Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 7.1s

Alternatives: 19

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[Log[N[Max[x, y], $MachinePrecision]], $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 68.5%

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

4. Applied rewrites 68.5%

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

Alternative 2: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 68.5%

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

Alternative 3: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[(N[(N[Log[N[Max[x, y], $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $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. Taylor expanded in y around 0

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

   1. lower-log.f64 68.6%

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

4. Applied rewrites 68.6%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 68.6%

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

   5. lift--.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. associate--l+ N/A

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

   8. add-flip N/A

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

   9. sub-negate-rev N/A

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

   10. lift--.f64 N/A

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

   11. lower--.f64 68.6%

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

6. Applied rewrites 68.6%

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

7. Taylor expanded in x around 0

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

   1. lower--.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-log.f64 N/A

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

   4. lower-log.f64 68.5%

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

9. Applied rewrites 68.5%

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

Alternative 4: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[(N[Log[N[Max[x, y], $MachinePrecision]], $MachinePrecision] - N[(t - N[Log[z], $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. Taylor expanded in y around 0

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

   1. lower-log.f64 68.6%

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

4. Applied rewrites 68.6%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 68.6%

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

   5. lift--.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. associate--l+ N/A

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

   8. add-flip N/A

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

   9. sub-negate-rev N/A

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

   10. lift--.f64 N/A

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

   11. lower--.f64 68.6%

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

6. Applied rewrites 68.6%

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

7. Taylor expanded in x around 0

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

   1. lower--.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-log.f64 N/A

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

   4. lower-log.f64 68.5%

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

9. Applied rewrites 68.5%

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

    1. lift--.f64 N/A

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

    2. lift-+.f64 N/A

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

    3. associate--l+ N/A

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

    4. add-flip N/A

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

    5. sub-negate-rev N/A

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

    6. lift--.f64 N/A

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

    7. lower--.f64 68.5%

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

11. Applied rewrites 68.5%

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

Alternative 5: 90.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_1 := \log \left(\mathsf{min}\left(x, y\right) + \mathsf{max}\left(x, y\right)\right) + \log z\\ \mathbf{if}\;t\_1 \leq -720:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log \left(\mathsf{min}\left(x, y\right)\right)\right) - \left(t - \log z\right)\\ \mathbf{elif}\;t\_1 \leq 740:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \left(-\log \left(\frac{\frac{1}{z}}{\mathsf{max}\left(x, y\right)}\right)\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(\mathsf{max}\left(x, y\right)\right) + \left(\log z + -0.5 \cdot \log t\right)\right) - t\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log (+ (fmin x y) (fmax x y))) (log z))))
   (if (<= t_1 -720.0)
     (- (fma (- a 0.5) (log t) (log (fmin x y))) (- t (log z)))
     (if (<= t_1 740.0)
       (fma (- a 0.5) (log t) (- (- (log (/ (/ 1.0 z) (fmax x y)))) t))
       (- (+ (log (fmax x y)) (+ (log z) (* -0.5 (log t)))) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((fmin(x, y) + fmax(x, y))) + log(z);
	double tmp;
	if (t_1 <= -720.0) {
		tmp = fma((a - 0.5), log(t), log(fmin(x, y))) - (t - log(z));
	} else if (t_1 <= 740.0) {
		tmp = fma((a - 0.5), log(t), (-log(((1.0 / z) / fmax(x, y))) - t));
	} else {
		tmp = (log(fmax(x, y)) + (log(z) + (-0.5 * log(t)))) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(Float64(fmin(x, y) + fmax(x, y))) + log(z))
	tmp = 0.0
	if (t_1 <= -720.0)
		tmp = Float64(fma(Float64(a - 0.5), log(t), log(fmin(x, y))) - Float64(t - log(z)));
	elseif (t_1 <= 740.0)
		tmp = fma(Float64(a - 0.5), log(t), Float64(Float64(-log(Float64(Float64(1.0 / z) / fmax(x, y)))) - t));
	else
		tmp = Float64(Float64(log(fmax(x, y)) + Float64(log(z) + Float64(-0.5 * log(t)))) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[N[(N[Min[x, y], $MachinePrecision] + N[Max[x, y], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -720.0], N[(N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[Log[N[Min[x, y], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(t - N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 740.0], N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[((-N[Log[N[(N[(1.0 / z), $MachinePrecision] / N[Max[x, y], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) - t), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[N[Max[x, y], $MachinePrecision]], $MachinePrecision] + N[(N[Log[z], $MachinePrecision] + N[(-0.5 * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -720

   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 y around 0

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

      1. lower-log.f64 68.6%

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

   4. Applied rewrites 68.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 68.6%

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

      5. lift--.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate--l+ N/A

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

      8. add-flip N/A

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

      9. sub-negate-rev N/A

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

      10. lift--.f64 N/A

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

      11. lower--.f64 68.6%

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

   6. Applied rewrites 68.6%

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

      1. lift-fma.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+r- N/A

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

      4. lower--.f64 N/A

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

      5. lower-fma.f64 68.6%

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

   8. Applied rewrites 68.6%

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

   if -720 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 740

   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 y around 0

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

      1. lower-log.f64 68.6%

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

   4. Applied rewrites 68.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 68.6%

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

      5. lift--.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate--l+ N/A

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

      8. add-flip N/A

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

      9. sub-negate-rev N/A

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

      10. lift--.f64 N/A

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

      11. lower--.f64 68.6%

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

   6. Applied rewrites 68.6%

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

   7. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-log.f64 68.5%

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

   9. Applied rewrites 68.5%

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

       1. lift-+.f64 N/A

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

       2. add-flip N/A

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

       3. sub-negate-rev N/A

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

       4. lower-neg.f64 N/A

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

       5. lift-log.f64 N/A

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

       6. neg-log N/A

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

       7. lift-log.f64 N/A

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

       8. diff-log N/A

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

       9. lower-log.f64 N/A

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

       10. lower-/.f64 N/A

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

       11. lower-/.f64 52.7%

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

   11. Applied rewrites 52.7%

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

   if 740 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z))

   1. Initial program 99.6%

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

   2. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-log.f64 62.5%

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

   4. Applied rewrites 62.5%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log.f64 41.1%

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

   7. Applied rewrites 41.1%

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

Alternative 6: 89.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_1 := \log \left(\mathsf{min}\left(x, y\right) + \mathsf{max}\left(x, y\right)\right) + \log z\\ \mathbf{if}\;t\_1 \leq -720:\\ \;\;\;\;\log \left(\mathsf{max}\left(x, y\right) + \mathsf{min}\left(x, y\right)\right) - -1 \cdot \left(a \cdot \log t\right)\\ \mathbf{elif}\;t\_1 \leq 740:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \left(-\log \left(\frac{\frac{1}{z}}{\mathsf{max}\left(x, y\right)}\right)\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(\mathsf{max}\left(x, y\right)\right) + \left(\log z + -0.5 \cdot \log t\right)\right) - t\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log (+ (fmin x y) (fmax x y))) (log z))))
   (if (<= t_1 -720.0)
     (- (log (+ (fmax x y) (fmin x y))) (* -1.0 (* a (log t))))
     (if (<= t_1 740.0)
       (fma (- a 0.5) (log t) (- (- (log (/ (/ 1.0 z) (fmax x y)))) t))
       (- (+ (log (fmax x y)) (+ (log z) (* -0.5 (log t)))) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((fmin(x, y) + fmax(x, y))) + log(z);
	double tmp;
	if (t_1 <= -720.0) {
		tmp = log((fmax(x, y) + fmin(x, y))) - (-1.0 * (a * log(t)));
	} else if (t_1 <= 740.0) {
		tmp = fma((a - 0.5), log(t), (-log(((1.0 / z) / fmax(x, y))) - t));
	} else {
		tmp = (log(fmax(x, y)) + (log(z) + (-0.5 * log(t)))) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(Float64(fmin(x, y) + fmax(x, y))) + log(z))
	tmp = 0.0
	if (t_1 <= -720.0)
		tmp = Float64(log(Float64(fmax(x, y) + fmin(x, y))) - Float64(-1.0 * Float64(a * log(t))));
	elseif (t_1 <= 740.0)
		tmp = fma(Float64(a - 0.5), log(t), Float64(Float64(-log(Float64(Float64(1.0 / z) / fmax(x, y)))) - t));
	else
		tmp = Float64(Float64(log(fmax(x, y)) + Float64(log(z) + Float64(-0.5 * log(t)))) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[N[(N[Min[x, y], $MachinePrecision] + N[Max[x, y], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -720.0], N[(N[Log[N[(N[Max[x, y], $MachinePrecision] + N[Min[x, y], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(-1.0 * N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 740.0], N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[((-N[Log[N[(N[(1.0 / z), $MachinePrecision] / N[Max[x, y], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) - t), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[N[Max[x, y], $MachinePrecision]], $MachinePrecision] + N[(N[Log[z], $MachinePrecision] + N[(-0.5 * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -720

   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. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lift-+.f64 N/A

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

      5. add-flip N/A

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

      6. associate--l- N/A

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

      7. sub-to-mult N/A

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

      8. lower-unsound-*.f64 N/A

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

   3. Applied rewrites 99.4%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. div-add N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. lower-/.f64 99.4%

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 99.4%

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

   5. Applied rewrites 99.4%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. add-to-fraction N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

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

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

      10. lower--.f64 N/A

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 41.3%

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

   10. Applied rewrites 41.3%

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

   if -720 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 740

   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 y around 0

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

      1. lower-log.f64 68.6%

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

   4. Applied rewrites 68.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 68.6%

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

      5. lift--.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate--l+ N/A

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

      8. add-flip N/A

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

      9. sub-negate-rev N/A

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

      10. lift--.f64 N/A

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

      11. lower--.f64 68.6%

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

   6. Applied rewrites 68.6%

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

   7. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-log.f64 68.5%

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

   9. Applied rewrites 68.5%

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

       1. lift-+.f64 N/A

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

       2. add-flip N/A

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

       3. sub-negate-rev N/A

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

       4. lower-neg.f64 N/A

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

       5. lift-log.f64 N/A

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

       6. neg-log N/A

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

       7. lift-log.f64 N/A

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

       8. diff-log N/A

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

       9. lower-log.f64 N/A

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

       10. lower-/.f64 N/A

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

       11. lower-/.f64 52.7%

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

   11. Applied rewrites 52.7%

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

   if 740 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z))

   1. Initial program 99.6%

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

   2. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-log.f64 62.5%

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

   4. Applied rewrites 62.5%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log.f64 41.1%

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

   7. Applied rewrites 41.1%

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

Alternative 7: 89.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{max}\left(x, y\right) + \mathsf{min}\left(x, y\right)\\ t_2 := \log \left(\mathsf{min}\left(x, y\right) + \mathsf{max}\left(x, y\right)\right) + \log z\\ \mathbf{if}\;t\_2 \leq -720:\\ \;\;\;\;\log t\_1 - -1 \cdot \left(a \cdot \log t\right)\\ \mathbf{elif}\;t\_2 \leq 720:\\ \;\;\;\;\mathsf{fma}\left(\log t, a - 0.5, \log \left(z \cdot t\_1\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(\mathsf{max}\left(x, y\right)\right) + \left(\log z + -0.5 \cdot \log t\right)\right) - t\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (fmax x y) (fmin x y)))
        (t_2 (+ (log (+ (fmin x y) (fmax x y))) (log z))))
   (if (<= t_2 -720.0)
     (- (log t_1) (* -1.0 (* a (log t))))
     (if (<= t_2 720.0)
       (- (fma (log t) (- a 0.5) (log (* z t_1))) t)
       (- (+ (log (fmax x y)) (+ (log z) (* -0.5 (log t)))) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fmax(x, y) + fmin(x, y);
	double t_2 = log((fmin(x, y) + fmax(x, y))) + log(z);
	double tmp;
	if (t_2 <= -720.0) {
		tmp = log(t_1) - (-1.0 * (a * log(t)));
	} else if (t_2 <= 720.0) {
		tmp = fma(log(t), (a - 0.5), log((z * t_1))) - t;
	} else {
		tmp = (log(fmax(x, y)) + (log(z) + (-0.5 * log(t)))) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(fmax(x, y) + fmin(x, y))
	t_2 = Float64(log(Float64(fmin(x, y) + fmax(x, y))) + log(z))
	tmp = 0.0
	if (t_2 <= -720.0)
		tmp = Float64(log(t_1) - Float64(-1.0 * Float64(a * log(t))));
	elseif (t_2 <= 720.0)
		tmp = Float64(fma(log(t), Float64(a - 0.5), log(Float64(z * t_1))) - t);
	else
		tmp = Float64(Float64(log(fmax(x, y)) + Float64(log(z) + Float64(-0.5 * log(t)))) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Max[x, y], $MachinePrecision] + N[Min[x, y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Log[N[(N[Min[x, y], $MachinePrecision] + N[Max[x, y], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -720.0], N[(N[Log[t$95$1], $MachinePrecision] - N[(-1.0 * N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 720.0], N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision] + N[Log[N[(z * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[N[Max[x, y], $MachinePrecision]], $MachinePrecision] + N[(N[Log[z], $MachinePrecision] + N[(-0.5 * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -720

   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. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lift-+.f64 N/A

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

      5. add-flip N/A

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

      6. associate--l- N/A

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

      7. sub-to-mult N/A

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

      8. lower-unsound-*.f64 N/A

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

   3. Applied rewrites 99.4%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. div-add N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. lower-/.f64 99.4%

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 99.4%

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

   5. Applied rewrites 99.4%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. add-to-fraction N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

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

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

      10. lower--.f64 N/A

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 41.3%

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

   10. Applied rewrites 41.3%

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

   if -720 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 720

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+r- N/A

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

      5. lower--.f64 N/A

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

   3. Applied rewrites 75.7%

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

   if 720 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z))

   1. Initial program 99.6%

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

   2. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-log.f64 62.5%

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

   4. Applied rewrites 62.5%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log.f64 41.1%

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

   7. Applied rewrites 41.1%

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

Alternative 8: 89.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{max}\left(x, y\right) + \mathsf{min}\left(x, y\right)\\ t_2 := \log \left(\mathsf{min}\left(x, y\right) + \mathsf{max}\left(x, y\right)\right) + \log z\\ \mathbf{if}\;t\_2 \leq -720:\\ \;\;\;\;\log t\_1 - -1 \cdot \left(a \cdot \log t\right)\\ \mathbf{elif}\;t\_2 \leq 720:\\ \;\;\;\;\mathsf{fma}\left(\log t, a - 0.5, \log \left(z \cdot t\_1\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log t, \left(\log \left(\mathsf{max}\left(x, y\right)\right) + \log z\right) - t\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (fmax x y) (fmin x y)))
        (t_2 (+ (log (+ (fmin x y) (fmax x y))) (log z))))
   (if (<= t_2 -720.0)
     (- (log t_1) (* -1.0 (* a (log t))))
     (if (<= t_2 720.0)
       (- (fma (log t) (- a 0.5) (log (* z t_1))) t)
       (fma -0.5 (log t) (- (+ (log (fmax x y)) (log z)) t))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fmax(x, y) + fmin(x, y);
	double t_2 = log((fmin(x, y) + fmax(x, y))) + log(z);
	double tmp;
	if (t_2 <= -720.0) {
		tmp = log(t_1) - (-1.0 * (a * log(t)));
	} else if (t_2 <= 720.0) {
		tmp = fma(log(t), (a - 0.5), log((z * t_1))) - t;
	} else {
		tmp = fma(-0.5, log(t), ((log(fmax(x, y)) + log(z)) - t));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(fmax(x, y) + fmin(x, y))
	t_2 = Float64(log(Float64(fmin(x, y) + fmax(x, y))) + log(z))
	tmp = 0.0
	if (t_2 <= -720.0)
		tmp = Float64(log(t_1) - Float64(-1.0 * Float64(a * log(t))));
	elseif (t_2 <= 720.0)
		tmp = Float64(fma(log(t), Float64(a - 0.5), log(Float64(z * t_1))) - t);
	else
		tmp = fma(-0.5, log(t), Float64(Float64(log(fmax(x, y)) + log(z)) - t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Max[x, y], $MachinePrecision] + N[Min[x, y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Log[N[(N[Min[x, y], $MachinePrecision] + N[Max[x, y], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -720.0], N[(N[Log[t$95$1], $MachinePrecision] - N[(-1.0 * N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 720.0], N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision] + N[Log[N[(z * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(-0.5 * N[Log[t], $MachinePrecision] + N[(N[(N[Log[N[Max[x, y], $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -720

   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. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lift-+.f64 N/A

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

      5. add-flip N/A

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

      6. associate--l- N/A

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

      7. sub-to-mult N/A

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

      8. lower-unsound-*.f64 N/A

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

   3. Applied rewrites 99.4%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. div-add N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. lower-/.f64 99.4%

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 99.4%

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

   5. Applied rewrites 99.4%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. add-to-fraction N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

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

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

      10. lower--.f64 N/A

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 41.3%

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

   10. Applied rewrites 41.3%

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

   if -720 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 720

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+r- N/A

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

      5. lower--.f64 N/A

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

   3. Applied rewrites 75.7%

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

   if 720 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z))

   1. Initial program 99.6%

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

   2. Taylor expanded in y around 0

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

      1. lower-log.f64 68.6%

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

   4. Applied rewrites 68.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 68.6%

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

      5. lift--.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate--l+ N/A

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

      8. add-flip N/A

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

      9. sub-negate-rev N/A

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

      10. lift--.f64 N/A

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

      11. lower--.f64 68.6%

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

   6. Applied rewrites 68.6%

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

   7. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-log.f64 68.5%

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

   9. Applied rewrites 68.5%

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

   10. Taylor expanded in a around 0

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

   12. Applied rewrites 41.1%

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

Alternative 9: 88.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_1 := \log \left(x + y\right)\\ t_2 := t\_1 + \log z\\ \mathbf{if}\;t\_2 \leq -720:\\ \;\;\;\;\log \left(y + x\right) - -1 \cdot \left(a \cdot \log t\right)\\ \mathbf{elif}\;t\_2 \leq 720:\\ \;\;\;\;\mathsf{fma}\left(\log t, a - 0.5, \log \left(z \cdot \left(y + x\right)\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 - \log \left(\frac{\sqrt{t}}{z}\right)\right) - t\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (log (+ x y))) (t_2 (+ t_1 (log z))))
   (if (<= t_2 -720.0)
     (- (log (+ y x)) (* -1.0 (* a (log t))))
     (if (<= t_2 720.0)
       (- (fma (log t) (- a 0.5) (log (* z (+ y x)))) t)
       (- (- t_1 (log (/ (sqrt t) z))) t)))))

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 tmp;
	if (t_2 <= -720.0) {
		tmp = log((y + x)) - (-1.0 * (a * log(t)));
	} else if (t_2 <= 720.0) {
		tmp = fma(log(t), (a - 0.5), log((z * (y + x)))) - t;
	} else {
		tmp = (t_1 - log((sqrt(t) / z))) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = log(Float64(x + y))
	t_2 = Float64(t_1 + log(z))
	tmp = 0.0
	if (t_2 <= -720.0)
		tmp = Float64(log(Float64(y + x)) - Float64(-1.0 * Float64(a * log(t))));
	elseif (t_2 <= 720.0)
		tmp = Float64(fma(log(t), Float64(a - 0.5), log(Float64(z * Float64(y + x)))) - t);
	else
		tmp = Float64(Float64(t_1 - log(Float64(sqrt(t) / z))) - t);
	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]}, If[LessEqual[t$95$2, -720.0], N[(N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision] - N[(-1.0 * N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 720.0], N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision] + N[Log[N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(t$95$1 - N[Log[N[(N[Sqrt[t], $MachinePrecision] / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -720

   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. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lift-+.f64 N/A

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

      5. add-flip N/A

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

      6. associate--l- N/A

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

      7. sub-to-mult N/A

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

      8. lower-unsound-*.f64 N/A

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

   3. Applied rewrites 99.4%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. div-add N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. lower-/.f64 99.4%

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 99.4%

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

   5. Applied rewrites 99.4%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. add-to-fraction N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

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

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

      10. lower--.f64 N/A

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 41.3%

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

   10. Applied rewrites 41.3%

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

   if -720 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 720

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+r- N/A

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

      5. lower--.f64 N/A

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

   3. Applied rewrites 75.7%

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

   if 720 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z))

   1. Initial program 99.6%

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

   2. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-log.f64 62.5%

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

   4. Applied rewrites 62.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. add-flip N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. associate-+l- N/A

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

      9. lower--.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-log.f64 N/A

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

      15. log-pow-rev N/A

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

      16. neg-log N/A

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

      17. lift-log.f64 N/A

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

      18. diff-log N/A

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

      19. lower-log.f64 N/A

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

      20. lower-/.f64 N/A

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

   6. Applied rewrites 55.3%

      \[\leadsto \left(\log \left(x + y\right) - \log \left(\frac{\sqrt{t}}{z}\right)\right) - t \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 87.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_1 := \log \left(\mathsf{min}\left(x, y\right) + \mathsf{max}\left(x, y\right)\right)\\ t_2 := t\_1 + \log z\\ \mathbf{if}\;t\_2 \leq -720:\\ \;\;\;\;\log \left(\mathsf{max}\left(x, y\right) + \mathsf{min}\left(x, y\right)\right) - -1 \cdot \left(a \cdot \log t\right)\\ \mathbf{elif}\;t\_2 \leq 720:\\ \;\;\;\;\left(\log \left(z \cdot \mathsf{max}\left(x, y\right)\right) - t\right) - \left(0.5 - a\right) \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 - \log \left(\frac{\sqrt{t}}{z}\right)\right) - t\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (log (+ (fmin x y) (fmax x y)))) (t_2 (+ t_1 (log z))))
   (if (<= t_2 -720.0)
     (- (log (+ (fmax x y) (fmin x y))) (* -1.0 (* a (log t))))
     (if (<= t_2 720.0)
       (- (- (log (* z (fmax x y))) t) (* (- 0.5 a) (log t)))
       (- (- t_1 (log (/ (sqrt t) z))) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((fmin(x, y) + fmax(x, y)));
	double t_2 = t_1 + log(z);
	double tmp;
	if (t_2 <= -720.0) {
		tmp = log((fmax(x, y) + fmin(x, y))) - (-1.0 * (a * log(t)));
	} else if (t_2 <= 720.0) {
		tmp = (log((z * fmax(x, y))) - t) - ((0.5 - a) * log(t));
	} else {
		tmp = (t_1 - log((sqrt(t) / z))) - 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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = log((fmin(x, y) + fmax(x, y)))
    t_2 = t_1 + log(z)
    if (t_2 <= (-720.0d0)) then
        tmp = log((fmax(x, y) + fmin(x, y))) - ((-1.0d0) * (a * log(t)))
    else if (t_2 <= 720.0d0) then
        tmp = (log((z * fmax(x, y))) - t) - ((0.5d0 - a) * log(t))
    else
        tmp = (t_1 - log((sqrt(t) / z))) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = Math.log((fmin(x, y) + fmax(x, y)));
	double t_2 = t_1 + Math.log(z);
	double tmp;
	if (t_2 <= -720.0) {
		tmp = Math.log((fmax(x, y) + fmin(x, y))) - (-1.0 * (a * Math.log(t)));
	} else if (t_2 <= 720.0) {
		tmp = (Math.log((z * fmax(x, y))) - t) - ((0.5 - a) * Math.log(t));
	} else {
		tmp = (t_1 - Math.log((Math.sqrt(t) / z))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = math.log((fmin(x, y) + fmax(x, y)))
	t_2 = t_1 + math.log(z)
	tmp = 0
	if t_2 <= -720.0:
		tmp = math.log((fmax(x, y) + fmin(x, y))) - (-1.0 * (a * math.log(t)))
	elif t_2 <= 720.0:
		tmp = (math.log((z * fmax(x, y))) - t) - ((0.5 - a) * math.log(t))
	else:
		tmp = (t_1 - math.log((math.sqrt(t) / z))) - t
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = log(Float64(fmin(x, y) + fmax(x, y)))
	t_2 = Float64(t_1 + log(z))
	tmp = 0.0
	if (t_2 <= -720.0)
		tmp = Float64(log(Float64(fmax(x, y) + fmin(x, y))) - Float64(-1.0 * Float64(a * log(t))));
	elseif (t_2 <= 720.0)
		tmp = Float64(Float64(log(Float64(z * fmax(x, y))) - t) - Float64(Float64(0.5 - a) * log(t)));
	else
		tmp = Float64(Float64(t_1 - log(Float64(sqrt(t) / z))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = log((min(x, y) + max(x, y)));
	t_2 = t_1 + log(z);
	tmp = 0.0;
	if (t_2 <= -720.0)
		tmp = log((max(x, y) + min(x, y))) - (-1.0 * (a * log(t)));
	elseif (t_2 <= 720.0)
		tmp = (log((z * max(x, y))) - t) - ((0.5 - a) * log(t));
	else
		tmp = (t_1 - log((sqrt(t) / z))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[Log[N[(N[Min[x, y], $MachinePrecision] + N[Max[x, y], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + N[Log[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -720.0], N[(N[Log[N[(N[Max[x, y], $MachinePrecision] + N[Min[x, y], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(-1.0 * N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 720.0], N[(N[(N[Log[N[(z * N[Max[x, y], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision] - N[(N[(0.5 - a), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 - N[Log[N[(N[Sqrt[t], $MachinePrecision] / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -720

   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. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lift-+.f64 N/A

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

      5. add-flip N/A

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

      6. associate--l- N/A

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

      7. sub-to-mult N/A

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

      8. lower-unsound-*.f64 N/A

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

   3. Applied rewrites 99.4%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. div-add N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. lower-/.f64 99.4%

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 99.4%

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

   5. Applied rewrites 99.4%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. add-to-fraction N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

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

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

      10. lower--.f64 N/A

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 41.3%

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

   10. Applied rewrites 41.3%

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

   if -720 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 720

   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 y around 0

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

      1. lower-log.f64 68.6%

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

   4. Applied rewrites 68.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 68.6%

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

      5. lift--.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate--l+ N/A

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

      8. add-flip N/A

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

      9. sub-negate-rev N/A

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

      10. lift--.f64 N/A

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

      11. lower--.f64 68.6%

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

   6. Applied rewrites 68.6%

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

   7. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-log.f64 68.5%

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

   9. Applied rewrites 68.5%

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

       1. lift-fma.f64 N/A

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

       2. +-commutative N/A

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

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

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

       4. lift--.f64 N/A

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

       5. sub-negate-rev N/A

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

       6. lift-log.f64 N/A

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

       7. lower--.f64 N/A

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

   11. Applied rewrites 52.5%

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

   if 720 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z))

   1. Initial program 99.6%

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

   2. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-log.f64 62.5%

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

   4. Applied rewrites 62.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. add-flip N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. associate-+l- N/A

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

      9. lower--.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-log.f64 N/A

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

      15. log-pow-rev N/A

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

      16. neg-log N/A

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

      17. lift-log.f64 N/A

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

      18. diff-log N/A

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

      19. lower-log.f64 N/A

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

      20. lower-/.f64 N/A

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

   6. Applied rewrites 55.3%

      \[\leadsto \left(\log \left(x + y\right) - \log \left(\frac{\sqrt{t}}{z}\right)\right) - t \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 87.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_1 := \log \left(\mathsf{min}\left(x, y\right) + \mathsf{max}\left(x, y\right)\right)\\ t_2 := t\_1 + \log z\\ \mathbf{if}\;t\_2 \leq -720:\\ \;\;\;\;\log \left(\mathsf{max}\left(x, y\right) + \mathsf{min}\left(x, y\right)\right) - -1 \cdot \left(a \cdot \log t\right)\\ \mathbf{elif}\;t\_2 \leq 720:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log \left(z \cdot \mathsf{max}\left(x, y\right)\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 - \log \left(\frac{\sqrt{t}}{z}\right)\right) - t\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (log (+ (fmin x y) (fmax x y)))) (t_2 (+ t_1 (log z))))
   (if (<= t_2 -720.0)
     (- (log (+ (fmax x y) (fmin x y))) (* -1.0 (* a (log t))))
     (if (<= t_2 720.0)
       (fma (- a 0.5) (log t) (- (log (* z (fmax x y))) t))
       (- (- t_1 (log (/ (sqrt t) z))) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((fmin(x, y) + fmax(x, y)));
	double t_2 = t_1 + log(z);
	double tmp;
	if (t_2 <= -720.0) {
		tmp = log((fmax(x, y) + fmin(x, y))) - (-1.0 * (a * log(t)));
	} else if (t_2 <= 720.0) {
		tmp = fma((a - 0.5), log(t), (log((z * fmax(x, y))) - t));
	} else {
		tmp = (t_1 - log((sqrt(t) / z))) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = log(Float64(fmin(x, y) + fmax(x, y)))
	t_2 = Float64(t_1 + log(z))
	tmp = 0.0
	if (t_2 <= -720.0)
		tmp = Float64(log(Float64(fmax(x, y) + fmin(x, y))) - Float64(-1.0 * Float64(a * log(t))));
	elseif (t_2 <= 720.0)
		tmp = fma(Float64(a - 0.5), log(t), Float64(log(Float64(z * fmax(x, y))) - t));
	else
		tmp = Float64(Float64(t_1 - log(Float64(sqrt(t) / z))) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[Log[N[(N[Min[x, y], $MachinePrecision] + N[Max[x, y], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + N[Log[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -720.0], N[(N[Log[N[(N[Max[x, y], $MachinePrecision] + N[Min[x, y], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(-1.0 * N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 720.0], N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[(N[Log[N[(z * N[Max[x, y], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 - N[Log[N[(N[Sqrt[t], $MachinePrecision] / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -720

   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. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lift-+.f64 N/A

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

      5. add-flip N/A

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

      6. associate--l- N/A

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

      7. sub-to-mult N/A

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

      8. lower-unsound-*.f64 N/A

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

   3. Applied rewrites 99.4%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. div-add N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. lower-/.f64 99.4%

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 99.4%

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

   5. Applied rewrites 99.4%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. add-to-fraction N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

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

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

      10. lower--.f64 N/A

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 41.3%

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

   10. Applied rewrites 41.3%

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

   if -720 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 720

   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 y around 0

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

      1. lower-log.f64 68.6%

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

   4. Applied rewrites 68.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 68.6%

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

      5. lift--.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate--l+ N/A

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

      8. add-flip N/A

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

      9. sub-negate-rev N/A

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

      10. lift--.f64 N/A

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

      11. lower--.f64 68.6%

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

   6. Applied rewrites 68.6%

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

   7. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-log.f64 68.5%

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

   9. Applied rewrites 68.5%

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

       1. lift-+.f64 N/A

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

       2. lift-log.f64 N/A

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

       3. lift-log.f64 N/A

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

       4. sum-log N/A

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

       5. lower-log.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 52.5%

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

   11. Applied rewrites 52.5%

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

   if 720 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z))

   1. Initial program 99.6%

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

   2. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-log.f64 62.5%

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

   4. Applied rewrites 62.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. add-flip N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. associate-+l- N/A

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

      9. lower--.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-log.f64 N/A

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

      15. log-pow-rev N/A

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

      16. neg-log N/A

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

      17. lift-log.f64 N/A

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

      18. diff-log N/A

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

      19. lower-log.f64 N/A

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

      20. lower-/.f64 N/A

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

   6. Applied rewrites 55.3%

      \[\leadsto \left(\log \left(x + y\right) - \log \left(\frac{\sqrt{t}}{z}\right)\right) - t \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 79.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_1 := a \cdot \log t\\ \mathbf{if}\;a - 0.5 \leq -5 \cdot 10^{+81}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a - 0.5 \leq 2 \cdot 10^{+30}:\\ \;\;\;\;\left(\log \left(x + y\right) - \log \left(\frac{\sqrt{t}}{z}\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* a (log t))))
   (if (<= (- a 0.5) -5e+81)
     t_1
     (if (<= (- a 0.5) 2e+30)
       (- (- (log (+ x y)) (log (/ (sqrt t) z))) t)
       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 - 0.5) <= -5e+81) {
		tmp = t_1;
	} else if ((a - 0.5) <= 2e+30) {
		tmp = (log((x + y)) - log((sqrt(t) / z))) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = a * log(t)
    if ((a - 0.5d0) <= (-5d+81)) then
        tmp = t_1
    else if ((a - 0.5d0) <= 2d+30) then
        tmp = (log((x + y)) - log((sqrt(t) / z))) - t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = a * Math.log(t);
	double tmp;
	if ((a - 0.5) <= -5e+81) {
		tmp = t_1;
	} else if ((a - 0.5) <= 2e+30) {
		tmp = (Math.log((x + y)) - Math.log((Math.sqrt(t) / z))) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = a * math.log(t)
	tmp = 0
	if (a - 0.5) <= -5e+81:
		tmp = t_1
	elif (a - 0.5) <= 2e+30:
		tmp = (math.log((x + y)) - math.log((math.sqrt(t) / z))) - t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(a * log(t))
	tmp = 0.0
	if (Float64(a - 0.5) <= -5e+81)
		tmp = t_1;
	elseif (Float64(a - 0.5) <= 2e+30)
		tmp = Float64(Float64(log(Float64(x + y)) - log(Float64(sqrt(t) / z))) - t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = a * log(t);
	tmp = 0.0;
	if ((a - 0.5) <= -5e+81)
		tmp = t_1;
	elseif ((a - 0.5) <= 2e+30)
		tmp = (log((x + y)) - log((sqrt(t) / z))) - t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a - 0.5), $MachinePrecision], -5e+81], t$95$1, If[LessEqual[N[(a - 0.5), $MachinePrecision], 2e+30], N[(N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] - N[Log[N[(N[Sqrt[t], $MachinePrecision] / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 a #s(literal 1/2 binary64)) < -4.9999999999999998e81 or 2e30 < (-.f64 a #s(literal 1/2 binary64))

   1. Initial program 99.6%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 38.5%

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

   4. Applied rewrites 38.5%

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

   if -4.9999999999999998e81 < (-.f64 a #s(literal 1/2 binary64)) < 2e30

   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 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-log.f64 62.5%

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

   4. Applied rewrites 62.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. add-flip N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. associate-+l- N/A

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

      9. lower--.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-log.f64 N/A

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

      15. log-pow-rev N/A

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

      16. neg-log N/A

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

      17. lift-log.f64 N/A

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

      18. diff-log N/A

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

      19. lower-log.f64 N/A

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

      20. lower-/.f64 N/A

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

   6. Applied rewrites 55.3%

      \[\leadsto \left(\log \left(x + y\right) - \log \left(\frac{\sqrt{t}}{z}\right)\right) - t \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 79.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_1 := a \cdot \log t\\ \mathbf{if}\;a - 0.5 \leq -5 \cdot 10^{+81}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a - 0.5 \leq 2 \cdot 10^{+30}:\\ \;\;\;\;\left(\log z - \log \left(\frac{\sqrt{t}}{x + y}\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* a (log t))))
   (if (<= (- a 0.5) -5e+81)
     t_1
     (if (<= (- a 0.5) 2e+30)
       (- (- (log z) (log (/ (sqrt t) (+ x y)))) t)
       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 - 0.5) <= -5e+81) {
		tmp = t_1;
	} else if ((a - 0.5) <= 2e+30) {
		tmp = (log(z) - log((sqrt(t) / (x + y)))) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = a * log(t)
    if ((a - 0.5d0) <= (-5d+81)) then
        tmp = t_1
    else if ((a - 0.5d0) <= 2d+30) then
        tmp = (log(z) - log((sqrt(t) / (x + y)))) - t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = a * Math.log(t);
	double tmp;
	if ((a - 0.5) <= -5e+81) {
		tmp = t_1;
	} else if ((a - 0.5) <= 2e+30) {
		tmp = (Math.log(z) - Math.log((Math.sqrt(t) / (x + y)))) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = a * math.log(t)
	tmp = 0
	if (a - 0.5) <= -5e+81:
		tmp = t_1
	elif (a - 0.5) <= 2e+30:
		tmp = (math.log(z) - math.log((math.sqrt(t) / (x + y)))) - t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(a * log(t))
	tmp = 0.0
	if (Float64(a - 0.5) <= -5e+81)
		tmp = t_1;
	elseif (Float64(a - 0.5) <= 2e+30)
		tmp = Float64(Float64(log(z) - log(Float64(sqrt(t) / Float64(x + y)))) - t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = a * log(t);
	tmp = 0.0;
	if ((a - 0.5) <= -5e+81)
		tmp = t_1;
	elseif ((a - 0.5) <= 2e+30)
		tmp = (log(z) - log((sqrt(t) / (x + y)))) - t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a - 0.5), $MachinePrecision], -5e+81], t$95$1, If[LessEqual[N[(a - 0.5), $MachinePrecision], 2e+30], N[(N[(N[Log[z], $MachinePrecision] - N[Log[N[(N[Sqrt[t], $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 a #s(literal 1/2 binary64)) < -4.9999999999999998e81 or 2e30 < (-.f64 a #s(literal 1/2 binary64))

   1. Initial program 99.6%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 38.5%

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

   4. Applied rewrites 38.5%

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

   if -4.9999999999999998e81 < (-.f64 a #s(literal 1/2 binary64)) < 2e30

   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 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-log.f64 62.5%

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

   4. Applied rewrites 62.5%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. add-flip N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. sub-negate N/A

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

      10. lift-*.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. log-pow-rev N/A

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

      13. neg-log N/A

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

      14. lift-log.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-+.f64 N/A

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

      18. diff-log N/A

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

      19. lower-log.f64 N/A

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

      20. lower-/.f64 N/A

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

   6. Applied rewrites 57.0%

      \[\leadsto \left(\log z - \log \left(\frac{\sqrt{t}}{x + y}\right)\right) - t \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 74.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+22}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t\_1 \leq 1020:\\ \;\;\;\;\left(\log \left(\left(x + y\right) \cdot z\right) - \log \left(\sqrt{t}\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log \left(y + x\right) - -1 \cdot \left(a \cdot \log t\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t)))))
   (if (<= t_1 -5e+22)
     (- t)
     (if (<= t_1 1020.0)
       (- (- (log (* (+ x y) z)) (log (sqrt t))) t)
       (- (log (+ y x)) (* -1.0 (* a (log t))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
	double tmp;
	if (t_1 <= -5e+22) {
		tmp = -t;
	} else if (t_1 <= 1020.0) {
		tmp = (log(((x + y) * z)) - log(sqrt(t))) - t;
	} else {
		tmp = log((y + x)) - (-1.0 * (a * log(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) :: t_1
    real(8) :: tmp
    t_1 = ((log((x + y)) + log(z)) - t) + ((a - 0.5d0) * log(t))
    if (t_1 <= (-5d+22)) then
        tmp = -t
    else if (t_1 <= 1020.0d0) then
        tmp = (log(((x + y) * z)) - log(sqrt(t))) - t
    else
        tmp = log((y + x)) - ((-1.0d0) * (a * log(t)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = ((Math.log((x + y)) + Math.log(z)) - t) + ((a - 0.5) * Math.log(t));
	double tmp;
	if (t_1 <= -5e+22) {
		tmp = -t;
	} else if (t_1 <= 1020.0) {
		tmp = (Math.log(((x + y) * z)) - Math.log(Math.sqrt(t))) - t;
	} else {
		tmp = Math.log((y + x)) - (-1.0 * (a * Math.log(t)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = ((math.log((x + y)) + math.log(z)) - t) + ((a - 0.5) * math.log(t))
	tmp = 0
	if t_1 <= -5e+22:
		tmp = -t
	elif t_1 <= 1020.0:
		tmp = (math.log(((x + y) * z)) - math.log(math.sqrt(t))) - t
	else:
		tmp = math.log((y + x)) - (-1.0 * (a * math.log(t)))
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + Float64(Float64(a - 0.5) * log(t)))
	tmp = 0.0
	if (t_1 <= -5e+22)
		tmp = Float64(-t);
	elseif (t_1 <= 1020.0)
		tmp = Float64(Float64(log(Float64(Float64(x + y) * z)) - log(sqrt(t))) - t);
	else
		tmp = Float64(log(Float64(y + x)) - Float64(-1.0 * Float64(a * log(t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
	tmp = 0.0;
	if (t_1 <= -5e+22)
		tmp = -t;
	elseif (t_1 <= 1020.0)
		tmp = (log(((x + y) * z)) - log(sqrt(t))) - t;
	else
		tmp = log((y + x)) - (-1.0 * (a * log(t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, -5e+22], (-t), If[LessEqual[t$95$1, 1020.0], N[(N[(N[Log[N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision] - N[Log[N[Sqrt[t], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision] - N[(-1.0 * N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < -4.9999999999999996e22

   1. Initial program 99.6%

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

   2. Taylor expanded in t around inf

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

      1. lower-*.f64 38.3%

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

   4. Applied rewrites 38.3%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 38.3%

         \[\leadsto -t \]

   6. Applied rewrites 38.3%

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

   if -4.9999999999999996e22 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 1020

   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 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-log.f64 62.5%

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

   4. Applied rewrites 62.5%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. lift-log.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. add-flip N/A

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

      9. lower--.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. sum-log N/A

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

      12. lower-log.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-log.f64 N/A

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

      16. log-pow-rev N/A

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

      17. neg-log N/A

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

      18. lower-log.f64 N/A

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

      19. pow-flip N/A

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

      20. metadata-eval N/A

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

      21. unpow1/2 N/A

          \[\leadsto \left(\log \left(\left(x + y\right) \cdot z\right) - \log \left(\sqrt{t}\right)\right) - t \]

      22. lower-sqrt.f64 47.2%

          \[\leadsto \left(\log \left(\left(x + y\right) \cdot z\right) - \log \left(\sqrt{t}\right)\right) - t \]

   6. Applied rewrites 47.2%

      \[\leadsto \left(\log \left(\left(x + y\right) \cdot z\right) - \log \left(\sqrt{t}\right)\right) - t \]

   if 1020 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t)))

   1. Initial program 99.6%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   2. 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. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lift-+.f64 N/A

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

      5. add-flip N/A

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

      6. associate--l- N/A

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

      7. sub-to-mult N/A

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

      8. lower-unsound-*.f64 N/A

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

   3. Applied rewrites 99.4%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. div-add N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. lower-/.f64 99.4%

          \[\leadsto \left(1 - \mathsf{fma}\left(0.5 - a, \frac{\log t}{\log \left(x + y\right)}, \color{blue}{\frac{t - \log z}{\log \left(y + x\right)}}\right)\right) \cdot \log \left(y + x\right) \]

      11. lift-+.f64 N/A

          \[\leadsto \left(1 - \mathsf{fma}\left(\frac{1}{2} - a, \frac{\log t}{\log \left(x + y\right)}, \frac{t - \log z}{\log \color{blue}{\left(y + x\right)}}\right)\right) \cdot \log \left(y + x\right) \]

      12. +-commutative N/A

          \[\leadsto \left(1 - \mathsf{fma}\left(\frac{1}{2} - a, \frac{\log t}{\log \left(x + y\right)}, \frac{t - \log z}{\log \color{blue}{\left(x + y\right)}}\right)\right) \cdot \log \left(y + x\right) \]

      13. lift-+.f64 99.4%

          \[\leadsto \left(1 - \mathsf{fma}\left(0.5 - a, \frac{\log t}{\log \left(x + y\right)}, \frac{t - \log z}{\log \color{blue}{\left(x + y\right)}}\right)\right) \cdot \log \left(y + x\right) \]

   5. Applied rewrites 99.4%

      \[\leadsto \left(1 - \color{blue}{\mathsf{fma}\left(0.5 - a, \frac{\log t}{\log \left(x + y\right)}, \frac{t - \log z}{\log \left(x + y\right)}\right)}\right) \cdot \log \left(y + x\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(1 - \mathsf{fma}\left(\frac{1}{2} - a, \frac{\log t}{\log \left(x + y\right)}, \frac{t - \log z}{\log \left(x + y\right)}\right)\right) \cdot \log \left(y + x\right)} \]

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(1 - \mathsf{fma}\left(\frac{1}{2} - a, \frac{\log t}{\log \left(x + y\right)}, \frac{t - \log z}{\log \left(x + y\right)}\right)\right)} \cdot \log \left(y + x\right) \]

      3. lift-fma.f64 N/A

         \[\leadsto \left(1 - \color{blue}{\left(\left(\frac{1}{2} - a\right) \cdot \frac{\log t}{\log \left(x + y\right)} + \frac{t - \log z}{\log \left(x + y\right)}\right)}\right) \cdot \log \left(y + x\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(1 - \left(\left(\frac{1}{2} - a\right) \cdot \frac{\log t}{\log \left(x + y\right)} + \color{blue}{\frac{t - \log z}{\log \left(x + y\right)}}\right)\right) \cdot \log \left(y + x\right) \]

      5. add-to-fraction N/A

         \[\leadsto \left(1 - \color{blue}{\frac{\left(\left(\frac{1}{2} - a\right) \cdot \frac{\log t}{\log \left(x + y\right)}\right) \cdot \log \left(x + y\right) + \left(t - \log z\right)}{\log \left(x + y\right)}}\right) \cdot \log \left(y + x\right) \]

      6. lift-+.f64 N/A

         \[\leadsto \left(1 - \frac{\left(\left(\frac{1}{2} - a\right) \cdot \frac{\log t}{\log \left(x + y\right)}\right) \cdot \log \left(x + y\right) + \left(t - \log z\right)}{\log \left(x + y\right)}\right) \cdot \log \color{blue}{\left(y + x\right)} \]

      7. +-commutative N/A

         \[\leadsto \left(1 - \frac{\left(\left(\frac{1}{2} - a\right) \cdot \frac{\log t}{\log \left(x + y\right)}\right) \cdot \log \left(x + y\right) + \left(t - \log z\right)}{\log \left(x + y\right)}\right) \cdot \log \color{blue}{\left(x + y\right)} \]

      8. lift-+.f64 N/A

         \[\leadsto \left(1 - \frac{\left(\left(\frac{1}{2} - a\right) \cdot \frac{\log t}{\log \left(x + y\right)}\right) \cdot \log \left(x + y\right) + \left(t - \log z\right)}{\log \left(x + y\right)}\right) \cdot \log \color{blue}{\left(x + y\right)} \]

      9. sub-to-mult-rev N/A

         \[\leadsto \color{blue}{\log \left(x + y\right) - \left(\left(\left(\frac{1}{2} - a\right) \cdot \frac{\log t}{\log \left(x + y\right)}\right) \cdot \log \left(x + y\right) + \left(t - \log z\right)\right)} \]

      10. lower--.f64 N/A

          \[\leadsto \color{blue}{\log \left(x + y\right) - \left(\left(\left(\frac{1}{2} - a\right) \cdot \frac{\log t}{\log \left(x + y\right)}\right) \cdot \log \left(x + y\right) + \left(t - \log z\right)\right)} \]

   7. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\log \left(y + x\right) - \mathsf{fma}\left(\frac{\log t}{\log \left(y + x\right)} \cdot \left(0.5 - a\right), \log \left(y + x\right), t - \log z\right)} \]

   8. Taylor expanded in a around inf

      \[\leadsto \log \left(y + x\right) - \color{blue}{-1 \cdot \left(a \cdot \log t\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \log \left(y + x\right) - -1 \cdot \color{blue}{\left(a \cdot \log t\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \log \left(y + x\right) - -1 \cdot \left(a \cdot \color{blue}{\log t}\right) \]

      3. lower-log.f64 41.3%

         \[\leadsto \log \left(y + x\right) - -1 \cdot \left(a \cdot \log t\right) \]

   10. Applied rewrites 41.3%

       \[\leadsto \log \left(y + x\right) - \color{blue}{-1 \cdot \left(a \cdot \log t\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 72.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_1 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+22}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t\_1 \leq 700:\\ \;\;\;\;\log \left(\frac{\left(x + y\right) \cdot z}{\sqrt{t}}\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log \left(y + x\right) - -1 \cdot \left(a \cdot \log t\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t)))))
   (if (<= t_1 -5e+22)
     (- t)
     (if (<= t_1 700.0)
       (- (log (/ (* (+ x y) z) (sqrt t))) t)
       (- (log (+ y x)) (* -1.0 (* a (log t))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
	double tmp;
	if (t_1 <= -5e+22) {
		tmp = -t;
	} else if (t_1 <= 700.0) {
		tmp = log((((x + y) * z) / sqrt(t))) - t;
	} else {
		tmp = log((y + x)) - (-1.0 * (a * log(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) :: t_1
    real(8) :: tmp
    t_1 = ((log((x + y)) + log(z)) - t) + ((a - 0.5d0) * log(t))
    if (t_1 <= (-5d+22)) then
        tmp = -t
    else if (t_1 <= 700.0d0) then
        tmp = log((((x + y) * z) / sqrt(t))) - t
    else
        tmp = log((y + x)) - ((-1.0d0) * (a * log(t)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = ((Math.log((x + y)) + Math.log(z)) - t) + ((a - 0.5) * Math.log(t));
	double tmp;
	if (t_1 <= -5e+22) {
		tmp = -t;
	} else if (t_1 <= 700.0) {
		tmp = Math.log((((x + y) * z) / Math.sqrt(t))) - t;
	} else {
		tmp = Math.log((y + x)) - (-1.0 * (a * Math.log(t)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = ((math.log((x + y)) + math.log(z)) - t) + ((a - 0.5) * math.log(t))
	tmp = 0
	if t_1 <= -5e+22:
		tmp = -t
	elif t_1 <= 700.0:
		tmp = math.log((((x + y) * z) / math.sqrt(t))) - t
	else:
		tmp = math.log((y + x)) - (-1.0 * (a * math.log(t)))
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + Float64(Float64(a - 0.5) * log(t)))
	tmp = 0.0
	if (t_1 <= -5e+22)
		tmp = Float64(-t);
	elseif (t_1 <= 700.0)
		tmp = Float64(log(Float64(Float64(Float64(x + y) * z) / sqrt(t))) - t);
	else
		tmp = Float64(log(Float64(y + x)) - Float64(-1.0 * Float64(a * log(t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
	tmp = 0.0;
	if (t_1 <= -5e+22)
		tmp = -t;
	elseif (t_1 <= 700.0)
		tmp = log((((x + y) * z) / sqrt(t))) - t;
	else
		tmp = log((y + x)) - (-1.0 * (a * log(t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, -5e+22], (-t), If[LessEqual[t$95$1, 700.0], N[(N[Log[N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] / N[Sqrt[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision], N[(N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision] - N[(-1.0 * N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < -4.9999999999999996e22

   1. Initial program 99.6%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]

   2. Taylor expanded in t around inf

      \[\leadsto \color{blue}{-1 \cdot t} \]
   3. Step-by-step derivation

      1. lower-*.f64 38.3%

         \[\leadsto -1 \cdot \color{blue}{t} \]

   4. Applied rewrites 38.3%

      \[\leadsto \color{blue}{-1 \cdot t} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{t} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(t\right) \]

      3. lower-neg.f64 38.3%

         \[\leadsto -t \]

   6. Applied rewrites 38.3%

      \[\leadsto \color{blue}{-t} \]

   if -4.9999999999999996e22 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 700

   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 0

      \[\leadsto \color{blue}{\left(\log z + \left(\log \left(x + y\right) + \frac{-1}{2} \cdot \log t\right)\right) - t} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(\log z + \left(\log \left(x + y\right) + \frac{-1}{2} \cdot \log t\right)\right) - \color{blue}{t} \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\log z + \left(\log \left(x + y\right) + \frac{-1}{2} \cdot \log t\right)\right) - t \]

      3. lower-log.f64 N/A

         \[\leadsto \left(\log z + \left(\log \left(x + y\right) + \frac{-1}{2} \cdot \log t\right)\right) - t \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\log z + \left(\log \left(x + y\right) + \frac{-1}{2} \cdot \log t\right)\right) - t \]

      5. lower-log.f64 N/A

         \[\leadsto \left(\log z + \left(\log \left(x + y\right) + \frac{-1}{2} \cdot \log t\right)\right) - t \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\log z + \left(\log \left(x + y\right) + \frac{-1}{2} \cdot \log t\right)\right) - t \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\log z + \left(\log \left(x + y\right) + \frac{-1}{2} \cdot \log t\right)\right) - t \]

      8. lower-log.f64 62.5%

         \[\leadsto \left(\log z + \left(\log \left(x + y\right) + -0.5 \cdot \log t\right)\right) - t \]

   4. Applied rewrites 62.5%

      \[\leadsto \color{blue}{\left(\log z + \left(\log \left(x + y\right) + -0.5 \cdot \log t\right)\right) - t} \]
   5. Step-by-step derivation

   6. Applied rewrites 43.4%

      \[\leadsto \log \left(\frac{\left(x + y\right) \cdot z}{\sqrt{t}}\right) - \color{blue}{t} \]

   if 700 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t)))

   1. Initial program 99.6%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   2. 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. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\log \left(x + y\right) + \log z\right) - t\right)} + \left(a - \frac{1}{2}\right) \cdot \log t \]

      3. associate-+l- N/A

         \[\leadsto \color{blue}{\left(\log \left(x + y\right) + \log z\right) - \left(t - \left(a - \frac{1}{2}\right) \cdot \log t\right)} \]

      4. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\log \left(x + y\right) + \log z\right)} - \left(t - \left(a - \frac{1}{2}\right) \cdot \log t\right) \]

      5. add-flip N/A

         \[\leadsto \color{blue}{\left(\log \left(x + y\right) - \left(\mathsf{neg}\left(\log z\right)\right)\right)} - \left(t - \left(a - \frac{1}{2}\right) \cdot \log t\right) \]

      6. associate--l- N/A

         \[\leadsto \color{blue}{\log \left(x + y\right) - \left(\left(\mathsf{neg}\left(\log z\right)\right) + \left(t - \left(a - \frac{1}{2}\right) \cdot \log t\right)\right)} \]

      7. sub-to-mult N/A

         \[\leadsto \color{blue}{\left(1 - \frac{\left(\mathsf{neg}\left(\log z\right)\right) + \left(t - \left(a - \frac{1}{2}\right) \cdot \log t\right)}{\log \left(x + y\right)}\right) \cdot \log \left(x + y\right)} \]

      8. lower-unsound-*.f64 N/A

         \[\leadsto \color{blue}{\left(1 - \frac{\left(\mathsf{neg}\left(\log z\right)\right) + \left(t - \left(a - \frac{1}{2}\right) \cdot \log t\right)}{\log \left(x + y\right)}\right) \cdot \log \left(x + y\right)} \]

   3. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\left(1 - \frac{\mathsf{fma}\left(0.5 - a, \log t, t - \log z\right)}{\log \left(y + x\right)}\right) \cdot \log \left(y + x\right)} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(1 - \color{blue}{\frac{\mathsf{fma}\left(\frac{1}{2} - a, \log t, t - \log z\right)}{\log \left(y + x\right)}}\right) \cdot \log \left(y + x\right) \]

      2. lift-fma.f64 N/A

         \[\leadsto \left(1 - \frac{\color{blue}{\left(\frac{1}{2} - a\right) \cdot \log t + \left(t - \log z\right)}}{\log \left(y + x\right)}\right) \cdot \log \left(y + x\right) \]

      3. div-add N/A

         \[\leadsto \left(1 - \color{blue}{\left(\frac{\left(\frac{1}{2} - a\right) \cdot \log t}{\log \left(y + x\right)} + \frac{t - \log z}{\log \left(y + x\right)}\right)}\right) \cdot \log \left(y + x\right) \]

      4. associate-/l* N/A

         \[\leadsto \left(1 - \left(\color{blue}{\left(\frac{1}{2} - a\right) \cdot \frac{\log t}{\log \left(y + x\right)}} + \frac{t - \log z}{\log \left(y + x\right)}\right)\right) \cdot \log \left(y + x\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(1 - \color{blue}{\mathsf{fma}\left(\frac{1}{2} - a, \frac{\log t}{\log \left(y + x\right)}, \frac{t - \log z}{\log \left(y + x\right)}\right)}\right) \cdot \log \left(y + x\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \left(1 - \mathsf{fma}\left(\frac{1}{2} - a, \color{blue}{\frac{\log t}{\log \left(y + x\right)}}, \frac{t - \log z}{\log \left(y + x\right)}\right)\right) \cdot \log \left(y + x\right) \]

      7. lift-+.f64 N/A

         \[\leadsto \left(1 - \mathsf{fma}\left(\frac{1}{2} - a, \frac{\log t}{\log \color{blue}{\left(y + x\right)}}, \frac{t - \log z}{\log \left(y + x\right)}\right)\right) \cdot \log \left(y + x\right) \]

      8. +-commutative N/A

         \[\leadsto \left(1 - \mathsf{fma}\left(\frac{1}{2} - a, \frac{\log t}{\log \color{blue}{\left(x + y\right)}}, \frac{t - \log z}{\log \left(y + x\right)}\right)\right) \cdot \log \left(y + x\right) \]

      9. lift-+.f64 N/A

         \[\leadsto \left(1 - \mathsf{fma}\left(\frac{1}{2} - a, \frac{\log t}{\log \color{blue}{\left(x + y\right)}}, \frac{t - \log z}{\log \left(y + x\right)}\right)\right) \cdot \log \left(y + x\right) \]

      10. lower-/.f64 99.4%

          \[\leadsto \left(1 - \mathsf{fma}\left(0.5 - a, \frac{\log t}{\log \left(x + y\right)}, \color{blue}{\frac{t - \log z}{\log \left(y + x\right)}}\right)\right) \cdot \log \left(y + x\right) \]

      11. lift-+.f64 N/A

          \[\leadsto \left(1 - \mathsf{fma}\left(\frac{1}{2} - a, \frac{\log t}{\log \left(x + y\right)}, \frac{t - \log z}{\log \color{blue}{\left(y + x\right)}}\right)\right) \cdot \log \left(y + x\right) \]

      12. +-commutative N/A

          \[\leadsto \left(1 - \mathsf{fma}\left(\frac{1}{2} - a, \frac{\log t}{\log \left(x + y\right)}, \frac{t - \log z}{\log \color{blue}{\left(x + y\right)}}\right)\right) \cdot \log \left(y + x\right) \]

      13. lift-+.f64 99.4%

          \[\leadsto \left(1 - \mathsf{fma}\left(0.5 - a, \frac{\log t}{\log \left(x + y\right)}, \frac{t - \log z}{\log \color{blue}{\left(x + y\right)}}\right)\right) \cdot \log \left(y + x\right) \]

   5. Applied rewrites 99.4%

      \[\leadsto \left(1 - \color{blue}{\mathsf{fma}\left(0.5 - a, \frac{\log t}{\log \left(x + y\right)}, \frac{t - \log z}{\log \left(x + y\right)}\right)}\right) \cdot \log \left(y + x\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(1 - \mathsf{fma}\left(\frac{1}{2} - a, \frac{\log t}{\log \left(x + y\right)}, \frac{t - \log z}{\log \left(x + y\right)}\right)\right) \cdot \log \left(y + x\right)} \]

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(1 - \mathsf{fma}\left(\frac{1}{2} - a, \frac{\log t}{\log \left(x + y\right)}, \frac{t - \log z}{\log \left(x + y\right)}\right)\right)} \cdot \log \left(y + x\right) \]

      3. lift-fma.f64 N/A

         \[\leadsto \left(1 - \color{blue}{\left(\left(\frac{1}{2} - a\right) \cdot \frac{\log t}{\log \left(x + y\right)} + \frac{t - \log z}{\log \left(x + y\right)}\right)}\right) \cdot \log \left(y + x\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(1 - \left(\left(\frac{1}{2} - a\right) \cdot \frac{\log t}{\log \left(x + y\right)} + \color{blue}{\frac{t - \log z}{\log \left(x + y\right)}}\right)\right) \cdot \log \left(y + x\right) \]

      5. add-to-fraction N/A

         \[\leadsto \left(1 - \color{blue}{\frac{\left(\left(\frac{1}{2} - a\right) \cdot \frac{\log t}{\log \left(x + y\right)}\right) \cdot \log \left(x + y\right) + \left(t - \log z\right)}{\log \left(x + y\right)}}\right) \cdot \log \left(y + x\right) \]

      6. lift-+.f64 N/A

         \[\leadsto \left(1 - \frac{\left(\left(\frac{1}{2} - a\right) \cdot \frac{\log t}{\log \left(x + y\right)}\right) \cdot \log \left(x + y\right) + \left(t - \log z\right)}{\log \left(x + y\right)}\right) \cdot \log \color{blue}{\left(y + x\right)} \]

      7. +-commutative N/A

         \[\leadsto \left(1 - \frac{\left(\left(\frac{1}{2} - a\right) \cdot \frac{\log t}{\log \left(x + y\right)}\right) \cdot \log \left(x + y\right) + \left(t - \log z\right)}{\log \left(x + y\right)}\right) \cdot \log \color{blue}{\left(x + y\right)} \]

      8. lift-+.f64 N/A

         \[\leadsto \left(1 - \frac{\left(\left(\frac{1}{2} - a\right) \cdot \frac{\log t}{\log \left(x + y\right)}\right) \cdot \log \left(x + y\right) + \left(t - \log z\right)}{\log \left(x + y\right)}\right) \cdot \log \color{blue}{\left(x + y\right)} \]

      9. sub-to-mult-rev N/A

         \[\leadsto \color{blue}{\log \left(x + y\right) - \left(\left(\left(\frac{1}{2} - a\right) \cdot \frac{\log t}{\log \left(x + y\right)}\right) \cdot \log \left(x + y\right) + \left(t - \log z\right)\right)} \]

      10. lower--.f64 N/A

          \[\leadsto \color{blue}{\log \left(x + y\right) - \left(\left(\left(\frac{1}{2} - a\right) \cdot \frac{\log t}{\log \left(x + y\right)}\right) \cdot \log \left(x + y\right) + \left(t - \log z\right)\right)} \]

   7. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\log \left(y + x\right) - \mathsf{fma}\left(\frac{\log t}{\log \left(y + x\right)} \cdot \left(0.5 - a\right), \log \left(y + x\right), t - \log z\right)} \]

   8. Taylor expanded in a around inf

      \[\leadsto \log \left(y + x\right) - \color{blue}{-1 \cdot \left(a \cdot \log t\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \log \left(y + x\right) - -1 \cdot \color{blue}{\left(a \cdot \log t\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \log \left(y + x\right) - -1 \cdot \left(a \cdot \color{blue}{\log t}\right) \]

      3. lower-log.f64 41.3%

         \[\leadsto \log \left(y + x\right) - -1 \cdot \left(a \cdot \log t\right) \]

   10. Applied rewrites 41.3%

       \[\leadsto \log \left(y + x\right) - \color{blue}{-1 \cdot \left(a \cdot \log t\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 71.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_1 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+22}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t\_1 \leq 712:\\ \;\;\;\;\log \left(\frac{\left(x + y\right) \cdot z}{\sqrt{t}}\right) - t\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t)))))
   (if (<= t_1 -5e+22)
     (- t)
     (if (<= t_1 712.0)
       (- (log (/ (* (+ x y) z) (sqrt t))) t)
       (* a (log t))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
	double tmp;
	if (t_1 <= -5e+22) {
		tmp = -t;
	} else if (t_1 <= 712.0) {
		tmp = log((((x + y) * z) / sqrt(t))) - t;
	} else {
		tmp = a * log(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) :: t_1
    real(8) :: tmp
    t_1 = ((log((x + y)) + log(z)) - t) + ((a - 0.5d0) * log(t))
    if (t_1 <= (-5d+22)) then
        tmp = -t
    else if (t_1 <= 712.0d0) then
        tmp = log((((x + y) * z) / sqrt(t))) - t
    else
        tmp = a * log(t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = ((Math.log((x + y)) + Math.log(z)) - t) + ((a - 0.5) * Math.log(t));
	double tmp;
	if (t_1 <= -5e+22) {
		tmp = -t;
	} else if (t_1 <= 712.0) {
		tmp = Math.log((((x + y) * z) / Math.sqrt(t))) - t;
	} else {
		tmp = a * Math.log(t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = ((math.log((x + y)) + math.log(z)) - t) + ((a - 0.5) * math.log(t))
	tmp = 0
	if t_1 <= -5e+22:
		tmp = -t
	elif t_1 <= 712.0:
		tmp = math.log((((x + y) * z) / math.sqrt(t))) - t
	else:
		tmp = a * math.log(t)
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + Float64(Float64(a - 0.5) * log(t)))
	tmp = 0.0
	if (t_1 <= -5e+22)
		tmp = Float64(-t);
	elseif (t_1 <= 712.0)
		tmp = Float64(log(Float64(Float64(Float64(x + y) * z) / sqrt(t))) - t);
	else
		tmp = Float64(a * log(t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
	tmp = 0.0;
	if (t_1 <= -5e+22)
		tmp = -t;
	elseif (t_1 <= 712.0)
		tmp = log((((x + y) * z) / sqrt(t))) - t;
	else
		tmp = a * log(t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, -5e+22], (-t), If[LessEqual[t$95$1, 712.0], N[(N[Log[N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] / N[Sqrt[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision], N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < -4.9999999999999996e22

   1. Initial program 99.6%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]

   2. Taylor expanded in t around inf

      \[\leadsto \color{blue}{-1 \cdot t} \]
   3. Step-by-step derivation

      1. lower-*.f64 38.3%

         \[\leadsto -1 \cdot \color{blue}{t} \]

   4. Applied rewrites 38.3%

      \[\leadsto \color{blue}{-1 \cdot t} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{t} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(t\right) \]

      3. lower-neg.f64 38.3%

         \[\leadsto -t \]

   6. Applied rewrites 38.3%

      \[\leadsto \color{blue}{-t} \]

   if -4.9999999999999996e22 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 712

   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 0

      \[\leadsto \color{blue}{\left(\log z + \left(\log \left(x + y\right) + \frac{-1}{2} \cdot \log t\right)\right) - t} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(\log z + \left(\log \left(x + y\right) + \frac{-1}{2} \cdot \log t\right)\right) - \color{blue}{t} \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\log z + \left(\log \left(x + y\right) + \frac{-1}{2} \cdot \log t\right)\right) - t \]

      3. lower-log.f64 N/A

         \[\leadsto \left(\log z + \left(\log \left(x + y\right) + \frac{-1}{2} \cdot \log t\right)\right) - t \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\log z + \left(\log \left(x + y\right) + \frac{-1}{2} \cdot \log t\right)\right) - t \]

      5. lower-log.f64 N/A

         \[\leadsto \left(\log z + \left(\log \left(x + y\right) + \frac{-1}{2} \cdot \log t\right)\right) - t \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\log z + \left(\log \left(x + y\right) + \frac{-1}{2} \cdot \log t\right)\right) - t \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\log z + \left(\log \left(x + y\right) + \frac{-1}{2} \cdot \log t\right)\right) - t \]

      8. lower-log.f64 62.5%

         \[\leadsto \left(\log z + \left(\log \left(x + y\right) + -0.5 \cdot \log t\right)\right) - t \]

   4. Applied rewrites 62.5%

      \[\leadsto \color{blue}{\left(\log z + \left(\log \left(x + y\right) + -0.5 \cdot \log t\right)\right) - t} \]
   5. Step-by-step derivation

   6. Applied rewrites 43.4%

      \[\leadsto \log \left(\frac{\left(x + y\right) \cdot z}{\sqrt{t}}\right) - \color{blue}{t} \]

   if 712 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t)))

   1. Initial program 99.6%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \log t} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto a \cdot \color{blue}{\log t} \]

      2. lower-log.f64 38.5%

         \[\leadsto a \cdot \log t \]

   4. Applied rewrites 38.5%

      \[\leadsto \color{blue}{a \cdot \log t} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 62.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;t \leq 67000000000:\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 67000000000.0) (* a (log t)) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 67000000000.0) {
		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 <= 67000000000.0d0) 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 <= 67000000000.0) {
		tmp = a * Math.log(t);
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 67000000000.0:
		tmp = a * math.log(t)
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 67000000000.0)
		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 <= 67000000000.0)
		tmp = a * log(t);
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 67000000000.0], N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if t < 6.7e10

   1. Initial program 99.6%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \log t} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto a \cdot \color{blue}{\log t} \]

      2. lower-log.f64 38.5%

         \[\leadsto a \cdot \log t \]

   4. Applied rewrites 38.5%

      \[\leadsto \color{blue}{a \cdot \log t} \]

   if 6.7e10 < t

   1. Initial program 99.6%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]

   2. Taylor expanded in t around inf

      \[\leadsto \color{blue}{-1 \cdot t} \]
   3. Step-by-step derivation

      1. lower-*.f64 38.3%

         \[\leadsto -1 \cdot \color{blue}{t} \]

   4. Applied rewrites 38.3%

      \[\leadsto \color{blue}{-1 \cdot t} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{t} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(t\right) \]

      3. lower-neg.f64 38.3%

         \[\leadsto -t \]

   6. Applied rewrites 38.3%

      \[\leadsto \color{blue}{-t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 38.3% accurate, 17.6× speedup

Math

\[-t \]

FPCore

(FPCore (x y z t a) :precision binary64 (- t))

C

double code(double x, double y, double z, double t, double a) {
	return -t;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = -t
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return -t;
}

Python

def code(x, y, z, t, a):
	return -t

Julia

function code(x, y, z, t, a)
	return Float64(-t)
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = -t;
end

Wolfram

code[x_, y_, z_, t_, a_] := (-t)

Derivation

1. Initial program 99.6%

   \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]

2. Taylor expanded in t around inf

   \[\leadsto \color{blue}{-1 \cdot t} \]
3. Step-by-step derivation

   1. lower-*.f64 38.3%

      \[\leadsto -1 \cdot \color{blue}{t} \]

4. Applied rewrites 38.3%

   \[\leadsto \color{blue}{-1 \cdot t} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto -1 \cdot \color{blue}{t} \]

   2. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(t\right) \]

   3. lower-neg.f64 38.3%

      \[\leadsto -t \]

6. Applied rewrites 38.3%

   \[\leadsto \color{blue}{-t} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025183 
(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))))