Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 11.8s

Alternatives: 16

Speedup: 0.8×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, y, z, t, a)
	return Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + Float64(fma(Float64(log(t) / a), -0.5, log(t)) * a))
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[(N[(N[Log[t], $MachinePrecision] / a), $MachinePrecision] * -0.5 + N[Log[t], $MachinePrecision]), $MachinePrecision] * a), $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 a around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lift-log.f64 N/A

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

   8. lift-log.f64 99.2

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

4. Applied rewrites 99.2%

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

Alternative 2: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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. Add Preprocessing

Alternative 3: 94.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[(N[Log[y], $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

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

4. Applied rewrites 68.8%

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

Alternative 4: 84.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   3. lower-+.f64 N/A

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

   4. +-commutative N/A

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

   5. sum-log N/A

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

   6. lower-log.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lift-log.f64 N/A

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

   10. lift--.f64 53.1

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

4. Applied rewrites 53.1%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-log.f64 N/A

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

   5. log-prod N/A

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

   6. +-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. lift-log.f64 N/A

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

   9. lift--.f64 N/A

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

   10. associate-+r+ N/A

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

   11. associate--l+ N/A

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

   12. lower-+.f64 N/A

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

   13. lower-log.f64 N/A

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

   14. lower--.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-fma.f64 N/A

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

   17. lift-log.f64 N/A

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

   18. lift--.f64 N/A

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

   19. lift-log.f64 68.8

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

6. Applied rewrites 68.8%

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

Alternative 5: 84.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.45:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, -t\right)\\ \mathbf{elif}\;a \leq 1.7:\\ \;\;\;\;\log y + \left(\mathsf{fma}\left(\log t, -0.5, \log z\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log t \cdot a + \log z\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -1.45)
   (fma (- a 0.5) (log t) (- t))
   (if (<= a 1.7)
     (+ (log y) (- (fma (log t) -0.5 (log z)) t))
     (- (+ (* (log t) a) (log z)) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -1.45) {
		tmp = fma((a - 0.5), log(t), -t);
	} else if (a <= 1.7) {
		tmp = log(y) + (fma(log(t), -0.5, log(z)) - t);
	} else {
		tmp = ((log(t) * a) + log(z)) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -1.45)
		tmp = fma(Float64(a - 0.5), log(t), Float64(-t));
	elseif (a <= 1.7)
		tmp = Float64(log(y) + Float64(fma(log(t), -0.5, log(z)) - t));
	else
		tmp = Float64(Float64(Float64(log(t) * a) + log(z)) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -1.45], N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + (-t)), $MachinePrecision], If[LessEqual[a, 1.7], N[(N[Log[y], $MachinePrecision] + N[(N[(N[Log[t], $MachinePrecision] * -0.5 + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.44999999999999996

   1. Initial program 99.7%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \frac{a \cdot a - \color{blue}{\frac{1}{4}}}{a + \frac{1}{2}} \cdot \log t \]

      7. lower-+.f64 51.8

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \frac{a \cdot a - 0.25}{\color{blue}{a + 0.5}} \cdot \log t \]

   3. Applied rewrites 51.8%

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \color{blue}{\frac{a \cdot a - 0.25}{a + 0.5}} \cdot \log t \]

   4. Taylor expanded in t around inf

      \[\leadsto \color{blue}{-1 \cdot t} + \frac{a \cdot a - \frac{1}{4}}{a + \frac{1}{2}} \cdot \log t \]
   5. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(t\right)\right) + \frac{a \cdot a - \frac{1}{4}}{a + \frac{1}{2}} \cdot \log t \]

      2. lift-neg.f64 51.0

         \[\leadsto \left(-t\right) + \frac{a \cdot a - 0.25}{a + 0.5} \cdot \log t \]

   6. Applied rewrites 51.0%

      \[\leadsto \color{blue}{\left(-t\right)} + \frac{a \cdot a - 0.25}{a + 0.5} \cdot \log t \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-t\right) + \frac{a \cdot a - \frac{1}{4}}{a + \frac{1}{2}} \cdot \log t} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(-t\right) + \color{blue}{\frac{a \cdot a - \frac{1}{4}}{a + \frac{1}{2}} \cdot \log t} \]

      3. lift-+.f64 N/A

         \[\leadsto \left(-t\right) + \frac{a \cdot a - \frac{1}{4}}{\color{blue}{a + \frac{1}{2}}} \cdot \log t \]

      4. lift-/.f64 N/A

         \[\leadsto \left(-t\right) + \color{blue}{\frac{a \cdot a - \frac{1}{4}}{a + \frac{1}{2}}} \cdot \log t \]

      5. lift-*.f64 N/A

         \[\leadsto \left(-t\right) + \frac{\color{blue}{a \cdot a} - \frac{1}{4}}{a + \frac{1}{2}} \cdot \log t \]

      6. lift--.f64 N/A

         \[\leadsto \left(-t\right) + \frac{\color{blue}{a \cdot a - \frac{1}{4}}}{a + \frac{1}{2}} \cdot \log t \]

      7. lift-log.f64 N/A

         \[\leadsto \left(-t\right) + \frac{a \cdot a - \frac{1}{4}}{a + \frac{1}{2}} \cdot \color{blue}{\log t} \]

      8. +-commutative N/A

         \[\leadsto \color{blue}{\frac{a \cdot a - \frac{1}{4}}{a + \frac{1}{2}} \cdot \log t + \left(-t\right)} \]

      9. pow2 N/A

         \[\leadsto \frac{\color{blue}{{a}^{2}} - \frac{1}{4}}{a + \frac{1}{2}} \cdot \log t + \left(-t\right) \]

      10. pow2 N/A

          \[\leadsto \frac{\color{blue}{a \cdot a} - \frac{1}{4}}{a + \frac{1}{2}} \cdot \log t + \left(-t\right) \]

      11. metadata-eval N/A

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

      12. flip-- N/A

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

   8. Applied rewrites 98.8%

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

   if -1.44999999999999996 < a < 1.69999999999999996

   1. Initial program 99.5%

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

   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. associate-+r+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. sum-log N/A

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

      6. lower-log.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. lift--.f64 48.2

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

   4. Applied rewrites 48.2%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. log-prod N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lift--.f64 N/A

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

      10. associate-+r+ N/A

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

      11. associate--l+ N/A

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

      12. lower-+.f64 N/A

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

      13. lower-log.f64 N/A

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

      14. lower--.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lift-log.f64 N/A

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

      18. lift--.f64 N/A

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

      19. lift-log.f64 63.4

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

   6. Applied rewrites 63.4%

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

   7. Taylor expanded in a around 0

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

      1. flip-- 63.0

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

      2. metadata-eval 63.0

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

   9. Applied rewrites 63.0%

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

   if 1.69999999999999996 < a

   1. Initial program 99.7%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift-log.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in y around inf

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

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

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

      2. flip-- N/A

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

      3. metadata-eval N/A

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

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

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

      5. lower--.f64 N/A

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

   7. Applied rewrites 72.6%

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

   8. Taylor expanded in a around inf

      \[\leadsto \left(a \cdot \log t + \log z\right) - t \]
   9. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 98.6

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

   10. Applied rewrites 98.6%

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

Alternative 6: 81.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 650:\\ \;\;\;\;\log y + \mathsf{fma}\left(\log t, a - 0.5, \log z\right)\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot a - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 650.0)
   (+ (log y) (fma (log t) (- a 0.5) (log z)))
   (- (* (log t) a) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 650.0) {
		tmp = log(y) + fma(log(t), (a - 0.5), log(z));
	} else {
		tmp = (log(t) * a) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 650.0)
		tmp = Float64(log(y) + fma(log(t), Float64(a - 0.5), log(z)));
	else
		tmp = Float64(Float64(log(t) * a) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 650.0], N[(N[Log[y], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 650

   1. Initial program 99.3%

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

   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. associate-+r+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. sum-log N/A

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

      6. lower-log.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. lift--.f64 48.5

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

   4. Applied rewrites 48.5%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. log-prod N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lift--.f64 N/A

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

      10. associate-+r+ N/A

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

      11. associate--l+ N/A

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

      12. lower-+.f64 N/A

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

      13. lower-log.f64 N/A

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

      14. lower--.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lift-log.f64 N/A

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

      18. lift--.f64 N/A

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

      19. lift-log.f64 63.5

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

   6. Applied rewrites 63.5%

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

   7. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. lift-log.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift--.f64 63.1

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

   9. Applied rewrites 63.1%

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

   if 650 < t

   1. Initial program 99.9%

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

   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. associate-+r+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. sum-log N/A

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

      6. lower-log.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. lift--.f64 57.7

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

   4. Applied rewrites 57.7%

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

   5. Taylor expanded in a around inf

      \[\leadsto a \cdot \log t - t \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 98.8

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

   7. Applied rewrites 98.8%

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

Alternative 7: 80.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(x + y\right) + \log z\\ t_2 := \left(\log t \cdot a + \log z\right) - t\\ \mathbf{if}\;t\_1 \leq -750:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 700:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log \left(z \cdot \left(y + x\right)\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log (+ x y)) (log z))) (t_2 (- (+ (* (log t) a) (log z)) t)))
   (if (<= t_1 -750.0)
     t_2
     (if (<= t_1 700.0)
       (fma (- a 0.5) (log t) (- (log (* z (+ y x))) t))
       t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((x + y)) + log(z);
	double t_2 = ((log(t) * a) + log(z)) - t;
	double tmp;
	if (t_1 <= -750.0) {
		tmp = t_2;
	} else if (t_1 <= 700.0) {
		tmp = fma((a - 0.5), log(t), (log((z * (y + x))) - t));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(Float64(x + y)) + log(z))
	t_2 = Float64(Float64(Float64(log(t) * a) + log(z)) - t)
	tmp = 0.0
	if (t_1 <= -750.0)
		tmp = t_2;
	elseif (t_1 <= 700.0)
		tmp = fma(Float64(a - 0.5), log(t), Float64(log(Float64(z * Float64(y + x))) - t));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[t$95$1, -750.0], t$95$2, If[LessEqual[t$95$1, 700.0], N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[(N[Log[N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -750 or 700 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z))

   1. Initial program 99.7%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift-log.f64 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in y around inf

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

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

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

      2. flip-- N/A

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

      3. metadata-eval N/A

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

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

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

      5. lower--.f64 N/A

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

   7. Applied rewrites 69.9%

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

   8. Taylor expanded in a around inf

      \[\leadsto \left(a \cdot \log t + \log z\right) - t \]
   9. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 78.2

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

   10. Applied rewrites 78.2%

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

   if -750 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 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. 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. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lift-log.f64 N/A

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

      14. lower--.f64 N/A

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

      15. +-commutative N/A

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

      16. sum-log N/A

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

      17. lower-log.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.6

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

   3. Applied rewrites 99.6%

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

Alternative 8: 77.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(x + y\right) + \log z\\ t_2 := \left(\log t \cdot a + \log z\right) - t\\ \mathbf{if}\;t\_1 \leq -750:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 700:\\ \;\;\;\;\mathsf{fma}\left(\log t, a - 0.5, \log \left(z \cdot y\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log (+ x y)) (log z))) (t_2 (- (+ (* (log t) a) (log z)) t)))
   (if (<= t_1 -750.0)
     t_2
     (if (<= t_1 700.0) (- (fma (log t) (- a 0.5) (log (* z y))) t) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((x + y)) + log(z);
	double t_2 = ((log(t) * a) + log(z)) - t;
	double tmp;
	if (t_1 <= -750.0) {
		tmp = t_2;
	} else if (t_1 <= 700.0) {
		tmp = fma(log(t), (a - 0.5), log((z * y))) - t;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(Float64(x + y)) + log(z))
	t_2 = Float64(Float64(Float64(log(t) * a) + log(z)) - t)
	tmp = 0.0
	if (t_1 <= -750.0)
		tmp = t_2;
	elseif (t_1 <= 700.0)
		tmp = Float64(fma(log(t), Float64(a - 0.5), log(Float64(z * y))) - t);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[t$95$1, -750.0], t$95$2, If[LessEqual[t$95$1, 700.0], N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision] + N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -750 or 700 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z))

   1. Initial program 99.7%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift-log.f64 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in y around inf

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

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

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

      2. flip-- N/A

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

      3. metadata-eval N/A

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

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

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

      5. lower--.f64 N/A

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

   7. Applied rewrites 69.9%

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

   8. Taylor expanded in a around inf

      \[\leadsto \left(a \cdot \log t + \log z\right) - t \]
   9. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 78.2

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

   10. Applied rewrites 78.2%

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

   if -750 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 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 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. associate-+r+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. sum-log N/A

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

      6. lower-log.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. lift--.f64 64.8

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

   4. Applied rewrites 64.8%

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

      1. fp-cancel-sign-sub-inv 64.8

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

      2. flip-- 64.8

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

      3. metadata-eval 64.8

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

      4. fp-cancel-sign-sub-inv 64.8

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

   6. Applied rewrites 64.8%

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

Alternative 9: 72.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ t_2 := \left(\log t \cdot a + \log z\right) - t\\ \mathbf{if}\;t\_1 \leq -50000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 1050:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log t, \log \left(z \cdot y\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))
        (t_2 (- (+ (* (log t) a) (log z)) t)))
   (if (<= t_1 -50000000000.0)
     t_2
     (if (<= t_1 1050.0) (- (fma -0.5 (log t) (log (* z y))) t) t_2))))

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 t_2 = ((log(t) * a) + log(z)) - t;
	double tmp;
	if (t_1 <= -50000000000.0) {
		tmp = t_2;
	} else if (t_1 <= 1050.0) {
		tmp = fma(-0.5, log(t), log((z * y))) - t;
	} else {
		tmp = t_2;
	}
	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)))
	t_2 = Float64(Float64(Float64(log(t) * a) + log(z)) - t)
	tmp = 0.0
	if (t_1 <= -50000000000.0)
		tmp = t_2;
	elseif (t_1 <= 1050.0)
		tmp = Float64(fma(-0.5, log(t), log(Float64(z * y))) - t);
	else
		tmp = t_2;
	end
	return 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]}, Block[{t$95$2 = N[(N[(N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[t$95$1, -50000000000.0], t$95$2, If[LessEqual[t$95$1, 1050.0], N[(N[(-0.5 * N[Log[t], $MachinePrecision] + N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < -5e10 or 1050 < (+.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 99.8%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift-log.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in y around inf

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

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

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

      2. flip-- N/A

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

      3. metadata-eval N/A

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

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

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

      5. lower--.f64 N/A

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

   7. Applied rewrites 73.3%

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

   8. Taylor expanded in a around inf

      \[\leadsto \left(a \cdot \log t + \log z\right) - t \]
   9. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 95.6

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

   10. Applied rewrites 95.6%

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

   if -5e10 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 1050

   1. Initial program 99.0%

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

   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. associate-+r+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. sum-log N/A

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

      6. lower-log.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. lift--.f64 45.7

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

   4. Applied rewrites 45.7%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-log.f64 N/A

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

      6. lift-*.f64 44.3

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

   7. Applied rewrites 44.3%

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

Alternative 10: 70.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ t_2 := \left(\log t \cdot a + \log z\right) - t\\ \mathbf{if}\;t\_1 \leq -700:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 1050:\\ \;\;\;\;\mathsf{fma}\left(\log t, a - 0.5, \log \left(z \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))
        (t_2 (- (+ (* (log t) a) (log z)) t)))
   (if (<= t_1 -700.0)
     t_2
     (if (<= t_1 1050.0) (fma (log t) (- a 0.5) (log (* z y))) t_2))))

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 t_2 = ((log(t) * a) + log(z)) - t;
	double tmp;
	if (t_1 <= -700.0) {
		tmp = t_2;
	} else if (t_1 <= 1050.0) {
		tmp = fma(log(t), (a - 0.5), log((z * y)));
	} else {
		tmp = t_2;
	}
	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)))
	t_2 = Float64(Float64(Float64(log(t) * a) + log(z)) - t)
	tmp = 0.0
	if (t_1 <= -700.0)
		tmp = t_2;
	elseif (t_1 <= 1050.0)
		tmp = fma(log(t), Float64(a - 0.5), log(Float64(z * y)));
	else
		tmp = t_2;
	end
	return 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]}, Block[{t$95$2 = N[(N[(N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[t$95$1, -700.0], t$95$2, If[LessEqual[t$95$1, 1050.0], N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision] + N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < -700 or 1050 < (+.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 99.8%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift-log.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in y around inf

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

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

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

      2. flip-- N/A

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

      3. metadata-eval N/A

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

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

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

      5. lower--.f64 N/A

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

   7. Applied rewrites 73.0%

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

   8. Taylor expanded in a around inf

      \[\leadsto \left(a \cdot \log t + \log z\right) - t \]
   9. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 94.5

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

   10. Applied rewrites 94.5%

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

   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))) < 1050

   1. Initial program 99.0%

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

   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. associate-+r+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. sum-log N/A

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

      6. lower-log.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. lift--.f64 46.4

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

   4. Applied rewrites 46.4%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift--.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-log.f64 N/A

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

      7. lift-*.f64 45.5

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

   7. Applied rewrites 45.5%

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

Alternative 11: 68.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log t \cdot a\\ \mathbf{if}\;\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \leq -4:\\ \;\;\;\;\left(t\_1 + \log z\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log y + t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (log t) a)))
   (if (<= (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))) -4.0)
     (- (+ t_1 (log z)) t)
     (+ (log y) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log(t) * a;
	double tmp;
	if ((((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t))) <= -4.0) {
		tmp = (t_1 + log(z)) - t;
	} else {
		tmp = log(y) + 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 = log(t) * a
    if ((((log((x + y)) + log(z)) - t) + ((a - 0.5d0) * log(t))) <= (-4.0d0)) then
        tmp = (t_1 + log(z)) - t
    else
        tmp = log(y) + 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 = Math.log(t) * a;
	double tmp;
	if ((((Math.log((x + y)) + Math.log(z)) - t) + ((a - 0.5) * Math.log(t))) <= -4.0) {
		tmp = (t_1 + Math.log(z)) - t;
	} else {
		tmp = Math.log(y) + t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = math.log(t) * a
	tmp = 0
	if (((math.log((x + y)) + math.log(z)) - t) + ((a - 0.5) * math.log(t))) <= -4.0:
		tmp = (t_1 + math.log(z)) - t
	else:
		tmp = math.log(y) + t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(t) * a)
	tmp = 0.0
	if (Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + Float64(Float64(a - 0.5) * log(t))) <= -4.0)
		tmp = Float64(Float64(t_1 + log(z)) - t);
	else
		tmp = Float64(log(y) + t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = log(t) * a;
	tmp = 0.0;
	if ((((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t))) <= -4.0)
		tmp = (t_1 + log(z)) - t;
	else
		tmp = log(y) + t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[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], -4.0], N[(N[(t$95$1 + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[Log[y], $MachinePrecision] + t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.7%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift-log.f64 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in y around inf

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

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

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

      2. flip-- N/A

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

      3. metadata-eval N/A

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

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

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

      5. lower--.f64 N/A

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

   7. Applied rewrites 71.7%

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

   8. Taylor expanded in a around inf

      \[\leadsto \left(a \cdot \log t + \log z\right) - t \]
   9. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 91.1

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

   10. Applied rewrites 91.1%

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

   if -4 < (+.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 99.4%

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

   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. associate-+r+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. sum-log N/A

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

      6. lower-log.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. lift--.f64 48.7

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

   4. Applied rewrites 48.7%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. log-prod N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lift--.f64 N/A

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

      10. associate-+r+ N/A

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

      11. associate--l+ N/A

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

      12. lower-+.f64 N/A

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

      13. lower-log.f64 N/A

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

      14. lower--.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lift-log.f64 N/A

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

      18. lift--.f64 N/A

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

      19. lift-log.f64 64.0

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

   6. Applied rewrites 64.0%

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

   7. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lift-log.f64 N/A

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

      3. lift-*.f64 41.8

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

   9. Applied rewrites 41.8%

      \[\leadsto \log y + \log t \cdot \color{blue}{a} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 68.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log t \cdot a - t\\ \mathbf{if}\;a \leq -1.6:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 0.52:\\ \;\;\;\;\log y + \left(-t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* (log t) a) t)))
   (if (<= a -1.6) t_1 (if (<= a 0.52) (+ (log y) (- t)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (log(t) * a) - t;
	double tmp;
	if (a <= -1.6) {
		tmp = t_1;
	} else if (a <= 0.52) {
		tmp = log(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 = (log(t) * a) - t
    if (a <= (-1.6d0)) then
        tmp = t_1
    else if (a <= 0.52d0) then
        tmp = log(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 = (Math.log(t) * a) - t;
	double tmp;
	if (a <= -1.6) {
		tmp = t_1;
	} else if (a <= 0.52) {
		tmp = Math.log(y) + -t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (math.log(t) * a) - t
	tmp = 0
	if a <= -1.6:
		tmp = t_1
	elif a <= 0.52:
		tmp = math.log(y) + -t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(log(t) * a) - t)
	tmp = 0.0
	if (a <= -1.6)
		tmp = t_1;
	elseif (a <= 0.52)
		tmp = Float64(log(y) + Float64(-t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (log(t) * a) - t;
	tmp = 0.0;
	if (a <= -1.6)
		tmp = t_1;
	elseif (a <= 0.52)
		tmp = log(y) + -t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[a, -1.6], t$95$1, If[LessEqual[a, 0.52], N[(N[Log[y], $MachinePrecision] + (-t)), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.6000000000000001 or 0.52000000000000002 < a

   1. Initial program 99.7%

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

   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. associate-+r+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. sum-log N/A

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

      6. lower-log.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. lift--.f64 57.7

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

   4. Applied rewrites 57.7%

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

   5. Taylor expanded in a around inf

      \[\leadsto a \cdot \log t - t \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 98.6

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

   7. Applied rewrites 98.6%

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

   if -1.6000000000000001 < a < 0.52000000000000002

   1. Initial program 99.5%

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

   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. associate-+r+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. sum-log N/A

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

      6. lower-log.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. lift--.f64 48.2

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

   4. Applied rewrites 48.2%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. log-prod N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lift--.f64 N/A

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

      10. associate-+r+ N/A

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

      11. associate--l+ N/A

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

      12. lower-+.f64 N/A

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

      13. lower-log.f64 N/A

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

      14. lower--.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lift-log.f64 N/A

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

      18. lift--.f64 N/A

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

      19. lift-log.f64 63.4

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

   6. Applied rewrites 63.4%

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

   7. Taylor expanded in t around inf

      \[\leadsto \log y + -1 \cdot \color{blue}{t} \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lift-neg.f64 41.3

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

   9. Applied rewrites 41.3%

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

Alternative 13: 68.1% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $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. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. lower-/.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. metadata-eval N/A

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \frac{a \cdot a - \color{blue}{\frac{1}{4}}}{a + \frac{1}{2}} \cdot \log t \]

   7. lower-+.f64 75.9

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \frac{a \cdot a - 0.25}{\color{blue}{a + 0.5}} \cdot \log t \]

3. Applied rewrites 75.9%

   \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \color{blue}{\frac{a \cdot a - 0.25}{a + 0.5}} \cdot \log t \]

4. Taylor expanded in t around inf

   \[\leadsto \color{blue}{-1 \cdot t} + \frac{a \cdot a - \frac{1}{4}}{a + \frac{1}{2}} \cdot \log t \]
5. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \left(\mathsf{neg}\left(t\right)\right) + \frac{a \cdot a - \frac{1}{4}}{a + \frac{1}{2}} \cdot \log t \]

   2. lift-neg.f64 54.0

      \[\leadsto \left(-t\right) + \frac{a \cdot a - 0.25}{a + 0.5} \cdot \log t \]

6. Applied rewrites 54.0%

   \[\leadsto \color{blue}{\left(-t\right)} + \frac{a \cdot a - 0.25}{a + 0.5} \cdot \log t \]
7. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(-t\right) + \frac{a \cdot a - \frac{1}{4}}{a + \frac{1}{2}} \cdot \log t} \]

   2. lift-*.f64 N/A

      \[\leadsto \left(-t\right) + \color{blue}{\frac{a \cdot a - \frac{1}{4}}{a + \frac{1}{2}} \cdot \log t} \]

   3. lift-+.f64 N/A

      \[\leadsto \left(-t\right) + \frac{a \cdot a - \frac{1}{4}}{\color{blue}{a + \frac{1}{2}}} \cdot \log t \]

   4. lift-/.f64 N/A

      \[\leadsto \left(-t\right) + \color{blue}{\frac{a \cdot a - \frac{1}{4}}{a + \frac{1}{2}}} \cdot \log t \]

   5. lift-*.f64 N/A

      \[\leadsto \left(-t\right) + \frac{\color{blue}{a \cdot a} - \frac{1}{4}}{a + \frac{1}{2}} \cdot \log t \]

   6. lift--.f64 N/A

      \[\leadsto \left(-t\right) + \frac{\color{blue}{a \cdot a - \frac{1}{4}}}{a + \frac{1}{2}} \cdot \log t \]

   7. lift-log.f64 N/A

      \[\leadsto \left(-t\right) + \frac{a \cdot a - \frac{1}{4}}{a + \frac{1}{2}} \cdot \color{blue}{\log t} \]

   8. +-commutative N/A

      \[\leadsto \color{blue}{\frac{a \cdot a - \frac{1}{4}}{a + \frac{1}{2}} \cdot \log t + \left(-t\right)} \]

   9. pow2 N/A

      \[\leadsto \frac{\color{blue}{{a}^{2}} - \frac{1}{4}}{a + \frac{1}{2}} \cdot \log t + \left(-t\right) \]

   10. pow2 N/A

       \[\leadsto \frac{\color{blue}{a \cdot a} - \frac{1}{4}}{a + \frac{1}{2}} \cdot \log t + \left(-t\right) \]

   11. metadata-eval N/A

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

   12. flip-- N/A

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

8. Applied rewrites 77.7%

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

Alternative 14: 63.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.5 \cdot 10^{+14}:\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;\log y + \left(-t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 1.5e+14) (* (log t) a) (+ (log y) (- t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 1.5e+14) {
		tmp = log(t) * a;
	} else {
		tmp = log(y) + -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 <= 1.5d+14) then
        tmp = log(t) * a
    else
        tmp = log(y) + -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 <= 1.5e+14) {
		tmp = Math.log(t) * a;
	} else {
		tmp = Math.log(y) + -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 1.5e+14:
		tmp = math.log(t) * a
	else:
		tmp = math.log(y) + -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 1.5e+14)
		tmp = Float64(log(t) * a);
	else
		tmp = Float64(log(y) + Float64(-t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 1.5e+14)
		tmp = log(t) * a;
	else
		tmp = log(y) + -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 1.5e+14], N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision], N[(N[Log[y], $MachinePrecision] + (-t)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.5e14

   1. Initial program 99.4%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 51.9

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

   4. Applied rewrites 51.9%

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

   if 1.5e14 < t

   1. Initial program 99.9%

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

   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. associate-+r+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. sum-log N/A

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

      6. lower-log.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. lift--.f64 58.1

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

   4. Applied rewrites 58.1%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. log-prod N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lift--.f64 N/A

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

      10. associate-+r+ N/A

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

      11. associate--l+ N/A

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

      12. lower-+.f64 N/A

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

      13. lower-log.f64 N/A

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

      14. lower--.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lift-log.f64 N/A

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

      18. lift--.f64 N/A

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

      19. lift-log.f64 74.6

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

   6. Applied rewrites 74.6%

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

   7. Taylor expanded in t around inf

      \[\leadsto \log y + -1 \cdot \color{blue}{t} \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lift-neg.f64 55.8

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

   9. Applied rewrites 55.8%

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

Alternative 15: 53.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.5 \cdot 10^{+14}:\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 1.5e+14) (* (log t) a) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 1.5e+14) {
		tmp = log(t) * a;
	} 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 <= 1.5d+14) then
        tmp = log(t) * a
    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 <= 1.5e+14) {
		tmp = Math.log(t) * a;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 1.5e+14:
		tmp = math.log(t) * a
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 1.5e+14)
		tmp = Float64(log(t) * a);
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 1.5e+14)
		tmp = log(t) * a;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 1.5e+14], N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if t < 1.5e14

   1. Initial program 99.4%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 51.9

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

   4. Applied rewrites 51.9%

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

   if 1.5e14 < t

   1. Initial program 99.9%

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

   2. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 75.0

         \[\leadsto -t \]

   4. Applied rewrites 75.0%

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

Alternative 16: 37.7% accurate, 17.6× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 37.7

      \[\leadsto -t \]

4. Applied rewrites 37.7%

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

Reproduce

herbie shell --seed 2025119 
(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))))