Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 7.8s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

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

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: 83.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 68.9%

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

Alternative 3: 83.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(x + y\right) + \log z\\ \mathbf{if}\;t\_1 \leq 720:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log \left(z \cdot \left(y + x\right)\right) - t\right)\\ \mathbf{elif}\;t\_1 \leq 861:\\ \;\;\;\;\left(\log \left(\left|x\right| \cdot z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\log y + \log z\right) - t\right) + -0.5 \cdot \log t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log (+ x y)) (log z))))
   (if (<= t_1 720.0)
     (fma (- a 0.5) (log t) (- (log (* z (+ y x))) t))
     (if (<= t_1 861.0)
       (+ (- (log (* (fabs x) z)) t) (* (- a 0.5) (log t)))
       (+ (- (+ (log y) (log z)) t) (* -0.5 (log t)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((x + y)) + log(z);
	double tmp;
	if (t_1 <= 720.0) {
		tmp = fma((a - 0.5), log(t), (log((z * (y + x))) - t));
	} else if (t_1 <= 861.0) {
		tmp = (log((fabs(x) * z)) - t) + ((a - 0.5) * log(t));
	} else {
		tmp = ((log(y) + log(z)) - t) + (-0.5 * log(t));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(Float64(x + y)) + log(z))
	tmp = 0.0
	if (t_1 <= 720.0)
		tmp = fma(Float64(a - 0.5), log(t), Float64(log(Float64(z * Float64(y + x))) - t));
	elseif (t_1 <= 861.0)
		tmp = Float64(Float64(log(Float64(abs(x) * z)) - t) + Float64(Float64(a - 0.5) * log(t)));
	else
		tmp = Float64(Float64(Float64(log(y) + log(z)) - t) + Float64(-0.5 * log(t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 720.0], N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[(N[Log[N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 861.0], N[(N[(N[Log[N[(N[Abs[x], $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Log[y], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + N[(-0.5 * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. 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

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift-log.f64 N/A

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

      16. lower--.f64 N/A

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

   3. Applied rewrites 76.0%

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

   if 720 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 861

   1. Initial program 99.6%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. sum-log N/A

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

      3. lower-log.f64 N/A

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

      4. lower-*.f64 53.6

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

   4. Applied rewrites 53.6%

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

      1. lift-*.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. sum-log N/A

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

      4. +-commutative N/A

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

      5. log-fabs N/A

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

      6. sum-log N/A

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

      7. lower-log.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-fabs.f64 68.1

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

   6. Applied rewrites 68.1%

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

   if 861 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z))

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 68.9%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 41.6%

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

Alternative 4: 82.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(x + y\right) + \log z\\ \mathbf{if}\;t\_1 \leq 720:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log \left(z \cdot \left(y + x\right)\right) - t\right)\\ \mathbf{elif}\;t\_1 \leq 915:\\ \;\;\;\;\left(\log \left(\left|x\right| \cdot z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log t, a - 0.5, \log y\right) + \log z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log (+ x y)) (log z))))
   (if (<= t_1 720.0)
     (fma (- a 0.5) (log t) (- (log (* z (+ y x))) t))
     (if (<= t_1 915.0)
       (+ (- (log (* (fabs x) z)) t) (* (- a 0.5) (log t)))
       (+ (fma (log t) (- a 0.5) (log y)) (log z))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((x + y)) + log(z);
	double tmp;
	if (t_1 <= 720.0) {
		tmp = fma((a - 0.5), log(t), (log((z * (y + x))) - t));
	} else if (t_1 <= 915.0) {
		tmp = (log((fabs(x) * z)) - t) + ((a - 0.5) * log(t));
	} else {
		tmp = fma(log(t), (a - 0.5), log(y)) + log(z);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(Float64(x + y)) + log(z))
	tmp = 0.0
	if (t_1 <= 720.0)
		tmp = fma(Float64(a - 0.5), log(t), Float64(log(Float64(z * Float64(y + x))) - t));
	elseif (t_1 <= 915.0)
		tmp = Float64(Float64(log(Float64(abs(x) * z)) - t) + Float64(Float64(a - 0.5) * log(t)));
	else
		tmp = Float64(fma(log(t), Float64(a - 0.5), log(y)) + log(z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 720.0], N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[(N[Log[N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 915.0], N[(N[(N[Log[N[(N[Abs[x], $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. 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

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift-log.f64 N/A

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

      16. lower--.f64 N/A

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

   3. Applied rewrites 76.0%

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

   if 720 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 915

   1. Initial program 99.6%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. sum-log N/A

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

      3. lower-log.f64 N/A

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

      4. lower-*.f64 53.6

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

   4. Applied rewrites 53.6%

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

      1. lift-*.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. sum-log N/A

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

      4. +-commutative N/A

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

      5. log-fabs N/A

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

      6. sum-log N/A

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

      7. lower-log.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-fabs.f64 68.1

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

   6. Applied rewrites 68.1%

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

   if 915 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z))

   1. Initial program 99.6%

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

   2. Taylor expanded in t around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. sum-log N/A

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

      6. lower-log.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. lift--.f64 47.5

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

   4. Applied rewrites 47.5%

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

      1. lift-+.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. sum-log N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lift--.f64 N/A

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

      10. associate-+r+ N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-log.f64 N/A

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

      16. lift--.f64 N/A

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

      17. lower-log.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-+.f64 N/A

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

      20. lift-log.f64 62.0

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

   6. Applied rewrites 62.0%

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

   7. Taylor expanded in x around 0

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

      1. lower-log.f64 41.4

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

   9. Applied rewrites 41.4%

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

Alternative 5: 68.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(x + y\right) + \log z \leq 720:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log \left(z \cdot \left(y + x\right)\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(\left|x\right| \cdot z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (+ (log (+ x y)) (log z)) 720.0)
   (fma (- a 0.5) (log t) (- (log (* z (+ y x))) t))
   (+ (- (log (* (fabs x) z)) t) (* (- a 0.5) (log t)))))

C

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

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(log(Float64(x + y)) + log(z)) <= 720.0)
		tmp = fma(Float64(a - 0.5), log(t), Float64(log(Float64(z * Float64(y + x))) - t));
	else
		tmp = Float64(Float64(log(Float64(abs(x) * z)) - t) + Float64(Float64(a - 0.5) * log(t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision], 720.0], 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[(N[Abs[x], $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. 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

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift-log.f64 N/A

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

      16. lower--.f64 N/A

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

   3. Applied rewrites 76.0%

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

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

   1. Initial program 99.6%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. sum-log N/A

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

      3. lower-log.f64 N/A

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

      4. lower-*.f64 53.6

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

   4. Applied rewrites 53.6%

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

      1. lift-*.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. sum-log N/A

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

      4. +-commutative N/A

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

      5. log-fabs N/A

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

      6. sum-log N/A

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

      7. lower-log.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-fabs.f64 68.1

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

   6. Applied rewrites 68.1%

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

Alternative 6: 65.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(x + y\right) + \log z \leq 720:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log \left(z \cdot y\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(\left|x\right| \cdot z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (+ (log (+ x y)) (log z)) 720.0)
   (fma (- a 0.5) (log t) (- (log (* z y)) t))
   (+ (- (log (* (fabs x) z)) t) (* (- a 0.5) (log t)))))

C

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

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(log(Float64(x + y)) + log(z)) <= 720.0)
		tmp = fma(Float64(a - 0.5), log(t), Float64(log(Float64(z * y)) - t));
	else
		tmp = Float64(Float64(log(Float64(abs(x) * z)) - t) + Float64(Float64(a - 0.5) * log(t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision], 720.0], N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[(N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[N[(N[Abs[x], $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 68.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-log.f64 68.9

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

      11. lift-+.f64 N/A

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

      12. lift-log.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-log.f64 N/A

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

      15. sum-log N/A

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

      16. lower-log.f64 N/A

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

      17. lower-*.f64 53.0

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

   6. Applied rewrites 53.0%

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

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

   1. Initial program 99.6%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. sum-log N/A

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

      3. lower-log.f64 N/A

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

      4. lower-*.f64 53.6

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

   4. Applied rewrites 53.6%

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

      1. lift-*.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. sum-log N/A

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

      4. +-commutative N/A

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

      5. log-fabs N/A

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

      6. sum-log N/A

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

      7. lower-log.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-fabs.f64 68.1

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

   6. Applied rewrites 68.1%

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

Alternative 7: 65.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(x + y\right) + \log z\\ \mathbf{if}\;t\_1 \leq 720:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log \left(z \cdot y\right) - t\right)\\ \mathbf{elif}\;t\_1 \leq 780:\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log (+ x y)) (log z))))
   (if (<= t_1 720.0)
     (fma (- a 0.5) (log t) (- (log (* z y)) t))
     (if (<= t_1 780.0) (* (log t) a) (- t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((x + y)) + log(z);
	double tmp;
	if (t_1 <= 720.0) {
		tmp = fma((a - 0.5), log(t), (log((z * y)) - t));
	} else if (t_1 <= 780.0) {
		tmp = log(t) * a;
	} else {
		tmp = -t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(Float64(x + y)) + log(z))
	tmp = 0.0
	if (t_1 <= 720.0)
		tmp = fma(Float64(a - 0.5), log(t), Float64(log(Float64(z * y)) - t));
	elseif (t_1 <= 780.0)
		tmp = Float64(log(t) * a);
	else
		tmp = Float64(-t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 720.0], N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[(N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 780.0], N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision], (-t)]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 68.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-log.f64 68.9

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

      11. lift-+.f64 N/A

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

      12. lift-log.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-log.f64 N/A

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

      15. sum-log N/A

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

      16. lower-log.f64 N/A

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

      17. lower-*.f64 53.0

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

   6. Applied rewrites 53.0%

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

   if 720 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 780

   1. Initial program 99.6%

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

   2. Taylor expanded in a around inf

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

      1. *-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 37.9

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

   4. Applied rewrites 37.9%

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

   if 780 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z))

   1. Initial program 99.6%

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

   2. Taylor expanded in 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 38.5

         \[\leadsto -t \]

   4. Applied rewrites 38.5%

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

Alternative 8: 61.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log t \cdot a\\ \mathbf{if}\;a - 0.5 \leq -5 \cdot 10^{+21}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a - 0.5 \leq 5 \cdot 10^{+98}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log t, \log \left(\left|y\right| \cdot z\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (log t) a)))
   (if (<= (- a 0.5) -5e+21)
     t_1
     (if (<= (- a 0.5) 5e+98)
       (fma -0.5 (log t) (- (log (* (fabs y) z)) t))
       t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log(t) * a;
	double tmp;
	if ((a - 0.5) <= -5e+21) {
		tmp = t_1;
	} else if ((a - 0.5) <= 5e+98) {
		tmp = fma(-0.5, log(t), (log((fabs(y) * z)) - t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(t) * a)
	tmp = 0.0
	if (Float64(a - 0.5) <= -5e+21)
		tmp = t_1;
	elseif (Float64(a - 0.5) <= 5e+98)
		tmp = fma(-0.5, log(t), Float64(log(Float64(abs(y) * z)) - t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[N[(a - 0.5), $MachinePrecision], -5e+21], t$95$1, If[LessEqual[N[(a - 0.5), $MachinePrecision], 5e+98], N[(-0.5 * N[Log[t], $MachinePrecision] + N[(N[Log[N[(N[Abs[y], $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 a #s(literal 1/2 binary64)) < -5e21 or 4.9999999999999998e98 < (-.f64 a #s(literal 1/2 binary64))

   1. Initial program 99.6%

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

   2. Taylor expanded in a around inf

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

      1. *-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 37.9

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

   4. Applied rewrites 37.9%

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

   if -5e21 < (-.f64 a #s(literal 1/2 binary64)) < 4.9999999999999998e98

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 68.9%

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

      1. lift-+.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. log-fabs N/A

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

      4. lift-log.f64 N/A

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

      5. sum-log N/A

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

      6. lower-log.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-fabs.f64 68.2

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

   6. Applied rewrites 68.2%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-log.f64 68.2

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

   8. Applied rewrites 68.2%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 40.6%

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

Alternative 9: 59.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 59000:\\ \;\;\;\;\mathsf{fma}\left(\log t, a - 0.5, \log \left(z \cdot y\right)\right)\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{+90}:\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 59000.0)
   (fma (log t) (- a 0.5) (log (* z y)))
   (if (<= t 8.5e+90) (* (log t) a) (- t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 59000.0) {
		tmp = fma(log(t), (a - 0.5), log((z * y)));
	} else if (t <= 8.5e+90) {
		tmp = log(t) * a;
	} else {
		tmp = -t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 59000.0)
		tmp = fma(log(t), Float64(a - 0.5), log(Float64(z * y)));
	elseif (t <= 8.5e+90)
		tmp = Float64(log(t) * a);
	else
		tmp = Float64(-t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 59000.0], N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision] + N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 8.5e+90], N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision], (-t)]]

Derivation

1. Split input into 3 regimes

2. if t < 59000

   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 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. sum-log N/A

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

      6. lower-log.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. lift--.f64 47.5

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

   4. Applied rewrites 47.5%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 32.0

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

   7. Applied rewrites 32.0%

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

   if 59000 < t < 8.5000000000000002e90

   1. Initial program 99.6%

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

   2. Taylor expanded in a around inf

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

      1. *-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 37.9

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

   4. Applied rewrites 37.9%

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

   if 8.5000000000000002e90 < t

   1. Initial program 99.6%

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

   2. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 38.5

         \[\leadsto -t \]

   4. Applied rewrites 38.5%

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

Alternative 10: 57.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ \mathbf{if}\;t\_1 \leq -500:\\ \;\;\;\;-t\\ \mathbf{elif}\;t\_1 \leq 1050:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log t, \log \left(z \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t)))))
   (if (<= t_1 -500.0)
     (- t)
     (if (<= t_1 1050.0) (fma -0.5 (log t) (log (* z y))) (* (log t) a)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
	double tmp;
	if (t_1 <= -500.0) {
		tmp = -t;
	} else if (t_1 <= 1050.0) {
		tmp = fma(-0.5, log(t), log((z * y)));
	} else {
		tmp = log(t) * a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + Float64(Float64(a - 0.5) * log(t)))
	tmp = 0.0
	if (t_1 <= -500.0)
		tmp = Float64(-t);
	elseif (t_1 <= 1050.0)
		tmp = fma(-0.5, log(t), log(Float64(z * y)));
	else
		tmp = Float64(log(t) * a);
	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]}, If[LessEqual[t$95$1, -500.0], (-t), If[LessEqual[t$95$1, 1050.0], N[(-0.5 * N[Log[t], $MachinePrecision] + N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   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 38.5

         \[\leadsto -t \]

   4. Applied rewrites 38.5%

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

   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))) < 1050

   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 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. sum-log N/A

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

      6. lower-log.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. lift--.f64 47.5

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

   4. Applied rewrites 47.5%

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

      1. lift-+.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. sum-log N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lift--.f64 N/A

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

      10. associate-+r+ N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-log.f64 N/A

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

      16. lift--.f64 N/A

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

      17. lower-log.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-+.f64 N/A

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

      20. lift-log.f64 62.0

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

   6. Applied rewrites 62.0%

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

   7. Taylor expanded in a around 0

      \[\leadsto \log z + \color{blue}{\left(\log \left(x + y\right) + \frac{-1}{2} \cdot \log t\right)} \]
   8. 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. lower-log.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 20.6

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

   9. Applied rewrites 20.6%

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

   10. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 11.9

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

   12. Applied rewrites 11.9%

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

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

   1. Initial program 99.6%

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

   2. Taylor expanded in a around inf

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

      1. *-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 37.9

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

   4. Applied rewrites 37.9%

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

Alternative 11: 56.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log t \cdot a\\ \mathbf{if}\;a - 0.5 \leq -2000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a - 0.5 \leq 5 \cdot 10^{+98}:\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (log t) a)))
   (if (<= (- a 0.5) -2000000.0) t_1 (if (<= (- a 0.5) 5e+98) (- t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log(t) * a;
	double tmp;
	if ((a - 0.5) <= -2000000.0) {
		tmp = t_1;
	} else if ((a - 0.5) <= 5e+98) {
		tmp = -t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = log(t) * a
    if ((a - 0.5d0) <= (-2000000.0d0)) then
        tmp = t_1
    else if ((a - 0.5d0) <= 5d+98) then
        tmp = -t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = Math.log(t) * a;
	double tmp;
	if ((a - 0.5) <= -2000000.0) {
		tmp = t_1;
	} else if ((a - 0.5) <= 5e+98) {
		tmp = -t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = math.log(t) * a
	tmp = 0
	if (a - 0.5) <= -2000000.0:
		tmp = t_1
	elif (a - 0.5) <= 5e+98:
		tmp = -t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(t) * a)
	tmp = 0.0
	if (Float64(a - 0.5) <= -2000000.0)
		tmp = t_1;
	elseif (Float64(a - 0.5) <= 5e+98)
		tmp = Float64(-t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = log(t) * a;
	tmp = 0.0;
	if ((a - 0.5) <= -2000000.0)
		tmp = t_1;
	elseif ((a - 0.5) <= 5e+98)
		tmp = -t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[N[(a - 0.5), $MachinePrecision], -2000000.0], t$95$1, If[LessEqual[N[(a - 0.5), $MachinePrecision], 5e+98], (-t), t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 a #s(literal 1/2 binary64)) < -2e6 or 4.9999999999999998e98 < (-.f64 a #s(literal 1/2 binary64))

   1. Initial program 99.6%

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

   2. Taylor expanded in a around inf

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

      1. *-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 37.9

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

   4. Applied rewrites 37.9%

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

   if -2e6 < (-.f64 a #s(literal 1/2 binary64)) < 4.9999999999999998e98

   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 38.5

         \[\leadsto -t \]

   4. Applied rewrites 38.5%

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

Alternative 12: 38.5% accurate, 17.6× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 38.5

      \[\leadsto -t \]

4. Applied rewrites 38.5%

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

Reproduce

herbie shell --seed 2025136 
(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))))