Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 6.8s

Alternatives: 18

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

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

   3. lift-+.f64 N/A

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

   4. lift-log.f64 N/A

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

   5. lift-log.f64 N/A

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

   6. associate--l+ N/A

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

   7. lower-+.f64 N/A

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

   8. lift-log.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lift-log.f64 99.6

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

3. Applied rewrites 99.6%

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

Alternative 3: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(x + y\right)\\ \mathbf{if}\;a \leq -8.2:\\ \;\;\;\;\left(\log \left(y + x\right) + -1 \cdot t\right) + \left(a - 0.5\right) \cdot \log t\\ \mathbf{elif}\;a \leq 1.75:\\ \;\;\;\;\left(\log z + \left(t\_1 + -0.5 \cdot \log t\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \log t, t\_1 + \left(-t\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (log (+ x y))))
   (if (<= a -8.2)
     (+ (+ (log (+ y x)) (* -1.0 t)) (* (- a 0.5) (log t)))
     (if (<= a 1.75)
       (- (+ (log z) (+ t_1 (* -0.5 (log t)))) t)
       (fma a (log t) (+ t_1 (- t)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((x + y));
	double tmp;
	if (a <= -8.2) {
		tmp = (log((y + x)) + (-1.0 * t)) + ((a - 0.5) * log(t));
	} else if (a <= 1.75) {
		tmp = (log(z) + (t_1 + (-0.5 * log(t)))) - t;
	} else {
		tmp = fma(a, log(t), (t_1 + -t));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = log(Float64(x + y))
	tmp = 0.0
	if (a <= -8.2)
		tmp = Float64(Float64(log(Float64(y + x)) + Float64(-1.0 * t)) + Float64(Float64(a - 0.5) * log(t)));
	elseif (a <= 1.75)
		tmp = Float64(Float64(log(z) + Float64(t_1 + Float64(-0.5 * log(t)))) - t);
	else
		tmp = fma(a, log(t), Float64(t_1 + Float64(-t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[a, -8.2], N[(N[(N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision] + N[(-1.0 * t), $MachinePrecision]), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.75], N[(N[(N[Log[z], $MachinePrecision] + N[(t$95$1 + N[(-0.5 * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(a * N[Log[t], $MachinePrecision] + N[(t$95$1 + (-t)), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -8.1999999999999993

   1. Initial program 99.7%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lift-log.f64 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in t around inf

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

      1. lower-*.f64 98.9

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

   6. Applied rewrites 98.9%

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

   if -8.1999999999999993 < a < 1.75

   1. Initial program 99.5%

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

   2. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 50.6

         \[\leadsto -t \]

   4. Applied rewrites 50.6%

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

   5. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-log.f64 98.3

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

   7. Applied rewrites 98.3%

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

   if 1.75 < 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. 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

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

      3. lift-+.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lift-log.f64 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in t around inf

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

      1. lower-*.f64 98.3

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

   6. Applied rewrites 98.3%

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

   7. Taylor expanded in a around inf

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

   9. Applied rewrites 98.3%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. lift-log.f64 N/A

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

       5. lower-fma.f64 N/A

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

       6. lift-log.f64 98.3

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

       7. lift-+.f64 N/A

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

       8. +-commutative N/A

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

       9. lower-+.f64 98.3

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

       10. lift-*.f64 N/A

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

       11. mul-1-neg N/A

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

       12. lift-neg.f64 98.3

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

   11. Applied rewrites 98.3%

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

Alternative 4: 94.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -8.2:\\ \;\;\;\;\left(\log \left(y + x\right) + -1 \cdot t\right) + \left(a - 0.5\right) \cdot \log t\\ \mathbf{elif}\;a \leq 1.75:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.5, \log t, \log y\right) + \log z\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \log t, \log \left(x + y\right) + \left(-t\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -8.2)
   (+ (+ (log (+ y x)) (* -1.0 t)) (* (- a 0.5) (log t)))
   (if (<= a 1.75)
     (- (+ (fma -0.5 (log t) (log y)) (log z)) t)
     (fma a (log t) (+ (log (+ x y)) (- t))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -8.2) {
		tmp = (log((y + x)) + (-1.0 * t)) + ((a - 0.5) * log(t));
	} else if (a <= 1.75) {
		tmp = (fma(-0.5, log(t), log(y)) + log(z)) - t;
	} else {
		tmp = fma(a, log(t), (log((x + y)) + -t));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -8.2)
		tmp = Float64(Float64(log(Float64(y + x)) + Float64(-1.0 * t)) + Float64(Float64(a - 0.5) * log(t)));
	elseif (a <= 1.75)
		tmp = Float64(Float64(fma(-0.5, log(t), log(y)) + log(z)) - t);
	else
		tmp = fma(a, log(t), Float64(log(Float64(x + y)) + Float64(-t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -8.2], N[(N[(N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision] + N[(-1.0 * t), $MachinePrecision]), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.75], N[(N[(N[(-0.5 * N[Log[t], $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(a * N[Log[t], $MachinePrecision] + N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + (-t)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -8.1999999999999993

   1. Initial program 99.7%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lift-log.f64 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in t around inf

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

      1. lower-*.f64 98.9

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

   6. Applied rewrites 98.9%

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

   if -8.1999999999999993 < a < 1.75

   1. Initial program 99.5%

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

   2. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 50.6

         \[\leadsto -t \]

   4. Applied rewrites 50.6%

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

   5. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-log.f64 98.3

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

   7. Applied rewrites 98.3%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 63.0%

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

       1. lift-+.f64 N/A

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

       2. lift-log.f64 N/A

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

       3. +-commutative N/A

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

       4. lower-+.f64 N/A

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

       5. lift-+.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-log.f64 N/A

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

       8. +-commutative N/A

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

       9. lower-fma.f64 N/A

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

       10. lift-log.f64 N/A

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

       11. +-commutative N/A

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

       12. +-commutative N/A

           \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, \log t, \log y\right) + \mathsf{Rewrite<=}\left(lift-log.f64, \log z\right)\right) - t \]

   12. Applied rewrites 63.0%

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

   if 1.75 < 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. 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

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

      3. lift-+.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lift-log.f64 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in t around inf

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

      1. lower-*.f64 98.3

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

   6. Applied rewrites 98.3%

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

   7. Taylor expanded in a around inf

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

   9. Applied rewrites 98.3%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. lift-log.f64 N/A

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

       5. lower-fma.f64 N/A

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

       6. lift-log.f64 98.3

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

       7. lift-+.f64 N/A

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

       8. +-commutative N/A

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

       9. lower-+.f64 98.3

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

       10. lift-*.f64 N/A

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

       11. mul-1-neg N/A

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

       12. lift-neg.f64 98.3

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

   11. Applied rewrites 98.3%

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

Alternative 5: 94.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(x + y\right)\\ t_2 := t\_1 + \log z\\ \mathbf{if}\;t\_2 \leq -750:\\ \;\;\;\;\mathsf{fma}\left(a, \log t, t\_1 + \left(-t\right)\right)\\ \mathbf{elif}\;t\_2 \leq 705:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log \left(z \cdot \left(y + x\right)\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(y + x\right) + -1 \cdot t\right) + \left(a - 0.5\right) \cdot \log t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (log (+ x y))) (t_2 (+ t_1 (log z))))
   (if (<= t_2 -750.0)
     (fma a (log t) (+ t_1 (- t)))
     (if (<= t_2 705.0)
       (fma (- a 0.5) (log t) (- (log (* z (+ y x))) t))
       (+ (+ (log (+ y x)) (* -1.0 t)) (* (- a 0.5) (log t)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((x + y));
	double t_2 = t_1 + log(z);
	double tmp;
	if (t_2 <= -750.0) {
		tmp = fma(a, log(t), (t_1 + -t));
	} else if (t_2 <= 705.0) {
		tmp = fma((a - 0.5), log(t), (log((z * (y + x))) - t));
	} else {
		tmp = (log((y + x)) + (-1.0 * t)) + ((a - 0.5) * log(t));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = log(Float64(x + y))
	t_2 = Float64(t_1 + log(z))
	tmp = 0.0
	if (t_2 <= -750.0)
		tmp = fma(a, log(t), Float64(t_1 + Float64(-t)));
	elseif (t_2 <= 705.0)
		tmp = fma(Float64(a - 0.5), log(t), Float64(log(Float64(z * Float64(y + x))) - t));
	else
		tmp = Float64(Float64(log(Float64(y + x)) + Float64(-1.0 * t)) + Float64(Float64(a - 0.5) * log(t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + N[Log[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -750.0], N[(a * N[Log[t], $MachinePrecision] + N[(t$95$1 + (-t)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 705.0], N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[(N[Log[N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision] + N[(-1.0 * t), $MachinePrecision]), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.6%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lift-log.f64 99.6

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in t around inf

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

      1. lower-*.f64 78.0

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

   6. Applied rewrites 78.0%

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

   7. Taylor expanded in a around inf

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

   9. Applied rewrites 78.7%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. lift-log.f64 N/A

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

       5. lower-fma.f64 N/A

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

       6. lift-log.f64 78.7

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

       7. lift-+.f64 N/A

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

       8. +-commutative N/A

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

       9. lower-+.f64 78.7

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

       10. lift-*.f64 N/A

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

       11. mul-1-neg N/A

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

       12. lift-neg.f64 78.7

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

   11. Applied rewrites 78.7%

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

   if -750 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 705

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lift-log.f64 N/A

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

      14. lower--.f64 N/A

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

      15. +-commutative N/A

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

      16. sum-log N/A

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

      17. lower-log.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.5

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

   3. Applied rewrites 99.5%

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

   if 705 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z))

   1. Initial program 99.7%

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

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

      3. lift-+.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lift-log.f64 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in t around inf

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

      1. lower-*.f64 78.1

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

   6. Applied rewrites 78.1%

      \[\leadsto \left(\log \left(y + x\right) + \color{blue}{-1 \cdot t}\right) + \left(a - 0.5\right) \cdot \log t \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 92.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(x + y\right)\\ t_2 := t\_1 + \log z\\ t_3 := \mathsf{fma}\left(a, \log t, t\_1 + \left(-t\right)\right)\\ \mathbf{if}\;t\_2 \leq -750:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 705:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log \left(z \cdot \left(y + x\right)\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (log (+ x y)))
        (t_2 (+ t_1 (log z)))
        (t_3 (fma a (log t) (+ t_1 (- t)))))
   (if (<= t_2 -750.0)
     t_3
     (if (<= t_2 705.0)
       (fma (- a 0.5) (log t) (- (log (* z (+ y x))) t))
       t_3))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((x + y));
	double t_2 = t_1 + log(z);
	double t_3 = fma(a, log(t), (t_1 + -t));
	double tmp;
	if (t_2 <= -750.0) {
		tmp = t_3;
	} else if (t_2 <= 705.0) {
		tmp = fma((a - 0.5), log(t), (log((z * (y + x))) - t));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = log(Float64(x + y))
	t_2 = Float64(t_1 + log(z))
	t_3 = fma(a, log(t), Float64(t_1 + Float64(-t)))
	tmp = 0.0
	if (t_2 <= -750.0)
		tmp = t_3;
	elseif (t_2 <= 705.0)
		tmp = fma(Float64(a - 0.5), log(t), Float64(log(Float64(z * Float64(y + x))) - t));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + N[Log[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(a * N[Log[t], $MachinePrecision] + N[(t$95$1 + (-t)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -750.0], t$95$3, If[LessEqual[t$95$2, 705.0], N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[(N[Log[N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -750 or 705 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z))

   1. Initial program 99.7%

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

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

      3. lift-+.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lift-log.f64 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in t around inf

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

      1. lower-*.f64 78.0

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

   6. Applied rewrites 78.0%

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

   7. Taylor expanded in a around inf

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

   9. Applied rewrites 77.9%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. lift-log.f64 N/A

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

       5. lower-fma.f64 N/A

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

       6. lift-log.f64 77.9

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

       7. lift-+.f64 N/A

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

       8. +-commutative N/A

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

       9. lower-+.f64 77.9

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

       10. lift-*.f64 N/A

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

       11. mul-1-neg N/A

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

       12. lift-neg.f64 77.9

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

   11. Applied rewrites 77.9%

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

   if -750 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 705

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lift-log.f64 N/A

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

      14. lower--.f64 N/A

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

      15. +-commutative N/A

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

      16. sum-log N/A

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

      17. lower-log.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.5

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

   3. Applied rewrites 99.5%

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

Alternative 7: 92.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(x + y\right)\\ t_2 := t\_1 + \log z\\ t_3 := \mathsf{fma}\left(a, \log t, t\_1 + \left(-t\right)\right)\\ \mathbf{if}\;t\_2 \leq -750:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 705:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log \left(z \cdot y\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (log (+ x y)))
        (t_2 (+ t_1 (log z)))
        (t_3 (fma a (log t) (+ t_1 (- t)))))
   (if (<= t_2 -750.0)
     t_3
     (if (<= t_2 705.0) (fma (- a 0.5) (log t) (- (log (* z y)) t)) t_3))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((x + y));
	double t_2 = t_1 + log(z);
	double t_3 = fma(a, log(t), (t_1 + -t));
	double tmp;
	if (t_2 <= -750.0) {
		tmp = t_3;
	} else if (t_2 <= 705.0) {
		tmp = fma((a - 0.5), log(t), (log((z * y)) - t));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = log(Float64(x + y))
	t_2 = Float64(t_1 + log(z))
	t_3 = fma(a, log(t), Float64(t_1 + Float64(-t)))
	tmp = 0.0
	if (t_2 <= -750.0)
		tmp = t_3;
	elseif (t_2 <= 705.0)
		tmp = fma(Float64(a - 0.5), log(t), Float64(log(Float64(z * y)) - t));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + N[Log[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(a * N[Log[t], $MachinePrecision] + N[(t$95$1 + (-t)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -750.0], t$95$3, If[LessEqual[t$95$2, 705.0], N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[(N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -750 or 705 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z))

   1. Initial program 99.7%

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

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

      3. lift-+.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lift-log.f64 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in t around inf

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

      1. lower-*.f64 78.0

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

   6. Applied rewrites 78.0%

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

   7. Taylor expanded in a around inf

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

   9. Applied rewrites 77.9%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. lift-log.f64 N/A

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

       5. lower-fma.f64 N/A

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

       6. lift-log.f64 77.9

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

       7. lift-+.f64 N/A

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

       8. +-commutative N/A

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

       9. lower-+.f64 77.9

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

       10. lift-*.f64 N/A

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

       11. mul-1-neg N/A

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

       12. lift-neg.f64 77.9

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

   11. Applied rewrites 77.9%

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

   if -750 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 705

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lift-log.f64 N/A

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

      14. lower--.f64 N/A

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

      15. +-commutative N/A

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

      16. sum-log N/A

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

      17. lower-log.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.5

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

   3. Applied rewrites 99.5%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 65.7%

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

Alternative 8: 92.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(x + y\right)\\ t_2 := \left(\left(t\_1 + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ t_3 := \mathsf{fma}\left(a, \log t, t\_1 + \left(-t\right)\right)\\ \mathbf{if}\;t\_2 \leq -500:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 1020:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log \left(z \cdot \left(x + y\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (log (+ x y)))
        (t_2 (+ (- (+ t_1 (log z)) t) (* (- a 0.5) (log t))))
        (t_3 (fma a (log t) (+ t_1 (- t)))))
   (if (<= t_2 -500.0)
     t_3
     (if (<= t_2 1020.0) (fma (- a 0.5) (log t) (log (* z (+ x y)))) t_3))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((x + y));
	double t_2 = ((t_1 + log(z)) - t) + ((a - 0.5) * log(t));
	double t_3 = fma(a, log(t), (t_1 + -t));
	double tmp;
	if (t_2 <= -500.0) {
		tmp = t_3;
	} else if (t_2 <= 1020.0) {
		tmp = fma((a - 0.5), log(t), log((z * (x + y))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = log(Float64(x + y))
	t_2 = Float64(Float64(Float64(t_1 + log(z)) - t) + Float64(Float64(a - 0.5) * log(t)))
	t_3 = fma(a, log(t), Float64(t_1 + Float64(-t)))
	tmp = 0.0
	if (t_2 <= -500.0)
		tmp = t_3;
	elseif (t_2 <= 1020.0)
		tmp = fma(Float64(a - 0.5), log(t), log(Float64(z * Float64(x + y))));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$1 + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(a * N[Log[t], $MachinePrecision] + N[(t$95$1 + (-t)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -500.0], t$95$3, If[LessEqual[t$95$2, 1020.0], N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[Log[N[(z * N[(x + y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$3]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < -500 or 1020 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t)))

   1. Initial program 99.8%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lift-log.f64 99.8

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in t around inf

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

      1. lower-*.f64 93.6

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

   6. Applied rewrites 93.6%

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

   7. Taylor expanded in a around inf

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

   9. Applied rewrites 93.6%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. lift-log.f64 N/A

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

       5. lower-fma.f64 N/A

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

       6. lift-log.f64 93.6

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

       7. lift-+.f64 N/A

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

       8. +-commutative N/A

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

       9. lower-+.f64 93.6

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

       10. lift-*.f64 N/A

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

       11. mul-1-neg N/A

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

       12. lift-neg.f64 93.6

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

   11. Applied rewrites 93.6%

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

   if -500 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 1020

   1. Initial program 98.9%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lift-log.f64 N/A

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

      14. lower--.f64 N/A

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

      15. +-commutative N/A

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

      16. sum-log N/A

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

      17. lower-log.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 90.9

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

   3. Applied rewrites 90.9%

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

   4. Taylor expanded in t around 0

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

      1. lower-log.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 88.7

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

   6. Applied rewrites 88.7%

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

Alternative 9: 84.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(x + y\right)\\ t_2 := \left(\left(t\_1 + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ t_3 := \mathsf{fma}\left(a, \log t, t\_1 + \left(-t\right)\right)\\ \mathbf{if}\;t\_2 \leq -5000000000:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 1020:\\ \;\;\;\;\left(\log \left(z \cdot y\right) + -0.5 \cdot \log t\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (log (+ x y)))
        (t_2 (+ (- (+ t_1 (log z)) t) (* (- a 0.5) (log t))))
        (t_3 (fma a (log t) (+ t_1 (- t)))))
   (if (<= t_2 -5000000000.0)
     t_3
     (if (<= t_2 1020.0) (- (+ (log (* z y)) (* -0.5 (log t))) t) t_3))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((x + y));
	double t_2 = ((t_1 + log(z)) - t) + ((a - 0.5) * log(t));
	double t_3 = fma(a, log(t), (t_1 + -t));
	double tmp;
	if (t_2 <= -5000000000.0) {
		tmp = t_3;
	} else if (t_2 <= 1020.0) {
		tmp = (log((z * y)) + (-0.5 * log(t))) - t;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = log(Float64(x + y))
	t_2 = Float64(Float64(Float64(t_1 + log(z)) - t) + Float64(Float64(a - 0.5) * log(t)))
	t_3 = fma(a, log(t), Float64(t_1 + Float64(-t)))
	tmp = 0.0
	if (t_2 <= -5000000000.0)
		tmp = t_3;
	elseif (t_2 <= 1020.0)
		tmp = Float64(Float64(log(Float64(z * y)) + Float64(-0.5 * log(t))) - t);
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$1 + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(a * N[Log[t], $MachinePrecision] + N[(t$95$1 + (-t)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5000000000.0], t$95$3, If[LessEqual[t$95$2, 1020.0], N[(N[(N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision] + N[(-0.5 * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], t$95$3]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < -5e9 or 1020 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t)))

   1. Initial program 99.8%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lift-log.f64 99.8

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in t around inf

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

      1. lower-*.f64 95.7

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

   6. Applied rewrites 95.7%

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

   7. Taylor expanded in a around inf

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

   9. Applied rewrites 95.6%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. lift-log.f64 N/A

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

       5. lower-fma.f64 N/A

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

       6. lift-log.f64 95.6

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

       7. lift-+.f64 N/A

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

       8. +-commutative N/A

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

       9. lower-+.f64 95.6

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

       10. lift-*.f64 N/A

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

       11. mul-1-neg N/A

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

       12. lift-neg.f64 95.6

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

   11. Applied rewrites 95.6%

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

   if -5e9 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 1020

   1. Initial program 99.0%

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

   2. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 4.3

         \[\leadsto -t \]

   4. Applied rewrites 4.3%

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

   5. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-log.f64 94.8

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

   7. Applied rewrites 94.8%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 50.7%

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

       1. lift-+.f64 N/A

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

       2. lift-log.f64 N/A

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

       3. lift-+.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-log.f64 N/A

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

       6. associate-+r+ N/A

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

       7. lift-log.f64 N/A

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

       8. +-commutative N/A

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

       9. log-prod N/A

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

       10. lift-*.f64 N/A

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

       11. lift-log.f64 N/A

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

       12. lower-+.f64 N/A

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

       13. lift-log.f64 N/A

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

       14. lift-*.f64 44.9

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

   12. Applied rewrites 44.9%

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

Alternative 10: 83.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(x + y\right)\\ t_2 := \left(\left(t\_1 + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ t_3 := \mathsf{fma}\left(a, \log t, t\_1 + \left(-t\right)\right)\\ \mathbf{if}\;t\_2 \leq -500:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 1020:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log t, \log \left(\left(x + y\right) \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (log (+ x y)))
        (t_2 (+ (- (+ t_1 (log z)) t) (* (- a 0.5) (log t))))
        (t_3 (fma a (log t) (+ t_1 (- t)))))
   (if (<= t_2 -500.0)
     t_3
     (if (<= t_2 1020.0) (fma -0.5 (log t) (log (* (+ x y) z))) t_3))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((x + y));
	double t_2 = ((t_1 + log(z)) - t) + ((a - 0.5) * log(t));
	double t_3 = fma(a, log(t), (t_1 + -t));
	double tmp;
	if (t_2 <= -500.0) {
		tmp = t_3;
	} else if (t_2 <= 1020.0) {
		tmp = fma(-0.5, log(t), log(((x + y) * z)));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = log(Float64(x + y))
	t_2 = Float64(Float64(Float64(t_1 + log(z)) - t) + Float64(Float64(a - 0.5) * log(t)))
	t_3 = fma(a, log(t), Float64(t_1 + Float64(-t)))
	tmp = 0.0
	if (t_2 <= -500.0)
		tmp = t_3;
	elseif (t_2 <= 1020.0)
		tmp = fma(-0.5, log(t), log(Float64(Float64(x + y) * z)));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$1 + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(a * N[Log[t], $MachinePrecision] + N[(t$95$1 + (-t)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -500.0], t$95$3, If[LessEqual[t$95$2, 1020.0], N[(-0.5 * N[Log[t], $MachinePrecision] + N[Log[N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$3]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < -500 or 1020 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t)))

   1. Initial program 99.8%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lift-log.f64 99.8

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in t around inf

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

      1. lower-*.f64 93.6

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

   6. Applied rewrites 93.6%

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

   7. Taylor expanded in a around inf

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

   9. Applied rewrites 93.6%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. lift-log.f64 N/A

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

       5. lower-fma.f64 N/A

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

       6. lift-log.f64 93.6

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

       7. lift-+.f64 N/A

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

       8. +-commutative N/A

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

       9. lower-+.f64 93.6

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

       10. lift-*.f64 N/A

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

       11. mul-1-neg N/A

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

       12. lift-neg.f64 93.6

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

   11. Applied rewrites 93.6%

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

   if -500 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 1020

   1. Initial program 98.9%

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

   2. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 3.1

         \[\leadsto -t \]

   4. Applied rewrites 3.1%

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

   5. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-log.f64 96.3

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

   7. Applied rewrites 96.3%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 51.0%

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

   11. Taylor expanded in t around 0

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

       1. associate-+r+ N/A

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

       2. sum-log N/A

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

       3. +-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. lift-log.f64 N/A

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

       6. sum-log N/A

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

       7. +-commutative N/A

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

       8. sum-log N/A

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

       9. lower-log.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. lower-+.f64 86.5

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

   13. Applied rewrites 86.5%

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

Alternative 11: 83.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ t_2 := \mathsf{fma}\left(a - 0.5, \log t, -t\right)\\ \mathbf{if}\;t\_1 \leq -10000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 1020:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log t, \log \left(\left(x + y\right) \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))
        (t_2 (fma (- a 0.5) (log t) (- t))))
   (if (<= t_1 -10000.0)
     t_2
     (if (<= t_1 1020.0) (fma -0.5 (log t) (log (* (+ x y) z))) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
	double t_2 = fma((a - 0.5), log(t), -t);
	double tmp;
	if (t_1 <= -10000.0) {
		tmp = t_2;
	} else if (t_1 <= 1020.0) {
		tmp = fma(-0.5, log(t), log(((x + y) * z)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + Float64(Float64(a - 0.5) * log(t)))
	t_2 = fma(Float64(a - 0.5), log(t), Float64(-t))
	tmp = 0.0
	if (t_1 <= -10000.0)
		tmp = t_2;
	elseif (t_1 <= 1020.0)
		tmp = fma(-0.5, log(t), log(Float64(Float64(x + y) * z)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + (-t)), $MachinePrecision]}, If[LessEqual[t$95$1, -10000.0], t$95$2, If[LessEqual[t$95$1, 1020.0], N[(-0.5 * N[Log[t], $MachinePrecision] + N[Log[N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < -1e4 or 1020 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t)))

   1. Initial program 99.8%

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

   2. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 95.0

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

   4. Applied rewrites 95.0%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-log.f64 95.0

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

      9. associate--l+ 95.0

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

      10. +-commutative 95.0

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

   6. Applied rewrites 95.0%

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

   if -1e4 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 1020

   1. Initial program 99.0%

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

   2. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 3.3

         \[\leadsto -t \]

   4. Applied rewrites 3.3%

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

   5. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-log.f64 95.9

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

   7. Applied rewrites 95.9%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 51.0%

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

   11. Taylor expanded in t around 0

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

       1. associate-+r+ N/A

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

       2. sum-log N/A

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

       3. +-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. lift-log.f64 N/A

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

       6. sum-log N/A

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

       7. +-commutative N/A

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

       8. sum-log N/A

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

       9. lower-log.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. lower-+.f64 82.8

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

   13. Applied rewrites 82.8%

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

Alternative 12: 83.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ t_2 := \mathsf{fma}\left(a - 0.5, \log t, -t\right)\\ \mathbf{if}\;t\_1 \leq -10000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 700:\\ \;\;\;\;\log \left(y \cdot \left(z \cdot {t}^{\left(a - 0.5\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))
        (t_2 (fma (- a 0.5) (log t) (- t))))
   (if (<= t_1 -10000.0)
     t_2
     (if (<= t_1 700.0) (log (* y (* z (pow t (- a 0.5))))) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
	double t_2 = fma((a - 0.5), log(t), -t);
	double tmp;
	if (t_1 <= -10000.0) {
		tmp = t_2;
	} else if (t_1 <= 700.0) {
		tmp = log((y * (z * pow(t, (a - 0.5)))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + Float64(Float64(a - 0.5) * log(t)))
	t_2 = fma(Float64(a - 0.5), log(t), Float64(-t))
	tmp = 0.0
	if (t_1 <= -10000.0)
		tmp = t_2;
	elseif (t_1 <= 700.0)
		tmp = log(Float64(y * Float64(z * (t ^ Float64(a - 0.5)))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + (-t)), $MachinePrecision]}, If[LessEqual[t$95$1, -10000.0], t$95$2, If[LessEqual[t$95$1, 700.0], N[Log[N[(y * N[(z * N[Power[t, N[(a - 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < -1e4 or 700 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t)))

   1. Initial program 99.8%

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

   2. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 91.1

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

   4. Applied rewrites 91.1%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-log.f64 91.1

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

      9. associate--l+ 91.1

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

      10. +-commutative 91.1

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

   6. Applied rewrites 91.1%

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

   if -1e4 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 700

   1. Initial program 98.9%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. associate-+r+ N/A

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

      3. sum-log N/A

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

      4. *-commutative N/A

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

      5. log-pow-rev N/A

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

      6. sum-log N/A

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

      7. lower-log.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lift--.f64 50.8

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

   4. Applied rewrites 50.8%

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

   5. Taylor expanded in t around 0

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

      1. lower-log.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lower-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift--.f64 48.6

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

   7. Applied rewrites 48.6%

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

Alternative 13: 83.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ t_2 := \mathsf{fma}\left(a - 0.5, \log t, -t\right)\\ \mathbf{if}\;t\_1 \leq -5000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 720:\\ \;\;\;\;\log \left(\left(y \cdot z\right) \cdot \frac{1}{\sqrt{t}}\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))
        (t_2 (fma (- a 0.5) (log t) (- t))))
   (if (<= t_1 -5000000000.0)
     t_2
     (if (<= t_1 720.0) (- (log (* (* y z) (/ 1.0 (sqrt t)))) t) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
	double t_2 = fma((a - 0.5), log(t), -t);
	double tmp;
	if (t_1 <= -5000000000.0) {
		tmp = t_2;
	} else if (t_1 <= 720.0) {
		tmp = log(((y * z) * (1.0 / sqrt(t)))) - t;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + Float64(Float64(a - 0.5) * log(t)))
	t_2 = fma(Float64(a - 0.5), log(t), Float64(-t))
	tmp = 0.0
	if (t_1 <= -5000000000.0)
		tmp = t_2;
	elseif (t_1 <= 720.0)
		tmp = Float64(log(Float64(Float64(y * z) * Float64(1.0 / sqrt(t)))) - t);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + (-t)), $MachinePrecision]}, If[LessEqual[t$95$1, -5000000000.0], t$95$2, If[LessEqual[t$95$1, 720.0], N[(N[Log[N[(N[(y * z), $MachinePrecision] * N[(1.0 / N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < -5e9 or 720 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t)))

   1. Initial program 99.8%

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

   2. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 91.8

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

   4. Applied rewrites 91.8%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-log.f64 91.8

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

      9. associate--l+ 91.8

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

      10. +-commutative 91.8

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

   6. Applied rewrites 91.8%

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

   if -5e9 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 720

   1. Initial program 98.9%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. associate-+r+ N/A

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

      3. sum-log N/A

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

      4. *-commutative N/A

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

      5. log-pow-rev N/A

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

      6. sum-log N/A

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

      7. lower-log.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lift--.f64 49.5

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

   4. Applied rewrites 49.5%

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

   5. Taylor expanded in a around 0

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

      1. sqrt-div N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 48.3

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

   7. Applied rewrites 48.3%

      \[\leadsto \log \left(\left(y \cdot z\right) \cdot \frac{1}{\sqrt{t}}\right) - t \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 80.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ t_2 := \mathsf{fma}\left(a - 0.5, \log t, -t\right)\\ \mathbf{if}\;t\_1 \leq -10000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 700:\\ \;\;\;\;\log \left(y \cdot \left(z \cdot \frac{1}{\sqrt{t}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))
        (t_2 (fma (- a 0.5) (log t) (- t))))
   (if (<= t_1 -10000.0)
     t_2
     (if (<= t_1 700.0) (log (* y (* z (/ 1.0 (sqrt t))))) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
	double t_2 = fma((a - 0.5), log(t), -t);
	double tmp;
	if (t_1 <= -10000.0) {
		tmp = t_2;
	} else if (t_1 <= 700.0) {
		tmp = log((y * (z * (1.0 / sqrt(t)))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + Float64(Float64(a - 0.5) * log(t)))
	t_2 = fma(Float64(a - 0.5), log(t), Float64(-t))
	tmp = 0.0
	if (t_1 <= -10000.0)
		tmp = t_2;
	elseif (t_1 <= 700.0)
		tmp = log(Float64(y * Float64(z * Float64(1.0 / sqrt(t)))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + (-t)), $MachinePrecision]}, If[LessEqual[t$95$1, -10000.0], t$95$2, If[LessEqual[t$95$1, 700.0], N[Log[N[(y * N[(z * N[(1.0 / N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < -1e4 or 700 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t)))

   1. Initial program 99.8%

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

   2. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 91.1

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

   4. Applied rewrites 91.1%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-log.f64 91.1

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

      9. associate--l+ 91.1

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

      10. +-commutative 91.1

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

   6. Applied rewrites 91.1%

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

   if -1e4 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 700

   1. Initial program 98.9%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. associate-+r+ N/A

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

      3. sum-log N/A

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

      4. *-commutative N/A

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

      5. log-pow-rev N/A

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

      6. sum-log N/A

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

      7. lower-log.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lift--.f64 50.8

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

   4. Applied rewrites 50.8%

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

   5. Taylor expanded in t around 0

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

      1. lower-log.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lower-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift--.f64 48.6

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

   7. Applied rewrites 48.6%

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

   8. Taylor expanded in a around 0

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

      1. sqrt-div N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 47.5

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

   10. Applied rewrites 47.5%

       \[\leadsto \log \left(y \cdot \left(z \cdot \frac{1}{\sqrt{t}}\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 77.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ t_2 := \mathsf{fma}\left(a - 0.5, \log t, -t\right)\\ \mathbf{if}\;t\_1 \leq -10000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 720:\\ \;\;\;\;\log \left(\frac{1}{\sqrt{t}} \cdot \left(y \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))
        (t_2 (fma (- a 0.5) (log t) (- t))))
   (if (<= t_1 -10000.0)
     t_2
     (if (<= t_1 720.0) (log (* (/ 1.0 (sqrt t)) (* y z))) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
	double t_2 = fma((a - 0.5), log(t), -t);
	double tmp;
	if (t_1 <= -10000.0) {
		tmp = t_2;
	} else if (t_1 <= 720.0) {
		tmp = log(((1.0 / sqrt(t)) * (y * z)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + Float64(Float64(a - 0.5) * log(t)))
	t_2 = fma(Float64(a - 0.5), log(t), Float64(-t))
	tmp = 0.0
	if (t_1 <= -10000.0)
		tmp = t_2;
	elseif (t_1 <= 720.0)
		tmp = log(Float64(Float64(1.0 / sqrt(t)) * Float64(y * z)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + (-t)), $MachinePrecision]}, If[LessEqual[t$95$1, -10000.0], t$95$2, If[LessEqual[t$95$1, 720.0], N[Log[N[(N[(1.0 / N[Sqrt[t], $MachinePrecision]), $MachinePrecision] * N[(y * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < -1e4 or 720 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t)))

   1. Initial program 99.8%

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

   2. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 91.3

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

   4. Applied rewrites 91.3%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-log.f64 91.3

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

      9. associate--l+ 91.3

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

      10. +-commutative 91.3

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

   6. Applied rewrites 91.3%

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

   if -1e4 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 720

   1. Initial program 98.9%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. associate-+r+ N/A

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

      3. sum-log N/A

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

      4. *-commutative N/A

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

      5. log-pow-rev N/A

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

      6. sum-log N/A

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

      7. lower-log.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lift--.f64 50.5

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

   4. Applied rewrites 50.5%

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

   5. Taylor expanded in t around 0

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

      1. lower-log.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lower-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift--.f64 48.3

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

   7. Applied rewrites 48.3%

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

   8. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 47.8

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

   10. Applied rewrites 47.8%

       \[\leadsto \log \left(\frac{1}{\sqrt{t}} \cdot \left(y \cdot z\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 68.6% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 77.2

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

4. Applied rewrites 77.2%

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

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-log.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lift--.f64 N/A

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

   8. lift-log.f64 77.2

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

   9. associate--l+ 77.2

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

   10. +-commutative 77.2

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

6. Applied rewrites 77.2%

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

Alternative 17: 61.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.4 \cdot 10^{+47}:\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 1.4e+47) (* (log t) a) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 1.4e+47) {
		tmp = log(t) * a;
	} else {
		tmp = -t;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 1.4d+47) then
        tmp = log(t) * a
    else
        tmp = -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 1.4e+47) {
		tmp = Math.log(t) * a;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 1.4e+47:
		tmp = math.log(t) * a
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 1.4e+47)
		tmp = Float64(log(t) * a);
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 1.4e+47)
		tmp = log(t) * a;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 1.4e+47], N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if t < 1.39999999999999994e47

   1. Initial program 99.4%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 50.1

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

   4. Applied rewrites 50.1%

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

   if 1.39999999999999994e47 < t

   1. Initial program 99.9%

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

   2. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 78.0

         \[\leadsto -t \]

   4. Applied rewrites 78.0%

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

Alternative 18: 38.0% 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. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 38.0

      \[\leadsto -t \]

4. Applied rewrites 38.0%

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

Reproduce

herbie shell --seed 2025101 
(FPCore (x y z t a)
  :name "Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2"
  :precision binary64
  (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))