Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 7.1s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

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 + x)));
}

Julia

function code(x, y, z, t, a)
	return Float64(fma(log(t), Float64(a - 0.5), log(z)) - Float64(t - log(Float64(y + x))))
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[N[(y + x), $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. +-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. sub-negate-rev N/A

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

   5. lift-+.f64 N/A

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

   6. associate--r+ N/A

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

   7. sub-negate N/A

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

   8. associate-+r- N/A

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

   9. lower--.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. lower--.f64 99.6

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 99.6

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

3. Applied rewrites 99.6%

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

Alternative 2: 85.9% 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. 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. sub-negate-rev N/A

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

   5. lift-+.f64 N/A

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

   6. associate--r+ N/A

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

   7. sub-negate N/A

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

   8. associate-+r- N/A

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

   9. lower--.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. lower--.f64 99.6

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 99.6

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

3. Applied rewrites 99.6%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 68.8%

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

Alternative 3: 85.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   5. lift-+.f64 N/A

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

   6. associate--r+ N/A

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

   7. sub-negate N/A

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

   8. associate-+r- N/A

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

   9. lower--.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. lower--.f64 99.6

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 99.6

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

3. Applied rewrites 99.6%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 68.8%

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

   1. lift--.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

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

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

   6. lift--.f64 N/A

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

   7. sub-negate-rev N/A

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

   8. lift--.f64 N/A

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

   9. associate--l- N/A

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

   10. lower--.f64 N/A

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

   11. lower-fma.f64 68.8

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

8. Applied rewrites 68.8%

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

Alternative 4: 68.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \log t\\ \mathbf{if}\;a \leq -1.2 \cdot 10^{+20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 14500000000:\\ \;\;\;\;\left(\log z + -0.5 \cdot \log t\right) - \left(t - \log y\right)\\ \mathbf{else}:\\ \;\;\;\;\log z - -1 \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* a (log t))))
   (if (<= a -1.2e+20)
     t_1
     (if (<= a 14500000000.0)
       (- (+ (log z) (* -0.5 (log t))) (- t (log y)))
       (- (log z) (* -1.0 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 <= -1.2e+20) {
		tmp = t_1;
	} else if (a <= 14500000000.0) {
		tmp = (log(z) + (-0.5 * log(t))) - (t - log(y));
	} else {
		tmp = log(z) - (-1.0 * 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 <= (-1.2d+20)) then
        tmp = t_1
    else if (a <= 14500000000.0d0) then
        tmp = (log(z) + ((-0.5d0) * log(t))) - (t - log(y))
    else
        tmp = log(z) - ((-1.0d0) * 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 <= -1.2e+20) {
		tmp = t_1;
	} else if (a <= 14500000000.0) {
		tmp = (Math.log(z) + (-0.5 * Math.log(t))) - (t - Math.log(y));
	} else {
		tmp = Math.log(z) - (-1.0 * t_1);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = a * math.log(t)
	tmp = 0
	if a <= -1.2e+20:
		tmp = t_1
	elif a <= 14500000000.0:
		tmp = (math.log(z) + (-0.5 * math.log(t))) - (t - math.log(y))
	else:
		tmp = math.log(z) - (-1.0 * t_1)
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(a * log(t))
	tmp = 0.0
	if (a <= -1.2e+20)
		tmp = t_1;
	elseif (a <= 14500000000.0)
		tmp = Float64(Float64(log(z) + Float64(-0.5 * log(t))) - Float64(t - log(y)));
	else
		tmp = Float64(log(z) - Float64(-1.0 * 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 <= -1.2e+20)
		tmp = t_1;
	elseif (a <= 14500000000.0)
		tmp = (log(z) + (-0.5 * log(t))) - (t - log(y));
	else
		tmp = log(z) - (-1.0 * 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[a, -1.2e+20], t$95$1, If[LessEqual[a, 14500000000.0], N[(N[(N[Log[z], $MachinePrecision] + N[(-0.5 * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Log[z], $MachinePrecision] - N[(-1.0 * t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.2e20

   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 38.8

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

   4. Applied rewrites 38.8%

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

   if -1.2e20 < a < 1.45e10

   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. sub-negate-rev N/A

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

      5. lift-+.f64 N/A

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

      6. associate--r+ N/A

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

      7. sub-negate N/A

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

      8. associate-+r- N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower--.f64 99.6

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 99.6

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 68.8%

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

   7. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-log.f64 40.8

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

   9. Applied rewrites 40.8%

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

   if 1.45e10 < a

   1. Initial program 99.6%

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

   2. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower--.f64 68.4

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

   4. Applied rewrites 68.4%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. +-commutative N/A

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

   6. Applied rewrites 68.4%

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

   7. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

         \[\leadsto \log z - -1 \cdot \left(a \cdot \log t\right) \]

      3. lower-log.f64 41.6

         \[\leadsto \log z - -1 \cdot \left(a \cdot \log t\right) \]

   9. Applied rewrites 41.6%

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

Alternative 5: 68.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \log t\\ \mathbf{if}\;a \leq -1.2 \cdot 10^{+20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 14500000000:\\ \;\;\;\;\log z - \mathsf{fma}\left(0.5, \log t, t - \log y\right)\\ \mathbf{else}:\\ \;\;\;\;\log z - -1 \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* a (log t))))
   (if (<= a -1.2e+20)
     t_1
     (if (<= a 14500000000.0)
       (- (log z) (fma 0.5 (log t) (- t (log y))))
       (- (log z) (* -1.0 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 <= -1.2e+20) {
		tmp = t_1;
	} else if (a <= 14500000000.0) {
		tmp = log(z) - fma(0.5, log(t), (t - log(y)));
	} else {
		tmp = log(z) - (-1.0 * t_1);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(a * log(t))
	tmp = 0.0
	if (a <= -1.2e+20)
		tmp = t_1;
	elseif (a <= 14500000000.0)
		tmp = Float64(log(z) - fma(0.5, log(t), Float64(t - log(y))));
	else
		tmp = Float64(log(z) - Float64(-1.0 * t_1));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -1.2e+20], t$95$1, If[LessEqual[a, 14500000000.0], N[(N[Log[z], $MachinePrecision] - N[(0.5 * N[Log[t], $MachinePrecision] + N[(t - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Log[z], $MachinePrecision] - N[(-1.0 * t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.2e20

   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 38.8

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

   4. Applied rewrites 38.8%

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

   if -1.2e20 < a < 1.45e10

   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. sub-negate-rev N/A

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

      5. lift-+.f64 N/A

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

      6. associate--r+ N/A

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

      7. sub-negate N/A

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

      8. associate-+r- N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower--.f64 99.6

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 99.6

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 68.8%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

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

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

      6. lift--.f64 N/A

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

      7. sub-negate-rev N/A

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

      8. lift--.f64 N/A

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

      9. associate--l- N/A

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

      10. lower--.f64 N/A

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

      11. lower-fma.f64 68.8

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

   8. Applied rewrites 68.8%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 40.8%

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

   if 1.45e10 < a

   1. Initial program 99.6%

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

   2. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower--.f64 68.4

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

   4. Applied rewrites 68.4%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. +-commutative N/A

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

   6. Applied rewrites 68.4%

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

   7. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

         \[\leadsto \log z - -1 \cdot \left(a \cdot \log t\right) \]

      3. lower-log.f64 41.6

         \[\leadsto \log z - -1 \cdot \left(a \cdot \log t\right) \]

   9. Applied rewrites 41.6%

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

Alternative 6: 68.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(x + y\right) + \log z\\ t_2 := \log z - -1 \cdot \left(a \cdot \log t\right)\\ \mathbf{if}\;t\_1 \leq -750:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 700:\\ \;\;\;\;\log \left(z \cdot \left(y + x\right)\right) - \mathsf{fma}\left(0.5 - a, \log t, t\right)\\ \mathbf{elif}\;t\_1 \leq 1035:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log (+ x y)) (log z)))
        (t_2 (- (log z) (* -1.0 (* a (log t))))))
   (if (<= t_1 -750.0)
     t_2
     (if (<= t_1 700.0)
       (- (log (* z (+ y x))) (fma (- 0.5 a) (log t) t))
       (if (<= t_1 1035.0) t_2 (- t))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((x + y)) + log(z);
	double t_2 = log(z) - (-1.0 * (a * log(t)));
	double tmp;
	if (t_1 <= -750.0) {
		tmp = t_2;
	} else if (t_1 <= 700.0) {
		tmp = log((z * (y + x))) - fma((0.5 - a), log(t), t);
	} else if (t_1 <= 1035.0) {
		tmp = t_2;
	} else {
		tmp = -t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(Float64(x + y)) + log(z))
	t_2 = Float64(log(z) - Float64(-1.0 * Float64(a * log(t))))
	tmp = 0.0
	if (t_1 <= -750.0)
		tmp = t_2;
	elseif (t_1 <= 700.0)
		tmp = Float64(log(Float64(z * Float64(y + x))) - fma(Float64(0.5 - a), log(t), t));
	elseif (t_1 <= 1035.0)
		tmp = t_2;
	else
		tmp = Float64(-t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Log[z], $MachinePrecision] - N[(-1.0 * N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -750.0], t$95$2, If[LessEqual[t$95$1, 700.0], N[(N[Log[N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(N[(0.5 - a), $MachinePrecision] * N[Log[t], $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1035.0], t$95$2, (-t)]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -750 or 700 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 1035

   1. Initial program 99.6%

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

   2. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower--.f64 68.4

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

   4. Applied rewrites 68.4%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. +-commutative N/A

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

   6. Applied rewrites 68.4%

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

   7. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

         \[\leadsto \log z - -1 \cdot \left(a \cdot \log t\right) \]

      3. lower-log.f64 41.6

         \[\leadsto \log z - -1 \cdot \left(a \cdot \log t\right) \]

   9. Applied rewrites 41.6%

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

   if -750 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 700

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. sum-log N/A

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

      9. lower-log.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lift-*.f64 N/A

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

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

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

      17. +-commutative N/A

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

      18. lift--.f64 N/A

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

      19. sub-negate-rev N/A

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

      20. lower-fma.f64 N/A

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

      21. lower--.f64 76.9

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

   3. Applied rewrites 76.9%

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

   if 1035 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z))

   1. Initial program 99.6%

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

   2. Taylor expanded in t around inf

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

      1. lower-*.f64 37.2

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

   4. Applied rewrites 37.2%

      \[\leadsto \color{blue}{-1 \cdot t} \]
   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. lower-neg.f64 37.2

         \[\leadsto -t \]

   6. Applied rewrites 37.2%

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

Alternative 7: 68.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(x + y\right) + \log z\\ t_2 := \log z - -1 \cdot \left(a \cdot \log t\right)\\ \mathbf{if}\;t\_1 \leq -750:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 700:\\ \;\;\;\;\mathsf{fma}\left(\log t, a - 0.5, \log \left(z \cdot \left(y + x\right)\right)\right) - t\\ \mathbf{elif}\;t\_1 \leq 1035:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log (+ x y)) (log z)))
        (t_2 (- (log z) (* -1.0 (* a (log t))))))
   (if (<= t_1 -750.0)
     t_2
     (if (<= t_1 700.0)
       (- (fma (log t) (- a 0.5) (log (* z (+ y x)))) t)
       (if (<= t_1 1035.0) t_2 (- t))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((x + y)) + log(z);
	double t_2 = log(z) - (-1.0 * (a * log(t)));
	double tmp;
	if (t_1 <= -750.0) {
		tmp = t_2;
	} else if (t_1 <= 700.0) {
		tmp = fma(log(t), (a - 0.5), log((z * (y + x)))) - t;
	} else if (t_1 <= 1035.0) {
		tmp = t_2;
	} else {
		tmp = -t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(Float64(x + y)) + log(z))
	t_2 = Float64(log(z) - Float64(-1.0 * Float64(a * log(t))))
	tmp = 0.0
	if (t_1 <= -750.0)
		tmp = t_2;
	elseif (t_1 <= 700.0)
		tmp = Float64(fma(log(t), Float64(a - 0.5), log(Float64(z * Float64(y + x)))) - t);
	elseif (t_1 <= 1035.0)
		tmp = t_2;
	else
		tmp = Float64(-t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Log[z], $MachinePrecision] - N[(-1.0 * N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -750.0], t$95$2, If[LessEqual[t$95$1, 700.0], N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision] + N[Log[N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[t$95$1, 1035.0], t$95$2, (-t)]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -750 or 700 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 1035

   1. Initial program 99.6%

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

   2. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower--.f64 68.4

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

   4. Applied rewrites 68.4%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. +-commutative N/A

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

   6. Applied rewrites 68.4%

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

   7. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

         \[\leadsto \log z - -1 \cdot \left(a \cdot \log t\right) \]

      3. lower-log.f64 41.6

         \[\leadsto \log z - -1 \cdot \left(a \cdot \log t\right) \]

   9. Applied rewrites 41.6%

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

   if -750 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 700

   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

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

   3. Applied rewrites 76.9%

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

   if 1035 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z))

   1. Initial program 99.6%

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

   2. Taylor expanded in t around inf

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

      1. lower-*.f64 37.2

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

   4. Applied rewrites 37.2%

      \[\leadsto \color{blue}{-1 \cdot t} \]
   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. lower-neg.f64 37.2

         \[\leadsto -t \]

   6. Applied rewrites 37.2%

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

Alternative 8: 65.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(x + y\right) + \log z\\ t_2 := \log z - -1 \cdot \left(a \cdot \log t\right)\\ \mathbf{if}\;t\_1 \leq -750:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 700:\\ \;\;\;\;\mathsf{fma}\left(\log t, a - 0.5, \log \left(z \cdot y\right)\right) - t\\ \mathbf{elif}\;t\_1 \leq 1035:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log (+ x y)) (log z)))
        (t_2 (- (log z) (* -1.0 (* a (log t))))))
   (if (<= t_1 -750.0)
     t_2
     (if (<= t_1 700.0)
       (- (fma (log t) (- a 0.5) (log (* z y))) t)
       (if (<= t_1 1035.0) t_2 (- t))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((x + y)) + log(z);
	double t_2 = log(z) - (-1.0 * (a * log(t)));
	double tmp;
	if (t_1 <= -750.0) {
		tmp = t_2;
	} else if (t_1 <= 700.0) {
		tmp = fma(log(t), (a - 0.5), log((z * y))) - t;
	} else if (t_1 <= 1035.0) {
		tmp = t_2;
	} else {
		tmp = -t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(Float64(x + y)) + log(z))
	t_2 = Float64(log(z) - Float64(-1.0 * Float64(a * log(t))))
	tmp = 0.0
	if (t_1 <= -750.0)
		tmp = t_2;
	elseif (t_1 <= 700.0)
		tmp = Float64(fma(log(t), Float64(a - 0.5), log(Float64(z * y))) - t);
	elseif (t_1 <= 1035.0)
		tmp = t_2;
	else
		tmp = Float64(-t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Log[z], $MachinePrecision] - N[(-1.0 * N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -750.0], t$95$2, If[LessEqual[t$95$1, 700.0], N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision] + N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[t$95$1, 1035.0], t$95$2, (-t)]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -750 or 700 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 1035

   1. Initial program 99.6%

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

   2. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower--.f64 68.4

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

   4. Applied rewrites 68.4%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. +-commutative N/A

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

   6. Applied rewrites 68.4%

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

   7. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

         \[\leadsto \log z - -1 \cdot \left(a \cdot \log t\right) \]

      3. lower-log.f64 41.6

         \[\leadsto \log z - -1 \cdot \left(a \cdot \log t\right) \]

   9. Applied rewrites 41.6%

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

   if -750 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 700

   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

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

   3. Applied rewrites 76.9%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 53.7%

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

   if 1035 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z))

   1. Initial program 99.6%

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

   2. Taylor expanded in t around inf

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

      1. lower-*.f64 37.2

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

   4. Applied rewrites 37.2%

      \[\leadsto \color{blue}{-1 \cdot t} \]
   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. lower-neg.f64 37.2

         \[\leadsto -t \]

   6. Applied rewrites 37.2%

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

Alternative 9: 64.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+32}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t\_1 \leq 900:\\ \;\;\;\;\mathsf{fma}\left(\log t, -0.5, \log \left(y \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log z - -1 \cdot \left(a \cdot \log t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t)))))
   (if (<= t_1 -5e+32)
     (- t)
     (if (<= t_1 900.0)
       (- (fma (log t) -0.5 (log (* y z))) t)
       (- (log z) (* -1.0 (* a (log t))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
	double tmp;
	if (t_1 <= -5e+32) {
		tmp = -t;
	} else if (t_1 <= 900.0) {
		tmp = fma(log(t), -0.5, log((y * z))) - t;
	} else {
		tmp = log(z) - (-1.0 * (a * log(t)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + Float64(Float64(a - 0.5) * log(t)))
	tmp = 0.0
	if (t_1 <= -5e+32)
		tmp = Float64(-t);
	elseif (t_1 <= 900.0)
		tmp = Float64(fma(log(t), -0.5, log(Float64(y * z))) - t);
	else
		tmp = Float64(log(z) - Float64(-1.0 * Float64(a * log(t))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+32], (-t), If[LessEqual[t$95$1, 900.0], N[(N[(N[Log[t], $MachinePrecision] * -0.5 + N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[Log[z], $MachinePrecision] - N[(-1.0 * N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < -4.9999999999999997e32

   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 37.2

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

   4. Applied rewrites 37.2%

      \[\leadsto \color{blue}{-1 \cdot t} \]
   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. lower-neg.f64 37.2

         \[\leadsto -t \]

   6. Applied rewrites 37.2%

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

   if -4.9999999999999997e32 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 900

   1. Initial program 99.6%

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

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

   3. Applied rewrites 76.9%

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

   4. Taylor expanded in a around 0

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

   6. Applied rewrites 48.2%

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

   7. Taylor expanded in x around 0

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

      1. lower-*.f64 32.0

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

   9. Applied rewrites 32.0%

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

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

   1. Initial program 99.6%

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

   2. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower--.f64 68.4

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

   4. Applied rewrites 68.4%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. +-commutative N/A

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

   6. Applied rewrites 68.4%

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

   7. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

         \[\leadsto \log z - -1 \cdot \left(a \cdot \log t\right) \]

      3. lower-log.f64 41.6

         \[\leadsto \log z - -1 \cdot \left(a \cdot \log t\right) \]

   9. Applied rewrites 41.6%

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

Alternative 10: 62.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \log t\\ \mathbf{if}\;a - 0.5 \leq -1 \cdot 10^{+32}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a - 0.5 \leq 5000000:\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* a (log t))))
   (if (<= (- a 0.5) -1e+32) t_1 (if (<= (- a 0.5) 5000000.0) (- 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) <= -1e+32) {
		tmp = t_1;
	} else if ((a - 0.5) <= 5000000.0) {
		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) <= (-1d+32)) then
        tmp = t_1
    else if ((a - 0.5d0) <= 5000000.0d0) 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) <= -1e+32) {
		tmp = t_1;
	} else if ((a - 0.5) <= 5000000.0) {
		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) <= -1e+32:
		tmp = t_1
	elif (a - 0.5) <= 5000000.0:
		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) <= -1e+32)
		tmp = t_1;
	elseif (Float64(a - 0.5) <= 5000000.0)
		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) <= -1e+32)
		tmp = t_1;
	elseif ((a - 0.5) <= 5000000.0)
		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], -1e+32], t$95$1, If[LessEqual[N[(a - 0.5), $MachinePrecision], 5000000.0], (-t), t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 a #s(literal 1/2 binary64)) < -1.00000000000000005e32 or 5e6 < (-.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 38.8

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

   4. Applied rewrites 38.8%

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

   if -1.00000000000000005e32 < (-.f64 a #s(literal 1/2 binary64)) < 5e6

   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 37.2

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

   4. Applied rewrites 37.2%

      \[\leadsto \color{blue}{-1 \cdot t} \]
   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. lower-neg.f64 37.2

         \[\leadsto -t \]

   6. Applied rewrites 37.2%

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

Alternative 11: 59.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.3 \cdot 10^{+41}:\\ \;\;\;\;\log z - -1 \cdot \left(a \cdot \log t\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 1.3e+41) (- (log z) (* -1.0 (* a (log t)))) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 1.3e+41) {
		tmp = log(z) - (-1.0 * (a * log(t)));
	} else {
		tmp = -t;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 1.3d+41) then
        tmp = log(z) - ((-1.0d0) * (a * log(t)))
    else
        tmp = -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 1.3e+41) {
		tmp = Math.log(z) - (-1.0 * (a * Math.log(t)));
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 1.3e+41:
		tmp = math.log(z) - (-1.0 * (a * math.log(t)))
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 1.3e+41)
		tmp = Float64(log(z) - Float64(-1.0 * Float64(a * log(t))));
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 1.3e+41)
		tmp = log(z) - (-1.0 * (a * log(t)));
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 1.3e+41], N[(N[Log[z], $MachinePrecision] - N[(-1.0 * N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if t < 1.3e41

   1. Initial program 99.6%

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

   2. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower--.f64 68.4

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

   4. Applied rewrites 68.4%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. +-commutative N/A

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

   6. Applied rewrites 68.4%

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

   7. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

         \[\leadsto \log z - -1 \cdot \left(a \cdot \log t\right) \]

      3. lower-log.f64 41.6

         \[\leadsto \log z - -1 \cdot \left(a \cdot \log t\right) \]

   9. Applied rewrites 41.6%

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

   if 1.3e41 < t

   1. Initial program 99.6%

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

   2. Taylor expanded in t around inf

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

      1. lower-*.f64 37.2

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

   4. Applied rewrites 37.2%

      \[\leadsto \color{blue}{-1 \cdot t} \]
   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. lower-neg.f64 37.2

         \[\leadsto -t \]

   6. Applied rewrites 37.2%

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

Alternative 12: 37.2% 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 37.2

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

4. Applied rewrites 37.2%

   \[\leadsto \color{blue}{-1 \cdot t} \]
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. lower-neg.f64 37.2

      \[\leadsto -t \]

6. Applied rewrites 37.2%

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

Reproduce

herbie shell --seed 2025143 
(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))))