Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 7.9s

Alternatives: 12

Speedup: 0.9×

Specification

Math

\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))

C

double code(double x, double y, double z, double t, double a) {
	return ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = ((log((x + y)) + log(z)) - t) + ((a - 0.5d0) * log(t))
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return ((Math.log((x + y)) + Math.log(z)) - t) + ((a - 0.5) * Math.log(t));
}

Python

def code(x, y, z, t, a):
	return ((math.log((x + y)) + math.log(z)) - t) + ((a - 0.5) * math.log(t))

Julia

function code(x, y, z, t, a)
	return Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + Float64(Float64(a - 0.5) * log(t)))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.6% accurate, 1.0× speedup

Math

\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))

C

double code(double x, double y, double z, double t, double a) {
	return ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = ((log((x + y)) + log(z)) - t) + ((a - 0.5d0) * log(t))
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return ((Math.log((x + y)) + Math.log(z)) - t) + ((a - 0.5) * Math.log(t));
}

Python

def code(x, y, z, t, a):
	return ((math.log((x + y)) + math.log(z)) - t) + ((a - 0.5) * math.log(t))

Julia

function code(x, y, z, t, a)
	return Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + Float64(Float64(a - 0.5) * log(t)))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.6% accurate, 0.9× speedup

Math

\[\mathsf{fma}\left(\frac{\log z}{t}, t, \log \left(x + y\right) - \mathsf{fma}\left(\log t, 0.5 - a, t\right)\right) \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (fma (/ (log z) t) t (- (log (+ x y)) (fma (log t) (- 0.5 a) t))))

C

double code(double x, double y, double z, double t, double a) {
	return fma((log(z) / t), t, (log((x + y)) - fma(log(t), (0.5 - a), t)));
}

Julia

function code(x, y, z, t, a)
	return fma(Float64(log(z) / t), t, Float64(log(Float64(x + y)) - fma(log(t), Float64(0.5 - a), t)))
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[Log[z], $MachinePrecision] / t), $MachinePrecision] * t + N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] - N[(N[Log[t], $MachinePrecision] * N[(0.5 - a), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]

3. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\left(1 - \frac{\mathsf{fma}\left(0.5 - a, \log t, t\right) - \log \left(y + x\right)}{\log z}\right) \cdot \log z} \]
4. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(1 - \frac{\mathsf{fma}\left(\frac{1}{2} - a, \log t, t\right) - \log \left(y + x\right)}{\log z}\right) \cdot \log z} \]

   2. lift--.f64 N/A

      \[\leadsto \color{blue}{\left(1 - \frac{\mathsf{fma}\left(\frac{1}{2} - a, \log t, t\right) - \log \left(y + x\right)}{\log z}\right)} \cdot \log z \]

   3. lift-/.f64 N/A

      \[\leadsto \left(1 - \color{blue}{\frac{\mathsf{fma}\left(\frac{1}{2} - a, \log t, t\right) - \log \left(y + x\right)}{\log z}}\right) \cdot \log z \]

   4. sub-to-mult-rev N/A

      \[\leadsto \color{blue}{\log z - \left(\mathsf{fma}\left(\frac{1}{2} - a, \log t, t\right) - \log \left(y + x\right)\right)} \]

   5. sub-flip N/A

      \[\leadsto \color{blue}{\log z + \left(\mathsf{neg}\left(\left(\mathsf{fma}\left(\frac{1}{2} - a, \log t, t\right) - \log \left(y + x\right)\right)\right)\right)} \]

   6. *-lft-identity N/A

      \[\leadsto \color{blue}{1 \cdot \log z} + \left(\mathsf{neg}\left(\left(\mathsf{fma}\left(\frac{1}{2} - a, \log t, t\right) - \log \left(y + x\right)\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \color{blue}{\log z \cdot 1} + \left(\mathsf{neg}\left(\left(\mathsf{fma}\left(\frac{1}{2} - a, \log t, t\right) - \log \left(y + x\right)\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \log z \cdot \color{blue}{{t}^{0}} + \left(\mathsf{neg}\left(\left(\mathsf{fma}\left(\frac{1}{2} - a, \log t, t\right) - \log \left(y + x\right)\right)\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \log z \cdot {t}^{\color{blue}{\left(-1 + 1\right)}} + \left(\mathsf{neg}\left(\left(\mathsf{fma}\left(\frac{1}{2} - a, \log t, t\right) - \log \left(y + x\right)\right)\right)\right) \]

   10. pow-plus N/A

       \[\leadsto \log z \cdot \color{blue}{\left({t}^{-1} \cdot t\right)} + \left(\mathsf{neg}\left(\left(\mathsf{fma}\left(\frac{1}{2} - a, \log t, t\right) - \log \left(y + x\right)\right)\right)\right) \]

   11. inv-pow N/A

       \[\leadsto \log z \cdot \left(\color{blue}{\frac{1}{t}} \cdot t\right) + \left(\mathsf{neg}\left(\left(\mathsf{fma}\left(\frac{1}{2} - a, \log t, t\right) - \log \left(y + x\right)\right)\right)\right) \]

   12. lift-/.f64 N/A

       \[\leadsto \log z \cdot \left(\color{blue}{\frac{1}{t}} \cdot t\right) + \left(\mathsf{neg}\left(\left(\mathsf{fma}\left(\frac{1}{2} - a, \log t, t\right) - \log \left(y + x\right)\right)\right)\right) \]

   13. associate-*l* N/A

       \[\leadsto \color{blue}{\left(\log z \cdot \frac{1}{t}\right) \cdot t} + \left(\mathsf{neg}\left(\left(\mathsf{fma}\left(\frac{1}{2} - a, \log t, t\right) - \log \left(y + x\right)\right)\right)\right) \]

   14. lift-/.f64 N/A

       \[\leadsto \left(\log z \cdot \color{blue}{\frac{1}{t}}\right) \cdot t + \left(\mathsf{neg}\left(\left(\mathsf{fma}\left(\frac{1}{2} - a, \log t, t\right) - \log \left(y + x\right)\right)\right)\right) \]

   15. mult-flip N/A

       \[\leadsto \color{blue}{\frac{\log z}{t}} \cdot t + \left(\mathsf{neg}\left(\left(\mathsf{fma}\left(\frac{1}{2} - a, \log t, t\right) - \log \left(y + x\right)\right)\right)\right) \]

   16. lift-/.f64 N/A

       \[\leadsto \color{blue}{\frac{\log z}{t}} \cdot t + \left(\mathsf{neg}\left(\left(\mathsf{fma}\left(\frac{1}{2} - a, \log t, t\right) - \log \left(y + x\right)\right)\right)\right) \]

5. Applied rewrites 99.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\log z}{t}, t, \log \left(x + y\right) - \mathsf{fma}\left(\log t, 0.5 - a, t\right)\right)} \]
6. Add Preprocessing

Alternative 2: 99.2% accurate, 1.0× speedup

Math

\[\left(\left(\log \left(\mathsf{max}\left(x, y\right)\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (+ (- (+ (log (fmax x y)) (log z)) t) (* (- a 0.5) (log t))))

C

double code(double x, double y, double z, double t, double a) {
	return ((log(fmax(x, y)) + log(z)) - t) + ((a - 0.5) * log(t));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = ((log(fmax(x, y)) + log(z)) - t) + ((a - 0.5d0) * log(t))
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return ((Math.log(fmax(x, y)) + Math.log(z)) - t) + ((a - 0.5) * Math.log(t));
}

Python

def code(x, y, z, t, a):
	return ((math.log(fmax(x, y)) + math.log(z)) - t) + ((a - 0.5) * math.log(t))

Julia

function code(x, y, z, t, a)
	return Float64(Float64(Float64(log(fmax(x, y)) + log(z)) - t) + Float64(Float64(a - 0.5) * log(t)))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = ((log(max(x, y)) + log(z)) - t) + ((a - 0.5) * log(t));
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[(N[Log[N[Max[x, y], $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]

2. Taylor expanded in x around 0

   \[\leadsto \left(\left(\log \color{blue}{y} + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
3. Step-by-step derivation

4. Applied rewrites 70.1%

   \[\leadsto \left(\left(\log \color{blue}{y} + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
5. Add Preprocessing

Alternative 3: 99.2% accurate, 1.0× speedup

Math

\[\log \left(\mathsf{max}\left(x, y\right)\right) + \left(\log z - \mathsf{fma}\left(\log t, 0.5 - a, t\right)\right) \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (+ (log (fmax x y)) (- (log z) (fma (log t) (- 0.5 a) t))))

C

double code(double x, double y, double z, double t, double a) {
	return log(fmax(x, y)) + (log(z) - fma(log(t), (0.5 - a), t));
}

Julia

function code(x, y, z, t, a)
	return Float64(log(fmax(x, y)) + Float64(log(z) - fma(log(t), Float64(0.5 - a), t)))
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[Log[N[Max[x, y], $MachinePrecision]], $MachinePrecision] + N[(N[Log[z], $MachinePrecision] - N[(N[Log[t], $MachinePrecision] * N[(0.5 - a), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]

2. Taylor expanded in x around 0

   \[\leadsto \left(\left(\log \color{blue}{y} + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
3. Step-by-step derivation

4. Applied rewrites 70.1%

   \[\leadsto \left(\left(\log \color{blue}{y} + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
5. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\log y + \log z\right) - t\right) + \left(a - \frac{1}{2}\right) \cdot \log t} \]

   2. lift--.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\log y + \log z\right) - t\right)} + \left(a - \frac{1}{2}\right) \cdot \log t \]

   3. associate--r- N/A

      \[\leadsto \color{blue}{\left(\log y + \log z\right) - \left(t - \left(a - \frac{1}{2}\right) \cdot \log t\right)} \]

   4. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(\log y + \log z\right)} - \left(t - \left(a - \frac{1}{2}\right) \cdot \log t\right) \]

   5. lift-*.f64 N/A

      \[\leadsto \left(\log y + \log z\right) - \left(t - \color{blue}{\left(a - \frac{1}{2}\right) \cdot \log t}\right) \]

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

      \[\leadsto \left(\log y + \log z\right) - \color{blue}{\left(t + \left(\mathsf{neg}\left(\left(a - \frac{1}{2}\right)\right)\right) \cdot \log t\right)} \]

   7. lift--.f64 N/A

      \[\leadsto \left(\log y + \log z\right) - \left(t + \left(\mathsf{neg}\left(\color{blue}{\left(a - \frac{1}{2}\right)}\right)\right) \cdot \log t\right) \]

   8. sub-negate-rev N/A

      \[\leadsto \left(\log y + \log z\right) - \left(t + \color{blue}{\left(\frac{1}{2} - a\right)} \cdot \log t\right) \]

   9. lift--.f64 N/A

      \[\leadsto \left(\log y + \log z\right) - \left(t + \color{blue}{\left(\frac{1}{2} - a\right)} \cdot \log t\right) \]

   10. +-commutative N/A

       \[\leadsto \left(\log y + \log z\right) - \color{blue}{\left(\left(\frac{1}{2} - a\right) \cdot \log t + t\right)} \]

   11. lift-fma.f64 N/A

       \[\leadsto \left(\log y + \log z\right) - \color{blue}{\mathsf{fma}\left(\frac{1}{2} - a, \log t, t\right)} \]

   12. associate--l+ N/A

       \[\leadsto \color{blue}{\log y + \left(\log z - \mathsf{fma}\left(\frac{1}{2} - a, \log t, t\right)\right)} \]

   13. lower-+.f64 N/A

       \[\leadsto \color{blue}{\log y + \left(\log z - \mathsf{fma}\left(\frac{1}{2} - a, \log t, t\right)\right)} \]

   14. lower--.f64 70.1%

       \[\leadsto \log y + \color{blue}{\left(\log z - \mathsf{fma}\left(0.5 - a, \log t, t\right)\right)} \]

   15. lift-fma.f64 N/A

       \[\leadsto \log y + \left(\log z - \color{blue}{\left(\left(\frac{1}{2} - a\right) \cdot \log t + t\right)}\right) \]

   16. *-commutative N/A

       \[\leadsto \log y + \left(\log z - \left(\color{blue}{\log t \cdot \left(\frac{1}{2} - a\right)} + t\right)\right) \]

   17. lower-fma.f64 70.1%

       \[\leadsto \log y + \left(\log z - \color{blue}{\mathsf{fma}\left(\log t, 0.5 - a, t\right)}\right) \]

6. Applied rewrites 70.1%

   \[\leadsto \color{blue}{\log y + \left(\log z - \mathsf{fma}\left(\log t, 0.5 - a, t\right)\right)} \]
7. Add Preprocessing

Alternative 4: 99.2% accurate, 1.0× speedup

Math

\[\log z - \left(\mathsf{fma}\left(0.5 - a, \log t, t\right) - \log \left(\mathsf{max}\left(x, y\right)\right)\right) \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (- (log z) (- (fma (- 0.5 a) (log t) t) (log (fmax x y)))))

C

double code(double x, double y, double z, double t, double a) {
	return log(z) - (fma((0.5 - a), log(t), t) - log(fmax(x, y)));
}

Julia

function code(x, y, z, t, a)
	return Float64(log(z) - Float64(fma(Float64(0.5 - a), log(t), t) - log(fmax(x, y))))
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[Log[z], $MachinePrecision] - N[(N[(N[(0.5 - a), $MachinePrecision] * N[Log[t], $MachinePrecision] + t), $MachinePrecision] - N[Log[N[Max[x, y], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]

3. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\left(1 - \frac{\mathsf{fma}\left(0.5 - a, \log t, t\right) - \log \left(y + x\right)}{\log z}\right) \cdot \log z} \]

4. Taylor expanded in x around 0

   \[\leadsto \left(1 - \frac{\mathsf{fma}\left(0.5 - a, \log t, t\right) - \log \color{blue}{y}}{\log z}\right) \cdot \log z \]
5. Step-by-step derivation

6. Applied rewrites 70.0%

   \[\leadsto \left(1 - \frac{\mathsf{fma}\left(0.5 - a, \log t, t\right) - \log \color{blue}{y}}{\log z}\right) \cdot \log z \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(1 - \frac{\mathsf{fma}\left(\frac{1}{2} - a, \log t, t\right) - \log y}{\log z}\right) \cdot \log z} \]

   2. lift--.f64 N/A

      \[\leadsto \color{blue}{\left(1 - \frac{\mathsf{fma}\left(\frac{1}{2} - a, \log t, t\right) - \log y}{\log z}\right)} \cdot \log z \]

   3. lift-/.f64 N/A

      \[\leadsto \left(1 - \color{blue}{\frac{\mathsf{fma}\left(\frac{1}{2} - a, \log t, t\right) - \log y}{\log z}}\right) \cdot \log z \]

   4. sub-to-mult-rev N/A

      \[\leadsto \color{blue}{\log z - \left(\mathsf{fma}\left(\frac{1}{2} - a, \log t, t\right) - \log y\right)} \]

   5. lower--.f64 70.1%

      \[\leadsto \color{blue}{\log z - \left(\mathsf{fma}\left(0.5 - a, \log t, t\right) - \log y\right)} \]

8. Applied rewrites 70.1%

   \[\leadsto \color{blue}{\log z - \left(\mathsf{fma}\left(0.5 - a, \log t, t\right) - \log y\right)} \]
9. Add Preprocessing

Alternative 5: 98.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \left(a - 0.5\right) \cdot \log t\\ t_2 := \left(\left(\log \left(\mathsf{min}\left(x, y\right) + \mathsf{max}\left(x, y\right)\right) + \log z\right) - t\right) + t\_1\\ t_3 := t \cdot -1 + t\_1\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+17}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 2000:\\ \;\;\;\;\left(\left(\log \left(\mathsf{max}\left(x, y\right)\right) + \log z\right) - t\right) + -0.5 \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) (log t)))
        (t_2 (+ (- (+ (log (+ (fmin x y) (fmax x y))) (log z)) t) t_1))
        (t_3 (+ (* t -1.0) t_1)))
   (if (<= t_2 -1e+17)
     t_3
     (if (<= t_2 2000.0)
       (+ (- (+ (log (fmax x y)) (log z)) t) (* -0.5 (log t)))
       t_3))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (a - 0.5) * log(t);
	double t_2 = ((log((fmin(x, y) + fmax(x, y))) + log(z)) - t) + t_1;
	double t_3 = (t * -1.0) + t_1;
	double tmp;
	if (t_2 <= -1e+17) {
		tmp = t_3;
	} else if (t_2 <= 2000.0) {
		tmp = ((log(fmax(x, y)) + log(z)) - t) + (-0.5 * log(t));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = (a - 0.5d0) * log(t)
    t_2 = ((log((fmin(x, y) + fmax(x, y))) + log(z)) - t) + t_1
    t_3 = (t * (-1.0d0)) + t_1
    if (t_2 <= (-1d+17)) then
        tmp = t_3
    else if (t_2 <= 2000.0d0) then
        tmp = ((log(fmax(x, y)) + log(z)) - t) + ((-0.5d0) * log(t))
    else
        tmp = t_3
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (a - 0.5) * Math.log(t);
	double t_2 = ((Math.log((fmin(x, y) + fmax(x, y))) + Math.log(z)) - t) + t_1;
	double t_3 = (t * -1.0) + t_1;
	double tmp;
	if (t_2 <= -1e+17) {
		tmp = t_3;
	} else if (t_2 <= 2000.0) {
		tmp = ((Math.log(fmax(x, y)) + Math.log(z)) - t) + (-0.5 * Math.log(t));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (a - 0.5) * math.log(t)
	t_2 = ((math.log((fmin(x, y) + fmax(x, y))) + math.log(z)) - t) + t_1
	t_3 = (t * -1.0) + t_1
	tmp = 0
	if t_2 <= -1e+17:
		tmp = t_3
	elif t_2 <= 2000.0:
		tmp = ((math.log(fmax(x, y)) + math.log(z)) - t) + (-0.5 * math.log(t))
	else:
		tmp = t_3
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(a - 0.5) * log(t))
	t_2 = Float64(Float64(Float64(log(Float64(fmin(x, y) + fmax(x, y))) + log(z)) - t) + t_1)
	t_3 = Float64(Float64(t * -1.0) + t_1)
	tmp = 0.0
	if (t_2 <= -1e+17)
		tmp = t_3;
	elseif (t_2 <= 2000.0)
		tmp = Float64(Float64(Float64(log(fmax(x, y)) + log(z)) - t) + Float64(-0.5 * log(t)));
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (a - 0.5) * log(t);
	t_2 = ((log((min(x, y) + max(x, y))) + log(z)) - t) + t_1;
	t_3 = (t * -1.0) + t_1;
	tmp = 0.0;
	if (t_2 <= -1e+17)
		tmp = t_3;
	elseif (t_2 <= 2000.0)
		tmp = ((log(max(x, y)) + log(z)) - t) + (-0.5 * log(t));
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Log[N[(N[Min[x, y], $MachinePrecision] + N[Max[x, y], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t * -1.0), $MachinePrecision] + t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+17], t$95$3, If[LessEqual[t$95$2, 2000.0], N[(N[(N[(N[Log[N[Max[x, y], $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + N[(-0.5 * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < -1e17 or 2e3 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t)))

   1. Initial program 99.6%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]

   2. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) - 1\right)} + \left(a - 0.5\right) \cdot \log t \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto t \cdot \color{blue}{\left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) - 1\right)} + \left(a - \frac{1}{2}\right) \cdot \log t \]

      2. lower--.f64 N/A

         \[\leadsto t \cdot \left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) - \color{blue}{1}\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      3. lower-+.f64 N/A

         \[\leadsto t \cdot \left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) - 1\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      4. lower-/.f64 N/A

         \[\leadsto t \cdot \left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) - 1\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      5. lower-log.f64 N/A

         \[\leadsto t \cdot \left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) - 1\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      6. lower-/.f64 N/A

         \[\leadsto t \cdot \left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) - 1\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      7. lower-log.f64 N/A

         \[\leadsto t \cdot \left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) - 1\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      8. lower-+.f64 99.2%

         \[\leadsto t \cdot \left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) - 1\right) + \left(a - 0.5\right) \cdot \log t \]

   4. Applied rewrites 99.2%

      \[\leadsto \color{blue}{t \cdot \left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) - 1\right)} + \left(a - 0.5\right) \cdot \log t \]

   5. Taylor expanded in t around inf

      \[\leadsto t \cdot -1 + \left(a - 0.5\right) \cdot \log t \]
   6. Step-by-step derivation

   7. Applied rewrites 77.4%

      \[\leadsto t \cdot -1 + \left(a - 0.5\right) \cdot \log t \]

   if -1e17 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 2e3

   1. Initial program 99.6%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]

   2. Taylor expanded in x around 0

      \[\leadsto \left(\left(\log \color{blue}{y} + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   3. Step-by-step derivation

   4. Applied rewrites 70.1%

      \[\leadsto \left(\left(\log \color{blue}{y} + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]

   5. Taylor expanded in a around 0

      \[\leadsto \left(\left(\log y + \log z\right) - t\right) + \color{blue}{\frac{-1}{2}} \cdot \log t \]
   6. Step-by-step derivation

   7. Applied rewrites 42.0%

      \[\leadsto \left(\left(\log y + \log z\right) - t\right) + \color{blue}{-0.5} \cdot \log t \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 91.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \left(a - 0.5\right) \cdot \log t\\ t_2 := \left(\left(\log \left(\mathsf{min}\left(x, y\right) + \mathsf{max}\left(x, y\right)\right) + \log z\right) - t\right) + t\_1\\ t_3 := t \cdot -1 + t\_1\\ \mathbf{if}\;t\_2 \leq -4000000:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 400000000:\\ \;\;\;\;\log \left(\mathsf{max}\left(x, y\right) \cdot z\right) + \log t \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) (log t)))
        (t_2 (+ (- (+ (log (+ (fmin x y) (fmax x y))) (log z)) t) t_1))
        (t_3 (+ (* t -1.0) t_1)))
   (if (<= t_2 -4000000.0)
     t_3
     (if (<= t_2 400000000.0)
       (+ (log (* (fmax x y) z)) (* (log t) (- a 0.5)))
       t_3))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (a - 0.5) * log(t);
	double t_2 = ((log((fmin(x, y) + fmax(x, y))) + log(z)) - t) + t_1;
	double t_3 = (t * -1.0) + t_1;
	double tmp;
	if (t_2 <= -4000000.0) {
		tmp = t_3;
	} else if (t_2 <= 400000000.0) {
		tmp = log((fmax(x, y) * z)) + (log(t) * (a - 0.5));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = (a - 0.5d0) * log(t)
    t_2 = ((log((fmin(x, y) + fmax(x, y))) + log(z)) - t) + t_1
    t_3 = (t * (-1.0d0)) + t_1
    if (t_2 <= (-4000000.0d0)) then
        tmp = t_3
    else if (t_2 <= 400000000.0d0) then
        tmp = log((fmax(x, y) * z)) + (log(t) * (a - 0.5d0))
    else
        tmp = t_3
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (a - 0.5) * Math.log(t);
	double t_2 = ((Math.log((fmin(x, y) + fmax(x, y))) + Math.log(z)) - t) + t_1;
	double t_3 = (t * -1.0) + t_1;
	double tmp;
	if (t_2 <= -4000000.0) {
		tmp = t_3;
	} else if (t_2 <= 400000000.0) {
		tmp = Math.log((fmax(x, y) * z)) + (Math.log(t) * (a - 0.5));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (a - 0.5) * math.log(t)
	t_2 = ((math.log((fmin(x, y) + fmax(x, y))) + math.log(z)) - t) + t_1
	t_3 = (t * -1.0) + t_1
	tmp = 0
	if t_2 <= -4000000.0:
		tmp = t_3
	elif t_2 <= 400000000.0:
		tmp = math.log((fmax(x, y) * z)) + (math.log(t) * (a - 0.5))
	else:
		tmp = t_3
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(a - 0.5) * log(t))
	t_2 = Float64(Float64(Float64(log(Float64(fmin(x, y) + fmax(x, y))) + log(z)) - t) + t_1)
	t_3 = Float64(Float64(t * -1.0) + t_1)
	tmp = 0.0
	if (t_2 <= -4000000.0)
		tmp = t_3;
	elseif (t_2 <= 400000000.0)
		tmp = Float64(log(Float64(fmax(x, y) * z)) + Float64(log(t) * Float64(a - 0.5)));
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (a - 0.5) * log(t);
	t_2 = ((log((min(x, y) + max(x, y))) + log(z)) - t) + t_1;
	t_3 = (t * -1.0) + t_1;
	tmp = 0.0;
	if (t_2 <= -4000000.0)
		tmp = t_3;
	elseif (t_2 <= 400000000.0)
		tmp = log((max(x, y) * z)) + (log(t) * (a - 0.5));
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Log[N[(N[Min[x, y], $MachinePrecision] + N[Max[x, y], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t * -1.0), $MachinePrecision] + t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -4000000.0], t$95$3, If[LessEqual[t$95$2, 400000000.0], N[(N[Log[N[(N[Max[x, y], $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < -4e6 or 4e8 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t)))

   1. Initial program 99.6%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]

   2. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) - 1\right)} + \left(a - 0.5\right) \cdot \log t \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto t \cdot \color{blue}{\left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) - 1\right)} + \left(a - \frac{1}{2}\right) \cdot \log t \]

      2. lower--.f64 N/A

         \[\leadsto t \cdot \left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) - \color{blue}{1}\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      3. lower-+.f64 N/A

         \[\leadsto t \cdot \left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) - 1\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      4. lower-/.f64 N/A

         \[\leadsto t \cdot \left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) - 1\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      5. lower-log.f64 N/A

         \[\leadsto t \cdot \left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) - 1\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      6. lower-/.f64 N/A

         \[\leadsto t \cdot \left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) - 1\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      7. lower-log.f64 N/A

         \[\leadsto t \cdot \left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) - 1\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      8. lower-+.f64 99.2%

         \[\leadsto t \cdot \left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) - 1\right) + \left(a - 0.5\right) \cdot \log t \]

   4. Applied rewrites 99.2%

      \[\leadsto \color{blue}{t \cdot \left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) - 1\right)} + \left(a - 0.5\right) \cdot \log t \]

   5. Taylor expanded in t around inf

      \[\leadsto t \cdot -1 + \left(a - 0.5\right) \cdot \log t \]
   6. Step-by-step derivation

   7. Applied rewrites 77.4%

      \[\leadsto t \cdot -1 + \left(a - 0.5\right) \cdot \log t \]

   if -4e6 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 4e8

   1. Initial program 99.6%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]

   3. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\left(1 - \frac{\mathsf{fma}\left(0.5 - a, \log t, t\right) - \log \left(y + x\right)}{\log z}\right) \cdot \log z} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(1 - \frac{\mathsf{fma}\left(\frac{1}{2} - a, \log t, t\right) - \log \left(y + x\right)}{\log z}\right) \cdot \log z} \]

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(1 - \frac{\mathsf{fma}\left(\frac{1}{2} - a, \log t, t\right) - \log \left(y + x\right)}{\log z}\right)} \cdot \log z \]

      3. lift-/.f64 N/A

         \[\leadsto \left(1 - \color{blue}{\frac{\mathsf{fma}\left(\frac{1}{2} - a, \log t, t\right) - \log \left(y + x\right)}{\log z}}\right) \cdot \log z \]

      4. sub-to-mult-rev N/A

         \[\leadsto \color{blue}{\log z - \left(\mathsf{fma}\left(\frac{1}{2} - a, \log t, t\right) - \log \left(y + x\right)\right)} \]

      5. sub-flip N/A

         \[\leadsto \color{blue}{\log z + \left(\mathsf{neg}\left(\left(\mathsf{fma}\left(\frac{1}{2} - a, \log t, t\right) - \log \left(y + x\right)\right)\right)\right)} \]

      6. lift--.f64 N/A

         \[\leadsto \log z + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{fma}\left(\frac{1}{2} - a, \log t, t\right) - \log \left(y + x\right)\right)}\right)\right) \]

      7. lift-+.f64 N/A

         \[\leadsto \log z + \left(\mathsf{neg}\left(\left(\mathsf{fma}\left(\frac{1}{2} - a, \log t, t\right) - \log \color{blue}{\left(y + x\right)}\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \log z + \left(\mathsf{neg}\left(\left(\mathsf{fma}\left(\frac{1}{2} - a, \log t, t\right) - \log \color{blue}{\left(x + y\right)}\right)\right)\right) \]

      9. lift-+.f64 N/A

         \[\leadsto \log z + \left(\mathsf{neg}\left(\left(\mathsf{fma}\left(\frac{1}{2} - a, \log t, t\right) - \log \color{blue}{\left(x + y\right)}\right)\right)\right) \]

      10. sub-negate-rev N/A

          \[\leadsto \log z + \color{blue}{\left(\log \left(x + y\right) - \mathsf{fma}\left(\frac{1}{2} - a, \log t, t\right)\right)} \]

      11. associate--l+ N/A

          \[\leadsto \color{blue}{\left(\log z + \log \left(x + y\right)\right) - \mathsf{fma}\left(\frac{1}{2} - a, \log t, t\right)} \]

   5. Applied rewrites 75.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a - 0.5, \log t, \log \left(\left(x + y\right) \cdot z\right)\right) - t} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\log \left(z \cdot \left(x + y\right)\right) + \log t \cdot \left(a - \frac{1}{2}\right)} \]
   7. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \log \left(z \cdot \left(x + y\right)\right) + \color{blue}{\log t \cdot \left(a - \frac{1}{2}\right)} \]

      2. lower-log.f64 N/A

         \[\leadsto \log \left(z \cdot \left(x + y\right)\right) + \color{blue}{\log t} \cdot \left(a - \frac{1}{2}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \log \left(z \cdot \left(x + y\right)\right) + \log \color{blue}{t} \cdot \left(a - \frac{1}{2}\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \log \left(z \cdot \left(x + y\right)\right) + \log t \cdot \left(a - \frac{1}{2}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \log \left(z \cdot \left(x + y\right)\right) + \log t \cdot \color{blue}{\left(a - \frac{1}{2}\right)} \]

      6. lower-log.f64 N/A

         \[\leadsto \log \left(z \cdot \left(x + y\right)\right) + \log t \cdot \left(\color{blue}{a} - \frac{1}{2}\right) \]

      7. lower--.f64 47.1%

         \[\leadsto \log \left(z \cdot \left(x + y\right)\right) + \log t \cdot \left(a - \color{blue}{0.5}\right) \]

   8. Applied rewrites 47.1%

      \[\leadsto \color{blue}{\log \left(z \cdot \left(x + y\right)\right) + \log t \cdot \left(a - 0.5\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \log \left(y \cdot z\right) + \log \color{blue}{t} \cdot \left(a - 0.5\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 32.2%

          \[\leadsto \log \left(y \cdot z\right) + \log t \cdot \left(a - 0.5\right) \]

   11. Applied rewrites 32.2%

       \[\leadsto \log \left(y \cdot z\right) + \log \color{blue}{t} \cdot \left(a - 0.5\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 91.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\log \left(x + y\right) + \log z \leq 680:\\ \;\;\;\;\mathsf{fma}\left(\log t, a - 0.5, \log \left(z \cdot \left(y + x\right)\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;t \cdot -1 + \left(a - 0.5\right) \cdot \log t\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (+ (log (+ x y)) (log z)) 680.0)
   (- (fma (log t) (- a 0.5) (log (* z (+ y x)))) t)
   (+ (* t -1.0) (* (- a 0.5) (log t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((log((x + y)) + log(z)) <= 680.0) {
		tmp = fma(log(t), (a - 0.5), log((z * (y + x)))) - t;
	} else {
		tmp = (t * -1.0) + ((a - 0.5) * log(t));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(log(Float64(x + y)) + log(z)) <= 680.0)
		tmp = Float64(fma(log(t), Float64(a - 0.5), log(Float64(z * Float64(y + x)))) - t);
	else
		tmp = Float64(Float64(t * -1.0) + Float64(Float64(a - 0.5) * log(t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision], 680.0], N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision] + N[Log[N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(t * -1.0), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 680

   1. Initial program 99.6%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - \frac{1}{2}\right) \cdot \log t} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(a - \frac{1}{2}\right) \cdot \log t + \left(\left(\log \left(x + y\right) + \log z\right) - t\right)} \]

      3. lift--.f64 N/A

         \[\leadsto \left(a - \frac{1}{2}\right) \cdot \log t + \color{blue}{\left(\left(\log \left(x + y\right) + \log z\right) - t\right)} \]

      4. associate-+r- N/A

         \[\leadsto \color{blue}{\left(\left(a - \frac{1}{2}\right) \cdot \log t + \left(\log \left(x + y\right) + \log z\right)\right) - t} \]

      5. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(a - \frac{1}{2}\right) \cdot \log t + \left(\log \left(x + y\right) + \log z\right)\right) - t} \]

   3. Applied rewrites 75.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log t, a - 0.5, \log \left(z \cdot \left(y + x\right)\right)\right) - t} \]

   if 680 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z))

   1. Initial program 99.6%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]

   2. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) - 1\right)} + \left(a - 0.5\right) \cdot \log t \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto t \cdot \color{blue}{\left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) - 1\right)} + \left(a - \frac{1}{2}\right) \cdot \log t \]

      2. lower--.f64 N/A

         \[\leadsto t \cdot \left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) - \color{blue}{1}\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      3. lower-+.f64 N/A

         \[\leadsto t \cdot \left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) - 1\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      4. lower-/.f64 N/A

         \[\leadsto t \cdot \left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) - 1\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      5. lower-log.f64 N/A

         \[\leadsto t \cdot \left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) - 1\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      6. lower-/.f64 N/A

         \[\leadsto t \cdot \left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) - 1\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      7. lower-log.f64 N/A

         \[\leadsto t \cdot \left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) - 1\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      8. lower-+.f64 99.2%

         \[\leadsto t \cdot \left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) - 1\right) + \left(a - 0.5\right) \cdot \log t \]

   4. Applied rewrites 99.2%

      \[\leadsto \color{blue}{t \cdot \left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) - 1\right)} + \left(a - 0.5\right) \cdot \log t \]

   5. Taylor expanded in t around inf

      \[\leadsto t \cdot -1 + \left(a - 0.5\right) \cdot \log t \]
   6. Step-by-step derivation

   7. Applied rewrites 77.4%

      \[\leadsto t \cdot -1 + \left(a - 0.5\right) \cdot \log t \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 90.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\log \left(\mathsf{min}\left(x, y\right) + \mathsf{max}\left(x, y\right)\right) + \log z \leq 680:\\ \;\;\;\;\log \left(\mathsf{max}\left(x, y\right) \cdot z\right) - \mathsf{fma}\left(\log t, 0.5 - a, t\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot -1 + \left(a - 0.5\right) \cdot \log t\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (+ (log (+ (fmin x y) (fmax x y))) (log z)) 680.0)
   (- (log (* (fmax x y) z)) (fma (log t) (- 0.5 a) t))
   (+ (* t -1.0) (* (- a 0.5) (log t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((log((fmin(x, y) + fmax(x, y))) + log(z)) <= 680.0) {
		tmp = log((fmax(x, y) * z)) - fma(log(t), (0.5 - a), t);
	} else {
		tmp = (t * -1.0) + ((a - 0.5) * log(t));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(log(Float64(fmin(x, y) + fmax(x, y))) + log(z)) <= 680.0)
		tmp = Float64(log(Float64(fmax(x, y) * z)) - fma(log(t), Float64(0.5 - a), t));
	else
		tmp = Float64(Float64(t * -1.0) + Float64(Float64(a - 0.5) * log(t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(N[Log[N[(N[Min[x, y], $MachinePrecision] + N[Max[x, y], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision], 680.0], N[(N[Log[N[(N[Max[x, y], $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision] - N[(N[Log[t], $MachinePrecision] * N[(0.5 - a), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision], N[(N[(t * -1.0), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 680

   1. Initial program 99.6%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]

   2. Taylor expanded in x around 0

      \[\leadsto \left(\left(\log \color{blue}{y} + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   3. Step-by-step derivation

   4. Applied rewrites 70.1%

      \[\leadsto \left(\left(\log \color{blue}{y} + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\log y + \log z\right) - t\right) + \left(a - \frac{1}{2}\right) \cdot \log t} \]

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\log y + \log z\right) - t\right)} + \left(a - \frac{1}{2}\right) \cdot \log t \]

      3. associate-+l- N/A

         \[\leadsto \color{blue}{\left(\log y + \log z\right) - \left(t - \left(a - \frac{1}{2}\right) \cdot \log t\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\log y + \log z\right) - \left(t - \color{blue}{\left(a - \frac{1}{2}\right) \cdot \log t}\right) \]

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

         \[\leadsto \left(\log y + \log z\right) - \color{blue}{\left(t + \left(\mathsf{neg}\left(\left(a - \frac{1}{2}\right)\right)\right) \cdot \log t\right)} \]

      6. lift--.f64 N/A

         \[\leadsto \left(\log y + \log z\right) - \left(t + \left(\mathsf{neg}\left(\color{blue}{\left(a - \frac{1}{2}\right)}\right)\right) \cdot \log t\right) \]

      7. sub-negate-rev N/A

         \[\leadsto \left(\log y + \log z\right) - \left(t + \color{blue}{\left(\frac{1}{2} - a\right)} \cdot \log t\right) \]

      8. lift--.f64 N/A

         \[\leadsto \left(\log y + \log z\right) - \left(t + \color{blue}{\left(\frac{1}{2} - a\right)} \cdot \log t\right) \]

      9. +-commutative N/A

         \[\leadsto \left(\log y + \log z\right) - \color{blue}{\left(\left(\frac{1}{2} - a\right) \cdot \log t + t\right)} \]

      10. lift-fma.f64 N/A

          \[\leadsto \left(\log y + \log z\right) - \color{blue}{\mathsf{fma}\left(\frac{1}{2} - a, \log t, t\right)} \]

      11. lower--.f64 70.1%

          \[\leadsto \color{blue}{\left(\log y + \log z\right) - \mathsf{fma}\left(0.5 - a, \log t, t\right)} \]

   6. Applied rewrites 53.8%

      \[\leadsto \color{blue}{\log \left(y \cdot z\right) - \mathsf{fma}\left(\log t, 0.5 - a, t\right)} \]

   if 680 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z))

   1. Initial program 99.6%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]

   2. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) - 1\right)} + \left(a - 0.5\right) \cdot \log t \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto t \cdot \color{blue}{\left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) - 1\right)} + \left(a - \frac{1}{2}\right) \cdot \log t \]

      2. lower--.f64 N/A

         \[\leadsto t \cdot \left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) - \color{blue}{1}\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      3. lower-+.f64 N/A

         \[\leadsto t \cdot \left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) - 1\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      4. lower-/.f64 N/A

         \[\leadsto t \cdot \left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) - 1\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      5. lower-log.f64 N/A

         \[\leadsto t \cdot \left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) - 1\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      6. lower-/.f64 N/A

         \[\leadsto t \cdot \left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) - 1\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      7. lower-log.f64 N/A

         \[\leadsto t \cdot \left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) - 1\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      8. lower-+.f64 99.2%

         \[\leadsto t \cdot \left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) - 1\right) + \left(a - 0.5\right) \cdot \log t \]

   4. Applied rewrites 99.2%

      \[\leadsto \color{blue}{t \cdot \left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) - 1\right)} + \left(a - 0.5\right) \cdot \log t \]

   5. Taylor expanded in t around inf

      \[\leadsto t \cdot -1 + \left(a - 0.5\right) \cdot \log t \]
   6. Step-by-step derivation

   7. Applied rewrites 77.4%

      \[\leadsto t \cdot -1 + \left(a - 0.5\right) \cdot \log t \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 77.4% accurate, 1.9× speedup

Math

\[t \cdot -1 + \left(a - 0.5\right) \cdot \log t \]

FPCore

(FPCore (x y z t a) :precision binary64 (+ (* t -1.0) (* (- a 0.5) (log t))))

C

double code(double x, double y, double z, double t, double a) {
	return (t * -1.0) + ((a - 0.5) * log(t));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = (t * (-1.0d0)) + ((a - 0.5d0) * log(t))
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return (t * -1.0) + ((a - 0.5) * Math.log(t));
}

Python

def code(x, y, z, t, a):
	return (t * -1.0) + ((a - 0.5) * math.log(t))

Julia

function code(x, y, z, t, a)
	return Float64(Float64(t * -1.0) + Float64(Float64(a - 0.5) * log(t)))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = (t * -1.0) + ((a - 0.5) * log(t));
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(t * -1.0), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]

2. Taylor expanded in t around inf

   \[\leadsto \color{blue}{t \cdot \left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) - 1\right)} + \left(a - 0.5\right) \cdot \log t \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto t \cdot \color{blue}{\left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) - 1\right)} + \left(a - \frac{1}{2}\right) \cdot \log t \]

   2. lower--.f64 N/A

      \[\leadsto t \cdot \left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) - \color{blue}{1}\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

   3. lower-+.f64 N/A

      \[\leadsto t \cdot \left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) - 1\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

   4. lower-/.f64 N/A

      \[\leadsto t \cdot \left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) - 1\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

   5. lower-log.f64 N/A

      \[\leadsto t \cdot \left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) - 1\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

   6. lower-/.f64 N/A

      \[\leadsto t \cdot \left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) - 1\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

   7. lower-log.f64 N/A

      \[\leadsto t \cdot \left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) - 1\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

   8. lower-+.f64 99.2%

      \[\leadsto t \cdot \left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) - 1\right) + \left(a - 0.5\right) \cdot \log t \]

4. Applied rewrites 99.2%

   \[\leadsto \color{blue}{t \cdot \left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) - 1\right)} + \left(a - 0.5\right) \cdot \log t \]

5. Taylor expanded in t around inf

   \[\leadsto t \cdot -1 + \left(a - 0.5\right) \cdot \log t \]
6. Step-by-step derivation

7. Applied rewrites 77.4%

   \[\leadsto t \cdot -1 + \left(a - 0.5\right) \cdot \log t \]
8. Add Preprocessing

Alternative 10: 62.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;t \leq 10^{+16}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (if (<= t 1e+16) (* a (log t)) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 1e+16) {
		tmp = a * log(t);
	} else {
		tmp = -t;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 1d+16) then
        tmp = a * log(t)
    else
        tmp = -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 1e+16) {
		tmp = a * Math.log(t);
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 1e+16:
		tmp = a * math.log(t)
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 1e+16)
		tmp = Float64(a * log(t));
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 1e+16)
		tmp = a * log(t);
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 1e+16], N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if t < 1e16

   1. Initial program 99.6%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \log t} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto a \cdot \color{blue}{\log t} \]

      2. lower-log.f64 38.3%

         \[\leadsto a \cdot \log t \]

   4. Applied rewrites 38.3%

      \[\leadsto \color{blue}{a \cdot \log t} \]

   if 1e16 < t

   1. Initial program 99.6%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]

   2. Taylor expanded in t around inf

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

      1. lower-*.f64 38.6%

         \[\leadsto -1 \cdot \color{blue}{t} \]

   4. Applied rewrites 38.6%

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

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{t} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(t\right) \]

      3. lower-neg.f64 38.6%

         \[\leadsto -t \]

   6. Applied rewrites 38.6%

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

Alternative 11: 38.6% accurate, 17.6× speedup

Math

\[-t \]

FPCore

(FPCore (x y z t a) :precision binary64 (- t))

C

double code(double x, double y, double z, double t, double a) {
	return -t;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = -t
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return -t;
}

Python

def code(x, y, z, t, a):
	return -t

Julia

function code(x, y, z, t, a)
	return Float64(-t)
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = -t;
end

Wolfram

code[x_, y_, z_, t_, a_] := (-t)

Derivation

1. Initial program 99.6%

   \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]

2. Taylor expanded in t around inf

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

   1. lower-*.f64 38.6%

      \[\leadsto -1 \cdot \color{blue}{t} \]

4. Applied rewrites 38.6%

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

   1. lift-*.f64 N/A

      \[\leadsto -1 \cdot \color{blue}{t} \]

   2. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(t\right) \]

   3. lower-neg.f64 38.6%

      \[\leadsto -t \]

6. Applied rewrites 38.6%

   \[\leadsto \color{blue}{-t} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025188 
(FPCore (x y z t a)
  :name "Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2"
  :precision binary64
  (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))