Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 8.0s

Alternatives: 13

Speedup: 0.7×

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, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. lift-+.f64 N/A

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

   2. add-flip N/A

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

   3. lift--.f64 N/A

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

   4. sub-flip N/A

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

   5. associate--l+ N/A

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. sum-to-mult N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-+.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 N/A

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

   15. lower--.f64 N/A

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

   16. lower-neg.f64 N/A

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

3. Applied rewrites 99.6%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

Alternative 3: 92.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in x around 0

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

   1. lower--.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-log.f64 N/A

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

   4. lower-+.f64 N/A

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

   5. lower-log.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-log.f64 N/A

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

   8. lower--.f64 69.7

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

4. Applied rewrites 69.7%

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

Alternative 4: 84.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   1. lower--.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-log.f64 N/A

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

   4. lower-+.f64 N/A

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

   5. lower-log.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-log.f64 N/A

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

   8. lower--.f64 69.7

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

4. Applied rewrites 69.7%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate--l+ N/A

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

   4. +-commutative N/A

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

   5. lower-+.f64 N/A

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

   6. lower--.f64 69.7

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 69.7

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

6. Applied rewrites 69.7%

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

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. associate-+l- N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. +-commutative N/A

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

   7. sum-to-mult-rev N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*r/ N/A

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

   10. lift-/.f64 N/A

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

   11. +-commutative N/A

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

   12. lift-fma.f64 N/A

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

   13. lower--.f64 N/A

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

8. Applied rewrites 69.7%

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

Alternative 5: 81.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(1, -t, \left(a - 0.5\right) \cdot \log t\right)\\ \mathbf{if}\;a \leq -0.82:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 1.7:\\ \;\;\;\;\left(\log y + \left(\log z + \log t \cdot -0.5\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma 1.0 (- t) (* (- a 0.5) (log t)))))
   (if (<= a -0.82)
     t_1
     (if (<= a 1.7) (- (+ (log y) (+ (log z) (* (log t) -0.5))) t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(1.0, -t, ((a - 0.5) * log(t)));
	double tmp;
	if (a <= -0.82) {
		tmp = t_1;
	} else if (a <= 1.7) {
		tmp = (log(y) + (log(z) + (log(t) * -0.5))) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(1.0, Float64(-t), Float64(Float64(a - 0.5) * log(t)))
	tmp = 0.0
	if (a <= -0.82)
		tmp = t_1;
	elseif (a <= 1.7)
		tmp = Float64(Float64(log(y) + Float64(log(z) + Float64(log(t) * -0.5))) - t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(1.0 * (-t) + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -0.82], t$95$1, If[LessEqual[a, 1.7], N[(N[(N[Log[y], $MachinePrecision] + N[(N[Log[z], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -0.819999999999999951 or 1.69999999999999996 < a

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

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

      3. lift--.f64 N/A

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

      4. sub-flip N/A

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

      5. associate--l+ N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. sum-to-mult N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-neg.f64 N/A

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

   3. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 + \frac{\log \left(y + x\right)}{\log z}, \log z, \left(-t\right) - \left(0.5 - a\right) \cdot \log t\right)} \]
   4. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sum-to-mult-rev N/A

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

      5. lift-log.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. log-prod N/A

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

      8. lift-*.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. lift--.f64 N/A

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

      11. associate-+r- N/A

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

      12. lift-neg.f64 N/A

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

      13. sub-flip N/A

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

      14. lift--.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift--.f64 N/A

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

      17. sub-negate-rev N/A

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

      18. lift--.f64 N/A

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

   5. Applied rewrites 75.9%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 77.7%

      \[\leadsto \mathsf{fma}\left(\color{blue}{1}, -t, \left(a - 0.5\right) \cdot \log t\right) \]

   if -0.819999999999999951 < a < 1.69999999999999996

   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

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower--.f64 69.7

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

   4. Applied rewrites 69.7%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 41.7%

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

Alternative 6: 81.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(1, -t, \left(a - 0.5\right) \cdot \log t\right)\\ \mathbf{if}\;a \leq -0.82:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 1.7:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.5, \log t, \log z\right) - t\right) + \log y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma 1.0 (- t) (* (- a 0.5) (log t)))))
   (if (<= a -0.82)
     t_1
     (if (<= a 1.7) (+ (- (fma -0.5 (log t) (log z)) t) (log y)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(1.0, -t, ((a - 0.5) * log(t)));
	double tmp;
	if (a <= -0.82) {
		tmp = t_1;
	} else if (a <= 1.7) {
		tmp = (fma(-0.5, log(t), log(z)) - t) + log(y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(1.0, Float64(-t), Float64(Float64(a - 0.5) * log(t)))
	tmp = 0.0
	if (a <= -0.82)
		tmp = t_1;
	elseif (a <= 1.7)
		tmp = Float64(Float64(fma(-0.5, log(t), log(z)) - t) + log(y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(1.0 * (-t) + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -0.82], t$95$1, If[LessEqual[a, 1.7], N[(N[(N[(-0.5 * N[Log[t], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -0.819999999999999951 or 1.69999999999999996 < a

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

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

      3. lift--.f64 N/A

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

      4. sub-flip N/A

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

      5. associate--l+ N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. sum-to-mult N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-neg.f64 N/A

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

   3. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 + \frac{\log \left(y + x\right)}{\log z}, \log z, \left(-t\right) - \left(0.5 - a\right) \cdot \log t\right)} \]
   4. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sum-to-mult-rev N/A

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

      5. lift-log.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. log-prod N/A

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

      8. lift-*.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. lift--.f64 N/A

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

      11. associate-+r- N/A

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

      12. lift-neg.f64 N/A

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

      13. sub-flip N/A

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

      14. lift--.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift--.f64 N/A

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

      17. sub-negate-rev N/A

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

      18. lift--.f64 N/A

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

   5. Applied rewrites 75.9%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 77.7%

      \[\leadsto \mathsf{fma}\left(\color{blue}{1}, -t, \left(a - 0.5\right) \cdot \log t\right) \]

   if -0.819999999999999951 < a < 1.69999999999999996

   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

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower--.f64 69.7

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

   4. Applied rewrites 69.7%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower--.f64 69.7

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 69.7

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

   6. Applied rewrites 69.7%

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

   7. Taylor expanded in a around 0

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

   9. Applied rewrites 41.7%

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

Alternative 7: 81.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(1, -t, \left(a - 0.5\right) \cdot \log t\right)\\ \mathbf{if}\;a \leq -0.82:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 1.7:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.5, \log t, \log y\right) - t\right) + \log z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma 1.0 (- t) (* (- a 0.5) (log t)))))
   (if (<= a -0.82)
     t_1
     (if (<= a 1.7) (+ (- (fma -0.5 (log t) (log y)) t) (log z)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(1.0, -t, ((a - 0.5) * log(t)));
	double tmp;
	if (a <= -0.82) {
		tmp = t_1;
	} else if (a <= 1.7) {
		tmp = (fma(-0.5, log(t), log(y)) - t) + log(z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(1.0, Float64(-t), Float64(Float64(a - 0.5) * log(t)))
	tmp = 0.0
	if (a <= -0.82)
		tmp = t_1;
	elseif (a <= 1.7)
		tmp = Float64(Float64(fma(-0.5, log(t), log(y)) - t) + log(z));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(1.0 * (-t) + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -0.82], t$95$1, If[LessEqual[a, 1.7], N[(N[(N[(-0.5 * N[Log[t], $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -0.819999999999999951 or 1.69999999999999996 < a

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

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

      3. lift--.f64 N/A

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

      4. sub-flip N/A

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

      5. associate--l+ N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. sum-to-mult N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-neg.f64 N/A

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

   3. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 + \frac{\log \left(y + x\right)}{\log z}, \log z, \left(-t\right) - \left(0.5 - a\right) \cdot \log t\right)} \]
   4. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sum-to-mult-rev N/A

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

      5. lift-log.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. log-prod N/A

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

      8. lift-*.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. lift--.f64 N/A

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

      11. associate-+r- N/A

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

      12. lift-neg.f64 N/A

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

      13. sub-flip N/A

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

      14. lift--.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift--.f64 N/A

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

      17. sub-negate-rev N/A

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

      18. lift--.f64 N/A

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

   5. Applied rewrites 75.9%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 77.7%

      \[\leadsto \mathsf{fma}\left(\color{blue}{1}, -t, \left(a - 0.5\right) \cdot \log t\right) \]

   if -0.819999999999999951 < a < 1.69999999999999996

   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

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower--.f64 69.7

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

   4. Applied rewrites 69.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. lower-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 69.7

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

   6. Applied rewrites 69.7%

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

   7. Taylor expanded in a around 0

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

   9. Applied rewrites 41.7%

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

       1. lift--.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. associate--l+ N/A

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

       4. +-commutative N/A

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

       5. lower-+.f64 N/A

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

       6. lower--.f64 41.7

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

   11. Applied rewrites 41.7%

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

Alternative 8: 77.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(x + y\right) + \log z\\ t_2 := \mathsf{fma}\left(1, -t, \left(a - 0.5\right) \cdot \log t\right)\\ \mathbf{if}\;t\_1 \leq -750:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 566:\\ \;\;\;\;\mathsf{fma}\left(\log t, a - 0.5, \log \left(z \cdot \left(y + x\right)\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log (+ x y)) (log z)))
        (t_2 (fma 1.0 (- t) (* (- a 0.5) (log t)))))
   (if (<= t_1 -750.0)
     t_2
     (if (<= t_1 566.0)
       (- (fma (log t) (- a 0.5) (log (* z (+ y x)))) t)
       t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((x + y)) + log(z);
	double t_2 = fma(1.0, -t, ((a - 0.5) * log(t)));
	double tmp;
	if (t_1 <= -750.0) {
		tmp = t_2;
	} else if (t_1 <= 566.0) {
		tmp = fma(log(t), (a - 0.5), log((z * (y + x)))) - t;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(Float64(x + y)) + log(z))
	t_2 = fma(1.0, Float64(-t), Float64(Float64(a - 0.5) * log(t)))
	tmp = 0.0
	if (t_1 <= -750.0)
		tmp = t_2;
	elseif (t_1 <= 566.0)
		tmp = Float64(fma(log(t), Float64(a - 0.5), log(Float64(z * Float64(y + x)))) - t);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 * (-t) + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -750.0], t$95$2, If[LessEqual[t$95$1, 566.0], N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision] + N[Log[N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -750 or 566 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z))

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

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

      3. lift--.f64 N/A

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

      4. sub-flip N/A

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

      5. associate--l+ N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. sum-to-mult N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-neg.f64 N/A

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

   3. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 + \frac{\log \left(y + x\right)}{\log z}, \log z, \left(-t\right) - \left(0.5 - a\right) \cdot \log t\right)} \]
   4. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sum-to-mult-rev N/A

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

      5. lift-log.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. log-prod N/A

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

      8. lift-*.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. lift--.f64 N/A

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

      11. associate-+r- N/A

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

      12. lift-neg.f64 N/A

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

      13. sub-flip N/A

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

      14. lift--.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift--.f64 N/A

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

      17. sub-negate-rev N/A

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

      18. lift--.f64 N/A

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

   5. Applied rewrites 75.9%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 77.7%

      \[\leadsto \mathsf{fma}\left(\color{blue}{1}, -t, \left(a - 0.5\right) \cdot \log t\right) \]

   if -750 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 566

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+r- N/A

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

      5. lower--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 99.6

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

      9. lift-+.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. sum-log N/A

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

      13. lower-log.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 76.2

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 76.2

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

   3. Applied rewrites 76.2%

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

Alternative 9: 69.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(x + y\right) + \log z\\ t_2 := \mathsf{fma}\left(1, -t, \left(a - 0.5\right) \cdot \log t\right)\\ \mathbf{if}\;t\_1 \leq -750:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 566:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log \left(z \cdot y\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log (+ x y)) (log z)))
        (t_2 (fma 1.0 (- t) (* (- a 0.5) (log t)))))
   (if (<= t_1 -750.0)
     t_2
     (if (<= t_1 566.0) (- (fma (- a 0.5) (log t) (log (* z y))) t) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((x + y)) + log(z);
	double t_2 = fma(1.0, -t, ((a - 0.5) * log(t)));
	double tmp;
	if (t_1 <= -750.0) {
		tmp = t_2;
	} else if (t_1 <= 566.0) {
		tmp = fma((a - 0.5), log(t), log((z * y))) - t;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(Float64(x + y)) + log(z))
	t_2 = fma(1.0, Float64(-t), Float64(Float64(a - 0.5) * log(t)))
	tmp = 0.0
	if (t_1 <= -750.0)
		tmp = t_2;
	elseif (t_1 <= 566.0)
		tmp = Float64(fma(Float64(a - 0.5), log(t), log(Float64(z * y))) - t);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 * (-t) + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -750.0], t$95$2, If[LessEqual[t$95$1, 566.0], N[(N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -750 or 566 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z))

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

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

      3. lift--.f64 N/A

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

      4. sub-flip N/A

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

      5. associate--l+ N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. sum-to-mult N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-neg.f64 N/A

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

   3. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 + \frac{\log \left(y + x\right)}{\log z}, \log z, \left(-t\right) - \left(0.5 - a\right) \cdot \log t\right)} \]
   4. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sum-to-mult-rev N/A

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

      5. lift-log.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. log-prod N/A

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

      8. lift-*.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. lift--.f64 N/A

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

      11. associate-+r- N/A

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

      12. lift-neg.f64 N/A

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

      13. sub-flip N/A

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

      14. lift--.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift--.f64 N/A

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

      17. sub-negate-rev N/A

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

      18. lift--.f64 N/A

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

   5. Applied rewrites 75.9%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 77.7%

      \[\leadsto \mathsf{fma}\left(\color{blue}{1}, -t, \left(a - 0.5\right) \cdot \log t\right) \]

   if -750 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 566

   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

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower--.f64 69.7

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

   4. Applied rewrites 69.7%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. sum-log N/A

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

      11. lower-log.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 53.5

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

   6. Applied rewrites 53.5%

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

Alternative 10: 69.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot \log t\\ t_2 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + t\_1\\ t_3 := \mathsf{fma}\left(1, -t, t\_1\right)\\ \mathbf{if}\;t\_2 \leq -2000:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 908:\\ \;\;\;\;\mathsf{fma}\left(\log t, -0.5, \log \left(z \cdot y\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) (log t)))
        (t_2 (+ (- (+ (log (+ x y)) (log z)) t) t_1))
        (t_3 (fma 1.0 (- t) t_1)))
   (if (<= t_2 -2000.0)
     t_3
     (if (<= t_2 908.0) (- (fma (log t) -0.5 (log (* z y))) t) t_3))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (a - 0.5) * log(t);
	double t_2 = ((log((x + y)) + log(z)) - t) + t_1;
	double t_3 = fma(1.0, -t, t_1);
	double tmp;
	if (t_2 <= -2000.0) {
		tmp = t_3;
	} else if (t_2 <= 908.0) {
		tmp = fma(log(t), -0.5, log((z * y))) - t;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(a - 0.5) * log(t))
	t_2 = Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + t_1)
	t_3 = fma(1.0, Float64(-t), t_1)
	tmp = 0.0
	if (t_2 <= -2000.0)
		tmp = t_3;
	elseif (t_2 <= 908.0)
		tmp = Float64(fma(log(t), -0.5, log(Float64(z * y))) - t);
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 * (-t) + t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -2000.0], t$95$3, If[LessEqual[t$95$2, 908.0], N[(N[(N[Log[t], $MachinePrecision] * -0.5 + N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], t$95$3]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < -2e3 or 908 < (+.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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. add-flip N/A

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

      3. lift--.f64 N/A

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

      4. sub-flip N/A

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

      5. associate--l+ N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. sum-to-mult N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-neg.f64 N/A

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

   3. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 + \frac{\log \left(y + x\right)}{\log z}, \log z, \left(-t\right) - \left(0.5 - a\right) \cdot \log t\right)} \]
   4. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sum-to-mult-rev N/A

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

      5. lift-log.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. log-prod N/A

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

      8. lift-*.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. lift--.f64 N/A

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

      11. associate-+r- N/A

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

      12. lift-neg.f64 N/A

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

      13. sub-flip N/A

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

      14. lift--.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift--.f64 N/A

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

      17. sub-negate-rev N/A

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

      18. lift--.f64 N/A

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

   5. Applied rewrites 75.9%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 77.7%

      \[\leadsto \mathsf{fma}\left(\color{blue}{1}, -t, \left(a - 0.5\right) \cdot \log t\right) \]

   if -2e3 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 908

   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

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower--.f64 69.7

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

   4. Applied rewrites 69.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. lower-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 69.7

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

   6. Applied rewrites 69.7%

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

   7. Taylor expanded in a around 0

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

   9. Applied rewrites 41.7%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lift-fma.f64 N/A

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

       4. associate-+l+ N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. lift-log.f64 N/A

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

       8. lift-log.f64 N/A

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

       9. sum-log N/A

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

       10. lower-log.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 32.3

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

   11. Applied rewrites 32.3%

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

Alternative 11: 69.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(1, -t, \left(a - 0.5\right) \cdot \log t\right) \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t, double a) {
	return fma(1.0, -t, ((a - 0.5) * log(t)));
}

Julia

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

Wolfram

code[x_, y_, z_, t_, a_] := N[(1.0 * (-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. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. add-flip N/A

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

   3. lift--.f64 N/A

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

   4. sub-flip N/A

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

   5. associate--l+ N/A

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. sum-to-mult N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-+.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 N/A

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

   15. lower--.f64 N/A

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

   16. lower-neg.f64 N/A

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

3. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(1 + \frac{\log \left(y + x\right)}{\log z}, \log z, \left(-t\right) - \left(0.5 - a\right) \cdot \log t\right)} \]
4. Step-by-step derivation

   1. lift-fma.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. sum-to-mult-rev N/A

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

   5. lift-log.f64 N/A

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

   6. lift-log.f64 N/A

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

   7. log-prod N/A

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

   8. lift-*.f64 N/A

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

   9. lift-log.f64 N/A

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

   10. lift--.f64 N/A

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

   11. associate-+r- N/A

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

   12. lift-neg.f64 N/A

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

   13. sub-flip N/A

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

   14. lift--.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. lift--.f64 N/A

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

   17. sub-negate-rev N/A

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

   18. lift--.f64 N/A

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

5. Applied rewrites 75.9%

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

6. Taylor expanded in t around inf

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

8. Applied rewrites 77.7%

   \[\leadsto \mathsf{fma}\left(\color{blue}{1}, -t, \left(a - 0.5\right) \cdot \log t\right) \]
9. Add Preprocessing

Alternative 12: 62.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \log t\\ \mathbf{if}\;a - 0.5 \leq -4 \cdot 10^{+23}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a - 0.5 \leq 5 \cdot 10^{+38}:\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* a (log t))))
   (if (<= (- a 0.5) -4e+23) t_1 (if (<= (- a 0.5) 5e+38) (- t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = a * log(t);
	double tmp;
	if ((a - 0.5) <= -4e+23) {
		tmp = t_1;
	} else if ((a - 0.5) <= 5e+38) {
		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 = a * log(t)
    if ((a - 0.5d0) <= (-4d+23)) then
        tmp = t_1
    else if ((a - 0.5d0) <= 5d+38) 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 = a * Math.log(t);
	double tmp;
	if ((a - 0.5) <= -4e+23) {
		tmp = t_1;
	} else if ((a - 0.5) <= 5e+38) {
		tmp = -t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = a * math.log(t)
	tmp = 0
	if (a - 0.5) <= -4e+23:
		tmp = t_1
	elif (a - 0.5) <= 5e+38:
		tmp = -t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(a * log(t))
	tmp = 0.0
	if (Float64(a - 0.5) <= -4e+23)
		tmp = t_1;
	elseif (Float64(a - 0.5) <= 5e+38)
		tmp = Float64(-t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = a * log(t);
	tmp = 0.0;
	if ((a - 0.5) <= -4e+23)
		tmp = t_1;
	elseif ((a - 0.5) <= 5e+38)
		tmp = -t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a - 0.5), $MachinePrecision], -4e+23], t$95$1, If[LessEqual[N[(a - 0.5), $MachinePrecision], 5e+38], (-t), t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 a #s(literal 1/2 binary64)) < -3.9999999999999997e23 or 4.9999999999999997e38 < (-.f64 a #s(literal 1/2 binary64))

   1. Initial program 99.6%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 38.4

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

   4. Applied rewrites 38.4%

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

   if -3.9999999999999997e23 < (-.f64 a #s(literal 1/2 binary64)) < 4.9999999999999997e38

   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. lower-*.f64 38.5

         \[\leadsto -1 \cdot \color{blue}{t} \]

   4. Applied rewrites 38.5%

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

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{t} \]

      2. mul-1-neg N/A

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

      3. lift-neg.f64 38.5

         \[\leadsto -t \]

   6. Applied rewrites 38.5%

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

Alternative 13: 38.5% accurate, 17.6× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in t around inf

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

   1. lower-*.f64 38.5

      \[\leadsto -1 \cdot \color{blue}{t} \]

4. Applied rewrites 38.5%

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

   1. lift-*.f64 N/A

      \[\leadsto -1 \cdot \color{blue}{t} \]

   2. mul-1-neg N/A

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

   3. lift-neg.f64 38.5

      \[\leadsto -t \]

6. Applied rewrites 38.5%

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

Reproduce

herbie shell --seed 2025156 
(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))))