Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 10.8s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. associate-+l- N/A

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

   4. lift-+.f64 N/A

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

   5. associate--l+ N/A

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

   6. lower-+.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lower--.f64 99.6

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 99.6

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

4. Applied rewrites 99.6%

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

Alternative 2: 71.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ \mathbf{if}\;t\_1 \leq -800:\\ \;\;\;\;\left(\log t - \frac{t}{a}\right) \cdot a\\ \mathbf{elif}\;t\_1 \leq 710:\\ \;\;\;\;\log \left({t}^{\left(a - 0.5\right)} \cdot \left(z \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(y + x\right) + \log t \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t)))))
   (if (<= t_1 -800.0)
     (* (- (log t) (/ t a)) a)
     (if (<= t_1 710.0)
       (log (* (pow t (- a 0.5)) (* z y)))
       (+ (log (+ y x)) (* (log t) a))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
	double tmp;
	if (t_1 <= -800.0) {
		tmp = (log(t) - (t / a)) * a;
	} else if (t_1 <= 710.0) {
		tmp = log((pow(t, (a - 0.5)) * (z * y)));
	} else {
		tmp = log((y + x)) + (log(t) * a);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((log((x + y)) + log(z)) - t) + ((a - 0.5d0) * log(t))
    if (t_1 <= (-800.0d0)) then
        tmp = (log(t) - (t / a)) * a
    else if (t_1 <= 710.0d0) then
        tmp = log(((t ** (a - 0.5d0)) * (z * y)))
    else
        tmp = log((y + x)) + (log(t) * a)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = ((Math.log((x + y)) + Math.log(z)) - t) + ((a - 0.5) * Math.log(t));
	double tmp;
	if (t_1 <= -800.0) {
		tmp = (Math.log(t) - (t / a)) * a;
	} else if (t_1 <= 710.0) {
		tmp = Math.log((Math.pow(t, (a - 0.5)) * (z * y)));
	} else {
		tmp = Math.log((y + x)) + (Math.log(t) * a);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = ((math.log((x + y)) + math.log(z)) - t) + ((a - 0.5) * math.log(t))
	tmp = 0
	if t_1 <= -800.0:
		tmp = (math.log(t) - (t / a)) * a
	elif t_1 <= 710.0:
		tmp = math.log((math.pow(t, (a - 0.5)) * (z * y)))
	else:
		tmp = math.log((y + x)) + (math.log(t) * a)
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + Float64(Float64(a - 0.5) * log(t)))
	tmp = 0.0
	if (t_1 <= -800.0)
		tmp = Float64(Float64(log(t) - Float64(t / a)) * a);
	elseif (t_1 <= 710.0)
		tmp = log(Float64((t ^ Float64(a - 0.5)) * Float64(z * y)));
	else
		tmp = Float64(log(Float64(y + x)) + Float64(log(t) * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
	tmp = 0.0;
	if (t_1 <= -800.0)
		tmp = (log(t) - (t / a)) * a;
	elseif (t_1 <= 710.0)
		tmp = log(((t ^ (a - 0.5)) * (z * y)));
	else
		tmp = log((y + x)) + (log(t) * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -800.0], N[(N[(N[Log[t], $MachinePrecision] - N[(t / a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[t$95$1, 710.0], N[Log[N[(N[Power[t, N[(a - 0.5), $MachinePrecision]], $MachinePrecision] * N[(z * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < -800

   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. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-log.f64 73.7

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

   5. Applied rewrites 73.7%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 60.1%

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

   9. Taylor expanded in a around inf

      \[\leadsto \left(\log t - \frac{t}{a}\right) \cdot a \]
   10. Step-by-step derivation

   11. Applied rewrites 78.7%

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

   if -800 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 710

   1. Initial program 98.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-log.f64 45.8

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

   5. Applied rewrites 45.8%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 44.8%

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

   10. Applied rewrites 42.3%

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

   if 710 < (+.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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lift-+.f64 N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower--.f64 99.6

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 80.7

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

   7. Applied rewrites 80.7%

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

Alternative 3: 88.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(x + y\right) + \log z \leq 710:\\ \;\;\;\;\mathsf{fma}\left(\log t, a - 0.5, \log \left(z \cdot \left(y + x\right)\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log t - \frac{t}{a}\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (+ (log (+ x y)) (log z)) 710.0)
   (- (fma (log t) (- a 0.5) (log (* z (+ y x)))) t)
   (* (- (log t) (/ t a)) a)))

C

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

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(log(Float64(x + y)) + log(z)) <= 710.0)
		tmp = Float64(fma(log(t), Float64(a - 0.5), log(Float64(z * Float64(y + x)))) - t);
	else
		tmp = Float64(Float64(log(t) - Float64(t / a)) * a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision], 710.0], N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision] + N[Log[N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[t], $MachinePrecision] - N[(t / a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+r- N/A

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

      5. lower--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 99.6

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

      9. lift-+.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. sum-log N/A

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

      13. lower-log.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 96.3

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 96.3

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

   4. Applied rewrites 96.3%

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

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

   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. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-log.f64 64.4

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

   5. Applied rewrites 64.4%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 57.8%

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

   9. Taylor expanded in a around inf

      \[\leadsto \left(\log t - \frac{t}{a}\right) \cdot a \]
   10. Step-by-step derivation

   11. Applied rewrites 70.8%

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

Alternative 4: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.45 \cdot 10^{+22} \lor \neg \left(a \leq 24000000\right):\\ \;\;\;\;\left(\log t - \frac{t}{a}\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log t, \log \left(y + x\right)\right) + \left(\log z - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -1.45e+22) (not (<= a 24000000.0)))
   (* (- (log t) (/ t a)) a)
   (+ (fma -0.5 (log t) (log (+ y x))) (- (log z) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -1.45e+22) || !(a <= 24000000.0)) {
		tmp = (log(t) - (t / a)) * a;
	} else {
		tmp = fma(-0.5, log(t), log((y + x))) + (log(z) - t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -1.45e+22) || !(a <= 24000000.0))
		tmp = Float64(Float64(log(t) - Float64(t / a)) * a);
	else
		tmp = Float64(fma(-0.5, log(t), log(Float64(y + x))) + Float64(log(z) - t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -1.45e+22], N[Not[LessEqual[a, 24000000.0]], $MachinePrecision]], N[(N[(N[Log[t], $MachinePrecision] - N[(t / a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], N[(N[(-0.5 * N[Log[t], $MachinePrecision] + N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.45e22 or 2.4e7 < a

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-log.f64 74.4

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

   5. Applied rewrites 74.4%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 74.4%

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

   9. Taylor expanded in a around inf

      \[\leadsto \left(\log t - \frac{t}{a}\right) \cdot a \]
   10. Step-by-step derivation

   11. Applied rewrites 99.5%

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

   if -1.45e22 < a < 2.4e7

   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. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-log.f64 98.9

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

   5. Applied rewrites 98.9%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.45 \cdot 10^{+22} \lor \neg \left(a \leq 24000000\right):\\ \;\;\;\;\left(\log t - \frac{t}{a}\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log t, \log \left(y + x\right)\right) + \left(\log z - t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.45 \cdot 10^{+22} \lor \neg \left(a \leq 24000000\right):\\ \;\;\;\;\left(\log t - \frac{t}{a}\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\log y + \left(\mathsf{fma}\left(-0.5, \log t, \log z\right) - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -1.45e+22) (not (<= a 24000000.0)))
   (* (- (log t) (/ t a)) a)
   (+ (log y) (- (fma -0.5 (log t) (log z)) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -1.45e+22) || !(a <= 24000000.0)) {
		tmp = (log(t) - (t / a)) * a;
	} else {
		tmp = log(y) + (fma(-0.5, log(t), log(z)) - t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -1.45e+22) || !(a <= 24000000.0))
		tmp = Float64(Float64(log(t) - Float64(t / a)) * a);
	else
		tmp = Float64(log(y) + Float64(fma(-0.5, log(t), log(z)) - t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -1.45e+22], N[Not[LessEqual[a, 24000000.0]], $MachinePrecision]], N[(N[(N[Log[t], $MachinePrecision] - N[(t / a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], N[(N[Log[y], $MachinePrecision] + N[(N[(-0.5 * N[Log[t], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.45e22 or 2.4e7 < a

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-log.f64 74.4

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

   5. Applied rewrites 74.4%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 74.4%

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

   9. Taylor expanded in a around inf

      \[\leadsto \left(\log t - \frac{t}{a}\right) \cdot a \]
   10. Step-by-step derivation

   11. Applied rewrites 99.5%

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

   if -1.45e22 < a < 2.4e7

   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. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-log.f64 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 64.3%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.45 \cdot 10^{+22} \lor \neg \left(a \leq 24000000\right):\\ \;\;\;\;\left(\log t - \frac{t}{a}\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\log y + \left(\mathsf{fma}\left(-0.5, \log t, \log z\right) - t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. lower-+.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower--.f64 N/A

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

   8. lower-log.f64 N/A

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

   9. lower-log.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lower-log.f64 69.1

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

5. Applied rewrites 69.1%

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

Alternative 7: 86.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -380000 \lor \neg \left(a \leq 24000000\right):\\ \;\;\;\;\left(\log t - \frac{t}{a}\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log t, \log \left(\left(x + y\right) \cdot z\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -380000.0) (not (<= a 24000000.0)))
   (* (- (log t) (/ t a)) a)
   (- (fma -0.5 (log t) (log (* (+ x y) z))) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -380000.0) || !(a <= 24000000.0)) {
		tmp = (log(t) - (t / a)) * a;
	} else {
		tmp = fma(-0.5, log(t), log(((x + y) * z))) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -380000.0) || !(a <= 24000000.0))
		tmp = Float64(Float64(log(t) - Float64(t / a)) * a);
	else
		tmp = Float64(fma(-0.5, log(t), log(Float64(Float64(x + y) * z))) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -380000.0], N[Not[LessEqual[a, 24000000.0]], $MachinePrecision]], N[(N[(N[Log[t], $MachinePrecision] - N[(t / a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], N[(N[(-0.5 * N[Log[t], $MachinePrecision] + N[Log[N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.8e5 or 2.4e7 < a

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-log.f64 73.8

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

   5. Applied rewrites 73.8%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 73.8%

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

   9. Taylor expanded in a around inf

      \[\leadsto \left(\log t - \frac{t}{a}\right) \cdot a \]
   10. Step-by-step derivation

   11. Applied rewrites 99.5%

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

   if -3.8e5 < a < 2.4e7

   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. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-log.f64 98.9

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

   5. Applied rewrites 98.9%

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

   7. Applied rewrites 74.4%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -380000 \lor \neg \left(a \leq 24000000\right):\\ \;\;\;\;\left(\log t - \frac{t}{a}\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log t, \log \left(\left(x + y\right) \cdot z\right)\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 0.026:\\ \;\;\;\;\log t \cdot \left(a - 0.5\right) + \log \left(z \cdot y\right)\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{+185}:\\ \;\;\;\;\left(\log t - \frac{t}{a}\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 0.026)
   (+ (* (log t) (- a 0.5)) (log (* z y)))
   (if (<= t 1.15e+185) (* (- (log t) (/ t a)) a) (- t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 0.026) {
		tmp = (log(t) * (a - 0.5)) + log((z * y));
	} else if (t <= 1.15e+185) {
		tmp = (log(t) - (t / a)) * 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 <= 0.026d0) then
        tmp = (log(t) * (a - 0.5d0)) + log((z * y))
    else if (t <= 1.15d+185) then
        tmp = (log(t) - (t / a)) * 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 <= 0.026) {
		tmp = (Math.log(t) * (a - 0.5)) + Math.log((z * y));
	} else if (t <= 1.15e+185) {
		tmp = (Math.log(t) - (t / a)) * a;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 0.026:
		tmp = (math.log(t) * (a - 0.5)) + math.log((z * y))
	elif t <= 1.15e+185:
		tmp = (math.log(t) - (t / a)) * a
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 0.026)
		tmp = Float64(Float64(log(t) * Float64(a - 0.5)) + log(Float64(z * y)));
	elseif (t <= 1.15e+185)
		tmp = Float64(Float64(log(t) - Float64(t / a)) * a);
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 0.026)
		tmp = (log(t) * (a - 0.5)) + log((z * y));
	elseif (t <= 1.15e+185)
		tmp = (log(t) - (t / a)) * a;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 0.026], N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision] + N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.15e+185], N[(N[(N[Log[t], $MachinePrecision] - N[(t / a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], (-t)]]

Derivation

1. Split input into 3 regimes

2. if t < 0.0259999999999999988

   1. Initial program 99.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-log.f64 64.4

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

   5. Applied rewrites 64.4%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 64.0%

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

   10. Applied rewrites 50.8%

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

   if 0.0259999999999999988 < t < 1.1500000000000001e185

   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. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-log.f64 80.6

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

   5. Applied rewrites 80.6%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 71.2%

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

   9. Taylor expanded in a around inf

      \[\leadsto \left(\log t - \frac{t}{a}\right) \cdot a \]
   10. Step-by-step derivation

   11. Applied rewrites 86.8%

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

   if 1.1500000000000001e185 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 90.3

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

   5. Applied rewrites 90.3%

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

Alternative 9: 67.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 0.026:\\ \;\;\;\;\mathsf{fma}\left(\log t, a - 0.5, \log \left(z \cdot y\right)\right)\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{+185}:\\ \;\;\;\;\left(\log t - \frac{t}{a}\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 0.026)
   (fma (log t) (- a 0.5) (log (* z y)))
   (if (<= t 1.15e+185) (* (- (log t) (/ t a)) a) (- t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 0.026) {
		tmp = fma(log(t), (a - 0.5), log((z * y)));
	} else if (t <= 1.15e+185) {
		tmp = (log(t) - (t / a)) * a;
	} else {
		tmp = -t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 0.026)
		tmp = fma(log(t), Float64(a - 0.5), log(Float64(z * y)));
	elseif (t <= 1.15e+185)
		tmp = Float64(Float64(log(t) - Float64(t / a)) * a);
	else
		tmp = Float64(-t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 0.026], N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision] + N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.15e+185], N[(N[(N[Log[t], $MachinePrecision] - N[(t / a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], (-t)]]

Derivation

1. Split input into 3 regimes

2. if t < 0.0259999999999999988

   1. Initial program 99.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-log.f64 64.4

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

   5. Applied rewrites 64.4%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 64.0%

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

   10. Applied rewrites 50.8%

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

   if 0.0259999999999999988 < t < 1.1500000000000001e185

   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. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-log.f64 80.6

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

   5. Applied rewrites 80.6%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 71.2%

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

   9. Taylor expanded in a around inf

      \[\leadsto \left(\log t - \frac{t}{a}\right) \cdot a \]
   10. Step-by-step derivation

   11. Applied rewrites 86.8%

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

   if 1.1500000000000001e185 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 90.3

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

   5. Applied rewrites 90.3%

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

Alternative 10: 75.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.65 \cdot 10^{-49} \lor \neg \left(a \leq 24000000\right):\\ \;\;\;\;\left(\log t - \frac{t}{a}\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -2.65e-49) (not (<= a 24000000.0)))
   (* (- (log t) (/ t a)) a)
   (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -2.65e-49) || !(a <= 24000000.0)) {
		tmp = (log(t) - (t / a)) * 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 ((a <= (-2.65d-49)) .or. (.not. (a <= 24000000.0d0))) then
        tmp = (log(t) - (t / a)) * 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 ((a <= -2.65e-49) || !(a <= 24000000.0)) {
		tmp = (Math.log(t) - (t / a)) * a;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -2.65e-49) or not (a <= 24000000.0):
		tmp = (math.log(t) - (t / a)) * a
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -2.65e-49) || !(a <= 24000000.0))
		tmp = Float64(Float64(log(t) - Float64(t / a)) * a);
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -2.65e-49) || ~((a <= 24000000.0)))
		tmp = (log(t) - (t / a)) * a;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -2.65e-49], N[Not[LessEqual[a, 24000000.0]], $MachinePrecision]], N[(N[(N[Log[t], $MachinePrecision] - N[(t / a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if a < -2.6500000000000001e-49 or 2.4e7 < 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. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-log.f64 69.8

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

   5. Applied rewrites 69.8%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 69.7%

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

   9. Taylor expanded in a around inf

      \[\leadsto \left(\log t - \frac{t}{a}\right) \cdot a \]
   10. Step-by-step derivation

   11. Applied rewrites 91.8%

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

   if -2.6500000000000001e-49 < a < 2.4e7

   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. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 59.0

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

   5. Applied rewrites 59.0%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.65 \cdot 10^{-49} \lor \neg \left(a \leq 24000000\right):\\ \;\;\;\;\left(\log t - \frac{t}{a}\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 63.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.5 \cdot 10^{+22} \lor \neg \left(a \leq 7.8 \cdot 10^{+17}\right):\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -3.5e+22) (not (<= a 7.8e+17))) (* (log t) a) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -3.5e+22) || !(a <= 7.8e+17)) {
		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 ((a <= (-3.5d+22)) .or. (.not. (a <= 7.8d+17))) 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 ((a <= -3.5e+22) || !(a <= 7.8e+17)) {
		tmp = Math.log(t) * a;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -3.5e+22) or not (a <= 7.8e+17):
		tmp = math.log(t) * a
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -3.5e+22) || !(a <= 7.8e+17))
		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 ((a <= -3.5e+22) || ~((a <= 7.8e+17)))
		tmp = log(t) * a;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -3.5e+22], N[Not[LessEqual[a, 7.8e+17]], $MachinePrecision]], N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if a < -3.5e22 or 7.8e17 < 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. Add Preprocessing

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 80.7

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

   5. Applied rewrites 80.7%

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

   if -3.5e22 < a < 7.8e17

   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. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 58.4

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

   5. Applied rewrites 58.4%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.5 \cdot 10^{+22} \lor \neg \left(a \leq 7.8 \cdot 10^{+17}\right):\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 38.4% accurate, 107.0× 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. Add Preprocessing

3. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 41.0

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

5. Applied rewrites 41.0%

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

Developer Target 1: 99.6% accurate, 1.0× speedup

Math

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

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(log(Float64(x + y)) + Float64(Float64(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[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2025015 
(FPCore (x y z t a)
  :name "Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (log (+ x y)) (+ (- (log z) t) (* (- a 1/2) (log t)))))

  (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))