Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 12.4s

Alternatives: 17

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

Alternative 2: 66.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ \mathbf{if}\;t\_1 \leq -10000000 \lor \neg \left(t\_1 \leq 2000\right):\\ \;\;\;\;\left(\mathsf{fma}\left(\log t, a - 0.5, \frac{x}{y}\right) + \log z\right) + \left(-t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log t, \log y\right) + \log z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t)))))
   (if (or (<= t_1 -10000000.0) (not (<= t_1 2000.0)))
     (+ (+ (fma (log t) (- a 0.5) (/ x y)) (log z)) (- t))
     (+ (fma -0.5 (log t) (log y)) (log z)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
	double tmp;
	if ((t_1 <= -10000000.0) || !(t_1 <= 2000.0)) {
		tmp = (fma(log(t), (a - 0.5), (x / y)) + log(z)) + -t;
	} else {
		tmp = fma(-0.5, log(t), log(y)) + log(z);
	}
	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 <= -10000000.0) || !(t_1 <= 2000.0))
		tmp = Float64(Float64(fma(log(t), Float64(a - 0.5), Float64(x / y)) + log(z)) + Float64(-t));
	else
		tmp = Float64(fma(-0.5, log(t), log(y)) + log(z));
	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[Or[LessEqual[t$95$1, -10000000.0], N[Not[LessEqual[t$95$1, 2000.0]], $MachinePrecision]], N[(N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision] + N[(x / y), $MachinePrecision]), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] + (-t)), $MachinePrecision], N[(N[(-0.5 * N[Log[t], $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < -1e7 or 2e3 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t)))

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-log.f64 57.8

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

   5. Applied rewrites 57.8%

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

   6. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 71.0

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

   8. Applied rewrites 71.0%

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

   if -1e7 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 2e3

   1. Initial program 99.2%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-log.f64 99.1

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 16.9

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

   8. Applied rewrites 16.9%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-log.f64 N/A

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

       4. lower-log.f64 11.2

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

   11. Applied rewrites 11.2%

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

   12. Taylor expanded in t around 0

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

       1. lower-log.f64 56.0

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

   14. Applied rewrites 56.0%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \leq -10000000 \lor \neg \left(\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \leq 2000\right):\\ \;\;\;\;\left(\mathsf{fma}\left(\log t, a - 0.5, \frac{x}{y}\right) + \log z\right) + \left(-t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log t, \log y\right) + \log z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ \mathbf{if}\;t\_1 \leq -640 \lor \neg \left(t\_1 \leq 1100\right):\\ \;\;\;\;a \cdot \log t + \left(\log y - t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log t, -0.5, \log \left(\left(y + x\right) \cdot z\right) - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t)))))
   (if (or (<= t_1 -640.0) (not (<= t_1 1100.0)))
     (+ (* a (log t)) (- (log y) t))
     (fma (log t) -0.5 (- (log (* (+ y x) z)) 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 <= -640.0) || !(t_1 <= 1100.0)) {
		tmp = (a * log(t)) + (log(y) - t);
	} else {
		tmp = fma(log(t), -0.5, (log(((y + x) * z)) - 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 <= -640.0) || !(t_1 <= 1100.0))
		tmp = Float64(Float64(a * log(t)) + Float64(log(y) - t));
	else
		tmp = fma(log(t), -0.5, Float64(log(Float64(Float64(y + x) * z)) - 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[Or[LessEqual[t$95$1, -640.0], N[Not[LessEqual[t$95$1, 1100.0]], $MachinePrecision]], N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] + N[(N[Log[y], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], N[(N[Log[t], $MachinePrecision] * -0.5 + N[(N[Log[N[(N[(y + x), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < -640 or 1100 < (+.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 99.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-log.f64 58.7

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

   5. Applied rewrites 58.7%

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

   6. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 68.2

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

   8. Applied rewrites 68.2%

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

   if -640 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 1100

   1. Initial program 99.2%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-log.f64 99.0

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

   5. Applied rewrites 99.0%

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

      1. lift-+.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. lift-log.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift--.f64 N/A

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

      9. associate--l+ N/A

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

      10. lift-log.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. sum-log N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. lift-log.f64 N/A

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

      18. lower--.f64 92.1

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

   7. Applied rewrites 92.1%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \leq -640 \lor \neg \left(\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \leq 1100\right):\\ \;\;\;\;a \cdot \log t + \left(\log y - t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log t, -0.5, \log \left(\left(y + x\right) \cdot z\right) - t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(x + y\right) + \log z\\ \mathbf{if}\;t\_1 \leq -750 \lor \neg \left(t\_1 \leq 710\right):\\ \;\;\;\;\left(\mathsf{fma}\left(\log t, a - 0.5, \frac{x}{y}\right) + \log z\right) + \left(-t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log \left(z \cdot \left(y + x\right)\right) - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log (+ x y)) (log z))))
   (if (or (<= t_1 -750.0) (not (<= t_1 710.0)))
     (+ (+ (fma (log t) (- a 0.5) (/ x y)) (log z)) (- t))
     (fma (- a 0.5) (log t) (- (log (* z (+ y x))) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((x + y)) + log(z);
	double tmp;
	if ((t_1 <= -750.0) || !(t_1 <= 710.0)) {
		tmp = (fma(log(t), (a - 0.5), (x / y)) + log(z)) + -t;
	} else {
		tmp = fma((a - 0.5), log(t), (log((z * (y + x))) - t));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(Float64(x + y)) + log(z))
	tmp = 0.0
	if ((t_1 <= -750.0) || !(t_1 <= 710.0))
		tmp = Float64(Float64(fma(log(t), Float64(a - 0.5), Float64(x / y)) + log(z)) + Float64(-t));
	else
		tmp = fma(Float64(a - 0.5), log(t), Float64(log(Float64(z * Float64(y + x))) - t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -750.0], N[Not[LessEqual[t$95$1, 710.0]], $MachinePrecision]], N[(N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision] + N[(x / y), $MachinePrecision]), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] + (-t)), $MachinePrecision], N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[(N[Log[N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -750 or 710 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z))

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-log.f64 55.1

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

   5. Applied rewrites 55.1%

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

   6. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 51.4

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

   8. Applied rewrites 51.4%

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

   if -750 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 710

   1. Initial program 99.7%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   2. Add Preprocessing
   3. 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

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

      4. lower-fma.f64 99.7

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

      5. lift-+.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. sum-log N/A

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

      9. lower-log.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.8

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(x + y\right) + \log z \leq -750 \lor \neg \left(\log \left(x + y\right) + \log z \leq 710\right):\\ \;\;\;\;\left(\mathsf{fma}\left(\log t, a - 0.5, \frac{x}{y}\right) + \log z\right) + \left(-t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log \left(z \cdot \left(y + x\right)\right) - t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(x + y\right) + \log z\\ \mathbf{if}\;t\_1 \leq -750 \lor \neg \left(t\_1 \leq 710\right):\\ \;\;\;\;a \cdot \log t + \left(\log y - t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log \left(z \cdot \left(y + x\right)\right) - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log (+ x y)) (log z))))
   (if (or (<= t_1 -750.0) (not (<= t_1 710.0)))
     (+ (* a (log t)) (- (log y) t))
     (fma (- a 0.5) (log t) (- (log (* z (+ y x))) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((x + y)) + log(z);
	double tmp;
	if ((t_1 <= -750.0) || !(t_1 <= 710.0)) {
		tmp = (a * log(t)) + (log(y) - t);
	} else {
		tmp = fma((a - 0.5), log(t), (log((z * (y + x))) - t));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(Float64(x + y)) + log(z))
	tmp = 0.0
	if ((t_1 <= -750.0) || !(t_1 <= 710.0))
		tmp = Float64(Float64(a * log(t)) + Float64(log(y) - t));
	else
		tmp = fma(Float64(a - 0.5), log(t), Float64(log(Float64(z * Float64(y + x))) - t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -750.0], N[Not[LessEqual[t$95$1, 710.0]], $MachinePrecision]], N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] + N[(N[Log[y], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[(N[Log[N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -750 or 710 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z))

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-log.f64 55.1

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

   5. Applied rewrites 55.1%

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

   6. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 58.8

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

   8. Applied rewrites 58.8%

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

   if -750 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 710

   1. Initial program 99.7%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   2. Add Preprocessing
   3. 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

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

      4. lower-fma.f64 99.7

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

      5. lift-+.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. sum-log N/A

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

      9. lower-log.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.8

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(x + y\right) + \log z \leq -750 \lor \neg \left(\log \left(x + y\right) + \log z \leq 710\right):\\ \;\;\;\;a \cdot \log t + \left(\log y - t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log \left(z \cdot \left(y + x\right)\right) - t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(x + y\right) + \log z\\ \mathbf{if}\;t\_1 \leq -750 \lor \neg \left(t\_1 \leq 710\right):\\ \;\;\;\;a \cdot \log t + \left(\log y - t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log \left(z \cdot y\right) - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log (+ x y)) (log z))))
   (if (or (<= t_1 -750.0) (not (<= t_1 710.0)))
     (+ (* a (log t)) (- (log y) t))
     (fma (- a 0.5) (log t) (- (log (* z y)) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((x + y)) + log(z);
	double tmp;
	if ((t_1 <= -750.0) || !(t_1 <= 710.0)) {
		tmp = (a * log(t)) + (log(y) - t);
	} else {
		tmp = fma((a - 0.5), log(t), (log((z * y)) - t));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(Float64(x + y)) + log(z))
	tmp = 0.0
	if ((t_1 <= -750.0) || !(t_1 <= 710.0))
		tmp = Float64(Float64(a * log(t)) + Float64(log(y) - t));
	else
		tmp = fma(Float64(a - 0.5), log(t), Float64(log(Float64(z * y)) - t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -750.0], N[Not[LessEqual[t$95$1, 710.0]], $MachinePrecision]], N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] + N[(N[Log[y], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[(N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -750 or 710 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z))

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-log.f64 55.1

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

   5. Applied rewrites 55.1%

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

   6. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 58.8

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

   8. Applied rewrites 58.8%

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

   if -750 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 710

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-log.f64 58.3

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

   5. Applied rewrites 58.3%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-log.f64 67.2

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

   8. Applied rewrites 67.2%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift-fma.f64 N/A

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

      5. lift--.f64 N/A

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

      6. associate-+l+ N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift--.f64 N/A

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

      10. associate-+r- N/A

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

      11. lower--.f64 N/A

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

      12. lift-log.f64 N/A

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

      13. lift-log.f64 N/A

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

      14. sum-log N/A

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

      15. lower-log.f64 N/A

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

      16. lower-*.f64 64.7

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

   10. Applied rewrites 64.7%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(x + y\right) + \log z \leq -750 \lor \neg \left(\log \left(x + y\right) + \log z \leq 710\right):\\ \;\;\;\;a \cdot \log t + \left(\log y - t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log \left(z \cdot y\right) - t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.82 \lor \neg \left(a \leq 0.98\right):\\ \;\;\;\;\left(\mathsf{fma}\left(\log t, a - 0.5, \frac{x}{y}\right) + \log z\right) + \left(-t\right)\\ \mathbf{else}:\\ \;\;\;\;\log y + \left(\mathsf{fma}\left(-0.5, \log t, \log z\right) - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -1.82) (not (<= a 0.98)))
   (+ (+ (fma (log t) (- a 0.5) (/ x y)) (log z)) (- t))
   (+ (log y) (- (fma -0.5 (log t) (log z)) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -1.82) || !(a <= 0.98)) {
		tmp = (fma(log(t), (a - 0.5), (x / y)) + log(z)) + -t;
	} else {
		tmp = log(y) + (fma(-0.5, log(t), log(z)) - t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -1.82) || !(a <= 0.98))
		tmp = Float64(Float64(fma(log(t), Float64(a - 0.5), Float64(x / y)) + log(z)) + Float64(-t));
	else
		tmp = Float64(log(y) + Float64(fma(-0.5, log(t), log(z)) - t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -1.82], N[Not[LessEqual[a, 0.98]], $MachinePrecision]], N[(N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision] + N[(x / y), $MachinePrecision]), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] + (-t)), $MachinePrecision], N[(N[Log[y], $MachinePrecision] + N[(N[(-0.5 * N[Log[t], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.82000000000000006 or 0.97999999999999998 < a

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-log.f64 60.0

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

   5. Applied rewrites 60.0%

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

   6. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 73.1

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

   8. Applied rewrites 73.1%

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

   if -1.82000000000000006 < a < 0.97999999999999998

   1. Initial program 99.6%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-log.f64 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower--.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower-log.f64 63.4

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

   8. Applied rewrites 63.4%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.82 \lor \neg \left(a \leq 0.98\right):\\ \;\;\;\;\left(\mathsf{fma}\left(\log t, a - 0.5, \frac{x}{y}\right) + \log z\right) + \left(-t\right)\\ \mathbf{else}:\\ \;\;\;\;\log y + \left(\mathsf{fma}\left(-0.5, \log t, \log z\right) - t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 9.2 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\log t, a - 0.5, \log z\right) + \log y\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \log t + \log z\right) + \left(\log y - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 9.2e-6)
   (+ (fma (log t) (- a 0.5) (log z)) (log y))
   (+ (+ (* a (log t)) (log z)) (- (log y) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 9.2e-6) {
		tmp = fma(log(t), (a - 0.5), log(z)) + log(y);
	} else {
		tmp = ((a * log(t)) + log(z)) + (log(y) - t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 9.2e-6)
		tmp = Float64(fma(log(t), Float64(a - 0.5), log(z)) + log(y));
	else
		tmp = Float64(Float64(Float64(a * log(t)) + log(z)) + Float64(log(y) - t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 9.2e-6], N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision], N[(N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] + N[(N[Log[y], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 9.2e-6

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-log.f64 57.9

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

   5. Applied rewrites 57.9%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-log.f64 68.8

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

   8. Applied rewrites 68.8%

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

   9. Taylor expanded in t around 0

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

       1. +-commutative N/A

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

       2. +-commutative N/A

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

       3. lower-+.f64 N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-log.f64 N/A

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

       6. lower--.f64 N/A

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

       7. lower-log.f64 N/A

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

       8. lower-log.f64 68.7

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

   11. Applied rewrites 68.7%

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

   if 9.2e-6 < t

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-log.f64 57.1

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

   5. Applied rewrites 57.1%

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

   6. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 66.5

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

   8. Applied rewrites 66.5%

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

Alternative 9: 81.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 450:\\ \;\;\;\;\mathsf{fma}\left(\log t, a - 0.5, \log z\right) + \log y\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) + \left(a - 0.5\right) \cdot \log t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 450.0)
   (+ (fma (log t) (- a 0.5) (log z)) (log y))
   (+ (- t) (* (- a 0.5) (log t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 450.0) {
		tmp = fma(log(t), (a - 0.5), log(z)) + log(y);
	} else {
		tmp = -t + ((a - 0.5) * log(t));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 450.0)
		tmp = Float64(fma(log(t), Float64(a - 0.5), log(z)) + log(y));
	else
		tmp = Float64(Float64(-t) + Float64(Float64(a - 0.5) * log(t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 450.0], N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision], N[((-t) + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 450

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-log.f64 58.1

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

   5. Applied rewrites 58.1%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-log.f64 68.8

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

   8. Applied rewrites 68.8%

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

   9. Taylor expanded in t around 0

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

       1. +-commutative N/A

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

       2. +-commutative N/A

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

       3. lower-+.f64 N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-log.f64 N/A

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

       6. lower--.f64 N/A

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

       7. lower-log.f64 N/A

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

       8. lower-log.f64 68.0

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

   11. Applied rewrites 68.0%

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

   if 450 < t

   1. Initial program 99.9%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 99.8

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

   5. Applied rewrites 99.8%

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

Alternative 10: 69.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. lower-+.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-+.f64 N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-log.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-log.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lower-log.f64 57.5

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

5. Applied rewrites 57.5%

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

6. Taylor expanded in x around 0

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

   1. lower--.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-log.f64 N/A

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

   7. lower--.f64 N/A

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

   8. lower-log.f64 N/A

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

   9. lower-log.f64 67.9

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

8. Applied rewrites 67.9%

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

Alternative 11: 69.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, y, z, t, a)
	return Float64(fma(log(t), Float64(a - 0.5), log(z)) + Float64(log(y) - t))
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[(N[Log[y], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. lower-+.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-log.f64 N/A

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

   7. lower--.f64 N/A

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

   8. lower-log.f64 N/A

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

   9. lower--.f64 N/A

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

   10. lower-log.f64 67.9

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

5. Applied rewrites 67.9%

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

Alternative 12: 57.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. lower-+.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-+.f64 N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-log.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-log.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lower-log.f64 57.5

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

5. Applied rewrites 57.5%

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

6. Taylor expanded in a around inf

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

   1. lower-*.f64 N/A

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

   2. lower-log.f64 56.3

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

8. Applied rewrites 56.3%

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

Alternative 13: 55.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -2 \cdot 10^{+34} \lor \neg \left(a - 0.5 \leq 50000000\right):\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= (- a 0.5) -2e+34) (not (<= (- a 0.5) 50000000.0)))
   (* (log t) a)
   (- (/ x y) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((a - 0.5) <= -2e+34) || !((a - 0.5) <= 50000000.0)) {
		tmp = log(t) * a;
	} else {
		tmp = (x / y) - t;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (((a - 0.5d0) <= (-2d+34)) .or. (.not. ((a - 0.5d0) <= 50000000.0d0))) then
        tmp = log(t) * a
    else
        tmp = (x / y) - 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 (((a - 0.5) <= -2e+34) || !((a - 0.5) <= 50000000.0)) {
		tmp = Math.log(t) * a;
	} else {
		tmp = (x / y) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if ((a - 0.5) <= -2e+34) or not ((a - 0.5) <= 50000000.0):
		tmp = math.log(t) * a
	else:
		tmp = (x / y) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((Float64(a - 0.5) <= -2e+34) || !(Float64(a - 0.5) <= 50000000.0))
		tmp = Float64(log(t) * a);
	else
		tmp = Float64(Float64(x / y) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (((a - 0.5) <= -2e+34) || ~(((a - 0.5) <= 50000000.0)))
		tmp = log(t) * a;
	else
		tmp = (x / y) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[N[(a - 0.5), $MachinePrecision], -2e+34], N[Not[LessEqual[N[(a - 0.5), $MachinePrecision], 50000000.0]], $MachinePrecision]], N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision], N[(N[(x / y), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 a #s(literal 1/2 binary64)) < -1.99999999999999989e34 or 5e7 < (-.f64 a #s(literal 1/2 binary64))

   1. Initial program 99.7%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 80.3

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

   5. Applied rewrites 80.3%

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

   if -1.99999999999999989e34 < (-.f64 a #s(literal 1/2 binary64)) < 5e7

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-log.f64 53.6

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

   5. Applied rewrites 53.6%

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

   6. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower--.f64 52.9

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

   8. Applied rewrites 52.9%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+r- N/A

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

      4. lower--.f64 N/A

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

   10. Applied rewrites 41.8%

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

   11. Taylor expanded in x around inf

       \[\leadsto \frac{x}{y} - t \]
   12. Step-by-step derivation

       1. lower-/.f64 36.3

          \[\leadsto \frac{x}{y} - t \]

   13. Applied rewrites 36.3%

       \[\leadsto \frac{x}{y} - t \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -2 \cdot 10^{+34} \lor \neg \left(a - 0.5 \leq 50000000\right):\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} - t\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 77.5% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 78.8

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

5. Applied rewrites 78.8%

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

Alternative 15: 74.9% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. lower-+.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-+.f64 N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-log.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-log.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lower-log.f64 57.5

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

5. Applied rewrites 57.5%

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

6. Taylor expanded in x around 0

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

   1. lower--.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-log.f64 N/A

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

   7. lower--.f64 N/A

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

   8. lower-log.f64 N/A

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

   9. lower-log.f64 67.9

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

8. Applied rewrites 67.9%

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

9. Taylor expanded in a around inf

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

    3. lower-log.f64 76.4

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

11. Applied rewrites 76.4%

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

Alternative 16: 28.2% accurate, 21.4× speedup

Math

\[\begin{array}{l} \\ \frac{x}{y} - t \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(x / y), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. lower-+.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-+.f64 N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-log.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-log.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lower-log.f64 57.5

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

5. Applied rewrites 57.5%

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

6. Taylor expanded in y around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. associate-*r* N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-log.f64 N/A

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

   7. lower--.f64 46.4

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

8. Applied rewrites 46.4%

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

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. associate-+r- N/A

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

   4. lower--.f64 N/A

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

10. Applied rewrites 37.5%

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

11. Taylor expanded in x around inf

    \[\leadsto \frac{x}{y} - t \]
12. Step-by-step derivation

    1. lower-/.f64 27.4

       \[\leadsto \frac{x}{y} - t \]

13. Applied rewrites 27.4%

    \[\leadsto \frac{x}{y} - t \]
14. Add Preprocessing

Alternative 17: 37.8% accurate, 107.0× 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 99.7%

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

3. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 37.2

      \[\leadsto -t \]

5. Applied rewrites 37.2%

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

Developer Target 1: 99.6% accurate, 1.0× speedup

Math

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

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(log(Float64(x + y)) + Float64(Float64(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[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2025026 
(FPCore (x y z t a)
  :name "Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (log (+ x y)) (+ (- (log z) t) (* (- a 1/2) (log t)))))

  (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))