Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 8.4s

Alternatives: 21

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, 0.8× speedup

Math

\[\begin{array}{l} \\ \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(\left(1 - \frac{0.5}{a}\right) \cdot a\right) \cdot \log t \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (+ (- (+ (log (+ x y)) (log z)) t) (* (* (- 1.0 (/ 0.5 a)) a) (log t))))

C

double code(double x, double y, double z, double t, double a) {
	return ((log((x + y)) + log(z)) - t) + (((1.0 - (0.5 / a)) * a) * 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) + (((1.0d0 - (0.5d0 / a)) * a) * 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) + (((1.0 - (0.5 / a)) * a) * Math.log(t));
}

Python

def code(x, y, z, t, a):
	return ((math.log((x + y)) + math.log(z)) - t) + (((1.0 - (0.5 / a)) * a) * math.log(t))

Julia

function code(x, y, z, t, a)
	return Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + Float64(Float64(Float64(1.0 - Float64(0.5 / a)) * a) * log(t)))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = ((log((x + y)) + log(z)) - t) + (((1.0 - (0.5 / a)) * a) * 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[(N[(1.0 - N[(0.5 / a), $MachinePrecision]), $MachinePrecision] * a), $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 a around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. associate-*r/ N/A

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

   5. metadata-eval N/A

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

   6. lower-/.f64 99.6

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

4. Applied rewrites 99.6%

   \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \color{blue}{\left(\left(1 - \frac{0.5}{a}\right) \cdot a\right)} \cdot \log t \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in a around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. associate-*r/ N/A

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

   5. metadata-eval N/A

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

   6. lower-/.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in z around inf

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

   1. lower--.f64 N/A

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

7. Applied rewrites 99.6%

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

Alternative 3: 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 4: 93.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in a around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. associate-*r/ N/A

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

   5. metadata-eval N/A

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

   6. lower-/.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in y around inf

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

   1. lower--.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lift-log.f64 N/A

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

   7. lift--.f64 N/A

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

   8. mul-1-neg N/A

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

   9. lower-neg.f64 N/A

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

   10. log-rec N/A

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

   11. lower-neg.f64 N/A

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

   12. lower-log.f64 N/A

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

   13. lift-log.f64 68.9

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

7. Applied rewrites 68.9%

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

   1. lift-neg.f64 N/A

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

   2. lift-log.f64 N/A

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

   3. lift-neg.f64 N/A

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

   4. remove-double-neg N/A

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

   5. lift-log.f64 68.9

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

9. Applied rewrites 68.9%

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

Alternative 5: 91.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in a around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. associate-*r/ N/A

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

   5. metadata-eval N/A

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

   6. lower-/.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in y around inf

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

   1. lower--.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lift-log.f64 N/A

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

   7. lift--.f64 N/A

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

   8. mul-1-neg N/A

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

   9. lower-neg.f64 N/A

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

   10. log-rec N/A

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

   11. lower-neg.f64 N/A

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

   12. lower-log.f64 N/A

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

   13. lift-log.f64 68.9

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

7. Applied rewrites 68.9%

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

   1. lift-neg.f64 N/A

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

   2. lift-log.f64 N/A

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

   3. lift-neg.f64 N/A

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

   4. remove-double-neg N/A

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

   5. lift-log.f64 68.9

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

9. Applied rewrites 68.9%

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

    1. lift--.f64 N/A

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

    2. lift--.f64 N/A

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

    3. lift-fma.f64 N/A

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

    4. lift-log.f64 N/A

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

    5. lift-log.f64 N/A

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

    6. lift-log.f64 N/A

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

    7. lower-+.f64 N/A

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

    8. associate-+l+ N/A

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

    9. sum-log N/A

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

    10. +-commutative N/A

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

    11. sum-log N/A

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

    12. associate-+r+ N/A

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

    13. associate--l+ N/A

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

    14. lower-+.f64 N/A

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

    15. lift-log.f64 N/A

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

    16. lower--.f64 N/A

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

11. Applied rewrites 68.9%

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

Alternative 6: 84.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 7.8 \cdot 10^{-5}:\\ \;\;\;\;\left(\log z + \log y\right) + \log t \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\log t, a, \log y\right) + \log z\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 7.8e-5)
   (+ (+ (log z) (log y)) (* (log t) (- a 0.5)))
   (- (+ (fma (log t) a (log y)) (log z)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 7.8e-5) {
		tmp = (log(z) + log(y)) + (log(t) * (a - 0.5));
	} else {
		tmp = (fma(log(t), a, log(y)) + log(z)) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 7.8e-5)
		tmp = Float64(Float64(log(z) + log(y)) + Float64(log(t) * Float64(a - 0.5)));
	else
		tmp = Float64(Float64(fma(log(t), a, log(y)) + log(z)) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 7.8e-5], N[(N[(N[Log[z], $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Log[t], $MachinePrecision] * a + N[Log[y], $MachinePrecision]), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 7.7999999999999999e-5

   1. Initial program 99.6%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. log-rec N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. lift-log.f64 68.9

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

   7. Applied rewrites 68.9%

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

   8. Taylor expanded in t around 0

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

      1. associate-+r+ N/A

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

      2. log-prod N/A

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

      3. lower-+.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 31.9

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

   10. Applied rewrites 31.9%

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

       1. lift-*.f64 N/A

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

       2. lift-log.f64 N/A

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

       3. sum-log N/A

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

       4. +-commutative N/A

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

       5. lower-+.f64 N/A

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

       6. lift-log.f64 N/A

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

       7. lift-log.f64 41.3

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

   12. Applied rewrites 41.3%

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

   if 7.7999999999999999e-5 < 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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. log-rec N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. lift-log.f64 68.9

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

   7. Applied rewrites 68.9%

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

   8. Taylor expanded in a around inf

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

   10. Applied rewrites 59.1%

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

       1. lift-neg.f64 N/A

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

       2. lift-log.f64 N/A

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

       3. lift-neg.f64 N/A

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

       4. remove-double-neg N/A

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

       5. lift-log.f64 59.1

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

   12. Applied rewrites 59.1%

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

Alternative 7: 84.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ t_2 := \left(\mathsf{fma}\left(\log t, a, \log y\right) + \log z\right) - t\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+16}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2000:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log t, \log z + \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))
        (t_2 (- (+ (fma (log t) a (log y)) (log z)) t)))
   (if (<= t_1 -4e+16)
     t_2
     (if (<= t_1 2000.0) (- (fma -0.5 (log t) (+ (log z) (log y))) t) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
	double t_2 = (fma(log(t), a, log(y)) + log(z)) - t;
	double tmp;
	if (t_1 <= -4e+16) {
		tmp = t_2;
	} else if (t_1 <= 2000.0) {
		tmp = fma(-0.5, log(t), (log(z) + log(y))) - t;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + Float64(Float64(a - 0.5) * log(t)))
	t_2 = Float64(Float64(fma(log(t), a, log(y)) + log(z)) - t)
	tmp = 0.0
	if (t_1 <= -4e+16)
		tmp = t_2;
	elseif (t_1 <= 2000.0)
		tmp = Float64(fma(-0.5, log(t), Float64(log(z) + log(y))) - t);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Log[t], $MachinePrecision] * a + N[Log[y], $MachinePrecision]), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+16], t$95$2, If[LessEqual[t$95$1, 2000.0], N[(N[(-0.5 * N[Log[t], $MachinePrecision] + N[(N[Log[z], $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < -4e16 or 2e3 < (+.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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. log-rec N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. lift-log.f64 68.9

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

   7. Applied rewrites 68.9%

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

   8. Taylor expanded in a around inf

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

   10. Applied rewrites 59.1%

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

       1. lift-neg.f64 N/A

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

       2. lift-log.f64 N/A

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

       3. lift-neg.f64 N/A

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

       4. remove-double-neg N/A

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

       5. lift-log.f64 59.1

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

   12. Applied rewrites 59.1%

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

   if -4e16 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 2e3

   1. Initial program 99.6%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. sum-log N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-log.f64 N/A

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

      9. *-commutative N/A

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

      10. +-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 46.8

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

   7. Applied rewrites 46.8%

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

   8. Taylor expanded in y around inf

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

      1. neg-log N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-log.f64 40.7

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

   10. Applied rewrites 40.7%

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

Alternative 8: 83.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\mathsf{fma}\left(\log t, a, \log y\right) + \log z\right) - t\\ \mathbf{if}\;a \leq -0.5:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 0.245:\\ \;\;\;\;\left(\mathsf{fma}\left(\log t, -0.5, \log y\right) + \log z\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (+ (fma (log t) a (log y)) (log z)) t)))
   (if (<= a -0.5)
     t_1
     (if (<= a 0.245) (- (+ (fma (log t) -0.5 (log y)) (log z)) t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (fma(log(t), a, log(y)) + log(z)) - t;
	double tmp;
	if (a <= -0.5) {
		tmp = t_1;
	} else if (a <= 0.245) {
		tmp = (fma(log(t), -0.5, log(y)) + log(z)) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(fma(log(t), a, log(y)) + log(z)) - t)
	tmp = 0.0
	if (a <= -0.5)
		tmp = t_1;
	elseif (a <= 0.245)
		tmp = Float64(Float64(fma(log(t), -0.5, log(y)) + log(z)) - t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(N[Log[t], $MachinePrecision] * a + N[Log[y], $MachinePrecision]), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[a, -0.5], t$95$1, If[LessEqual[a, 0.245], N[(N[(N[(N[Log[t], $MachinePrecision] * -0.5 + N[Log[y], $MachinePrecision]), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -0.5 or 0.245 < a

   1. Initial program 99.6%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. log-rec N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. lift-log.f64 68.9

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

   7. Applied rewrites 68.9%

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

   8. Taylor expanded in a around inf

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

   10. Applied rewrites 59.1%

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

       1. lift-neg.f64 N/A

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

       2. lift-log.f64 N/A

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

       3. lift-neg.f64 N/A

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

       4. remove-double-neg N/A

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

       5. lift-log.f64 59.1

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

   12. Applied rewrites 59.1%

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

   if -0.5 < a < 0.245

   1. Initial program 99.6%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. log-rec N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. lift-log.f64 68.9

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

   7. Applied rewrites 68.9%

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

      1. lift-neg.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. remove-double-neg N/A

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

      5. lift-log.f64 68.9

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

   9. Applied rewrites 68.9%

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

   10. Taylor expanded in a around 0

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

   12. Applied rewrites 40.7%

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

Alternative 9: 80.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \log t\\ \mathbf{if}\;a \leq -0.85:\\ \;\;\;\;\left(t\_1 + \log \left(y + x\right)\right) - t\\ \mathbf{elif}\;a \leq 0.245:\\ \;\;\;\;\left(\mathsf{fma}\left(\log t, -0.5, \log y\right) + \log z\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 + \log z\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* a (log t))))
   (if (<= a -0.85)
     (- (+ t_1 (log (+ y x))) t)
     (if (<= a 0.245)
       (- (+ (fma (log t) -0.5 (log y)) (log z)) t)
       (- (+ t_1 (log z)) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = a * log(t);
	double tmp;
	if (a <= -0.85) {
		tmp = (t_1 + log((y + x))) - t;
	} else if (a <= 0.245) {
		tmp = (fma(log(t), -0.5, log(y)) + log(z)) - t;
	} else {
		tmp = (t_1 + log(z)) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(a * log(t))
	tmp = 0.0
	if (a <= -0.85)
		tmp = Float64(Float64(t_1 + log(Float64(y + x))) - t);
	elseif (a <= 0.245)
		tmp = Float64(Float64(fma(log(t), -0.5, log(y)) + log(z)) - t);
	else
		tmp = Float64(Float64(t_1 + log(z)) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -0.85], N[(N[(t$95$1 + N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[a, 0.245], N[(N[(N[(N[Log[t], $MachinePrecision] * -0.5 + N[Log[y], $MachinePrecision]), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(t$95$1 + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -0.849999999999999978

   1. Initial program 99.6%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lift-log.f64 78.4

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

   10. Applied rewrites 78.4%

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

   if -0.849999999999999978 < a < 0.245

   1. Initial program 99.6%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. log-rec N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. lift-log.f64 68.9

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

   7. Applied rewrites 68.9%

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

      1. lift-neg.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. remove-double-neg N/A

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

      5. lift-log.f64 68.9

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

   9. Applied rewrites 68.9%

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

   10. Taylor expanded in a around 0

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

   12. Applied rewrites 40.7%

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

   if 0.245 < a

   1. Initial program 99.6%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. log-rec N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. lift-log.f64 68.9

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

   7. Applied rewrites 68.9%

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

   8. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lift-log.f64 77.9

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

   10. Applied rewrites 77.9%

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

Alternative 10: 78.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(x + y\right) + \log z\\ t_2 := \left(a \cdot \log t + \log z\right) - t\\ \mathbf{if}\;t\_1 \leq -750:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 720:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log \left(z \cdot \left(y + x\right)\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log (+ x y)) (log z))) (t_2 (- (+ (* a (log t)) (log z)) t)))
   (if (<= t_1 -750.0)
     t_2
     (if (<= t_1 720.0)
       (fma (- a 0.5) (log t) (- (log (* z (+ y x))) t))
       t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((x + y)) + log(z);
	double t_2 = ((a * log(t)) + log(z)) - t;
	double tmp;
	if (t_1 <= -750.0) {
		tmp = t_2;
	} else if (t_1 <= 720.0) {
		tmp = fma((a - 0.5), log(t), (log((z * (y + x))) - t));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(Float64(x + y)) + log(z))
	t_2 = Float64(Float64(Float64(a * log(t)) + log(z)) - t)
	tmp = 0.0
	if (t_1 <= -750.0)
		tmp = t_2;
	elseif (t_1 <= 720.0)
		tmp = fma(Float64(a - 0.5), log(t), Float64(log(Float64(z * Float64(y + x))) - t));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -750 or 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 a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. log-rec N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. lift-log.f64 68.9

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

   7. Applied rewrites 68.9%

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

   8. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lift-log.f64 77.9

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

   10. Applied rewrites 77.9%

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

   if -750 < (+.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 75.3%

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

Alternative 11: 77.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(x + y\right) + \log z\\ t_2 := \left(a \cdot \log t + \log z\right) - t\\ \mathbf{if}\;t\_1 \leq -750:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 720:\\ \;\;\;\;\left(\log \left(z \cdot y\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log (+ x y)) (log z))) (t_2 (- (+ (* a (log t)) (log z)) t)))
   (if (<= t_1 -750.0)
     t_2
     (if (<= t_1 720.0) (+ (- (log (* z y)) t) (* (- a 0.5) (log t))) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((x + y)) + log(z);
	double t_2 = ((a * log(t)) + log(z)) - t;
	double tmp;
	if (t_1 <= -750.0) {
		tmp = t_2;
	} else if (t_1 <= 720.0) {
		tmp = (log((z * y)) - t) + ((a - 0.5) * log(t));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = log((x + y)) + log(z)
    t_2 = ((a * log(t)) + log(z)) - t
    if (t_1 <= (-750.0d0)) then
        tmp = t_2
    else if (t_1 <= 720.0d0) then
        tmp = (log((z * y)) - t) + ((a - 0.5d0) * log(t))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = Math.log((x + y)) + Math.log(z);
	double t_2 = ((a * Math.log(t)) + Math.log(z)) - t;
	double tmp;
	if (t_1 <= -750.0) {
		tmp = t_2;
	} else if (t_1 <= 720.0) {
		tmp = (Math.log((z * y)) - t) + ((a - 0.5) * Math.log(t));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = math.log((x + y)) + math.log(z)
	t_2 = ((a * math.log(t)) + math.log(z)) - t
	tmp = 0
	if t_1 <= -750.0:
		tmp = t_2
	elif t_1 <= 720.0:
		tmp = (math.log((z * y)) - t) + ((a - 0.5) * math.log(t))
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(Float64(x + y)) + log(z))
	t_2 = Float64(Float64(Float64(a * log(t)) + log(z)) - t)
	tmp = 0.0
	if (t_1 <= -750.0)
		tmp = t_2;
	elseif (t_1 <= 720.0)
		tmp = Float64(Float64(log(Float64(z * y)) - t) + Float64(Float64(a - 0.5) * log(t)));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = log((x + y)) + log(z);
	t_2 = ((a * log(t)) + log(z)) - t;
	tmp = 0.0;
	if (t_1 <= -750.0)
		tmp = t_2;
	elseif (t_1 <= 720.0)
		tmp = (log((z * y)) - t) + ((a - 0.5) * log(t));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[t$95$1, -750.0], t$95$2, If[LessEqual[t$95$1, 720.0], N[(N[(N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -750 or 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 a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. log-rec N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. lift-log.f64 68.9

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

   7. Applied rewrites 68.9%

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

   8. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lift-log.f64 77.9

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

   10. Applied rewrites 77.9%

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

   if -750 < (+.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

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

      1. +-commutative N/A

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

      2. sum-log N/A

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

      3. lower-log.f64 N/A

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

      4. lower-*.f64 52.8

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

   4. Applied rewrites 52.8%

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

Alternative 12: 75.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(x + y\right) + \log z\\ t_2 := \left(a \cdot \log t + \log z\right) - t\\ \mathbf{if}\;t\_1 \leq -750:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 720:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log \left(z \cdot y\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log (+ x y)) (log z))) (t_2 (- (+ (* a (log t)) (log z)) t)))
   (if (<= t_1 -750.0)
     t_2
     (if (<= t_1 720.0) (fma (- a 0.5) (log t) (- (log (* z y)) t)) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((x + y)) + log(z);
	double t_2 = ((a * log(t)) + log(z)) - t;
	double tmp;
	if (t_1 <= -750.0) {
		tmp = t_2;
	} else if (t_1 <= 720.0) {
		tmp = fma((a - 0.5), log(t), (log((z * y)) - t));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(Float64(x + y)) + log(z))
	t_2 = Float64(Float64(Float64(a * log(t)) + log(z)) - t)
	tmp = 0.0
	if (t_1 <= -750.0)
		tmp = t_2;
	elseif (t_1 <= 720.0)
		tmp = fma(Float64(a - 0.5), log(t), Float64(log(Float64(z * y)) - t));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -750 or 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 a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. log-rec N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. lift-log.f64 68.9

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

   7. Applied rewrites 68.9%

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

   8. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lift-log.f64 77.9

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

   10. Applied rewrites 77.9%

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

   if -750 < (+.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 75.3%

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

   4. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 52.8

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

   6. Applied rewrites 52.8%

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

Alternative 13: 68.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot \log t\\ t_2 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + t\_1\\ \mathbf{if}\;t\_2 \leq -500000000:\\ \;\;\;\;\left(-t\right) + t\_1\\ \mathbf{elif}\;t\_2 \leq 900:\\ \;\;\;\;\log \left(z \cdot \left(x + y\right)\right) - 0.5 \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \log t + \log \left(y + x\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) (log t)))
        (t_2 (+ (- (+ (log (+ x y)) (log z)) t) t_1)))
   (if (<= t_2 -500000000.0)
     (+ (- t) t_1)
     (if (<= t_2 900.0)
       (- (log (* z (+ x y))) (* 0.5 (log t)))
       (- (+ (* a (log t)) (log (+ y x))) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (a - 0.5) * log(t);
	double t_2 = ((log((x + y)) + log(z)) - t) + t_1;
	double tmp;
	if (t_2 <= -500000000.0) {
		tmp = -t + t_1;
	} else if (t_2 <= 900.0) {
		tmp = log((z * (x + y))) - (0.5 * log(t));
	} else {
		tmp = ((a * log(t)) + log((y + x))) - t;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (a - 0.5d0) * log(t)
    t_2 = ((log((x + y)) + log(z)) - t) + t_1
    if (t_2 <= (-500000000.0d0)) then
        tmp = -t + t_1
    else if (t_2 <= 900.0d0) then
        tmp = log((z * (x + y))) - (0.5d0 * log(t))
    else
        tmp = ((a * log(t)) + log((y + x))) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (a - 0.5) * Math.log(t);
	double t_2 = ((Math.log((x + y)) + Math.log(z)) - t) + t_1;
	double tmp;
	if (t_2 <= -500000000.0) {
		tmp = -t + t_1;
	} else if (t_2 <= 900.0) {
		tmp = Math.log((z * (x + y))) - (0.5 * Math.log(t));
	} else {
		tmp = ((a * Math.log(t)) + Math.log((y + x))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (a - 0.5) * math.log(t)
	t_2 = ((math.log((x + y)) + math.log(z)) - t) + t_1
	tmp = 0
	if t_2 <= -500000000.0:
		tmp = -t + t_1
	elif t_2 <= 900.0:
		tmp = math.log((z * (x + y))) - (0.5 * math.log(t))
	else:
		tmp = ((a * math.log(t)) + math.log((y + x))) - t
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(a - 0.5) * log(t))
	t_2 = Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + t_1)
	tmp = 0.0
	if (t_2 <= -500000000.0)
		tmp = Float64(Float64(-t) + t_1);
	elseif (t_2 <= 900.0)
		tmp = Float64(log(Float64(z * Float64(x + y))) - Float64(0.5 * log(t)));
	else
		tmp = Float64(Float64(Float64(a * log(t)) + log(Float64(y + x))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (a - 0.5) * log(t);
	t_2 = ((log((x + y)) + log(z)) - t) + t_1;
	tmp = 0.0;
	if (t_2 <= -500000000.0)
		tmp = -t + t_1;
	elseif (t_2 <= 900.0)
		tmp = log((z * (x + y))) - (0.5 * log(t));
	else
		tmp = ((a * log(t)) + log((y + x))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -500000000.0], N[((-t) + t$95$1), $MachinePrecision], If[LessEqual[t$95$2, 900.0], N[(N[Log[N[(z * N[(x + y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(0.5 * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] + N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < -5e8

   1. Initial program 99.6%

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

   2. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 78.1

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

   4. Applied rewrites 78.1%

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

   if -5e8 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 900

   1. Initial program 99.6%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. sum-log N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-log.f64 N/A

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

      9. *-commutative N/A

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

      10. +-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 46.8

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

   7. Applied rewrites 46.8%

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

   8. Taylor expanded in t around 0

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

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

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

      2. lower--.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 N/A

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

      8. lift-log.f64 19.3

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

   10. Applied rewrites 19.3%

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

   if 900 < (+.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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lift-log.f64 78.4

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

   10. Applied rewrites 78.4%

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

Alternative 14: 68.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot \log t\\ t_2 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + t\_1\\ \mathbf{if}\;t\_2 \leq -500000000:\\ \;\;\;\;\left(-t\right) + t\_1\\ \mathbf{elif}\;t\_2 \leq 900:\\ \;\;\;\;\mathsf{fma}\left(\log t, a - 0.5, \log \left(z \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \log t + \log \left(y + x\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) (log t)))
        (t_2 (+ (- (+ (log (+ x y)) (log z)) t) t_1)))
   (if (<= t_2 -500000000.0)
     (+ (- t) t_1)
     (if (<= t_2 900.0)
       (fma (log t) (- a 0.5) (log (* z y)))
       (- (+ (* a (log t)) (log (+ y x))) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (a - 0.5) * log(t);
	double t_2 = ((log((x + y)) + log(z)) - t) + t_1;
	double tmp;
	if (t_2 <= -500000000.0) {
		tmp = -t + t_1;
	} else if (t_2 <= 900.0) {
		tmp = fma(log(t), (a - 0.5), log((z * y)));
	} else {
		tmp = ((a * log(t)) + log((y + x))) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(a - 0.5) * log(t))
	t_2 = Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + t_1)
	tmp = 0.0
	if (t_2 <= -500000000.0)
		tmp = Float64(Float64(-t) + t_1);
	elseif (t_2 <= 900.0)
		tmp = fma(log(t), Float64(a - 0.5), log(Float64(z * y)));
	else
		tmp = Float64(Float64(Float64(a * log(t)) + log(Float64(y + x))) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -500000000.0], N[((-t) + t$95$1), $MachinePrecision], If[LessEqual[t$95$2, 900.0], N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision] + N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] + N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < -5e8

   1. Initial program 99.6%

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

   2. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 78.1

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

   4. Applied rewrites 78.1%

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

   if -5e8 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 900

   1. Initial program 99.6%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. log-rec N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. lift-log.f64 68.9

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

   7. Applied rewrites 68.9%

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

   8. Taylor expanded in t around 0

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

      1. associate-+r+ N/A

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

      2. log-prod N/A

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

      3. lower-+.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 31.9

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

   10. Applied rewrites 31.9%

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

       1. lift-+.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-log.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift--.f64 N/A

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

       6. lift-log.f64 N/A

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

       7. +-commutative N/A

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

       8. lower-fma.f64 N/A

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

       9. lift-log.f64 N/A

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

       10. lift--.f64 N/A

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

       11. lift-log.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 31.9

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

   12. Applied rewrites 31.9%

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

   if 900 < (+.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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lift-log.f64 78.4

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

   10. Applied rewrites 78.4%

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

Alternative 15: 68.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot \log t\\ t_2 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + t\_1\\ \mathbf{if}\;t\_2 \leq -500000000:\\ \;\;\;\;\left(-t\right) + t\_1\\ \mathbf{elif}\;t\_2 \leq 900:\\ \;\;\;\;\mathsf{fma}\left(\log t, a - 0.5, \log \left(z \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \log t + \log z\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) (log t)))
        (t_2 (+ (- (+ (log (+ x y)) (log z)) t) t_1)))
   (if (<= t_2 -500000000.0)
     (+ (- t) t_1)
     (if (<= t_2 900.0)
       (fma (log t) (- a 0.5) (log (* z y)))
       (- (+ (* a (log t)) (log z)) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (a - 0.5) * log(t);
	double t_2 = ((log((x + y)) + log(z)) - t) + t_1;
	double tmp;
	if (t_2 <= -500000000.0) {
		tmp = -t + t_1;
	} else if (t_2 <= 900.0) {
		tmp = fma(log(t), (a - 0.5), log((z * y)));
	} else {
		tmp = ((a * log(t)) + log(z)) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(a - 0.5) * log(t))
	t_2 = Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + t_1)
	tmp = 0.0
	if (t_2 <= -500000000.0)
		tmp = Float64(Float64(-t) + t_1);
	elseif (t_2 <= 900.0)
		tmp = fma(log(t), Float64(a - 0.5), log(Float64(z * y)));
	else
		tmp = Float64(Float64(Float64(a * log(t)) + log(z)) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -500000000.0], N[((-t) + t$95$1), $MachinePrecision], If[LessEqual[t$95$2, 900.0], N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision] + N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < -5e8

   1. Initial program 99.6%

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

   2. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 78.1

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

   4. Applied rewrites 78.1%

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

   if -5e8 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 900

   1. Initial program 99.6%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. log-rec N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. lift-log.f64 68.9

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

   7. Applied rewrites 68.9%

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

   8. Taylor expanded in t around 0

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

      1. associate-+r+ N/A

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

      2. log-prod N/A

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

      3. lower-+.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 31.9

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

   10. Applied rewrites 31.9%

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

       1. lift-+.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-log.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift--.f64 N/A

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

       6. lift-log.f64 N/A

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

       7. +-commutative N/A

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

       8. lower-fma.f64 N/A

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

       9. lift-log.f64 N/A

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

       10. lift--.f64 N/A

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

       11. lift-log.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 31.9

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

   12. Applied rewrites 31.9%

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

   if 900 < (+.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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. log-rec N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. lift-log.f64 68.9

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

   7. Applied rewrites 68.9%

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

   8. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lift-log.f64 77.9

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

   10. Applied rewrites 77.9%

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

Alternative 16: 68.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot \log t\\ t_2 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + t\_1\\ \mathbf{if}\;t\_2 \leq -500000000:\\ \;\;\;\;\left(-t\right) + t\_1\\ \mathbf{elif}\;t\_2 \leq 900:\\ \;\;\;\;\log \left(y \cdot z\right) + \log t \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \log t + \log z\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) (log t)))
        (t_2 (+ (- (+ (log (+ x y)) (log z)) t) t_1)))
   (if (<= t_2 -500000000.0)
     (+ (- t) t_1)
     (if (<= t_2 900.0)
       (+ (log (* y z)) (* (log t) -0.5))
       (- (+ (* a (log t)) (log z)) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (a - 0.5) * log(t);
	double t_2 = ((log((x + y)) + log(z)) - t) + t_1;
	double tmp;
	if (t_2 <= -500000000.0) {
		tmp = -t + t_1;
	} else if (t_2 <= 900.0) {
		tmp = log((y * z)) + (log(t) * -0.5);
	} else {
		tmp = ((a * log(t)) + log(z)) - t;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (a - 0.5d0) * log(t)
    t_2 = ((log((x + y)) + log(z)) - t) + t_1
    if (t_2 <= (-500000000.0d0)) then
        tmp = -t + t_1
    else if (t_2 <= 900.0d0) then
        tmp = log((y * z)) + (log(t) * (-0.5d0))
    else
        tmp = ((a * log(t)) + log(z)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (a - 0.5) * Math.log(t);
	double t_2 = ((Math.log((x + y)) + Math.log(z)) - t) + t_1;
	double tmp;
	if (t_2 <= -500000000.0) {
		tmp = -t + t_1;
	} else if (t_2 <= 900.0) {
		tmp = Math.log((y * z)) + (Math.log(t) * -0.5);
	} else {
		tmp = ((a * Math.log(t)) + Math.log(z)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (a - 0.5) * math.log(t)
	t_2 = ((math.log((x + y)) + math.log(z)) - t) + t_1
	tmp = 0
	if t_2 <= -500000000.0:
		tmp = -t + t_1
	elif t_2 <= 900.0:
		tmp = math.log((y * z)) + (math.log(t) * -0.5)
	else:
		tmp = ((a * math.log(t)) + math.log(z)) - t
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(a - 0.5) * log(t))
	t_2 = Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + t_1)
	tmp = 0.0
	if (t_2 <= -500000000.0)
		tmp = Float64(Float64(-t) + t_1);
	elseif (t_2 <= 900.0)
		tmp = Float64(log(Float64(y * z)) + Float64(log(t) * -0.5));
	else
		tmp = Float64(Float64(Float64(a * log(t)) + log(z)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (a - 0.5) * log(t);
	t_2 = ((log((x + y)) + log(z)) - t) + t_1;
	tmp = 0.0;
	if (t_2 <= -500000000.0)
		tmp = -t + t_1;
	elseif (t_2 <= 900.0)
		tmp = log((y * z)) + (log(t) * -0.5);
	else
		tmp = ((a * log(t)) + log(z)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -500000000.0], N[((-t) + t$95$1), $MachinePrecision], If[LessEqual[t$95$2, 900.0], N[(N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < -5e8

   1. Initial program 99.6%

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

   2. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 78.1

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

   4. Applied rewrites 78.1%

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

   if -5e8 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 900

   1. Initial program 99.6%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. log-rec N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. lift-log.f64 68.9

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

   7. Applied rewrites 68.9%

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

   8. Taylor expanded in t around 0

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

      1. associate-+r+ N/A

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

      2. log-prod N/A

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

      3. lower-+.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 31.9

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

   10. Applied rewrites 31.9%

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

   11. Taylor expanded in a around 0

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

   13. Applied rewrites 10.9%

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

   if 900 < (+.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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. log-rec N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. lift-log.f64 68.9

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

   7. Applied rewrites 68.9%

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

   8. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lift-log.f64 77.9

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

   10. Applied rewrites 77.9%

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

Alternative 17: 68.4% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in a around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. associate-*r/ N/A

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

   5. metadata-eval N/A

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

   6. lower-/.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in y around inf

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

   1. lower--.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lift-log.f64 N/A

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

   7. lift--.f64 N/A

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

   8. mul-1-neg N/A

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

   9. lower-neg.f64 N/A

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

   10. log-rec N/A

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

   11. lower-neg.f64 N/A

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

   12. lower-log.f64 N/A

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

   13. lift-log.f64 68.9

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

7. Applied rewrites 68.9%

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

8. Taylor expanded in a around inf

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

   1. lower-*.f64 N/A

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

   2. lift-log.f64 77.9

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

10. Applied rewrites 77.9%

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

Alternative 18: 68.4% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(x, y, z, t, a):
	return -t + ((a - 0.5) * math.log(t))

Julia

function code(x, y, z, t, a)
	return Float64(Float64(-t) + Float64(Float64(a - 0.5) * log(t)))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = -t + ((a - 0.5) * log(t));
end

Wolfram

code[x_, y_, z_, t_, a_] := N[((-t) + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]

2. Taylor expanded in t around inf

   \[\leadsto \color{blue}{-1 \cdot t} + \left(a - \frac{1}{2}\right) \cdot \log t \]
3. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \left(\mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

   2. lower-neg.f64 78.1

      \[\leadsto \left(-t\right) + \left(a - 0.5\right) \cdot \log t \]

4. Applied rewrites 78.1%

   \[\leadsto \color{blue}{\left(-t\right)} + \left(a - 0.5\right) \cdot \log t \]
5. Add Preprocessing

Alternative 19: 68.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ a \cdot \log t - t \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (- (* a (log t)) t))

C

double code(double x, double y, double z, double t, double a) {
	return (a * log(t)) - t;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = (a * log(t)) - t
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return (a * Math.log(t)) - t;
}

Python

def code(x, y, z, t, a):
	return (a * math.log(t)) - t

Julia

function code(x, y, z, t, a)
	return Float64(Float64(a * log(t)) - t)
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = (a * log(t)) - t;
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]

2. Taylor expanded in a around inf

   \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \color{blue}{\left(a \cdot \left(1 - \frac{1}{2} \cdot \frac{1}{a}\right)\right)} \cdot \log t \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(\left(1 - \frac{1}{2} \cdot \frac{1}{a}\right) \cdot \color{blue}{a}\right) \cdot \log t \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(\left(1 - \frac{1}{2} \cdot \frac{1}{a}\right) \cdot \color{blue}{a}\right) \cdot \log t \]

   3. lower--.f64 N/A

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(\left(1 - \frac{1}{2} \cdot \frac{1}{a}\right) \cdot a\right) \cdot \log t \]

   4. associate-*r/ N/A

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(\left(1 - \frac{\frac{1}{2} \cdot 1}{a}\right) \cdot a\right) \cdot \log t \]

   5. metadata-eval N/A

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(\left(1 - \frac{\frac{1}{2}}{a}\right) \cdot a\right) \cdot \log t \]

   6. lower-/.f64 99.6

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(\left(1 - \frac{0.5}{a}\right) \cdot a\right) \cdot \log t \]

4. Applied rewrites 99.6%

   \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \color{blue}{\left(\left(1 - \frac{0.5}{a}\right) \cdot a\right)} \cdot \log t \]

5. Taylor expanded in z around inf

   \[\leadsto \color{blue}{\left(\log \left(x + y\right) + \left(-1 \cdot \log \left(\frac{1}{z}\right) + \log t \cdot \left(a - \frac{1}{2}\right)\right)\right) - t} \]
6. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto \left(\log \left(x + y\right) + \left(-1 \cdot \log \left(\frac{1}{z}\right) + \log t \cdot \left(a - \frac{1}{2}\right)\right)\right) - \color{blue}{t} \]

7. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\log t, a - 0.5, -\left(-\log z\right)\right) + \log \left(y + x\right)\right) - t} \]

8. Taylor expanded in a around inf

   \[\leadsto a \cdot \log t - t \]
9. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto a \cdot \log t - t \]

   2. lift-log.f64 75.6

      \[\leadsto a \cdot \log t - t \]

10. Applied rewrites 75.6%

    \[\leadsto a \cdot \log t - t \]
11. Add Preprocessing

Alternative 20: 63.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.02 \cdot 10^{+52}:\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 1.02e+52) (* (log t) a) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 1.02e+52) {
		tmp = log(t) * a;
	} else {
		tmp = -t;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 1.02d+52) then
        tmp = log(t) * a
    else
        tmp = -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 1.02e+52) {
		tmp = Math.log(t) * a;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 1.02e+52:
		tmp = math.log(t) * a
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 1.02e+52)
		tmp = Float64(log(t) * a);
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 1.02e+52)
		tmp = log(t) * a;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 1.02e+52], N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if t < 1.02000000000000002e52

   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 39.2

         \[\leadsto \log t \cdot a \]

   4. Applied rewrites 39.2%

      \[\leadsto \color{blue}{\log t \cdot a} \]

   if 1.02000000000000002e52 < 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 38.3

         \[\leadsto -t \]

   4. Applied rewrites 38.3%

      \[\leadsto \color{blue}{-t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 38.3% accurate, 17.6× speedup

Math

\[\begin{array}{l} \\ -t \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (- t))

C

double code(double x, double y, double z, double t, double a) {
	return -t;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = -t
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return -t;
}

Python

def code(x, y, z, t, a):
	return -t

Julia

function code(x, y, z, t, a)
	return Float64(-t)
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = -t;
end

Wolfram

code[x_, y_, z_, t_, a_] := (-t)

Derivation

1. Initial program 99.6%

   \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]

2. Taylor expanded in t around inf

   \[\leadsto \color{blue}{-1 \cdot t} \]
3. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(t\right) \]

   2. lower-neg.f64 38.3

      \[\leadsto -t \]

4. Applied rewrites 38.3%

   \[\leadsto \color{blue}{-t} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025123 
(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))))