \]

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

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

   10. metadata-eval N/A

       \[\leadsto \tan^{-1}_* \frac{\left(N - \color{blue}{-1} \cdot 1\right) - N}{1 + \left(N + 1\right) \cdot N} \]

   11. metadata-eval N/A

       \[\leadsto \tan^{-1}_* \frac{\left(N - \color{blue}{-1}\right) - N}{1 + \left(N + 1\right) \cdot N} \]

   12. lower--.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(N - -1\right)} - N}{1 + \left(N + 1\right) \cdot N} \]

   13. +-commutative N/A

       \[\leadsto \tan^{-1}_* \frac{\left(N - -1\right) - N}{\color{blue}{\left(N + 1\right) \cdot N + 1}} \]

   14. lower-fma.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\left(N - -1\right) - N}{\color{blue}{\mathsf{fma}\left(N + 1, N, 1\right)}} \]

   15. metadata-eval N/A

       \[\leadsto \tan^{-1}_* \frac{\left(N - -1\right) - N}{\mathsf{fma}\left(N + \color{blue}{1 \cdot 1}, N, 1\right)} \]

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

       \[\leadsto \tan^{-1}_* \frac{\left(N - -1\right) - N}{\mathsf{fma}\left(\color{blue}{N - \left(\mathsf{neg}\left(1\right)\right) \cdot 1}, N, 1\right)} \]

   17. metadata-eval N/A

       \[\leadsto \tan^{-1}_* \frac{\left(N - -1\right) - N}{\mathsf{fma}\left(N - \color{blue}{-1} \cdot 1, N, 1\right)} \]

   18. metadata-eval N/A

       \[\leadsto \tan^{-1}_* \frac{\left(N - -1\right) - N}{\mathsf{fma}\left(N - \color{blue}{-1}, N, 1\right)} \]

   19. lower--.f64 15.8

       \[\leadsto \tan^{-1}_* \frac{\left(N - -1\right) - N}{\mathsf{fma}\left(\color{blue}{N - -1}, N, 1\right)} \]

4. Applied rewrites 15.8%

   \[\leadsto \color{blue}{\tan^{-1}_* \frac{\left(N - -1\right) - N}{\mathsf{fma}\left(N - -1, N, 1\right)}} \]

5. Taylor expanded in N around 0

   \[\leadsto \tan^{-1}_* \frac{\color{blue}{1}}{\mathsf{fma}\left(N - -1, N, 1\right)} \]
6. Step-by-step derivation

7. Applied rewrites 99.6%

   \[\leadsto \tan^{-1}_* \frac{\color{blue}{1}}{\mathsf{fma}\left(N - -1, N, 1\right)} \]
8. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{1}{\mathsf{fma}\left(\color{blue}{N - -1}, N, 1\right)} \]

   2. lift-fma.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{\left(N - -1\right) \cdot N + 1}} \]

   3. metadata-eval N/A

      \[\leadsto \tan^{-1}_* \frac{1}{\left(N - -1\right) \cdot N + \color{blue}{-1 \cdot -1}} \]

   4. metadata-eval N/A

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

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

      \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{\left(N - -1\right) \cdot N - 1 \cdot -1}} \]

   6. *-commutative N/A

      \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{N \cdot \left(N - -1\right)} - 1 \cdot -1} \]

   7. distribute-rgt-out-- N/A

      \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{\left(N \cdot N - -1 \cdot N\right)} - 1 \cdot -1} \]

   8. pow2 N/A

      \[\leadsto \tan^{-1}_* \frac{1}{\left(\color{blue}{{N}^{2}} - -1 \cdot N\right) - 1 \cdot -1} \]

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

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

   10. metadata-eval N/A

       \[\leadsto \tan^{-1}_* \frac{1}{\left({N}^{2} + \color{blue}{1} \cdot N\right) - 1 \cdot -1} \]

   11. *-lft-identity N/A

       \[\leadsto \tan^{-1}_* \frac{1}{\left({N}^{2} + \color{blue}{N}\right) - 1 \cdot -1} \]

   12. metadata-eval N/A

       \[\leadsto \tan^{-1}_* \frac{1}{\left({N}^{2} + N\right) - \color{blue}{-1}} \]

   13. associate-+r- N/A

       \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{{N}^{2} + \left(N - -1\right)}} \]

   14. +-commutative N/A

       \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{\left(N - -1\right) + {N}^{2}}} \]

   15. pow2 N/A

       \[\leadsto \tan^{-1}_* \frac{1}{\left(N - -1\right) + \color{blue}{N \cdot N}} \]

   16. lift--.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{\left(N - -1\right)} + N \cdot N} \]

   17. lift-*.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{1}{\left(N - -1\right) + \color{blue}{N \cdot N}} \]

   18. lift-+.f64 99.6

       \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{\left(N - -1\right) + N \cdot N}} \]

9. Applied rewrites 99.6%

   \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{\left(N - -1\right) + N \cdot N}} \]
10. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{1}{\left(N - -1\right) + \color{blue}{N \cdot N}} \]

    2. pow2 N/A

       \[\leadsto \tan^{-1}_* \frac{1}{\left(N - -1\right) + \color{blue}{{N}^{2}}} \]

    3. metadata-eval N/A

       \[\leadsto \tan^{-1}_* \frac{1}{\left(N - -1\right) + {N}^{\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)}}} \]

    4. pow-neg N/A

       \[\leadsto \tan^{-1}_* \frac{1}{\left(N - -1\right) + \color{blue}{\frac{1}{{N}^{-2}}}} \]

    5. metadata-eval N/A

       \[\leadsto \tan^{-1}_* \frac{1}{\left(N - -1\right) + \frac{1}{{N}^{\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}}}} \]

    6. pow-flip N/A

       \[\leadsto \tan^{-1}_* \frac{1}{\left(N - -1\right) + \frac{1}{\color{blue}{\frac{1}{{N}^{2}}}}} \]

    7. lower-/.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{1}{\left(N - -1\right) + \color{blue}{\frac{1}{\frac{1}{{N}^{2}}}}} \]

    8. pow-flip N/A

       \[\leadsto \tan^{-1}_* \frac{1}{\left(N - -1\right) + \frac{1}{\color{blue}{{N}^{\left(\mathsf{neg}\left(2\right)\right)}}}} \]

    9. metadata-eval N/A

       \[\leadsto \tan^{-1}_* \frac{1}{\left(N - -1\right) + \frac{1}{{N}^{\color{blue}{-2}}}} \]

    10. lower-pow.f64 99.8

        \[\leadsto \tan^{-1}_* \frac{1}{\left(N - -1\right) + \frac{1}{\color{blue}{{N}^{-2}}}} \]

11. Applied rewrites 99.8%

    \[\leadsto \tan^{-1}_* \frac{1}{\left(N - -1\right) + \color{blue}{\frac{1}{{N}^{-2}}}} \]
12. Add Preprocessing

Alternative 2: 99.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \tan^{-1}_* \frac{1}{\left(N - -1\right) + N \cdot N} \end{array} \]

FPCore

(FPCore (N) :precision binary64 (atan2 1.0 (+ (- N -1.0) (* N N))))

C

double code(double N) {
	return atan2(1.0, ((N - -1.0) + (N * N)));
}

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(n)
use fmin_fmax_functions
    real(8), intent (in) :: n
    code = atan2(1.0d0, ((n - (-1.0d0)) + (n * n)))
end function

Java

public static double code(double N) {
	return Math.atan2(1.0, ((N - -1.0) + (N * N)));
}

Python

def code(N):
	return math.atan2(1.0, ((N - -1.0) + (N * N)))

Julia

function code(N)
	return atan(1.0, Float64(Float64(N - -1.0) + Float64(N * N)))
end

MATLAB

function tmp = code(N)
	tmp = atan2(1.0, ((N - -1.0) + (N * N)));
end

Wolfram

code[N_] := N[ArcTan[1.0 / N[(N[(N - -1.0), $MachinePrecision] + N[(N * N), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 6.9%

   \[\tan^{-1} \left(N + 1\right) - \tan^{-1} N \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\tan^{-1} \left(N + 1\right) - \tan^{-1} N} \]

   2. lift-+.f64 N/A

      \[\leadsto \tan^{-1} \color{blue}{\left(N + 1\right)} - \tan^{-1} N \]

   3. lift-atan.f64 N/A

      \[\leadsto \color{blue}{\tan^{-1} \left(N + 1\right)} - \tan^{-1} N \]

   4. lift-atan.f64 N/A

      \[\leadsto \tan^{-1} \left(N + 1\right) - \color{blue}{\tan^{-1} N} \]

   5. diff-atan N/A

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\left(N + 1\right) - N}{1 + \left(N + 1\right) \cdot N}} \]

   6. lower-atan2.f64 N/A

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\left(N + 1\right) - N}{1 + \left(N + 1\right) \cdot N}} \]

   7. lower--.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(N + 1\right) - N}}{1 + \left(N + 1\right) \cdot N} \]

   8. metadata-eval N/A

      \[\leadsto \tan^{-1}_* \frac{\left(N + \color{blue}{1 \cdot 1}\right) - N}{1 + \left(N + 1\right) \cdot N} \]

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

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

   10. metadata-eval N/A

       \[\leadsto \tan^{-1}_* \frac{\left(N - \color{blue}{-1} \cdot 1\right) - N}{1 + \left(N + 1\right) \cdot N} \]

   11. metadata-eval N/A

       \[\leadsto \tan^{-1}_* \frac{\left(N - \color{blue}{-1}\right) - N}{1 + \left(N + 1\right) \cdot N} \]

   12. lower--.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(N - -1\right)} - N}{1 + \left(N + 1\right) \cdot N} \]

   13. +-commutative N/A

       \[\leadsto \tan^{-1}_* \frac{\left(N - -1\right) - N}{\color{blue}{\left(N + 1\right) \cdot N + 1}} \]

   14. lower-fma.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\left(N - -1\right) - N}{\color{blue}{\mathsf{fma}\left(N + 1, N, 1\right)}} \]

   15. metadata-eval N/A

       \[\leadsto \tan^{-1}_* \frac{\left(N - -1\right) - N}{\mathsf{fma}\left(N + \color{blue}{1 \cdot 1}, N, 1\right)} \]

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

       \[\leadsto \tan^{-1}_* \frac{\left(N - -1\right) - N}{\mathsf{fma}\left(\color{blue}{N - \left(\mathsf{neg}\left(1\right)\right) \cdot 1}, N, 1\right)} \]

   17. metadata-eval N/A

       \[\leadsto \tan^{-1}_* \frac{\left(N - -1\right) - N}{\mathsf{fma}\left(N - \color{blue}{-1} \cdot 1, N, 1\right)} \]

   18. metadata-eval N/A

       \[\leadsto \tan^{-1}_* \frac{\left(N - -1\right) - N}{\mathsf{fma}\left(N - \color{blue}{-1}, N, 1\right)} \]

   19. lower--.f64 15.8

       \[\leadsto \tan^{-1}_* \frac{\left(N - -1\right) - N}{\mathsf{fma}\left(\color{blue}{N - -1}, N, 1\right)} \]

4. Applied rewrites 15.8%

   \[\leadsto \color{blue}{\tan^{-1}_* \frac{\left(N - -1\right) - N}{\mathsf{fma}\left(N - -1, N, 1\right)}} \]

5. Taylor expanded in N around 0

   \[\leadsto \tan^{-1}_* \frac{\color{blue}{1}}{\mathsf{fma}\left(N - -1, N, 1\right)} \]
6. Step-by-step derivation

7. Applied rewrites 99.6%

   \[\leadsto \tan^{-1}_* \frac{\color{blue}{1}}{\mathsf{fma}\left(N - -1, N, 1\right)} \]
8. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{1}{\mathsf{fma}\left(\color{blue}{N - -1}, N, 1\right)} \]

   2. lift-fma.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{\left(N - -1\right) \cdot N + 1}} \]

   3. metadata-eval N/A

      \[\leadsto \tan^{-1}_* \frac{1}{\left(N - -1\right) \cdot N + \color{blue}{-1 \cdot -1}} \]

   4. metadata-eval N/A

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

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

      \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{\left(N - -1\right) \cdot N - 1 \cdot -1}} \]

   6. *-commutative N/A

      \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{N \cdot \left(N - -1\right)} - 1 \cdot -1} \]

   7. distribute-rgt-out-- N/A

      \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{\left(N \cdot N - -1 \cdot N\right)} - 1 \cdot -1} \]

   8. pow2 N/A

      \[\leadsto \tan^{-1}_* \frac{1}{\left(\color{blue}{{N}^{2}} - -1 \cdot N\right) - 1 \cdot -1} \]

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

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

   10. metadata-eval N/A

       \[\leadsto \tan^{-1}_* \frac{1}{\left({N}^{2} + \color{blue}{1} \cdot N\right) - 1 \cdot -1} \]

   11. *-lft-identity N/A

       \[\leadsto \tan^{-1}_* \frac{1}{\left({N}^{2} + \color{blue}{N}\right) - 1 \cdot -1} \]

   12. metadata-eval N/A

       \[\leadsto \tan^{-1}_* \frac{1}{\left({N}^{2} + N\right) - \color{blue}{-1}} \]

   13. associate-+r- N/A

       \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{{N}^{2} + \left(N - -1\right)}} \]

   14. +-commutative N/A

       \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{\left(N - -1\right) + {N}^{2}}} \]

   15. pow2 N/A

       \[\leadsto \tan^{-1}_* \frac{1}{\left(N - -1\right) + \color{blue}{N \cdot N}} \]

   16. lift--.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{\left(N - -1\right)} + N \cdot N} \]

   17. lift-*.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{1}{\left(N - -1\right) + \color{blue}{N \cdot N}} \]

   18. lift-+.f64 99.6

       \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{\left(N - -1\right) + N \cdot N}} \]

9. Applied rewrites 99.6%

   \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{\left(N - -1\right) + N \cdot N}} \]
10. Add Preprocessing

Alternative 3: 99.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, N, 1\right) + N} \end{array} \]

FPCore

(FPCore (N) :precision binary64 (atan2 1.0 (+ (fma N N 1.0) N)))

C

double code(double N) {
	return atan2(1.0, (fma(N, N, 1.0) + N));
}

Julia

function code(N)
	return atan(1.0, Float64(fma(N, N, 1.0) + N))
end

Wolfram

code[N_] := N[ArcTan[1.0 / N[(N[(N * N + 1.0), $MachinePrecision] + N), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 6.9%

   \[\tan^{-1} \left(N + 1\right) - \tan^{-1} N \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\tan^{-1} \left(N + 1\right) - \tan^{-1} N} \]

   2. lift-+.f64 N/A

      \[\leadsto \tan^{-1} \color{blue}{\left(N + 1\right)} - \tan^{-1} N \]

   3. lift-atan.f64 N/A

      \[\leadsto \color{blue}{\tan^{-1} \left(N + 1\right)} - \tan^{-1} N \]

   4. lift-atan.f64 N/A

      \[\leadsto \tan^{-1} \left(N + 1\right) - \color{blue}{\tan^{-1} N} \]

   5. diff-atan N/A

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\left(N + 1\right) - N}{1 + \left(N + 1\right) \cdot N}} \]

   6. lower-atan2.f64 N/A

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\left(N + 1\right) - N}{1 + \left(N + 1\right) \cdot N}} \]

   7. lower--.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(N + 1\right) - N}}{1 + \left(N + 1\right) \cdot N} \]

   8. metadata-eval N/A

      \[\leadsto \tan^{-1}_* \frac{\left(N + \color{blue}{1 \cdot 1}\right) - N}{1 + \left(N + 1\right) \cdot N} \]

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

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

   10. metadata-eval N/A

       \[\leadsto \tan^{-1}_* \frac{\left(N - \color{blue}{-1} \cdot 1\right) - N}{1 + \left(N + 1\right) \cdot N} \]

   11. metadata-eval N/A

       \[\leadsto \tan^{-1}_* \frac{\left(N - \color{blue}{-1}\right) - N}{1 + \left(N + 1\right) \cdot N} \]

   12. lower--.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(N - -1\right)} - N}{1 + \left(N + 1\right) \cdot N} \]

   13. +-commutative N/A

       \[\leadsto \tan^{-1}_* \frac{\left(N - -1\right) - N}{\color{blue}{\left(N + 1\right) \cdot N + 1}} \]

   14. lower-fma.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\left(N - -1\right) - N}{\color{blue}{\mathsf{fma}\left(N + 1, N, 1\right)}} \]

   15. metadata-eval N/A

       \[\leadsto \tan^{-1}_* \frac{\left(N - -1\right) - N}{\mathsf{fma}\left(N + \color{blue}{1 \cdot 1}, N, 1\right)} \]

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

       \[\leadsto \tan^{-1}_* \frac{\left(N - -1\right) - N}{\mathsf{fma}\left(\color{blue}{N - \left(\mathsf{neg}\left(1\right)\right) \cdot 1}, N, 1\right)} \]

   17. metadata-eval N/A

       \[\leadsto \tan^{-1}_* \frac{\left(N - -1\right) - N}{\mathsf{fma}\left(N - \color{blue}{-1} \cdot 1, N, 1\right)} \]

   18. metadata-eval N/A

       \[\leadsto \tan^{-1}_* \frac{\left(N - -1\right) - N}{\mathsf{fma}\left(N - \color{blue}{-1}, N, 1\right)} \]

   19. lower--.f64 15.8

       \[\leadsto \tan^{-1}_* \frac{\left(N - -1\right) - N}{\mathsf{fma}\left(\color{blue}{N - -1}, N, 1\right)} \]

4. Applied rewrites 15.8%

   \[\leadsto \color{blue}{\tan^{-1}_* \frac{\left(N - -1\right) - N}{\mathsf{fma}\left(N - -1, N, 1\right)}} \]

5. Taylor expanded in N around 0

   \[\leadsto \tan^{-1}_* \frac{\color{blue}{1}}{\mathsf{fma}\left(N - -1, N, 1\right)} \]
6. Step-by-step derivation

7. Applied rewrites 99.6%

   \[\leadsto \tan^{-1}_* \frac{\color{blue}{1}}{\mathsf{fma}\left(N - -1, N, 1\right)} \]
8. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{1}{\mathsf{fma}\left(\color{blue}{N - -1}, N, 1\right)} \]

   2. lift-fma.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{\left(N - -1\right) \cdot N + 1}} \]

   3. +-commutative N/A

      \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{1 + \left(N - -1\right) \cdot N}} \]

   4. *-commutative N/A

      \[\leadsto \tan^{-1}_* \frac{1}{1 + \color{blue}{N \cdot \left(N - -1\right)}} \]

   5. distribute-rgt-out-- N/A

      \[\leadsto \tan^{-1}_* \frac{1}{1 + \color{blue}{\left(N \cdot N - -1 \cdot N\right)}} \]

   6. pow2 N/A

      \[\leadsto \tan^{-1}_* \frac{1}{1 + \left(\color{blue}{{N}^{2}} - -1 \cdot N\right)} \]

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

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

   8. metadata-eval N/A

      \[\leadsto \tan^{-1}_* \frac{1}{1 + \left({N}^{2} + \color{blue}{1} \cdot N\right)} \]

   9. *-lft-identity N/A

      \[\leadsto \tan^{-1}_* \frac{1}{1 + \left({N}^{2} + \color{blue}{N}\right)} \]

   10. associate-+r+ N/A

       \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{\left(1 + {N}^{2}\right) + N}} \]

   11. +-commutative N/A

       \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{\left({N}^{2} + 1\right)} + N} \]

   12. pow2 N/A

       \[\leadsto \tan^{-1}_* \frac{1}{\left(\color{blue}{N \cdot N} + 1\right) + N} \]

   13. lower-+.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{\left(N \cdot N + 1\right) + N}} \]

   14. lift-fma.f64 99.6

       \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{\mathsf{fma}\left(N, N, 1\right)} + N} \]

9. Applied rewrites 99.6%

   \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{\mathsf{fma}\left(N, N, 1\right) + N}} \]
10. Add Preprocessing

Alternative 4: 96.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \tan^{-1}_* \frac{1}{N + N \cdot N} \end{array} \]

FPCore

(FPCore (N) :precision binary64 (atan2 1.0 (+ N (* N N))))

C

double code(double N) {
	return atan2(1.0, (N + (N * N)));
}

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(n)
use fmin_fmax_functions
    real(8), intent (in) :: n
    code = atan2(1.0d0, (n + (n * n)))
end function

Java

public static double code(double N) {
	return Math.atan2(1.0, (N + (N * N)));
}

Python

def code(N):
	return math.atan2(1.0, (N + (N * N)))

Julia

function code(N)
	return atan(1.0, Float64(N + Float64(N * N)))
end

MATLAB

function tmp = code(N)
	tmp = atan2(1.0, (N + (N * N)));
end

Wolfram

code[N_] := N[ArcTan[1.0 / N[(N + N[(N * N), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 6.9%

   \[\tan^{-1} \left(N + 1\right) - \tan^{-1} N \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\tan^{-1} \left(N + 1\right) - \tan^{-1} N} \]

   2. lift-+.f64 N/A

      \[\leadsto \tan^{-1} \color{blue}{\left(N + 1\right)} - \tan^{-1} N \]

   3. lift-atan.f64 N/A

      \[\leadsto \color{blue}{\tan^{-1} \left(N + 1\right)} - \tan^{-1} N \]

   4. lift-atan.f64 N/A

      \[\leadsto \tan^{-1} \left(N + 1\right) - \color{blue}{\tan^{-1} N} \]

   5. diff-atan N/A

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\left(N + 1\right) - N}{1 + \left(N + 1\right) \cdot N}} \]

   6. lower-atan2.f64 N/A

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\left(N + 1\right) - N}{1 + \left(N + 1\right) \cdot N}} \]

   7. lower--.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(N + 1\right) - N}}{1 + \left(N + 1\right) \cdot N} \]

   8. metadata-eval N/A

      \[\leadsto \tan^{-1}_* \frac{\left(N + \color{blue}{1 \cdot 1}\right) - N}{1 + \left(N + 1\right) \cdot N} \]

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

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

   10. metadata-eval N/A

       \[\leadsto \tan^{-1}_* \frac{\left(N - \color{blue}{-1} \cdot 1\right) - N}{1 + \left(N + 1\right) \cdot N} \]

   11. metadata-eval N/A

       \[\leadsto \tan^{-1}_* \frac{\left(N - \color{blue}{-1}\right) - N}{1 + \left(N + 1\right) \cdot N} \]

   12. lower--.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(N - -1\right)} - N}{1 + \left(N + 1\right) \cdot N} \]

   13. +-commutative N/A

       \[\leadsto \tan^{-1}_* \frac{\left(N - -1\right) - N}{\color{blue}{\left(N + 1\right) \cdot N + 1}} \]

   14. lower-fma.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\left(N - -1\right) - N}{\color{blue}{\mathsf{fma}\left(N + 1, N, 1\right)}} \]

   15. metadata-eval N/A

       \[\leadsto \tan^{-1}_* \frac{\left(N - -1\right) - N}{\mathsf{fma}\left(N + \color{blue}{1 \cdot 1}, N, 1\right)} \]

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

       \[\leadsto \tan^{-1}_* \frac{\left(N - -1\right) - N}{\mathsf{fma}\left(\color{blue}{N - \left(\mathsf{neg}\left(1\right)\right) \cdot 1}, N, 1\right)} \]

   17. metadata-eval N/A

       \[\leadsto \tan^{-1}_* \frac{\left(N - -1\right) - N}{\mathsf{fma}\left(N - \color{blue}{-1} \cdot 1, N, 1\right)} \]

   18. metadata-eval N/A

       \[\leadsto \tan^{-1}_* \frac{\left(N - -1\right) - N}{\mathsf{fma}\left(N - \color{blue}{-1}, N, 1\right)} \]

   19. lower--.f64 15.8

       \[\leadsto \tan^{-1}_* \frac{\left(N - -1\right) - N}{\mathsf{fma}\left(\color{blue}{N - -1}, N, 1\right)} \]

4. Applied rewrites 15.8%

   \[\leadsto \color{blue}{\tan^{-1}_* \frac{\left(N - -1\right) - N}{\mathsf{fma}\left(N - -1, N, 1\right)}} \]

5. Taylor expanded in N around 0

   \[\leadsto \tan^{-1}_* \frac{\color{blue}{1}}{\mathsf{fma}\left(N - -1, N, 1\right)} \]
6. Step-by-step derivation

7. Applied rewrites 99.6%

   \[\leadsto \tan^{-1}_* \frac{\color{blue}{1}}{\mathsf{fma}\left(N - -1, N, 1\right)} \]
8. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{1}{\mathsf{fma}\left(\color{blue}{N - -1}, N, 1\right)} \]

   2. lift-fma.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{\left(N - -1\right) \cdot N + 1}} \]

   3. metadata-eval N/A

      \[\leadsto \tan^{-1}_* \frac{1}{\left(N - -1\right) \cdot N + \color{blue}{-1 \cdot -1}} \]

   4. metadata-eval N/A

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

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

      \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{\left(N - -1\right) \cdot N - 1 \cdot -1}} \]

   6. *-commutative N/A

      \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{N \cdot \left(N - -1\right)} - 1 \cdot -1} \]

   7. distribute-rgt-out-- N/A

      \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{\left(N \cdot N - -1 \cdot N\right)} - 1 \cdot -1} \]

   8. pow2 N/A

      \[\leadsto \tan^{-1}_* \frac{1}{\left(\color{blue}{{N}^{2}} - -1 \cdot N\right) - 1 \cdot -1} \]

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

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

   10. metadata-eval N/A

       \[\leadsto \tan^{-1}_* \frac{1}{\left({N}^{2} + \color{blue}{1} \cdot N\right) - 1 \cdot -1} \]

   11. *-lft-identity N/A

       \[\leadsto \tan^{-1}_* \frac{1}{\left({N}^{2} + \color{blue}{N}\right) - 1 \cdot -1} \]

   12. metadata-eval N/A

       \[\leadsto \tan^{-1}_* \frac{1}{\left({N}^{2} + N\right) - \color{blue}{-1}} \]

   13. associate-+r- N/A

       \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{{N}^{2} + \left(N - -1\right)}} \]

   14. +-commutative N/A

       \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{\left(N - -1\right) + {N}^{2}}} \]

   15. pow2 N/A

       \[\leadsto \tan^{-1}_* \frac{1}{\left(N - -1\right) + \color{blue}{N \cdot N}} \]

   16. lift--.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{\left(N - -1\right)} + N \cdot N} \]

   17. lift-*.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{1}{\left(N - -1\right) + \color{blue}{N \cdot N}} \]

   18. lift-+.f64 99.6

       \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{\left(N - -1\right) + N \cdot N}} \]

9. Applied rewrites 99.6%

   \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{\left(N - -1\right) + N \cdot N}} \]

10. Taylor expanded in N around inf

    \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{N} + N \cdot N} \]

12. Applied rewrites 97.8%

    \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{N} + N \cdot N} \]
13. Add Preprocessing

Alternative 5: 96.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, N, N\right)} \end{array} \]

FPCore

(FPCore (N) :precision binary64 (atan2 1.0 (fma N N N)))

C

double code(double N) {
	return atan2(1.0, fma(N, N, N));
}

Julia

function code(N)
	return atan(1.0, fma(N, N, N))
end

Wolfram

code[N_] := N[ArcTan[1.0 / N[(N * N + N), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 6.9%

   \[\tan^{-1} \left(N + 1\right) - \tan^{-1} N \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\tan^{-1} \left(N + 1\right) - \tan^{-1} N} \]

   2. lift-+.f64 N/A

      \[\leadsto \tan^{-1} \color{blue}{\left(N + 1\right)} - \tan^{-1} N \]

   3. lift-atan.f64 N/A

      \[\leadsto \color{blue}{\tan^{-1} \left(N + 1\right)} - \tan^{-1} N \]

   4. lift-atan.f64 N/A

      \[\leadsto \tan^{-1} \left(N + 1\right) - \color{blue}{\tan^{-1} N} \]

   5. diff-atan N/A

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\left(N + 1\right) - N}{1 + \left(N + 1\right) \cdot N}} \]

   6. lower-atan2.f64 N/A

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\left(N + 1\right) - N}{1 + \left(N + 1\right) \cdot N}} \]

   7. lower--.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(N + 1\right) - N}}{1 + \left(N + 1\right) \cdot N} \]

   8. metadata-eval N/A

      \[\leadsto \tan^{-1}_* \frac{\left(N + \color{blue}{1 \cdot 1}\right) - N}{1 + \left(N + 1\right) \cdot N} \]

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

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

   10. metadata-eval N/A

       \[\leadsto \tan^{-1}_* \frac{\left(N - \color{blue}{-1} \cdot 1\right) - N}{1 + \left(N + 1\right) \cdot N} \]

   11. metadata-eval N/A

       \[\leadsto \tan^{-1}_* \frac{\left(N - \color{blue}{-1}\right) - N}{1 + \left(N + 1\right) \cdot N} \]

   12. lower--.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(N - -1\right)} - N}{1 + \left(N + 1\right) \cdot N} \]

   13. +-commutative N/A

       \[\leadsto \tan^{-1}_* \frac{\left(N - -1\right) - N}{\color{blue}{\left(N + 1\right) \cdot N + 1}} \]

   14. lower-fma.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\left(N - -1\right) - N}{\color{blue}{\mathsf{fma}\left(N + 1, N, 1\right)}} \]

   15. metadata-eval N/A

       \[\leadsto \tan^{-1}_* \frac{\left(N - -1\right) - N}{\mathsf{fma}\left(N + \color{blue}{1 \cdot 1}, N, 1\right)} \]

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

       \[\leadsto \tan^{-1}_* \frac{\left(N - -1\right) - N}{\mathsf{fma}\left(\color{blue}{N - \left(\mathsf{neg}\left(1\right)\right) \cdot 1}, N, 1\right)} \]

   17. metadata-eval N/A

       \[\leadsto \tan^{-1}_* \frac{\left(N - -1\right) - N}{\mathsf{fma}\left(N - \color{blue}{-1} \cdot 1, N, 1\right)} \]

   18. metadata-eval N/A

       \[\leadsto \tan^{-1}_* \frac{\left(N - -1\right) - N}{\mathsf{fma}\left(N - \color{blue}{-1}, N, 1\right)} \]

   19. lower--.f64 15.8

       \[\leadsto \tan^{-1}_* \frac{\left(N - -1\right) - N}{\mathsf{fma}\left(\color{blue}{N - -1}, N, 1\right)} \]

4. Applied rewrites 15.8%

   \[\leadsto \color{blue}{\tan^{-1}_* \frac{\left(N - -1\right) - N}{\mathsf{fma}\left(N - -1, N, 1\right)}} \]

5. Taylor expanded in N around 0

   \[\leadsto \tan^{-1}_* \frac{\color{blue}{1}}{\mathsf{fma}\left(N - -1, N, 1\right)} \]
6. Step-by-step derivation

7. Applied rewrites 99.6%

   \[\leadsto \tan^{-1}_* \frac{\color{blue}{1}}{\mathsf{fma}\left(N - -1, N, 1\right)} \]

8. Taylor expanded in N around inf

   \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{{N}^{2} \cdot \left(1 + \frac{1}{N}\right)}} \]
9. Step-by-step derivation

   1. distribute-rgt-in N/A

      \[\leadsto \tan^{-1}_* \frac{1}{1 \cdot {N}^{2} + \color{blue}{\frac{1}{N} \cdot {N}^{2}}} \]

   2. inv-pow N/A

      \[\leadsto \tan^{-1}_* \frac{1}{1 \cdot {N}^{2} + {N}^{-1} \cdot {\color{blue}{N}}^{2}} \]

   3. pow-prod-up N/A

      \[\leadsto \tan^{-1}_* \frac{1}{1 \cdot {N}^{2} + {N}^{\color{blue}{\left(-1 + 2\right)}}} \]

   4. metadata-eval N/A

      \[\leadsto \tan^{-1}_* \frac{1}{1 \cdot {N}^{2} + {N}^{1}} \]

   5. unpow1 N/A

      \[\leadsto \tan^{-1}_* \frac{1}{1 \cdot {N}^{2} + N} \]

   6. *-lft-identity N/A

      \[\leadsto \tan^{-1}_* \frac{1}{{N}^{2} + N} \]

   7. pow2 N/A

      \[\leadsto \tan^{-1}_* \frac{1}{N \cdot N + N} \]

   8. lower-fma.f64 97.8

      \[\leadsto \tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, \color{blue}{N}, N\right)} \]

10. Applied rewrites 97.8%

    \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{\mathsf{fma}\left(N, N, N\right)}} \]
11. Add Preprocessing

Alternative 6: 93.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \tan^{-1}_* \frac{1}{N \cdot N} \end{array} \]

FPCore

(FPCore (N) :precision binary64 (atan2 1.0 (* N N)))

C

double code(double N) {
	return atan2(1.0, (N * N));
}

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(n)
use fmin_fmax_functions
    real(8), intent (in) :: n
    code = atan2(1.0d0, (n * n))
end function

Java

public static double code(double N) {
	return Math.atan2(1.0, (N * N));
}

Python

def code(N):
	return math.atan2(1.0, (N * N))

Julia

function code(N)
	return atan(1.0, Float64(N * N))
end

MATLAB

function tmp = code(N)
	tmp = atan2(1.0, (N * N));
end

Wolfram

code[N_] := N[ArcTan[1.0 / N[(N * N), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 6.9%

   \[\tan^{-1} \left(N + 1\right) - \tan^{-1} N \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\tan^{-1} \left(N + 1\right) - \tan^{-1} N} \]

   2. lift-+.f64 N/A

      \[\leadsto \tan^{-1} \color{blue}{\left(N + 1\right)} - \tan^{-1} N \]

   3. lift-atan.f64 N/A

      \[\leadsto \color{blue}{\tan^{-1} \left(N + 1\right)} - \tan^{-1} N \]

   4. lift-atan.f64 N/A

      \[\leadsto \tan^{-1} \left(N + 1\right) - \color{blue}{\tan^{-1} N} \]

   5. diff-atan N/A

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\left(N + 1\right) - N}{1 + \left(N + 1\right) \cdot N}} \]

   6. lower-atan2.f64 N/A

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\left(N + 1\right) - N}{1 + \left(N + 1\right) \cdot N}} \]

   7. lower--.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(N + 1\right) - N}}{1 + \left(N + 1\right) \cdot N} \]

   8. metadata-eval N/A

      \[\leadsto \tan^{-1}_* \frac{\left(N + \color{blue}{1 \cdot 1}\right) - N}{1 + \left(N + 1\right) \cdot N} \]

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

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

   10. metadata-eval N/A

       \[\leadsto \tan^{-1}_* \frac{\left(N - \color{blue}{-1} \cdot 1\right) - N}{1 + \left(N + 1\right) \cdot N} \]

   11. metadata-eval N/A

       \[\leadsto \tan^{-1}_* \frac{\left(N - \color{blue}{-1}\right) - N}{1 + \left(N + 1\right) \cdot N} \]

   12. lower--.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(N - -1\right)} - N}{1 + \left(N + 1\right) \cdot N} \]

   13. +-commutative N/A

       \[\leadsto \tan^{-1}_* \frac{\left(N - -1\right) - N}{\color{blue}{\left(N + 1\right) \cdot N + 1}} \]

   14. lower-fma.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\left(N - -1\right) - N}{\color{blue}{\mathsf{fma}\left(N + 1, N, 1\right)}} \]

   15. metadata-eval N/A

       \[\leadsto \tan^{-1}_* \frac{\left(N - -1\right) - N}{\mathsf{fma}\left(N + \color{blue}{1 \cdot 1}, N, 1\right)} \]

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

       \[\leadsto \tan^{-1}_* \frac{\left(N - -1\right) - N}{\mathsf{fma}\left(\color{blue}{N - \left(\mathsf{neg}\left(1\right)\right) \cdot 1}, N, 1\right)} \]

   17. metadata-eval N/A

       \[\leadsto \tan^{-1}_* \frac{\left(N - -1\right) - N}{\mathsf{fma}\left(N - \color{blue}{-1} \cdot 1, N, 1\right)} \]

   18. metadata-eval N/A

       \[\leadsto \tan^{-1}_* \frac{\left(N - -1\right) - N}{\mathsf{fma}\left(N - \color{blue}{-1}, N, 1\right)} \]

   19. lower--.f64 15.8

       \[\leadsto \tan^{-1}_* \frac{\left(N - -1\right) - N}{\mathsf{fma}\left(\color{blue}{N - -1}, N, 1\right)} \]

4. Applied rewrites 15.8%

   \[\leadsto \color{blue}{\tan^{-1}_* \frac{\left(N - -1\right) - N}{\mathsf{fma}\left(N - -1, N, 1\right)}} \]

5. Taylor expanded in N around 0

   \[\leadsto \tan^{-1}_* \frac{\color{blue}{1}}{\mathsf{fma}\left(N - -1, N, 1\right)} \]
6. Step-by-step derivation

7. Applied rewrites 99.6%

   \[\leadsto \tan^{-1}_* \frac{\color{blue}{1}}{\mathsf{fma}\left(N - -1, N, 1\right)} \]
8. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{1}{\mathsf{fma}\left(\color{blue}{N - -1}, N, 1\right)} \]

   2. lift-fma.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{\left(N - -1\right) \cdot N + 1}} \]

   3. metadata-eval N/A

      \[\leadsto \tan^{-1}_* \frac{1}{\left(N - -1\right) \cdot N + \color{blue}{-1 \cdot -1}} \]

   4. metadata-eval N/A

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

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

      \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{\left(N - -1\right) \cdot N - 1 \cdot -1}} \]

   6. *-commutative N/A

      \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{N \cdot \left(N - -1\right)} - 1 \cdot -1} \]

   7. distribute-rgt-out-- N/A

      \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{\left(N \cdot N - -1 \cdot N\right)} - 1 \cdot -1} \]

   8. pow2 N/A

      \[\leadsto \tan^{-1}_* \frac{1}{\left(\color{blue}{{N}^{2}} - -1 \cdot N\right) - 1 \cdot -1} \]

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

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

   10. metadata-eval N/A

       \[\leadsto \tan^{-1}_* \frac{1}{\left({N}^{2} + \color{blue}{1} \cdot N\right) - 1 \cdot -1} \]

   11. *-lft-identity N/A

       \[\leadsto \tan^{-1}_* \frac{1}{\left({N}^{2} + \color{blue}{N}\right) - 1 \cdot -1} \]

   12. metadata-eval N/A

       \[\leadsto \tan^{-1}_* \frac{1}{\left({N}^{2} + N\right) - \color{blue}{-1}} \]

   13. associate-+r- N/A

       \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{{N}^{2} + \left(N - -1\right)}} \]

   14. +-commutative N/A

       \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{\left(N - -1\right) + {N}^{2}}} \]

   15. pow2 N/A

       \[\leadsto \tan^{-1}_* \frac{1}{\left(N - -1\right) + \color{blue}{N \cdot N}} \]

   16. lift--.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{\left(N - -1\right)} + N \cdot N} \]

   17. lift-*.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{1}{\left(N - -1\right) + \color{blue}{N \cdot N}} \]

   18. lift-+.f64 99.6

       \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{\left(N - -1\right) + N \cdot N}} \]

9. Applied rewrites 99.6%

   \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{\left(N - -1\right) + N \cdot N}} \]

10. Taylor expanded in N around inf

    \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{{N}^{2}}} \]
11. Step-by-step derivation

    1. pow2 N/A

       \[\leadsto \tan^{-1}_* \frac{1}{{N}^{2}} \]

    2. +-commutative N/A

       \[\leadsto \tan^{-1}_* \frac{1}{{\color{blue}{N}}^{2}} \]

    3. metadata-eval N/A

       \[\leadsto \tan^{-1}_* \frac{1}{{N}^{2}} \]

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

       \[\leadsto \tan^{-1}_* \frac{1}{{N}^{2}} \]

    5. metadata-eval N/A

       \[\leadsto \tan^{-1}_* \frac{1}{{N}^{2}} \]

    6. metadata-eval N/A

       \[\leadsto \tan^{-1}_* \frac{1}{{N}^{2}} \]

    7. +-commutative N/A

       \[\leadsto \tan^{-1}_* \frac{1}{{N}^{2}} \]

    8. associate-+l+ N/A

       \[\leadsto \tan^{-1}_* \frac{1}{{\color{blue}{N}}^{2}} \]

    9. pow2 N/A

       \[\leadsto \tan^{-1}_* \frac{1}{{N}^{2}} \]

    10. pow2 N/A

        \[\leadsto \tan^{-1}_* \frac{1}{N \cdot \color{blue}{N}} \]

    11. lift-*.f64 95.3

        \[\leadsto \tan^{-1}_* \frac{1}{N \cdot \color{blue}{N}} \]

12. Applied rewrites 95.3%

    \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{N \cdot N}} \]
13. Add Preprocessing

Alternative 7: 7.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \tan^{-1}_* \frac{1}{N - -1} \end{array} \]

FPCore

(FPCore (N) :precision binary64 (atan2 1.0 (- N -1.0)))

C

double code(double N) {
	return atan2(1.0, (N - -1.0));
}

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(n)
use fmin_fmax_functions
    real(8), intent (in) :: n
    code = atan2(1.0d0, (n - (-1.0d0)))
end function

Java

public static double code(double N) {
	return Math.atan2(1.0, (N - -1.0));
}

Python

def code(N):
	return math.atan2(1.0, (N - -1.0))

Julia

function code(N)
	return atan(1.0, Float64(N - -1.0))
end

MATLAB

function tmp = code(N)
	tmp = atan2(1.0, (N - -1.0));
end

Wolfram

code[N_] := N[ArcTan[1.0 / N[(N - -1.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 6.9%

   \[\tan^{-1} \left(N + 1\right) - \tan^{-1} N \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\tan^{-1} \left(N + 1\right) - \tan^{-1} N} \]

   2. lift-+.f64 N/A

      \[\leadsto \tan^{-1} \color{blue}{\left(N + 1\right)} - \tan^{-1} N \]

   3. lift-atan.f64 N/A

      \[\leadsto \color{blue}{\tan^{-1} \left(N + 1\right)} - \tan^{-1} N \]

   4. lift-atan.f64 N/A

      \[\leadsto \tan^{-1} \left(N + 1\right) - \color{blue}{\tan^{-1} N} \]

   5. diff-atan N/A

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\left(N + 1\right) - N}{1 + \left(N + 1\right) \cdot N}} \]

   6. lower-atan2.f64 N/A

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\left(N + 1\right) - N}{1 + \left(N + 1\right) \cdot N}} \]

   7. lower--.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(N + 1\right) - N}}{1 + \left(N + 1\right) \cdot N} \]

   8. metadata-eval N/A

      \[\leadsto \tan^{-1}_* \frac{\left(N + \color{blue}{1 \cdot 1}\right) - N}{1 + \left(N + 1\right) \cdot N} \]

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

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

   10. metadata-eval N/A

       \[\leadsto \tan^{-1}_* \frac{\left(N - \color{blue}{-1} \cdot 1\right) - N}{1 + \left(N + 1\right) \cdot N} \]

   11. metadata-eval N/A

       \[\leadsto \tan^{-1}_* \frac{\left(N - \color{blue}{-1}\right) - N}{1 + \left(N + 1\right) \cdot N} \]

   12. lower--.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(N - -1\right)} - N}{1 + \left(N + 1\right) \cdot N} \]

   13. +-commutative N/A

       \[\leadsto \tan^{-1}_* \frac{\left(N - -1\right) - N}{\color{blue}{\left(N + 1\right) \cdot N + 1}} \]

   14. lower-fma.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\left(N - -1\right) - N}{\color{blue}{\mathsf{fma}\left(N + 1, N, 1\right)}} \]

   15. metadata-eval N/A

       \[\leadsto \tan^{-1}_* \frac{\left(N - -1\right) - N}{\mathsf{fma}\left(N + \color{blue}{1 \cdot 1}, N, 1\right)} \]

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

       \[\leadsto \tan^{-1}_* \frac{\left(N - -1\right) - N}{\mathsf{fma}\left(\color{blue}{N - \left(\mathsf{neg}\left(1\right)\right) \cdot 1}, N, 1\right)} \]

   17. metadata-eval N/A

       \[\leadsto \tan^{-1}_* \frac{\left(N - -1\right) - N}{\mathsf{fma}\left(N - \color{blue}{-1} \cdot 1, N, 1\right)} \]

   18. metadata-eval N/A

       \[\leadsto \tan^{-1}_* \frac{\left(N - -1\right) - N}{\mathsf{fma}\left(N - \color{blue}{-1}, N, 1\right)} \]

   19. lower--.f64 15.8

       \[\leadsto \tan^{-1}_* \frac{\left(N - -1\right) - N}{\mathsf{fma}\left(\color{blue}{N - -1}, N, 1\right)} \]

4. Applied rewrites 15.8%

   \[\leadsto \color{blue}{\tan^{-1}_* \frac{\left(N - -1\right) - N}{\mathsf{fma}\left(N - -1, N, 1\right)}} \]

5. Taylor expanded in N around 0

   \[\leadsto \tan^{-1}_* \frac{\color{blue}{1}}{\mathsf{fma}\left(N - -1, N, 1\right)} \]
6. Step-by-step derivation

7. Applied rewrites 99.6%

   \[\leadsto \tan^{-1}_* \frac{\color{blue}{1}}{\mathsf{fma}\left(N - -1, N, 1\right)} \]

8. Taylor expanded in N around 0

   \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{1 + N}} \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \tan^{-1}_* \frac{1}{N + \color{blue}{1}} \]

   2. metadata-eval N/A

      \[\leadsto \tan^{-1}_* \frac{1}{N + 1 \cdot \color{blue}{1}} \]

   3. metadata-eval N/A

      \[\leadsto \tan^{-1}_* \frac{1}{N + \left(\mathsf{neg}\left(-1\right)\right) \cdot 1} \]

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

      \[\leadsto \tan^{-1}_* \frac{1}{N - \color{blue}{-1 \cdot 1}} \]

   5. metadata-eval N/A

      \[\leadsto \tan^{-1}_* \frac{1}{N - -1} \]

   6. lift--.f64 7.7

      \[\leadsto \tan^{-1}_* \frac{1}{N - \color{blue}{-1}} \]

10. Applied rewrites 7.7%

    \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{N - -1}} \]
11. Add Preprocessing

Alternative 8: 7.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \tan^{-1}_* \frac{1}{N} \end{array} \]

FPCore

(FPCore (N) :precision binary64 (atan2 1.0 N))

C

double code(double N) {
	return atan2(1.0, N);
}

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(n)
use fmin_fmax_functions
    real(8), intent (in) :: n
    code = atan2(1.0d0, n)
end function

Java

public static double code(double N) {
	return Math.atan2(1.0, N);
}

Python

def code(N):
	return math.atan2(1.0, N)

Julia

function code(N)
	return atan(1.0, N)
end

MATLAB

function tmp = code(N)
	tmp = atan2(1.0, N);
end

Wolfram

code[N_] := N[ArcTan[1.0 / N], $MachinePrecision]

Derivation

1. Initial program 6.9%

   \[\tan^{-1} \left(N + 1\right) - \tan^{-1} N \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\tan^{-1} \left(N + 1\right) - \tan^{-1} N} \]

   2. lift-+.f64 N/A

      \[\leadsto \tan^{-1} \color{blue}{\left(N + 1\right)} - \tan^{-1} N \]

   3. lift-atan.f64 N/A

      \[\leadsto \color{blue}{\tan^{-1} \left(N + 1\right)} - \tan^{-1} N \]

   4. lift-atan.f64 N/A

      \[\leadsto \tan^{-1} \left(N + 1\right) - \color{blue}{\tan^{-1} N} \]

   5. diff-atan N/A

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\left(N + 1\right) - N}{1 + \left(N + 1\right) \cdot N}} \]

   6. lower-atan2.f64 N/A

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\left(N + 1\right) - N}{1 + \left(N + 1\right) \cdot N}} \]

   7. lower--.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(N + 1\right) - N}}{1 + \left(N + 1\right) \cdot N} \]

   8. metadata-eval N/A

      \[\leadsto \tan^{-1}_* \frac{\left(N + \color{blue}{1 \cdot 1}\right) - N}{1 + \left(N + 1\right) \cdot N} \]

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

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

   10. metadata-eval N/A

       \[\leadsto \tan^{-1}_* \frac{\left(N - \color{blue}{-1} \cdot 1\right) - N}{1 + \left(N + 1\right) \cdot N} \]

   11. metadata-eval N/A

       \[\leadsto \tan^{-1}_* \frac{\left(N - \color{blue}{-1}\right) - N}{1 + \left(N + 1\right) \cdot N} \]

   12. lower--.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(N - -1\right)} - N}{1 + \left(N + 1\right) \cdot N} \]

   13. +-commutative N/A

       \[\leadsto \tan^{-1}_* \frac{\left(N - -1\right) - N}{\color{blue}{\left(N + 1\right) \cdot N + 1}} \]

   14. lower-fma.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\left(N - -1\right) - N}{\color{blue}{\mathsf{fma}\left(N + 1, N, 1\right)}} \]

   15. metadata-eval N/A

       \[\leadsto \tan^{-1}_* \frac{\left(N - -1\right) - N}{\mathsf{fma}\left(N + \color{blue}{1 \cdot 1}, N, 1\right)} \]

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

       \[\leadsto \tan^{-1}_* \frac{\left(N - -1\right) - N}{\mathsf{fma}\left(\color{blue}{N - \left(\mathsf{neg}\left(1\right)\right) \cdot 1}, N, 1\right)} \]

   17. metadata-eval N/A

       \[\leadsto \tan^{-1}_* \frac{\left(N - -1\right) - N}{\mathsf{fma}\left(N - \color{blue}{-1} \cdot 1, N, 1\right)} \]

   18. metadata-eval N/A

       \[\leadsto \tan^{-1}_* \frac{\left(N - -1\right) - N}{\mathsf{fma}\left(N - \color{blue}{-1}, N, 1\right)} \]

   19. lower--.f64 15.8

       \[\leadsto \tan^{-1}_* \frac{\left(N - -1\right) - N}{\mathsf{fma}\left(\color{blue}{N - -1}, N, 1\right)} \]

4. Applied rewrites 15.8%

   \[\leadsto \color{blue}{\tan^{-1}_* \frac{\left(N - -1\right) - N}{\mathsf{fma}\left(N - -1, N, 1\right)}} \]

5. Taylor expanded in N around 0

   \[\leadsto \tan^{-1}_* \frac{\color{blue}{1}}{\mathsf{fma}\left(N - -1, N, 1\right)} \]
6. Step-by-step derivation

7. Applied rewrites 99.6%

   \[\leadsto \tan^{-1}_* \frac{\color{blue}{1}}{\mathsf{fma}\left(N - -1, N, 1\right)} \]

8. Taylor expanded in N around 0

   \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{1 + N}} \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \tan^{-1}_* \frac{1}{N + \color{blue}{1}} \]

   2. metadata-eval N/A

      \[\leadsto \tan^{-1}_* \frac{1}{N + 1 \cdot \color{blue}{1}} \]

   3. metadata-eval N/A

      \[\leadsto \tan^{-1}_* \frac{1}{N + \left(\mathsf{neg}\left(-1\right)\right) \cdot 1} \]

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

      \[\leadsto \tan^{-1}_* \frac{1}{N - \color{blue}{-1 \cdot 1}} \]

   5. metadata-eval N/A

      \[\leadsto \tan^{-1}_* \frac{1}{N - -1} \]

   6. lift--.f64 7.7

      \[\leadsto \tan^{-1}_* \frac{1}{N - \color{blue}{-1}} \]

10. Applied rewrites 7.7%

    \[\leadsto \tan^{-1}_* \frac{1}{\color{blue}{N - -1}} \]

11. Taylor expanded in N around inf

    \[\leadsto \tan^{-1}_* \frac{1}{N} \]

13. Applied rewrites 7.7%

    \[\leadsto \tan^{-1}_* \frac{1}{N} \]
14. Add Preprocessing

Developer Target 1: 99.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \tan^{-1}_* \frac{1}{\mathsf{fma}\left(N, 1 + N, 1\right)} \end{array} \]

FPCore

(FPCore (N) :precision binary64 (atan2 1.0 (fma N (+ 1.0 N) 1.0)))

C

double code(double N) {
	return atan2(1.0, fma(N, (1.0 + N), 1.0));
}

Julia

function code(N)
	return atan(1.0, fma(N, Float64(1.0 + N), 1.0))
end

Wolfram

code[N_] := N[ArcTan[1.0 / N[(N * N[(1.0 + N), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]

Reproduce

herbie shell --seed 2025080 
(FPCore (N)
  :name "2atan (example 3.5)"
  :precision binary64
  :pre (and (> N 1.0) (< N 1e+100))

  :alt
  (! :herbie-platform default (atan2 1 (fma N (+ 1 N) 1)))

  (- (atan (+ N 1.0)) (atan N)))