Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 13.1s

Alternatives: 12

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \end{array} \]

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

\[\begin{array}{l} \\ \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \end{array} \]

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, 1.0× speedup

Math

\[\begin{array}{l} \\ \log \left(y + x\right) + \left(\left(\log z - t\right) - \left(0.5 - a\right) \cdot \log t\right) \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t, double a) {
	return log((y + x)) + ((log(z) - t) - ((0.5 - a) * 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((y + x)) + ((log(z) - t) - ((0.5d0 - a) * log(t)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

   \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
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. add-flip N/A

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

   3. lift--.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. associate--l+ N/A

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

   6. associate--l+ N/A

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

   7. lower-+.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lower--.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. distribute-lft-neg-out N/A

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

   15. lift--.f64 N/A

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

   16. sub-negate-rev N/A

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

   17. lower-*.f64 N/A

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

   18. lower--.f64 99.6

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

3. Applied rewrites 99.6%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision] + N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Log[z], $MachinePrecision] - t), $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. 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. lift-+.f64 N/A

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

   5. associate--l+ N/A

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

   6. associate-+r+ N/A

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

   7. lower-+.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 N/A

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

   14. lower--.f64 99.6

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

3. Applied rewrites 99.6%

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

Alternative 3: 85.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(\log y + \left(\log z + \log t \cdot \left(a - 0.5\right)\right)\right) - t \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t, double a) {
	return (log(y) + (log(z) + (log(t) * (a - 0.5)))) - 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(y) + (log(z) + (log(t) * (a - 0.5d0)))) - t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. lower--.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-log.f64 N/A

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

   4. lower-+.f64 N/A

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

   5. lower-log.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-log.f64 N/A

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

   8. lower--.f64 69.1

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

4. Applied rewrites 69.1%

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

Alternative 4: 85.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \log y + \left(\left(\log z - t\right) - \left(0.5 - a\right) \cdot \log t\right) \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t, double a) {
	return log(y) + ((log(z) - t) - ((0.5 - a) * 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(y) + ((log(z) - t) - ((0.5d0 - a) * log(t)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

   \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
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. add-flip N/A

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

   3. lift--.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. associate--l+ N/A

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

   6. associate--l+ N/A

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

   7. lower-+.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lower--.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. distribute-lft-neg-out N/A

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

   15. lift--.f64 N/A

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

   16. sub-negate-rev N/A

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

   17. lower-*.f64 N/A

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

   18. lower--.f64 99.6

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

3. Applied rewrites 99.6%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 69.1%

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

Alternative 5: 69.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \log z - \left(\mathsf{fma}\left(0.5 - a, \log t, t\right) - \log y\right) \end{array} \]

FPCore

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

Julia

function code(x, y, z, t, a)
	return Float64(log(z) - Float64(fma(Float64(0.5 - a), log(t), t) - log(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[y], $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. 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. add-flip N/A

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

   3. lift--.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. associate--l+ N/A

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

   6. associate--l+ N/A

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

   7. lower-+.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lower--.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. distribute-lft-neg-out N/A

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

   15. lift--.f64 N/A

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

   16. sub-negate-rev N/A

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

   17. lower-*.f64 N/A

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

   18. lower--.f64 99.6

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

3. Applied rewrites 99.6%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 69.1%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift--.f64 N/A

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

   4. lift--.f64 N/A

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

   5. associate--l- N/A

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

   6. associate-+l- N/A

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

   7. lower--.f64 N/A

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

   8. lower--.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. lower-fma.f64 69.1

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

8. Applied rewrites 69.1%

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

Alternative 6: 69.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 2.2 \cdot 10^{-13}:\\ \;\;\;\;\log y + \left(\log z - \log t \cdot \left(0.5 - a\right)\right)\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{+185}:\\ \;\;\;\;\log \left(y \cdot z\right) - \mathsf{fma}\left(0.5 - a, \log t, t\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 2.2e-13)
   (+ (log y) (- (log z) (* (log t) (- 0.5 a))))
   (if (<= t 1.5e+185) (- (log (* y z)) (fma (- 0.5 a) (log t) t)) (- t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 2.2e-13) {
		tmp = log(y) + (log(z) - (log(t) * (0.5 - a)));
	} else if (t <= 1.5e+185) {
		tmp = log((y * z)) - fma((0.5 - a), log(t), t);
	} else {
		tmp = -t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 2.2e-13)
		tmp = Float64(log(y) + Float64(log(z) - Float64(log(t) * Float64(0.5 - a))));
	elseif (t <= 1.5e+185)
		tmp = Float64(log(Float64(y * z)) - fma(Float64(0.5 - a), log(t), t));
	else
		tmp = Float64(-t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 2.2e-13], N[(N[Log[y], $MachinePrecision] + N[(N[Log[z], $MachinePrecision] - N[(N[Log[t], $MachinePrecision] * N[(0.5 - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.5e+185], N[(N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision] - N[(N[(0.5 - a), $MachinePrecision] * N[Log[t], $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision], (-t)]]

Derivation

1. Split input into 3 regimes

2. if t < 2.19999999999999997e-13

   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. add-flip N/A

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

      3. lift--.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. associate--l+ N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. distribute-lft-neg-out N/A

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

      15. lift--.f64 N/A

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

      16. sub-negate-rev N/A

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

      17. lower-*.f64 N/A

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

      18. lower--.f64 99.6

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 69.1%

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

   7. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower--.f64 41.8

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

   9. Applied rewrites 41.8%

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

   if 2.19999999999999997e-13 < t < 1.49999999999999997e185

   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. add-flip N/A

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

      3. lift--.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. associate--l+ N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. distribute-lft-neg-out N/A

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

      15. lift--.f64 N/A

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

      16. sub-negate-rev N/A

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

      17. lower-*.f64 N/A

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

      18. lower--.f64 99.6

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 69.1%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. associate--l- N/A

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

      5. associate-+r- N/A

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

      6. lower--.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. sum-log N/A

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

      10. lower-log.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 53.3

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

   8. Applied rewrites 53.3%

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

   if 1.49999999999999997e185 < 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 37.6

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

   4. Applied rewrites 37.6%

      \[\leadsto \color{blue}{-1 \cdot t} \]
   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. lift-neg.f64 37.6

         \[\leadsto -t \]

   6. Applied rewrites 37.6%

      \[\leadsto -t \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 69.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(x + y\right)\\ t_2 := t\_1 + \log z\\ t_3 := \mathsf{fma}\left(\log z - t, 1, t\_1\right)\\ \mathbf{if}\;t\_2 \leq -750:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 710:\\ \;\;\;\;\log \left(z \cdot \left(x + y\right)\right) - \mathsf{fma}\left(\log t, 0.5 - a, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (log (+ x y)))
        (t_2 (+ t_1 (log z)))
        (t_3 (fma (- (log z) t) 1.0 t_1)))
   (if (<= t_2 -750.0)
     t_3
     (if (<= t_2 710.0)
       (- (log (* z (+ x y))) (fma (log t) (- 0.5 a) t))
       t_3))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((x + y));
	double t_2 = t_1 + log(z);
	double t_3 = fma((log(z) - t), 1.0, t_1);
	double tmp;
	if (t_2 <= -750.0) {
		tmp = t_3;
	} else if (t_2 <= 710.0) {
		tmp = log((z * (x + y))) - fma(log(t), (0.5 - a), t);
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = log(Float64(x + y))
	t_2 = Float64(t_1 + log(z))
	t_3 = fma(Float64(log(z) - t), 1.0, t_1)
	tmp = 0.0
	if (t_2 <= -750.0)
		tmp = t_3;
	elseif (t_2 <= 710.0)
		tmp = Float64(log(Float64(z * Float64(x + y))) - fma(log(t), Float64(0.5 - a), t));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + N[Log[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] * 1.0 + t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -750.0], t$95$3, If[LessEqual[t$95$2, 710.0], N[(N[Log[N[(z * N[(x + y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(N[Log[t], $MachinePrecision] * N[(0.5 - a), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -750 or 710 < (+.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. 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. add-flip N/A

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

      3. lift--.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. associate--l+ N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. distribute-lft-neg-out N/A

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

      15. lift--.f64 N/A

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

      16. sub-negate-rev N/A

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

      17. lower-*.f64 N/A

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

      18. lower--.f64 99.6

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

   3. Applied rewrites 99.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-to-mult N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 41.9%

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

   if -750 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 710

   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. add-flip N/A

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

      3. lift--.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. associate--l+ N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. distribute-lft-neg-out N/A

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

      15. lift--.f64 N/A

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

      16. sub-negate-rev N/A

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

      17. lower-*.f64 N/A

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

      18. lower--.f64 99.6

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

   3. Applied rewrites 99.6%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. sub-negate-rev N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-+l- N/A

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

      11. associate--l+ N/A

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

      12. +-commutative N/A

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

   5. Applied rewrites 76.0%

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

Alternative 8: 66.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(x + y\right)\\ t_2 := t\_1 + \log z\\ t_3 := \mathsf{fma}\left(\log z - t, 1, t\_1\right)\\ \mathbf{if}\;t\_2 \leq -750:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 710:\\ \;\;\;\;\mathsf{fma}\left(\log t, a - 0.5, \log \left(z \cdot \left(y + x\right)\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (log (+ x y)))
        (t_2 (+ t_1 (log z)))
        (t_3 (fma (- (log z) t) 1.0 t_1)))
   (if (<= t_2 -750.0)
     t_3
     (if (<= t_2 710.0)
       (- (fma (log t) (- a 0.5) (log (* z (+ y x)))) t)
       t_3))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((x + y));
	double t_2 = t_1 + log(z);
	double t_3 = fma((log(z) - t), 1.0, t_1);
	double tmp;
	if (t_2 <= -750.0) {
		tmp = t_3;
	} else if (t_2 <= 710.0) {
		tmp = fma(log(t), (a - 0.5), log((z * (y + x)))) - t;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = log(Float64(x + y))
	t_2 = Float64(t_1 + log(z))
	t_3 = fma(Float64(log(z) - t), 1.0, t_1)
	tmp = 0.0
	if (t_2 <= -750.0)
		tmp = t_3;
	elseif (t_2 <= 710.0)
		tmp = Float64(fma(log(t), Float64(a - 0.5), log(Float64(z * Float64(y + x)))) - t);
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + N[Log[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] * 1.0 + t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -750.0], t$95$3, If[LessEqual[t$95$2, 710.0], N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision] + N[Log[N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], t$95$3]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -750 or 710 < (+.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. 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. add-flip N/A

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

      3. lift--.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. associate--l+ N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. distribute-lft-neg-out N/A

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

      15. lift--.f64 N/A

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

      16. sub-negate-rev N/A

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

      17. lower-*.f64 N/A

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

      18. lower--.f64 99.6

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

   3. Applied rewrites 99.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-to-mult N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 41.9%

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

   if -750 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 710

   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

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 99.6

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

      9. lift-+.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. sum-log N/A

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

      13. lower-log.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 76.0

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 76.0

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

   3. Applied rewrites 76.0%

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

Alternative 9: 66.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(x + y\right)\\ t_2 := t\_1 + \log z\\ t_3 := \mathsf{fma}\left(\log z - t, 1, t\_1\right)\\ \mathbf{if}\;t\_2 \leq -750:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 710:\\ \;\;\;\;\log \left(y \cdot z\right) - \mathsf{fma}\left(0.5 - a, \log t, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (log (+ x y)))
        (t_2 (+ t_1 (log z)))
        (t_3 (fma (- (log z) t) 1.0 t_1)))
   (if (<= t_2 -750.0)
     t_3
     (if (<= t_2 710.0) (- (log (* y z)) (fma (- 0.5 a) (log t) t)) t_3))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((x + y));
	double t_2 = t_1 + log(z);
	double t_3 = fma((log(z) - t), 1.0, t_1);
	double tmp;
	if (t_2 <= -750.0) {
		tmp = t_3;
	} else if (t_2 <= 710.0) {
		tmp = log((y * z)) - fma((0.5 - a), log(t), t);
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = log(Float64(x + y))
	t_2 = Float64(t_1 + log(z))
	t_3 = fma(Float64(log(z) - t), 1.0, t_1)
	tmp = 0.0
	if (t_2 <= -750.0)
		tmp = t_3;
	elseif (t_2 <= 710.0)
		tmp = Float64(log(Float64(y * z)) - fma(Float64(0.5 - a), log(t), t));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + N[Log[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] * 1.0 + t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -750.0], t$95$3, If[LessEqual[t$95$2, 710.0], N[(N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision] - N[(N[(0.5 - a), $MachinePrecision] * N[Log[t], $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -750 or 710 < (+.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. 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. add-flip N/A

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

      3. lift--.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. associate--l+ N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. distribute-lft-neg-out N/A

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

      15. lift--.f64 N/A

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

      16. sub-negate-rev N/A

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

      17. lower-*.f64 N/A

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

      18. lower--.f64 99.6

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

   3. Applied rewrites 99.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-to-mult N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 41.9%

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

   if -750 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 710

   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. add-flip N/A

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

      3. lift--.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. associate--l+ N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. distribute-lft-neg-out N/A

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

      15. lift--.f64 N/A

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

      16. sub-negate-rev N/A

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

      17. lower-*.f64 N/A

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

      18. lower--.f64 99.6

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 69.1%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. associate--l- N/A

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

      5. associate-+r- N/A

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

      6. lower--.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. sum-log N/A

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

      10. lower-log.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 53.3

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

   8. Applied rewrites 53.3%

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

Alternative 10: 62.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \log t\\ \mathbf{if}\;a \leq -2.9 \cdot 10^{+26}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 3.7 \cdot 10^{+71}:\\ \;\;\;\;\mathsf{fma}\left(\log z - t, 1, \log \left(x + y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* a (log t))))
   (if (<= a -2.9e+26)
     t_1
     (if (<= a 3.7e+71) (fma (- (log z) t) 1.0 (log (+ x y))) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = a * log(t);
	double tmp;
	if (a <= -2.9e+26) {
		tmp = t_1;
	} else if (a <= 3.7e+71) {
		tmp = fma((log(z) - t), 1.0, log((x + y)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(a * log(t))
	tmp = 0.0
	if (a <= -2.9e+26)
		tmp = t_1;
	elseif (a <= 3.7e+71)
		tmp = fma(Float64(log(z) - t), 1.0, log(Float64(x + y)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -2.9e+26], t$95$1, If[LessEqual[a, 3.7e+71], N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] * 1.0 + N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -2.9e26 or 3.7e71 < a

   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 39.0

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

   4. Applied rewrites 39.0%

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

   if -2.9e26 < a < 3.7e71

   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. add-flip N/A

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

      3. lift--.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. associate--l+ N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. distribute-lft-neg-out N/A

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

      15. lift--.f64 N/A

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

      16. sub-negate-rev N/A

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

      17. lower-*.f64 N/A

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

      18. lower--.f64 99.6

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

   3. Applied rewrites 99.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-to-mult N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 41.9%

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

Alternative 11: 59.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 7.8 \cdot 10^{+25}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 7.8e+25) (* a (log t)) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 7.8e+25) {
		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 <= 7.8d+25) 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 <= 7.8e+25) {
		tmp = a * Math.log(t);
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 7.8e+25:
		tmp = a * math.log(t)
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 7.8e+25)
		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 <= 7.8e+25)
		tmp = a * log(t);
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 7.8e+25], N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if t < 7.8000000000000004e25

   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 39.0

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

   4. Applied rewrites 39.0%

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

   if 7.8000000000000004e25 < 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 37.6

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

   4. Applied rewrites 37.6%

      \[\leadsto \color{blue}{-1 \cdot t} \]
   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. lift-neg.f64 37.6

         \[\leadsto -t \]

   6. Applied rewrites 37.6%

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

Alternative 12: 37.6% accurate, 17.6× speedup

Math

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

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 37.6

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

4. Applied rewrites 37.6%

   \[\leadsto \color{blue}{-1 \cdot t} \]
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. lift-neg.f64 37.6

      \[\leadsto -t \]

6. Applied rewrites 37.6%

   \[\leadsto -t \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025142 
(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))))