Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 7.2s

Alternatives: 13

Speedup: N/A×

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: 79.3% accurate, 1.0× speedup

Math

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

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), Float64(-Float64(-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 69.0

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

7. Applied rewrites 69.0%

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

Alternative 3: 79.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 69.0%

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

Alternative 4: 77.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\log t \cdot a + \log y\right) - t\\ \mathbf{if}\;a \leq -11:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 0.72:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.5, \log t, \log z\right) + \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (+ (* (log t) a) (log y)) t)))
   (if (<= a -11.0)
     t_1
     (if (<= a 0.72) (- (+ (fma -0.5 (log t) (log z)) (log y)) t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((log(t) * a) + log(y)) - t;
	double tmp;
	if (a <= -11.0) {
		tmp = t_1;
	} else if (a <= 0.72) {
		tmp = (fma(-0.5, log(t), log(z)) + log(y)) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(log(t) * a) + log(y)) - t)
	tmp = 0.0
	if (a <= -11.0)
		tmp = t_1;
	elseif (a <= 0.72)
		tmp = Float64(Float64(fma(-0.5, log(t), log(z)) + log(y)) - 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), $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[a, -11.0], t$95$1, If[LessEqual[a, 0.72], N[(N[(N[(-0.5 * N[Log[t], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -11 or 0.71999999999999997 < a

   1. Initial program 99.7%

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

   2. Taylor expanded in 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 99.7

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

   4. Applied rewrites 99.7%

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

   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 99.7%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 75.1%

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

   11. Taylor expanded in a around inf

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

       1. *-commutative N/A

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

       2. lift-log.f64 N/A

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

       3. lift-*.f64 74.6

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

   13. Applied rewrites 74.6%

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

   if -11 < a < 0.71999999999999997

   1. Initial program 99.5%

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

   2. Taylor expanded in 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 99.5

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

   4. Applied rewrites 99.5%

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

   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 99.5%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 62.9%

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

   11. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. lift-log.f64 N/A

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

       4. lift-log.f64 62.5

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

   13. Applied rewrites 62.5%

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

Alternative 5: 74.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\log t \cdot a + \log y\right) - t\\ \mathbf{if}\;a \leq -11:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 0.72:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.5, \log t, \log y\right) + \log z\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (+ (* (log t) a) (log y)) t)))
   (if (<= a -11.0)
     t_1
     (if (<= a 0.72) (- (+ (fma -0.5 (log t) (log y)) (log z)) t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((log(t) * a) + log(y)) - t;
	double tmp;
	if (a <= -11.0) {
		tmp = t_1;
	} else if (a <= 0.72) {
		tmp = (fma(-0.5, log(t), log(y)) + log(z)) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(log(t) * a) + log(y)) - t)
	tmp = 0.0
	if (a <= -11.0)
		tmp = t_1;
	elseif (a <= 0.72)
		tmp = Float64(Float64(fma(-0.5, log(t), 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), $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[a, -11.0], t$95$1, If[LessEqual[a, 0.72], N[(N[(N[(-0.5 * N[Log[t], $MachinePrecision] + 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 < -11 or 0.71999999999999997 < a

   1. Initial program 99.7%

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

   2. Taylor expanded in 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 99.7

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

   4. Applied rewrites 99.7%

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

   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 99.7%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 75.1%

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

   11. Taylor expanded in a around inf

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

       1. *-commutative N/A

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

       2. lift-log.f64 N/A

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

       3. lift-*.f64 74.6

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

   13. Applied rewrites 74.6%

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

   if -11 < a < 0.71999999999999997

   1. Initial program 99.5%

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

   2. Taylor expanded in 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 99.5

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

   4. Applied rewrites 99.5%

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

   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 62.9

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

   7. Applied rewrites 62.9%

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

   8. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-log.f64 62.4

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

   10. Applied rewrites 62.4%

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

Alternative 6: 69.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log t \cdot a\\ t_2 := \log \left(x + y\right) + \log z\\ \mathbf{if}\;t\_2 \leq -750:\\ \;\;\;\;\left(t\_1 + \log z\right) - t\\ \mathbf{elif}\;t\_2 \leq 656.5:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log \left(y \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 + \log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (log t) a)) (t_2 (+ (log (+ x y)) (log z))))
   (if (<= t_2 -750.0)
     (- (+ t_1 (log z)) t)
     (if (<= t_2 656.5)
       (- (fma (- a 0.5) (log t) (log (* y z))) t)
       (- (+ t_1 (log y)) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log(t) * a;
	double t_2 = log((x + y)) + log(z);
	double tmp;
	if (t_2 <= -750.0) {
		tmp = (t_1 + log(z)) - t;
	} else if (t_2 <= 656.5) {
		tmp = fma((a - 0.5), log(t), log((y * z))) - t;
	} else {
		tmp = (t_1 + log(y)) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(t) * a)
	t_2 = Float64(log(Float64(x + y)) + log(z))
	tmp = 0.0
	if (t_2 <= -750.0)
		tmp = Float64(Float64(t_1 + log(z)) - t);
	elseif (t_2 <= 656.5)
		tmp = Float64(fma(Float64(a - 0.5), log(t), log(Float64(y * z))) - t);
	else
		tmp = Float64(Float64(t_1 + log(y)) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]}, Block[{t$95$2 = N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -750.0], N[(N[(t$95$1 + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[t$95$2, 656.5], N[(N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(t$95$1 + N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.7%

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

   2. Taylor expanded in 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 99.7

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

   4. Applied rewrites 99.7%

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

   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 68.8

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

   7. Applied rewrites 68.8%

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

   8. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lift-log.f64 N/A

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

      3. lift-*.f64 82.6

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

   10. Applied rewrites 82.6%

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

   if -750 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 656.5

   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 99.5

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

   4. Applied rewrites 99.5%

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

   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 68.5

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

   7. Applied rewrites 68.5%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. remove-double-neg N/A

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

      9. lift-log.f64 N/A

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

      10. associate-+l+ N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lift--.f64 N/A

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

      14. lift-log.f64 N/A

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

      15. sum-log N/A

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

      16. lower-log.f64 N/A

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

      17. lower-*.f64 64.8

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

   9. Applied rewrites 64.8%

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

   if 656.5 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z))

   1. Initial program 99.7%

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

   2. Taylor expanded in 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 99.7

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

   4. Applied rewrites 99.7%

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

   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 99.7%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 70.7%

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

   11. Taylor expanded in a around inf

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

       1. *-commutative N/A

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

       2. lift-log.f64 N/A

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

       3. lift-*.f64 58.6

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

   13. Applied rewrites 58.6%

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

Alternative 7: 69.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ t_2 := \log t \cdot a\\ \mathbf{if}\;t\_1 \leq -600:\\ \;\;\;\;\left(t\_2 + \log z\right) - t\\ \mathbf{elif}\;t\_1 \leq 1000:\\ \;\;\;\;\log \left(y \cdot z\right) + \log t \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_2 + \log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))
        (t_2 (* (log t) a)))
   (if (<= t_1 -600.0)
     (- (+ t_2 (log z)) t)
     (if (<= t_1 1000.0)
       (+ (log (* y z)) (* (log t) (- a 0.5)))
       (- (+ t_2 (log y)) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
	double t_2 = log(t) * a;
	double tmp;
	if (t_1 <= -600.0) {
		tmp = (t_2 + log(z)) - t;
	} else if (t_1 <= 1000.0) {
		tmp = log((y * z)) + (log(t) * (a - 0.5));
	} else {
		tmp = (t_2 + log(y)) - t;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = ((log((x + y)) + log(z)) - t) + ((a - 0.5d0) * log(t))
    t_2 = log(t) * a
    if (t_1 <= (-600.0d0)) then
        tmp = (t_2 + log(z)) - t
    else if (t_1 <= 1000.0d0) then
        tmp = log((y * z)) + (log(t) * (a - 0.5d0))
    else
        tmp = (t_2 + log(y)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = ((Math.log((x + y)) + Math.log(z)) - t) + ((a - 0.5) * Math.log(t));
	double t_2 = Math.log(t) * a;
	double tmp;
	if (t_1 <= -600.0) {
		tmp = (t_2 + Math.log(z)) - t;
	} else if (t_1 <= 1000.0) {
		tmp = Math.log((y * z)) + (Math.log(t) * (a - 0.5));
	} else {
		tmp = (t_2 + Math.log(y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = ((math.log((x + y)) + math.log(z)) - t) + ((a - 0.5) * math.log(t))
	t_2 = math.log(t) * a
	tmp = 0
	if t_1 <= -600.0:
		tmp = (t_2 + math.log(z)) - t
	elif t_1 <= 1000.0:
		tmp = math.log((y * z)) + (math.log(t) * (a - 0.5))
	else:
		tmp = (t_2 + math.log(y)) - t
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + Float64(Float64(a - 0.5) * log(t)))
	t_2 = Float64(log(t) * a)
	tmp = 0.0
	if (t_1 <= -600.0)
		tmp = Float64(Float64(t_2 + log(z)) - t);
	elseif (t_1 <= 1000.0)
		tmp = Float64(log(Float64(y * z)) + Float64(log(t) * Float64(a - 0.5)));
	else
		tmp = Float64(Float64(t_2 + log(y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
	t_2 = log(t) * a;
	tmp = 0.0;
	if (t_1 <= -600.0)
		tmp = (t_2 + log(z)) - t;
	elseif (t_1 <= 1000.0)
		tmp = log((y * z)) + (log(t) * (a - 0.5));
	else
		tmp = (t_2 + log(y)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[t$95$1, -600.0], N[(N[(t$95$2 + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[t$95$1, 1000.0], N[(N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 + N[Log[y], $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))) < -600

   1. Initial program 99.8%

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

   2. Taylor expanded in 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 99.8

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

   4. Applied rewrites 99.8%

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

   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 74.1

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

   7. Applied rewrites 74.1%

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

   8. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lift-log.f64 N/A

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

      3. lift-*.f64 97.3

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

   10. Applied rewrites 97.3%

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

   if -600 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 1e3

   1. Initial program 99.0%

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

   2. Taylor expanded in 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 98.9

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

   4. Applied rewrites 98.9%

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

   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 51.6

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

   7. Applied rewrites 51.6%

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

   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. lower-+.f64 N/A

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

      3. sum-log N/A

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

      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 46.8

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

   10. Applied rewrites 46.8%

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

   if 1e3 < (+.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 x around 0

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

   10. Applied rewrites 71.8%

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

   11. Taylor expanded in a around inf

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

       1. *-commutative N/A

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

       2. lift-log.f64 N/A

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

       3. lift-*.f64 63.0

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

   13. Applied rewrites 63.0%

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

Alternative 8: 68.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ t_2 := \log t \cdot a\\ \mathbf{if}\;t\_1 \leq -600:\\ \;\;\;\;\left(t\_2 + \log z\right) - t\\ \mathbf{elif}\;t\_1 \leq 1000:\\ \;\;\;\;\mathsf{fma}\left(\log t, a - 0.5, \log \left(z \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_2 + \log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))
        (t_2 (* (log t) a)))
   (if (<= t_1 -600.0)
     (- (+ t_2 (log z)) t)
     (if (<= t_1 1000.0)
       (fma (log t) (- a 0.5) (log (* z y)))
       (- (+ t_2 (log y)) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
	double t_2 = log(t) * a;
	double tmp;
	if (t_1 <= -600.0) {
		tmp = (t_2 + log(z)) - t;
	} else if (t_1 <= 1000.0) {
		tmp = fma(log(t), (a - 0.5), log((z * y)));
	} else {
		tmp = (t_2 + log(y)) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + Float64(Float64(a - 0.5) * log(t)))
	t_2 = Float64(log(t) * a)
	tmp = 0.0
	if (t_1 <= -600.0)
		tmp = Float64(Float64(t_2 + log(z)) - t);
	elseif (t_1 <= 1000.0)
		tmp = fma(log(t), Float64(a - 0.5), log(Float64(z * y)));
	else
		tmp = Float64(Float64(t_2 + log(y)) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[t$95$1, -600.0], N[(N[(t$95$2 + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[t$95$1, 1000.0], N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision] + N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 + N[Log[y], $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))) < -600

   1. Initial program 99.8%

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

   2. Taylor expanded in 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 99.8

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

   4. Applied rewrites 99.8%

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

   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 74.1

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

   7. Applied rewrites 74.1%

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

   8. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lift-log.f64 N/A

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

      3. lift-*.f64 97.3

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

   10. Applied rewrites 97.3%

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

   if -600 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 1e3

   1. Initial program 99.0%

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

   2. Taylor expanded in 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 98.9

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

   4. Applied rewrites 98.9%

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

   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 51.6

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

   7. Applied rewrites 51.6%

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

   8. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lift-log.f64 N/A

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

      3. lift-*.f64 12.1

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

   10. Applied rewrites 12.1%

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

   11. Taylor expanded in t around 0

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

       1. associate-+r+ N/A

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

       2. sum-log N/A

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

       3. +-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. lift-log.f64 N/A

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

       6. lift--.f64 N/A

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

       7. lift-log.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 46.8

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

   13. Applied rewrites 46.8%

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

   if 1e3 < (+.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 x around 0

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

   10. Applied rewrites 71.8%

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

   11. Taylor expanded in a around inf

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

       1. *-commutative N/A

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

       2. lift-log.f64 N/A

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

       3. lift-*.f64 63.0

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

   13. Applied rewrites 63.0%

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

Alternative 9: 68.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision] + N[Log[y], $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 x around 0

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

10. Applied rewrites 69.0%

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

11. Taylor expanded in a around inf

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

    1. *-commutative N/A

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

    2. lift-log.f64 N/A

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

    3. lift-*.f64 57.8

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

13. Applied rewrites 57.8%

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

Alternative 10: 63.9% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 77.1

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

4. Applied rewrites 77.1%

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

Alternative 11: 61.4% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. associate-+r+ N/A

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

   2. +-commutative N/A

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

   3. log-pow-rev N/A

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

   4. inv-pow N/A

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

   5. pow-pow N/A

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

   6. metadata-eval N/A

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

   7. unpow1 N/A

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

   8. associate-+r+ N/A

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

   9. lower--.f64 N/A

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

4. Applied rewrites 52.9%

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

5. Taylor expanded in a around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lift-log.f64 74.5

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

7. Applied rewrites 74.5%

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

Alternative 12: 57.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log t \cdot a\\ \mathbf{if}\;a \leq -5 \cdot 10^{+76}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 7.2 \cdot 10^{+24}:\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (log t) a)))
   (if (<= a -5e+76) t_1 (if (<= a 7.2e+24) (- t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log(t) * a;
	double tmp;
	if (a <= -5e+76) {
		tmp = t_1;
	} else if (a <= 7.2e+24) {
		tmp = -t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = log(t) * a
    if (a <= (-5d+76)) then
        tmp = t_1
    else if (a <= 7.2d+24) then
        tmp = -t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = Math.log(t) * a;
	double tmp;
	if (a <= -5e+76) {
		tmp = t_1;
	} else if (a <= 7.2e+24) {
		tmp = -t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = math.log(t) * a
	tmp = 0
	if a <= -5e+76:
		tmp = t_1
	elif a <= 7.2e+24:
		tmp = -t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(t) * a)
	tmp = 0.0
	if (a <= -5e+76)
		tmp = t_1;
	elseif (a <= 7.2e+24)
		tmp = Float64(-t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = log(t) * a;
	tmp = 0.0;
	if (a <= -5e+76)
		tmp = t_1;
	elseif (a <= 7.2e+24)
		tmp = -t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[a, -5e+76], t$95$1, If[LessEqual[a, 7.2e+24], (-t), t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -4.99999999999999991e76 or 7.19999999999999966e24 < a

   1. Initial program 99.7%

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

   2. Taylor expanded in 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 78.3

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

   4. Applied rewrites 78.3%

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

   if -4.99999999999999991e76 < a < 7.19999999999999966e24

   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 49.3

         \[\leadsto -t \]

   4. Applied rewrites 49.3%

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

Alternative 13: 37.9% 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 37.9

      \[\leadsto -t \]

4. Applied rewrites 37.9%

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

Reproduce

herbie shell --seed 2025120 
(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))))