Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 9.4s

Alternatives: 14

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

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

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 2: 88.6% 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 69.2%

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

Alternative 3: 87.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\log y + a \cdot \log t\right) - t\\ \mathbf{if}\;a \leq -0.64:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 1.8:\\ \;\;\;\;\left(\left(\log y + \log z\right) - t\right) + -0.5 \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (+ (log y) (* a (log t))) t)))
   (if (<= a -0.64)
     t_1
     (if (<= a 1.8) (+ (- (+ (log y) (log z)) t) (* -0.5 (log t))) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (log(y) + (a * log(t))) - t;
	double tmp;
	if (a <= -0.64) {
		tmp = t_1;
	} else if (a <= 1.8) {
		tmp = ((log(y) + log(z)) - t) + (-0.5 * log(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(y) + (a * log(t))) - t
    if (a <= (-0.64d0)) then
        tmp = t_1
    else if (a <= 1.8d0) then
        tmp = ((log(y) + log(z)) - t) + ((-0.5d0) * log(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(y) + (a * Math.log(t))) - t;
	double tmp;
	if (a <= -0.64) {
		tmp = t_1;
	} else if (a <= 1.8) {
		tmp = ((Math.log(y) + Math.log(z)) - t) + (-0.5 * Math.log(t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (math.log(y) + (a * math.log(t))) - t
	tmp = 0
	if a <= -0.64:
		tmp = t_1
	elif a <= 1.8:
		tmp = ((math.log(y) + math.log(z)) - t) + (-0.5 * math.log(t))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(log(y) + Float64(a * log(t))) - t)
	tmp = 0.0
	if (a <= -0.64)
		tmp = t_1;
	elseif (a <= 1.8)
		tmp = Float64(Float64(Float64(log(y) + log(z)) - t) + Float64(-0.5 * log(t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (log(y) + (a * log(t))) - t;
	tmp = 0.0;
	if (a <= -0.64)
		tmp = t_1;
	elseif (a <= 1.8)
		tmp = ((log(y) + log(z)) - t) + (-0.5 * log(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[y], $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[a, -0.64], t$95$1, If[LessEqual[a, 1.8], N[(N[(N[(N[Log[y], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + N[(-0.5 * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -0.640000000000000013 or 1.80000000000000004 < 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 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 25.9

         \[\leadsto -t \]

   4. Applied rewrites 25.9%

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

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 74.7

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

   7. Applied rewrites 74.7%

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

   8. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lift-log.f64 74.2

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

   10. Applied rewrites 74.2%

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

   if -0.640000000000000013 < a < 1.80000000000000004

   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

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

   4. Applied rewrites 63.6%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 63.2%

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

Alternative 4: 79.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\log y + a \cdot \log t\right) - t\\ \mathbf{if}\;a \leq -0.64:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 1.8:\\ \;\;\;\;\left(\log y + \mathsf{fma}\left(-0.5, \log t, \log z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (+ (log y) (* a (log t))) t)))
   (if (<= a -0.64)
     t_1
     (if (<= a 1.8) (- (+ (log y) (fma -0.5 (log t) (log z))) t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (log(y) + (a * log(t))) - t;
	double tmp;
	if (a <= -0.64) {
		tmp = t_1;
	} else if (a <= 1.8) {
		tmp = (log(y) + fma(-0.5, log(t), log(z))) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(log(y) + Float64(a * log(t))) - t)
	tmp = 0.0
	if (a <= -0.64)
		tmp = t_1;
	elseif (a <= 1.8)
		tmp = Float64(Float64(log(y) + fma(-0.5, log(t), log(z))) - t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[Log[y], $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[a, -0.64], t$95$1, If[LessEqual[a, 1.8], N[(N[(N[Log[y], $MachinePrecision] + N[(-0.5 * N[Log[t], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -0.640000000000000013 or 1.80000000000000004 < 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 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 25.9

         \[\leadsto -t \]

   4. Applied rewrites 25.9%

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

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 74.7

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

   7. Applied rewrites 74.7%

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

   8. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lift-log.f64 74.2

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

   10. Applied rewrites 74.2%

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

   if -0.640000000000000013 < a < 1.80000000000000004

   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 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 50.8

         \[\leadsto -t \]

   4. Applied rewrites 50.8%

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

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 63.6

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

   7. Applied rewrites 63.6%

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

   8. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-log.f64 63.2

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

   10. Applied rewrites 63.2%

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

Alternative 5: 78.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(x + y\right) + \log z\\ t_2 := \left(\log y + a \cdot \log t\right) - t\\ \mathbf{if}\;t\_1 \leq -750:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 670:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log \left(z \cdot \left(y + x\right)\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log (+ x y)) (log z))) (t_2 (- (+ (log y) (* a (log t))) t)))
   (if (<= t_1 -750.0)
     t_2
     (if (<= t_1 670.0)
       (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(y) + (a * log(t))) - t;
	double tmp;
	if (t_1 <= -750.0) {
		tmp = t_2;
	} else if (t_1 <= 670.0) {
		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(log(y) + Float64(a * log(t))) - t)
	tmp = 0.0
	if (t_1 <= -750.0)
		tmp = t_2;
	elseif (t_1 <= 670.0)
		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[Log[y], $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[t$95$1, -750.0], t$95$2, If[LessEqual[t$95$1, 670.0], 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 670 < (+.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 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 38.8

         \[\leadsto -t \]

   4. Applied rewrites 38.8%

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

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 69.7

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

   7. Applied rewrites 69.7%

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

   8. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lift-log.f64 58.9

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

   10. Applied rewrites 58.9%

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

   if -750 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 670

   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. 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 6: 77.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(x + y\right) + \log z\\ t_2 := \left(\log y + a \cdot \log t\right) - t\\ \mathbf{if}\;t\_1 \leq -750:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 670:\\ \;\;\;\;\left(\log \left(z \cdot y\right) + \log t \cdot \left(a - 0.5\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log (+ x y)) (log z))) (t_2 (- (+ (log y) (* a (log t))) t)))
   (if (<= t_1 -750.0)
     t_2
     (if (<= t_1 670.0) (- (+ (log (* z y)) (* (log t) (- a 0.5))) 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(y) + (a * log(t))) - t;
	double tmp;
	if (t_1 <= -750.0) {
		tmp = t_2;
	} else if (t_1 <= 670.0) {
		tmp = (log((z * y)) + (log(t) * (a - 0.5))) - t;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = log((x + y)) + log(z)
    t_2 = (log(y) + (a * log(t))) - t
    if (t_1 <= (-750.0d0)) then
        tmp = t_2
    else if (t_1 <= 670.0d0) then
        tmp = (log((z * y)) + (log(t) * (a - 0.5d0))) - t
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = Math.log((x + y)) + Math.log(z);
	double t_2 = (Math.log(y) + (a * Math.log(t))) - t;
	double tmp;
	if (t_1 <= -750.0) {
		tmp = t_2;
	} else if (t_1 <= 670.0) {
		tmp = (Math.log((z * y)) + (Math.log(t) * (a - 0.5))) - t;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = math.log((x + y)) + math.log(z)
	t_2 = (math.log(y) + (a * math.log(t))) - t
	tmp = 0
	if t_1 <= -750.0:
		tmp = t_2
	elif t_1 <= 670.0:
		tmp = (math.log((z * y)) + (math.log(t) * (a - 0.5))) - 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(log(y) + Float64(a * log(t))) - t)
	tmp = 0.0
	if (t_1 <= -750.0)
		tmp = t_2;
	elseif (t_1 <= 670.0)
		tmp = Float64(Float64(log(Float64(z * y)) + Float64(log(t) * Float64(a - 0.5))) - t);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = log((x + y)) + log(z);
	t_2 = (log(y) + (a * log(t))) - t;
	tmp = 0.0;
	if (t_1 <= -750.0)
		tmp = t_2;
	elseif (t_1 <= 670.0)
		tmp = (log((z * y)) + (log(t) * (a - 0.5))) - t;
	else
		tmp = t_2;
	end
	tmp_2 = 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[Log[y], $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[t$95$1, -750.0], t$95$2, If[LessEqual[t$95$1, 670.0], N[(N[(N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $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 670 < (+.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 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 38.8

         \[\leadsto -t \]

   4. Applied rewrites 38.8%

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

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 69.7

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

   7. Applied rewrites 69.7%

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

   8. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lift-log.f64 58.9

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

   10. Applied rewrites 58.9%

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

   if -750 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 670

   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 65.4

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

   4. Applied rewrites 65.4%

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

Alternative 7: 74.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ t_2 := a \cdot \log t\\ \mathbf{if}\;t\_1 \leq -20000000000:\\ \;\;\;\;t\_2 - t\\ \mathbf{elif}\;t\_1 \leq 890:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log t, \log \left(\left(y + x\right) \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log y + t\_2\right) - t\\ \end{array} \end{array} \]

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 (* a (log t))))
   (if (<= t_1 -20000000000.0)
     (- t_2 t)
     (if (<= t_1 890.0)
       (- (fma -0.5 (log t) (log (* (+ y x) z))) t)
       (- (+ (log y) t_2) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
	double t_2 = a * log(t);
	double tmp;
	if (t_1 <= -20000000000.0) {
		tmp = t_2 - t;
	} else if (t_1 <= 890.0) {
		tmp = fma(-0.5, log(t), log(((y + x) * z))) - t;
	} else {
		tmp = (log(y) + t_2) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + Float64(Float64(a - 0.5) * log(t)))
	t_2 = Float64(a * log(t))
	tmp = 0.0
	if (t_1 <= -20000000000.0)
		tmp = Float64(t_2 - t);
	elseif (t_1 <= 890.0)
		tmp = Float64(fma(-0.5, log(t), log(Float64(Float64(y + x) * z))) - t);
	else
		tmp = Float64(Float64(log(y) + t_2) - t);
	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[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -20000000000.0], N[(t$95$2 - t), $MachinePrecision], If[LessEqual[t$95$1, 890.0], N[(N[(-0.5 * N[Log[t], $MachinePrecision] + N[Log[N[(N[(y + x), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] + t$95$2), $MachinePrecision] - t), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   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 67.1

         \[\leadsto -t \]

   4. Applied rewrites 67.1%

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

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 75.3

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

   7. Applied rewrites 75.3%

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

   8. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lift-log.f64 99.5

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

   10. Applied rewrites 99.5%

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

   if -2e10 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 890

   1. Initial program 98.9%

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

   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 4.2

         \[\leadsto -t \]

   4. Applied rewrites 4.2%

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

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 51.5

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

   7. Applied rewrites 51.5%

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

   8. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. sum-log N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. sum-log N/A

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

      8. +-commutative N/A

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

      9. sum-log N/A

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

      10. lower-log.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 88.9

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

   10. Applied rewrites 88.9%

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

   if 890 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t)))

   1. Initial program 99.6%

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

   2. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 1.5

         \[\leadsto -t \]

   4. Applied rewrites 1.5%

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

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 70.1

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

   7. Applied rewrites 70.1%

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

   8. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lift-log.f64 59.9

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

   10. Applied rewrites 59.9%

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

Alternative 8: 69.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ t_2 := a \cdot \log t\\ \mathbf{if}\;t\_1 \leq -20000000000:\\ \;\;\;\;t\_2 - t\\ \mathbf{elif}\;t\_1 \leq 890:\\ \;\;\;\;\left(\log \left(y \cdot z\right) + -0.5 \cdot \log t\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log y + t\_2\right) - t\\ \end{array} \end{array} \]

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 (* a (log t))))
   (if (<= t_1 -20000000000.0)
     (- t_2 t)
     (if (<= t_1 890.0)
       (- (+ (log (* y z)) (* -0.5 (log t))) t)
       (- (+ (log y) t_2) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
	double t_2 = a * log(t);
	double tmp;
	if (t_1 <= -20000000000.0) {
		tmp = t_2 - t;
	} else if (t_1 <= 890.0) {
		tmp = (log((y * z)) + (-0.5 * log(t))) - t;
	} else {
		tmp = (log(y) + t_2) - t;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = ((log((x + y)) + log(z)) - t) + ((a - 0.5d0) * log(t))
    t_2 = a * log(t)
    if (t_1 <= (-20000000000.0d0)) then
        tmp = t_2 - t
    else if (t_1 <= 890.0d0) then
        tmp = (log((y * z)) + ((-0.5d0) * log(t))) - t
    else
        tmp = (log(y) + t_2) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = ((Math.log((x + y)) + Math.log(z)) - t) + ((a - 0.5) * Math.log(t));
	double t_2 = a * Math.log(t);
	double tmp;
	if (t_1 <= -20000000000.0) {
		tmp = t_2 - t;
	} else if (t_1 <= 890.0) {
		tmp = (Math.log((y * z)) + (-0.5 * Math.log(t))) - t;
	} else {
		tmp = (Math.log(y) + t_2) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = ((math.log((x + y)) + math.log(z)) - t) + ((a - 0.5) * math.log(t))
	t_2 = a * math.log(t)
	tmp = 0
	if t_1 <= -20000000000.0:
		tmp = t_2 - t
	elif t_1 <= 890.0:
		tmp = (math.log((y * z)) + (-0.5 * math.log(t))) - t
	else:
		tmp = (math.log(y) + t_2) - t
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + Float64(Float64(a - 0.5) * log(t)))
	t_2 = Float64(a * log(t))
	tmp = 0.0
	if (t_1 <= -20000000000.0)
		tmp = Float64(t_2 - t);
	elseif (t_1 <= 890.0)
		tmp = Float64(Float64(log(Float64(y * z)) + Float64(-0.5 * log(t))) - t);
	else
		tmp = Float64(Float64(log(y) + t_2) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
	t_2 = a * log(t);
	tmp = 0.0;
	if (t_1 <= -20000000000.0)
		tmp = t_2 - t;
	elseif (t_1 <= 890.0)
		tmp = (log((y * z)) + (-0.5 * log(t))) - t;
	else
		tmp = (log(y) + t_2) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -20000000000.0], N[(t$95$2 - t), $MachinePrecision], If[LessEqual[t$95$1, 890.0], N[(N[(N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision] + N[(-0.5 * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] + t$95$2), $MachinePrecision] - t), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   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 67.1

         \[\leadsto -t \]

   4. Applied rewrites 67.1%

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

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 75.3

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

   7. Applied rewrites 75.3%

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

   8. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lift-log.f64 99.5

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

   10. Applied rewrites 99.5%

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

   if -2e10 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 890

   1. Initial program 98.9%

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

   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 4.2

         \[\leadsto -t \]

   4. Applied rewrites 4.2%

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

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 51.5

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

   7. Applied rewrites 51.5%

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

   8. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. sum-log N/A

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

      3. lower-+.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-log.f64 46.3

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

   10. Applied rewrites 46.3%

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

   if 890 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t)))

   1. Initial program 99.6%

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

   2. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 1.5

         \[\leadsto -t \]

   4. Applied rewrites 1.5%

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

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 70.1

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

   7. Applied rewrites 70.1%

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

   8. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lift-log.f64 59.9

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

   10. Applied rewrites 59.9%

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

Alternative 9: 68.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ t_2 := a \cdot \log t\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+17}:\\ \;\;\;\;t\_2 - t\\ \mathbf{elif}\;t\_1 \leq 890:\\ \;\;\;\;\log \left(y \cdot z\right) + \log t \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log y + t\_2\right) - t\\ \end{array} \end{array} \]

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 (* a (log t))))
   (if (<= t_1 -5e+17)
     (- t_2 t)
     (if (<= t_1 890.0)
       (+ (log (* y z)) (* (log t) (- a 0.5)))
       (- (+ (log y) t_2) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
	double t_2 = a * log(t);
	double tmp;
	if (t_1 <= -5e+17) {
		tmp = t_2 - t;
	} else if (t_1 <= 890.0) {
		tmp = log((y * z)) + (log(t) * (a - 0.5));
	} else {
		tmp = (log(y) + t_2) - t;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = ((log((x + y)) + log(z)) - t) + ((a - 0.5d0) * log(t))
    t_2 = a * log(t)
    if (t_1 <= (-5d+17)) then
        tmp = t_2 - t
    else if (t_1 <= 890.0d0) then
        tmp = log((y * z)) + (log(t) * (a - 0.5d0))
    else
        tmp = (log(y) + t_2) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = ((Math.log((x + y)) + Math.log(z)) - t) + ((a - 0.5) * Math.log(t));
	double t_2 = a * Math.log(t);
	double tmp;
	if (t_1 <= -5e+17) {
		tmp = t_2 - t;
	} else if (t_1 <= 890.0) {
		tmp = Math.log((y * z)) + (Math.log(t) * (a - 0.5));
	} else {
		tmp = (Math.log(y) + t_2) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = ((math.log((x + y)) + math.log(z)) - t) + ((a - 0.5) * math.log(t))
	t_2 = a * math.log(t)
	tmp = 0
	if t_1 <= -5e+17:
		tmp = t_2 - t
	elif t_1 <= 890.0:
		tmp = math.log((y * z)) + (math.log(t) * (a - 0.5))
	else:
		tmp = (math.log(y) + t_2) - t
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + Float64(Float64(a - 0.5) * log(t)))
	t_2 = Float64(a * log(t))
	tmp = 0.0
	if (t_1 <= -5e+17)
		tmp = Float64(t_2 - t);
	elseif (t_1 <= 890.0)
		tmp = Float64(log(Float64(y * z)) + Float64(log(t) * Float64(a - 0.5)));
	else
		tmp = Float64(Float64(log(y) + t_2) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
	t_2 = a * log(t);
	tmp = 0.0;
	if (t_1 <= -5e+17)
		tmp = t_2 - t;
	elseif (t_1 <= 890.0)
		tmp = log((y * z)) + (log(t) * (a - 0.5));
	else
		tmp = (log(y) + t_2) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+17], N[(t$95$2 - t), $MachinePrecision], If[LessEqual[t$95$1, 890.0], N[(N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] + t$95$2), $MachinePrecision] - t), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < -5e17

   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 67.4

         \[\leadsto -t \]

   4. Applied rewrites 67.4%

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

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 75.4

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

   7. Applied rewrites 75.4%

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

   8. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lift-log.f64 99.9

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

   10. Applied rewrites 99.9%

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

   if -5e17 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 890

   1. Initial program 98.9%

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

   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 6.1

         \[\leadsto -t \]

   4. Applied rewrites 6.1%

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

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 52.3

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

   7. Applied rewrites 52.3%

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

   8. Taylor expanded in t around 0

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

      1. associate-+r+ N/A

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

      2. sum-log N/A

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

      3. lower-+.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 44.9

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

   10. Applied rewrites 44.9%

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

   if 890 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t)))

   1. Initial program 99.6%

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

   2. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 1.5

         \[\leadsto -t \]

   4. Applied rewrites 1.5%

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

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 70.1

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

   7. Applied rewrites 70.1%

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

   8. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lift-log.f64 59.9

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

   10. Applied rewrites 59.9%

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

Alternative 10: 68.7% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in 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 38.3

      \[\leadsto -t \]

4. Applied rewrites 38.3%

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

5. Taylor expanded in x around 0

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

   1. lower--.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-log.f64 N/A

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

   4. lower-+.f64 N/A

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

   5. lift-log.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lift-log.f64 N/A

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

   8. lift--.f64 69.2

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

7. Applied rewrites 69.2%

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

8. Taylor expanded in a around inf

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

   1. lower-*.f64 N/A

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

   2. lift-log.f64 58.3

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

10. Applied rewrites 58.3%

    \[\leadsto \left(\log y + a \cdot \log t\right) - t \]
11. Add Preprocessing

Alternative 11: 63.7% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 77.5

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

4. Applied rewrites 77.5%

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

Alternative 12: 62.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ a \cdot \log t - t \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in 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 38.3

      \[\leadsto -t \]

4. Applied rewrites 38.3%

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

5. Taylor expanded in x around 0

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

   1. lower--.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-log.f64 N/A

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

   4. lower-+.f64 N/A

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

   5. lift-log.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lift-log.f64 N/A

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

   8. lift--.f64 69.2

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

7. Applied rewrites 69.2%

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

8. Taylor expanded in a around inf

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

   1. lower-*.f64 N/A

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

   2. lift-log.f64 74.9

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

10. Applied rewrites 74.9%

    \[\leadsto a \cdot \log t - t \]
11. Add Preprocessing

Alternative 13: 58.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.4 \cdot 10^{+43}:\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 1.4e+43) (* (log t) a) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 1.4e+43) {
		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 <= 1.4d+43) 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 <= 1.4e+43) {
		tmp = Math.log(t) * a;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 1.4e+43:
		tmp = math.log(t) * a
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 1.4e+43)
		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 <= 1.4e+43)
		tmp = log(t) * a;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 1.4e+43], N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if t < 1.40000000000000009e43

   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 50.6

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

   4. Applied rewrites 50.6%

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

   if 1.40000000000000009e43 < 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 78.5

         \[\leadsto -t \]

   4. Applied rewrites 78.5%

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

Alternative 14: 38.3% 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 38.3

      \[\leadsto -t \]

4. Applied rewrites 38.3%

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

Reproduce

herbie shell --seed 2025114 
(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))))