Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 8.1s

Alternatives: 20

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 99.6%

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

Alternative 2: 71.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ t_2 := \log t \cdot a\\ \mathbf{if}\;t\_1 \leq -600:\\ \;\;\;\;\left(t\_2 + \log y\right) - t\\ \mathbf{elif}\;t\_1 \leq 950:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log t, \log \left(z \cdot y\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(t\_2 + \log z\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))
        (t_2 (* (log t) a)))
   (if (<= t_1 -600.0)
     (- (+ t_2 (log y)) t)
     (if (<= t_1 950.0)
       (- (fma -0.5 (log t) (log (* z y))) t)
       (- (+ t_2 (log 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 t_2 = log(t) * a;
	double tmp;
	if (t_1 <= -600.0) {
		tmp = (t_2 + log(y)) - t;
	} else if (t_1 <= 950.0) {
		tmp = fma(-0.5, log(t), log((z * y))) - t;
	} else {
		tmp = (t_2 + log(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)))
	t_2 = Float64(log(t) * a)
	tmp = 0.0
	if (t_1 <= -600.0)
		tmp = Float64(Float64(t_2 + log(y)) - t);
	elseif (t_1 <= 950.0)
		tmp = Float64(fma(-0.5, log(t), log(Float64(z * y))) - t);
	else
		tmp = Float64(Float64(t_2 + log(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]}, Block[{t$95$2 = N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[t$95$1, -600.0], N[(N[(t$95$2 + N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[t$95$1, 950.0], N[(N[(-0.5 * N[Log[t], $MachinePrecision] + N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(t$95$2 + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.8%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   2. 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. 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. associate-+r+ N/A

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

      3. sum-log N/A

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

      4. *-commutative N/A

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

      5. log-pow-rev N/A

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

      6. sum-log N/A

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

      7. lower-log.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lift--.f64 24.6

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

   5. Applied rewrites 24.6%

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

      1. lift-log.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. log-prod N/A

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

      7. log-pow-rev N/A

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

      8. *-commutative N/A

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

      9. sum-log N/A

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

      10. associate-+r+ N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-log.f64 N/A

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

      16. lift--.f64 N/A

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

      17. lift-log.f64 N/A

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

      18. lower-log.f64 73.8

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

   7. Applied rewrites 73.8%

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

   8. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 72.4

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

   10. Applied rewrites 72.4%

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

   if -600 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 950

   1. Initial program 99.0%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   2. 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. 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. associate-+r+ N/A

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

      3. sum-log N/A

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

      4. *-commutative N/A

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

      5. log-pow-rev N/A

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

      6. sum-log N/A

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

      7. lower-log.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lift--.f64 41.7

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

   5. Applied rewrites 41.7%

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

      1. lift-log.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. sum-log N/A

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

      8. log-pow-rev N/A

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

      9. lower-fma.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 47.2

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

   7. Applied rewrites 47.2%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 46.7%

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

   if 950 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t)))

   1. Initial program 99.6%

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

   3. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lift-log.f64 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 85.5

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

   8. Applied rewrites 85.5%

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

Alternative 3: 71.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ t_2 := \log t \cdot a\\ \mathbf{if}\;t\_1 \leq -600:\\ \;\;\;\;\left(t\_2 + \log y\right) - t\\ \mathbf{elif}\;t\_1 \leq 950:\\ \;\;\;\;\mathsf{fma}\left(\log t, a - 0.5, \log \left(z \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_2 + \log z\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))
        (t_2 (* (log t) a)))
   (if (<= t_1 -600.0)
     (- (+ t_2 (log y)) t)
     (if (<= t_1 950.0)
       (fma (log t) (- a 0.5) (log (* z y)))
       (- (+ t_2 (log 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 t_2 = log(t) * a;
	double tmp;
	if (t_1 <= -600.0) {
		tmp = (t_2 + log(y)) - t;
	} else if (t_1 <= 950.0) {
		tmp = fma(log(t), (a - 0.5), log((z * y)));
	} else {
		tmp = (t_2 + log(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)))
	t_2 = Float64(log(t) * a)
	tmp = 0.0
	if (t_1 <= -600.0)
		tmp = Float64(Float64(t_2 + log(y)) - t);
	elseif (t_1 <= 950.0)
		tmp = fma(log(t), Float64(a - 0.5), log(Float64(z * y)));
	else
		tmp = Float64(Float64(t_2 + log(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]}, Block[{t$95$2 = N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[t$95$1, -600.0], N[(N[(t$95$2 + N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[t$95$1, 950.0], N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision] + N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.8%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   2. 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. 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. associate-+r+ N/A

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

      3. sum-log N/A

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

      4. *-commutative N/A

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

      5. log-pow-rev N/A

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

      6. sum-log N/A

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

      7. lower-log.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lift--.f64 24.6

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

   5. Applied rewrites 24.6%

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

      1. lift-log.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. log-prod N/A

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

      7. log-pow-rev N/A

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

      8. *-commutative N/A

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

      9. sum-log N/A

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

      10. associate-+r+ N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-log.f64 N/A

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

      16. lift--.f64 N/A

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

      17. lift-log.f64 N/A

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

      18. lower-log.f64 73.8

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

   7. Applied rewrites 73.8%

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

   8. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 72.4

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

   10. Applied rewrites 72.4%

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

   if -600 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 950

   1. Initial program 99.0%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   2. 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. 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. associate-+r+ N/A

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

      3. sum-log N/A

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

      4. *-commutative N/A

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

      5. log-pow-rev N/A

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

      6. sum-log N/A

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

      7. lower-log.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lift--.f64 41.7

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

   5. Applied rewrites 41.7%

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

   6. Taylor expanded in t around 0

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

      1. pow-to-exp N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. sum-log N/A

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

      5. log-pow-rev N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 46.8

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

   8. Applied rewrites 46.8%

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

   if 950 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t)))

   1. Initial program 99.6%

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

   3. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lift-log.f64 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 85.5

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

   8. Applied rewrites 85.5%

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

Alternative 4: 71.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ \mathbf{if}\;t\_1 \leq -600 \lor \neg \left(t\_1 \leq 700\right):\\ \;\;\;\;\left(\log t \cdot a + \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(\left(y + x\right) \cdot z\right) \cdot \frac{1}{\sqrt{t}}\right)\\ \end{array} \end{array} \]

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 -600.0) (not (<= t_1 700.0)))
     (- (+ (* (log t) a) (log y)) t)
     (log (* (* (+ y x) z) (/ 1.0 (sqrt 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 <= -600.0) || !(t_1 <= 700.0)) {
		tmp = ((log(t) * a) + log(y)) - t;
	} else {
		tmp = log((((y + x) * z) * (1.0 / sqrt(t))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((log((x + y)) + log(z)) - t) + ((a - 0.5d0) * log(t))
    if ((t_1 <= (-600.0d0)) .or. (.not. (t_1 <= 700.0d0))) then
        tmp = ((log(t) * a) + log(y)) - t
    else
        tmp = log((((y + x) * z) * (1.0d0 / sqrt(t))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = ((Math.log((x + y)) + Math.log(z)) - t) + ((a - 0.5) * Math.log(t));
	double tmp;
	if ((t_1 <= -600.0) || !(t_1 <= 700.0)) {
		tmp = ((Math.log(t) * a) + Math.log(y)) - t;
	} else {
		tmp = Math.log((((y + x) * z) * (1.0 / Math.sqrt(t))));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = ((math.log((x + y)) + math.log(z)) - t) + ((a - 0.5) * math.log(t))
	tmp = 0
	if (t_1 <= -600.0) or not (t_1 <= 700.0):
		tmp = ((math.log(t) * a) + math.log(y)) - t
	else:
		tmp = math.log((((y + x) * z) * (1.0 / math.sqrt(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 <= -600.0) || !(t_1 <= 700.0))
		tmp = Float64(Float64(Float64(log(t) * a) + log(y)) - t);
	else
		tmp = log(Float64(Float64(Float64(y + x) * z) * Float64(1.0 / sqrt(t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
	tmp = 0.0;
	if ((t_1 <= -600.0) || ~((t_1 <= 700.0)))
		tmp = ((log(t) * a) + log(y)) - t;
	else
		tmp = log((((y + x) * z) * (1.0 / sqrt(t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -600.0], N[Not[LessEqual[t$95$1, 700.0]], $MachinePrecision]], N[(N[(N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[Log[N[(N[(N[(y + x), $MachinePrecision] * z), $MachinePrecision] * N[(1.0 / N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $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))) < -600 or 700 < (+.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 99.7%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   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. 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. associate-+r+ N/A

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

      3. sum-log N/A

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

      4. *-commutative N/A

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

      5. log-pow-rev N/A

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

      6. sum-log N/A

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

      7. lower-log.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lift--.f64 18.3

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

   5. Applied rewrites 18.3%

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

      1. lift-log.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. log-prod N/A

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

      7. log-pow-rev N/A

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

      8. *-commutative N/A

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

      9. sum-log N/A

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

      10. associate-+r+ N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-log.f64 N/A

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

      16. lift--.f64 N/A

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

      17. lift-log.f64 N/A

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

      18. lower-log.f64 71.4

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

   7. Applied rewrites 71.4%

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

   8. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 66.8

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

   10. Applied rewrites 66.8%

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

   if -600 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 700

   1. Initial program 99.0%

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

   3. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lift-log.f64 98.7

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in t around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. sum-log N/A

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

      4. *-commutative N/A

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

      5. log-pow-rev N/A

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

      6. sum-log N/A

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

      7. lower-log.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. lift--.f64 97.2

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

   8. Applied rewrites 97.2%

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

   9. Taylor expanded in a around 0

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

       1. sqrt-div N/A

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

       2. metadata-eval N/A

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

       3. lower-/.f64 N/A

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

       4. lower-sqrt.f64 96.6

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

   11. Applied rewrites 96.6%

       \[\leadsto \log \left(\left(\left(y + x\right) \cdot z\right) \cdot \frac{1}{\sqrt{t}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \leq -600 \lor \neg \left(\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \leq 700\right):\\ \;\;\;\;\left(\log t \cdot a + \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(\left(y + x\right) \cdot z\right) \cdot \frac{1}{\sqrt{t}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot \log t\\ t_2 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + t\_1\\ \mathbf{if}\;t\_2 \leq -600:\\ \;\;\;\;\left(\log t \cdot a + \log z\right) - t\\ \mathbf{elif}\;t\_2 \leq 700:\\ \;\;\;\;\log \left(\left(\left(y + x\right) \cdot z\right) \cdot \frac{1}{\sqrt{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) + t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) (log t)))
        (t_2 (+ (- (+ (log (+ x y)) (log z)) t) t_1)))
   (if (<= t_2 -600.0)
     (- (+ (* (log t) a) (log z)) t)
     (if (<= t_2 700.0)
       (log (* (* (+ y x) z) (/ 1.0 (sqrt t))))
       (+ (- t) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (a - 0.5) * log(t);
	double t_2 = ((log((x + y)) + log(z)) - t) + t_1;
	double tmp;
	if (t_2 <= -600.0) {
		tmp = ((log(t) * a) + log(z)) - t;
	} else if (t_2 <= 700.0) {
		tmp = log((((y + x) * z) * (1.0 / sqrt(t))));
	} else {
		tmp = -t + t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (a - 0.5d0) * log(t)
    t_2 = ((log((x + y)) + log(z)) - t) + t_1
    if (t_2 <= (-600.0d0)) then
        tmp = ((log(t) * a) + log(z)) - t
    else if (t_2 <= 700.0d0) then
        tmp = log((((y + x) * z) * (1.0d0 / sqrt(t))))
    else
        tmp = -t + t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (a - 0.5) * Math.log(t);
	double t_2 = ((Math.log((x + y)) + Math.log(z)) - t) + t_1;
	double tmp;
	if (t_2 <= -600.0) {
		tmp = ((Math.log(t) * a) + Math.log(z)) - t;
	} else if (t_2 <= 700.0) {
		tmp = Math.log((((y + x) * z) * (1.0 / Math.sqrt(t))));
	} else {
		tmp = -t + t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (a - 0.5) * math.log(t)
	t_2 = ((math.log((x + y)) + math.log(z)) - t) + t_1
	tmp = 0
	if t_2 <= -600.0:
		tmp = ((math.log(t) * a) + math.log(z)) - t
	elif t_2 <= 700.0:
		tmp = math.log((((y + x) * z) * (1.0 / math.sqrt(t))))
	else:
		tmp = -t + t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(a - 0.5) * log(t))
	t_2 = Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + t_1)
	tmp = 0.0
	if (t_2 <= -600.0)
		tmp = Float64(Float64(Float64(log(t) * a) + log(z)) - t);
	elseif (t_2 <= 700.0)
		tmp = log(Float64(Float64(Float64(y + x) * z) * Float64(1.0 / sqrt(t))));
	else
		tmp = Float64(Float64(-t) + t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (a - 0.5) * log(t);
	t_2 = ((log((x + y)) + log(z)) - t) + t_1;
	tmp = 0.0;
	if (t_2 <= -600.0)
		tmp = ((log(t) * a) + log(z)) - t;
	elseif (t_2 <= 700.0)
		tmp = log((((y + x) * z) * (1.0 / sqrt(t))));
	else
		tmp = -t + t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -600.0], N[(N[(N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[t$95$2, 700.0], N[Log[N[(N[(N[(y + x), $MachinePrecision] * z), $MachinePrecision] * N[(1.0 / N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[((-t) + t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.8%

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

   3. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lift-log.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 97.2

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

   8. Applied rewrites 97.2%

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

   if -600 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 700

   1. Initial program 99.0%

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

   3. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lift-log.f64 98.7

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in t around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. sum-log N/A

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

      4. *-commutative N/A

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

      5. log-pow-rev N/A

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

      6. sum-log N/A

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

      7. lower-log.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. lift--.f64 97.2

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

   8. Applied rewrites 97.2%

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

   9. Taylor expanded in a around 0

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

       1. sqrt-div N/A

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

       2. metadata-eval N/A

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

       3. lower-/.f64 N/A

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

       4. lower-sqrt.f64 96.6

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

   11. Applied rewrites 96.6%

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

   if 700 < (+.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 99.5%

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

   3. Taylor expanded in 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 77.1

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

   5. Applied rewrites 77.1%

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

Alternative 6: 90.0% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (a - 0.5d0) * log(t)
    t_2 = ((log((x + y)) + log(z)) - t) + t_1
    if (t_2 <= (-700.0d0)) then
        tmp = (log(t) * a) - t
    else if (t_2 <= 700.0d0) then
        tmp = log((((y + x) * z) * (1.0d0 / sqrt(t))))
    else
        tmp = -t + t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (a - 0.5) * Math.log(t);
	double t_2 = ((Math.log((x + y)) + Math.log(z)) - t) + t_1;
	double tmp;
	if (t_2 <= -700.0) {
		tmp = (Math.log(t) * a) - t;
	} else if (t_2 <= 700.0) {
		tmp = Math.log((((y + x) * z) * (1.0 / Math.sqrt(t))));
	} else {
		tmp = -t + t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (a - 0.5) * math.log(t)
	t_2 = ((math.log((x + y)) + math.log(z)) - t) + t_1
	tmp = 0
	if t_2 <= -700.0:
		tmp = (math.log(t) * a) - t
	elif t_2 <= 700.0:
		tmp = math.log((((y + x) * z) * (1.0 / math.sqrt(t))))
	else:
		tmp = -t + t_1
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (a - 0.5) * log(t);
	t_2 = ((log((x + y)) + log(z)) - t) + t_1;
	tmp = 0.0;
	if (t_2 <= -700.0)
		tmp = (log(t) * a) - t;
	elseif (t_2 <= 700.0)
		tmp = log((((y + x) * z) * (1.0 / sqrt(t))));
	else
		tmp = -t + t_1;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < -700

   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 a around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lift-log.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 97.5

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

   8. Applied rewrites 97.5%

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

   if -700 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 700

   1. Initial program 99.0%

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

   3. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lift-log.f64 98.7

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in t around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. sum-log N/A

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

      4. *-commutative N/A

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

      5. log-pow-rev N/A

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

      6. sum-log N/A

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

      7. lower-log.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. lift--.f64 95.2

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

   8. Applied rewrites 95.2%

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

   9. Taylor expanded in a around 0

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

       1. sqrt-div N/A

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

       2. metadata-eval N/A

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

       3. lower-/.f64 N/A

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

       4. lower-sqrt.f64 94.7

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

   11. Applied rewrites 94.7%

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

   if 700 < (+.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 99.5%

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

   3. Taylor expanded in 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 77.1

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

   5. Applied rewrites 77.1%

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

Alternative 7: 60.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log t \cdot a\\ t_2 := \log \left(x + y\right) + \log z\\ \mathbf{if}\;t\_2 \leq -720:\\ \;\;\;\;\left(t\_1 + \log z\right) - t\\ \mathbf{elif}\;t\_2 \leq 680:\\ \;\;\;\;\left(\log \left(z \cdot y\right) + \mathsf{fma}\left(\log t, a - 0.5, \frac{x}{y}\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 + \log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (log t) a)) (t_2 (+ (log (+ x y)) (log z))))
   (if (<= t_2 -720.0)
     (- (+ t_1 (log z)) t)
     (if (<= t_2 680.0)
       (- (+ (log (* z y)) (fma (log t) (- a 0.5) (/ x y))) t)
       (- (+ t_1 (log y)) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log(t) * a;
	double t_2 = log((x + y)) + log(z);
	double tmp;
	if (t_2 <= -720.0) {
		tmp = (t_1 + log(z)) - t;
	} else if (t_2 <= 680.0) {
		tmp = (log((z * y)) + fma(log(t), (a - 0.5), (x / y))) - t;
	} else {
		tmp = (t_1 + log(y)) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(t) * a)
	t_2 = Float64(log(Float64(x + y)) + log(z))
	tmp = 0.0
	if (t_2 <= -720.0)
		tmp = Float64(Float64(t_1 + log(z)) - t);
	elseif (t_2 <= 680.0)
		tmp = Float64(Float64(log(Float64(z * y)) + fma(log(t), Float64(a - 0.5), Float64(x / y))) - t);
	else
		tmp = Float64(Float64(t_1 + log(y)) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]}, Block[{t$95$2 = N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -720.0], N[(N[(t$95$1 + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[t$95$2, 680.0], N[(N[(N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision] + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(t$95$1 + N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -720

   1. Initial program 99.4%

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

   3. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lift-log.f64 99.4

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 75.7

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

   8. Applied rewrites 75.7%

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

   if -720 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 680

   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 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. lower--.f64 N/A

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

      2. associate-+r+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. sum-log N/A

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

      6. lower-log.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lower-/.f64 57.0

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

   5. Applied rewrites 57.0%

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

   if 680 < (+.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

      \[\leadsto \color{blue}{\left(\log y + \left(\log z + \log t \cdot \left(a - \frac{1}{2}\right)\right)\right) - t} \]
   4. 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. associate-+r+ N/A

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

      3. sum-log N/A

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

      4. *-commutative N/A

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

      5. log-pow-rev N/A

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

      6. sum-log N/A

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

      7. lower-log.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lift--.f64 8.9

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

   5. Applied rewrites 8.9%

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

      1. lift-log.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. log-prod N/A

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

      7. log-pow-rev N/A

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

      8. *-commutative N/A

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

      9. sum-log N/A

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

      10. associate-+r+ N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-log.f64 N/A

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

      16. lift--.f64 N/A

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

      17. lift-log.f64 N/A

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

      18. lower-log.f64 64.9

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

   7. Applied rewrites 64.9%

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

   8. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 54.5

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

   10. Applied rewrites 54.5%

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

Alternative 8: 65.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log t \cdot a\\ t_2 := \log \left(x + y\right) + \log z\\ \mathbf{if}\;t\_2 \leq -720:\\ \;\;\;\;\left(t\_1 + \log z\right) - t\\ \mathbf{elif}\;t\_2 \leq 680:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log \left(z \cdot y\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 + \log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (log t) a)) (t_2 (+ (log (+ x y)) (log z))))
   (if (<= t_2 -720.0)
     (- (+ t_1 (log z)) t)
     (if (<= t_2 680.0)
       (- (fma (- a 0.5) (log t) (log (* z y))) t)
       (- (+ t_1 (log y)) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log(t) * a;
	double t_2 = log((x + y)) + log(z);
	double tmp;
	if (t_2 <= -720.0) {
		tmp = (t_1 + log(z)) - t;
	} else if (t_2 <= 680.0) {
		tmp = fma((a - 0.5), log(t), log((z * y))) - t;
	} else {
		tmp = (t_1 + log(y)) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(t) * a)
	t_2 = Float64(log(Float64(x + y)) + log(z))
	tmp = 0.0
	if (t_2 <= -720.0)
		tmp = Float64(Float64(t_1 + log(z)) - t);
	elseif (t_2 <= 680.0)
		tmp = Float64(fma(Float64(a - 0.5), log(t), log(Float64(z * y))) - t);
	else
		tmp = Float64(Float64(t_1 + log(y)) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]}, Block[{t$95$2 = N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -720.0], N[(N[(t$95$1 + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[t$95$2, 680.0], N[(N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(t$95$1 + N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -720

   1. Initial program 99.4%

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

   3. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lift-log.f64 99.4

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 75.7

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

   8. Applied rewrites 75.7%

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

   if -720 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 680

   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 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. 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. associate-+r+ N/A

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

      3. sum-log N/A

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

      4. *-commutative N/A

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

      5. log-pow-rev N/A

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

      6. sum-log N/A

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

      7. lower-log.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lift--.f64 28.3

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

   5. Applied rewrites 28.3%

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

      1. lift-log.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. sum-log N/A

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

      8. log-pow-rev N/A

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

      9. lower-fma.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 63.8

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

   7. Applied rewrites 63.8%

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

   if 680 < (+.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

      \[\leadsto \color{blue}{\left(\log y + \left(\log z + \log t \cdot \left(a - \frac{1}{2}\right)\right)\right) - t} \]
   4. 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. associate-+r+ N/A

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

      3. sum-log N/A

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

      4. *-commutative N/A

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

      5. log-pow-rev N/A

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

      6. sum-log N/A

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

      7. lower-log.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lift--.f64 8.9

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

   5. Applied rewrites 8.9%

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

      1. lift-log.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. log-prod N/A

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

      7. log-pow-rev N/A

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

      8. *-commutative N/A

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

      9. sum-log N/A

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

      10. associate-+r+ N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-log.f64 N/A

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

      16. lift--.f64 N/A

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

      17. lift-log.f64 N/A

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

      18. lower-log.f64 64.9

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

   7. Applied rewrites 64.9%

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

   8. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 54.5

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

   10. Applied rewrites 54.5%

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

Alternative 9: 69.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -7.8 \lor \neg \left(a \leq 8.8 \cdot 10^{-13}\right):\\ \;\;\;\;\left(\mathsf{fma}\left(\log t, a, \log y\right) + \log z\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.5, \log t, \log z\right) + \log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -7.8) (not (<= a 8.8e-13)))
   (- (+ (fma (log t) a (log y)) (log z)) t)
   (- (+ (fma -0.5 (log t) (log z)) (log y)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -7.8) || !(a <= 8.8e-13)) {
		tmp = (fma(log(t), a, log(y)) + log(z)) - t;
	} else {
		tmp = (fma(-0.5, log(t), log(z)) + log(y)) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -7.8) || !(a <= 8.8e-13))
		tmp = Float64(Float64(fma(log(t), a, log(y)) + log(z)) - t);
	else
		tmp = Float64(Float64(fma(-0.5, log(t), log(z)) + log(y)) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -7.8], N[Not[LessEqual[a, 8.8e-13]], $MachinePrecision]], N[(N[(N[(N[Log[t], $MachinePrecision] * a + N[Log[y], $MachinePrecision]), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[(-0.5 * N[Log[t], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -7.79999999999999982 or 8.79999999999999986e-13 < 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 a around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lift-log.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 74.5%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 74.4%

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

   if -7.79999999999999982 < a < 8.79999999999999986e-13

   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

      \[\leadsto \color{blue}{\left(\log y + \left(\log z + \log t \cdot \left(a - \frac{1}{2}\right)\right)\right) - t} \]
   4. 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. associate-+r+ N/A

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

      3. sum-log N/A

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

      4. *-commutative N/A

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

      5. log-pow-rev N/A

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

      6. sum-log N/A

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

      7. lower-log.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lift--.f64 42.4

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

   5. Applied rewrites 42.4%

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

      1. lift-log.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. log-prod N/A

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

      7. log-pow-rev N/A

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

      8. *-commutative N/A

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

      9. sum-log N/A

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

      10. associate-+r+ N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-log.f64 N/A

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

      16. lift--.f64 N/A

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

      17. lift-log.f64 N/A

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

      18. lower-log.f64 60.1

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

   7. Applied rewrites 60.1%

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

   8. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-log.f64 59.9

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

   10. Applied rewrites 59.9%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -7.8 \lor \neg \left(a \leq 8.8 \cdot 10^{-13}\right):\\ \;\;\;\;\left(\mathsf{fma}\left(\log t, a, \log y\right) + \log z\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.5, \log t, \log z\right) + \log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 68.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -7.8 \lor \neg \left(a \leq 8.8 \cdot 10^{-13}\right):\\ \;\;\;\;\left(\log t \cdot a + \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.5, \log t, \log z\right) + \log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -7.8) (not (<= a 8.8e-13)))
   (- (+ (* (log t) a) (log y)) t)
   (- (+ (fma -0.5 (log t) (log z)) (log y)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -7.8) || !(a <= 8.8e-13)) {
		tmp = ((log(t) * a) + log(y)) - t;
	} else {
		tmp = (fma(-0.5, log(t), log(z)) + log(y)) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -7.8) || !(a <= 8.8e-13))
		tmp = Float64(Float64(Float64(log(t) * a) + log(y)) - t);
	else
		tmp = Float64(Float64(fma(-0.5, log(t), log(z)) + log(y)) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -7.8], N[Not[LessEqual[a, 8.8e-13]], $MachinePrecision]], N[(N[(N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[(-0.5 * N[Log[t], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -7.79999999999999982 or 8.79999999999999986e-13 < 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

      \[\leadsto \color{blue}{\left(\log y + \left(\log z + \log t \cdot \left(a - \frac{1}{2}\right)\right)\right) - t} \]
   4. 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. associate-+r+ N/A

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

      3. sum-log N/A

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

      4. *-commutative N/A

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

      5. log-pow-rev N/A

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

      6. sum-log N/A

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

      7. lower-log.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lift--.f64 4.2

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

   5. Applied rewrites 4.2%

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

      1. lift-log.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. log-prod N/A

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

      7. log-pow-rev N/A

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

      8. *-commutative N/A

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

      9. sum-log N/A

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

      10. associate-+r+ N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-log.f64 N/A

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

      16. lift--.f64 N/A

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

      17. lift-log.f64 N/A

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

      18. lower-log.f64 74.5

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

   7. Applied rewrites 74.5%

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

   8. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 74.4

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

   10. Applied rewrites 74.4%

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

   if -7.79999999999999982 < a < 8.79999999999999986e-13

   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

      \[\leadsto \color{blue}{\left(\log y + \left(\log z + \log t \cdot \left(a - \frac{1}{2}\right)\right)\right) - t} \]
   4. 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. associate-+r+ N/A

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

      3. sum-log N/A

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

      4. *-commutative N/A

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

      5. log-pow-rev N/A

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

      6. sum-log N/A

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

      7. lower-log.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lift--.f64 42.4

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

   5. Applied rewrites 42.4%

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

      1. lift-log.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. log-prod N/A

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

      7. log-pow-rev N/A

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

      8. *-commutative N/A

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

      9. sum-log N/A

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

      10. associate-+r+ N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-log.f64 N/A

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

      16. lift--.f64 N/A

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

      17. lift-log.f64 N/A

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

      18. lower-log.f64 60.1

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

   7. Applied rewrites 60.1%

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

   8. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-log.f64 59.9

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

   10. Applied rewrites 59.9%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -7.8 \lor \neg \left(a \leq 8.8 \cdot 10^{-13}\right):\\ \;\;\;\;\left(\log t \cdot a + \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.5, \log t, \log z\right) + \log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 86.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 4.5 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\log t, a - 0.5, \log \left(y + x\right)\right) + \log z\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\log t, a, \log y\right) + \log z\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 4.5e-7)
   (+ (fma (log t) (- a 0.5) (log (+ y x))) (log z))
   (- (+ (fma (log t) a (log y)) (log z)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 4.5e-7) {
		tmp = fma(log(t), (a - 0.5), log((y + x))) + log(z);
	} else {
		tmp = (fma(log(t), a, log(y)) + log(z)) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 4.5e-7)
		tmp = Float64(fma(log(t), Float64(a - 0.5), log(Float64(y + x))) + log(z));
	else
		tmp = Float64(Float64(fma(log(t), a, log(y)) + log(z)) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 4.5e-7], N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision] + N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Log[t], $MachinePrecision] * a + N[Log[y], $MachinePrecision]), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 4.4999999999999998e-7

   1. Initial program 99.3%

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

   3. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lift-log.f64 99.3

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in t around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. sum-log N/A

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

      4. *-commutative N/A

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

      5. log-pow-rev N/A

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

      6. sum-log N/A

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

      7. lower-log.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. lift--.f64 38.2

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

   8. Applied rewrites 38.2%

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

      1. lift-log.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. log-prod N/A

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

      8. +-commutative N/A

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

      9. sum-log N/A

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

      10. +-commutative N/A

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

      11. log-pow-rev N/A

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

      12. *-commutative N/A

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

      13. associate-+r+ N/A

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

      14. +-commutative N/A

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

   10. Applied rewrites 99.0%

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

   if 4.4999999999999998e-7 < 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 a around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lift-log.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 72.8%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 72.5%

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

Alternative 12: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in a around 0

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

   1. lower--.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 N/A

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

   4. +-commutative N/A

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

   5. distribute-rgt-out N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-log.f64 N/A

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

   8. lower-+.f64 N/A

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

   9. lift-log.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 N/A

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

   12. lift-log.f64 99.6

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

5. Applied rewrites 99.6%

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

Alternative 13: 69.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in a around 0

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

   1. lower--.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 N/A

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

   4. +-commutative N/A

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

   5. distribute-rgt-out N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-log.f64 N/A

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

   8. lower-+.f64 N/A

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

   9. lift-log.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 N/A

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

   12. lift-log.f64 99.6

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

5. Applied rewrites 99.6%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 67.2%

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

Alternative 14: 77.2% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = -t + (((1.0d0 - (0.5d0 / a)) * a) * log(t))
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = -t + (((1.0 - (0.5 / a)) * a) * log(t));
end

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in a around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. associate-*r/ N/A

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

   5. metadata-eval N/A

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

   6. lower-/.f64 99.6

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

5. Applied rewrites 99.6%

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

6. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. lift-neg.f64 76.5

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

8. Applied rewrites 76.5%

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

Alternative 15: 53.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -6.8 \cdot 10^{+49} \lor \neg \left(a \leq 2.5 \cdot 10^{+92}\right):\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -6.8e+49) (not (<= a 2.5e+92))) (* (log t) a) (- (/ x y) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -6.8e+49) || !(a <= 2.5e+92)) {
		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 <= (-6.8d+49)) .or. (.not. (a <= 2.5d+92))) 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 <= -6.8e+49) || !(a <= 2.5e+92)) {
		tmp = Math.log(t) * a;
	} else {
		tmp = (x / y) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -6.8e+49) or not (a <= 2.5e+92):
		tmp = math.log(t) * a
	else:
		tmp = (x / y) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -6.8e+49) || !(a <= 2.5e+92))
		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 <= -6.8e+49) || ~((a <= 2.5e+92)))
		tmp = log(t) * a;
	else
		tmp = (x / y) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -6.8e+49], N[Not[LessEqual[a, 2.5e+92]], $MachinePrecision]], N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision], N[(N[(x / y), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -6.8000000000000001e49 or 2.50000000000000011e92 < a

   1. Initial program 99.6%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   2. 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. lift-log.f64 86.2

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

   5. Applied rewrites 86.2%

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

   if -6.8000000000000001e49 < a < 2.50000000000000011e92

   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 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. lower--.f64 N/A

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

      2. associate-+r+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. sum-log N/A

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

      6. lower-log.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lower-/.f64 39.3

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

   5. Applied rewrites 39.3%

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

   6. Taylor expanded in x around inf

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

      1. lift-/.f64 33.0

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

   8. Applied rewrites 33.0%

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

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6.8 \cdot 10^{+49} \lor \neg \left(a \leq 2.5 \cdot 10^{+92}\right):\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} - t\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 77.2% 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 99.6%

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

3. Taylor expanded in 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 76.5

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

5. Applied rewrites 76.5%

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

Alternative 17: 77.2% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in a around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. associate-*r/ N/A

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

   5. metadata-eval N/A

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

   6. lower-/.f64 99.6

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

5. Applied rewrites 99.6%

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

6. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. lift-neg.f64 76.5

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

8. Applied rewrites 76.5%

   \[\leadsto \color{blue}{\left(-t\right)} + \left(\left(1 - \frac{0.5}{a}\right) \cdot a\right) \cdot \log t \]
9. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(1 - \frac{\frac{1}{2}}{a}\right) \cdot a\right) \cdot \log t} + \left(-t\right) \]

   4. lift-log.f64 N/A

      \[\leadsto \left(\left(1 - \frac{\frac{1}{2}}{a}\right) \cdot a\right) \cdot \color{blue}{\log t} + \left(-t\right) \]

   5. *-commutative N/A

      \[\leadsto \color{blue}{\log t \cdot \left(\left(1 - \frac{\frac{1}{2}}{a}\right) \cdot a\right)} + \left(-t\right) \]

   6. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log t, \left(1 - \frac{\frac{1}{2}}{a}\right) \cdot a, -t\right)} \]

   7. lift-log.f64 76.5

      \[\leadsto \mathsf{fma}\left(\color{blue}{\log t}, \left(1 - \frac{0.5}{a}\right) \cdot a, -t\right) \]

10. Applied rewrites 76.5%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\log t, \left(1 - \frac{0.5}{a}\right) \cdot a, -t\right)} \]

11. Taylor expanded in a around 0

    \[\leadsto \mathsf{fma}\left(\log t, a - \color{blue}{\frac{1}{2}}, -t\right) \]
12. Step-by-step derivation

    1. lift--.f64 76.5

       \[\leadsto \mathsf{fma}\left(\log t, a - 0.5, -t\right) \]

13. Applied rewrites 76.5%

    \[\leadsto \mathsf{fma}\left(\log t, a - \color{blue}{0.5}, -t\right) \]
14. Add Preprocessing

Alternative 18: 74.6% 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 99.6%

   \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
2. Add Preprocessing

3. Taylor expanded in a around 0

   \[\leadsto \color{blue}{\left(\log z + \left(\log \left(x + y\right) + \left(\frac{-1}{2} \cdot \log t + a \cdot \log t\right)\right)\right) - t} \]
4. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto \left(\log z + \left(\log \left(x + y\right) + \left(\frac{-1}{2} \cdot \log t + a \cdot \log t\right)\right)\right) - \color{blue}{t} \]

   2. +-commutative N/A

      \[\leadsto \left(\left(\log \left(x + y\right) + \left(\frac{-1}{2} \cdot \log t + a \cdot \log t\right)\right) + \log z\right) - t \]

   3. lower-+.f64 N/A

      \[\leadsto \left(\left(\log \left(x + y\right) + \left(\frac{-1}{2} \cdot \log t + a \cdot \log t\right)\right) + \log z\right) - t \]

   4. +-commutative N/A

      \[\leadsto \left(\left(\left(\frac{-1}{2} \cdot \log t + a \cdot \log t\right) + \log \left(x + y\right)\right) + \log z\right) - t \]

   5. distribute-rgt-out N/A

      \[\leadsto \left(\left(\log t \cdot \left(\frac{-1}{2} + a\right) + \log \left(x + y\right)\right) + \log z\right) - t \]

   6. lower-fma.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\log t, \frac{-1}{2} + a, \log \left(x + y\right)\right) + \log z\right) - t \]

   7. lift-log.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\log t, \frac{-1}{2} + a, \log \left(x + y\right)\right) + \log z\right) - t \]

   8. lower-+.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\log t, \frac{-1}{2} + a, \log \left(x + y\right)\right) + \log z\right) - t \]

   9. lift-log.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\log t, \frac{-1}{2} + a, \log \left(x + y\right)\right) + \log z\right) - t \]

   10. +-commutative N/A

       \[\leadsto \left(\mathsf{fma}\left(\log t, \frac{-1}{2} + a, \log \left(y + x\right)\right) + \log z\right) - t \]

   11. lower-+.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\log t, \frac{-1}{2} + a, \log \left(y + x\right)\right) + \log z\right) - t \]

   12. lift-log.f64 99.6

       \[\leadsto \left(\mathsf{fma}\left(\log t, -0.5 + a, \log \left(y + x\right)\right) + \log z\right) - t \]

5. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\log t, -0.5 + a, \log \left(y + x\right)\right) + \log z\right) - t} \]

6. Taylor expanded in a around inf

   \[\leadsto a \cdot \log t - t \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \log t \cdot a - t \]

   2. lower-*.f64 N/A

      \[\leadsto \log t \cdot a - t \]

   3. lift-log.f64 74.1

      \[\leadsto \log t \cdot a - t \]

8. Applied rewrites 74.1%

   \[\leadsto \log t \cdot a - t \]
9. Add Preprocessing

Alternative 19: 28.6% 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 99.6%

   \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
2. Add Preprocessing

3. Taylor expanded in 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. lower--.f64 N/A

      \[\leadsto \left(\log y + \left(\log z + \left(\log t \cdot \left(a - \frac{1}{2}\right) + \frac{x}{y}\right)\right)\right) - \color{blue}{t} \]

   2. associate-+r+ N/A

      \[\leadsto \left(\left(\log y + \log z\right) + \left(\log t \cdot \left(a - \frac{1}{2}\right) + \frac{x}{y}\right)\right) - t \]

   3. lower-+.f64 N/A

      \[\leadsto \left(\left(\log y + \log z\right) + \left(\log t \cdot \left(a - \frac{1}{2}\right) + \frac{x}{y}\right)\right) - t \]

   4. +-commutative N/A

      \[\leadsto \left(\left(\log z + \log y\right) + \left(\log t \cdot \left(a - \frac{1}{2}\right) + \frac{x}{y}\right)\right) - t \]

   5. sum-log N/A

      \[\leadsto \left(\log \left(z \cdot y\right) + \left(\log t \cdot \left(a - \frac{1}{2}\right) + \frac{x}{y}\right)\right) - t \]

   6. lower-log.f64 N/A

      \[\leadsto \left(\log \left(z \cdot y\right) + \left(\log t \cdot \left(a - \frac{1}{2}\right) + \frac{x}{y}\right)\right) - t \]

   7. lower-*.f64 N/A

      \[\leadsto \left(\log \left(z \cdot y\right) + \left(\log t \cdot \left(a - \frac{1}{2}\right) + \frac{x}{y}\right)\right) - t \]

   8. lower-fma.f64 N/A

      \[\leadsto \left(\log \left(z \cdot y\right) + \mathsf{fma}\left(\log t, a - \frac{1}{2}, \frac{x}{y}\right)\right) - t \]

   9. lift-log.f64 N/A

      \[\leadsto \left(\log \left(z \cdot y\right) + \mathsf{fma}\left(\log t, a - \frac{1}{2}, \frac{x}{y}\right)\right) - t \]

   10. lift--.f64 N/A

       \[\leadsto \left(\log \left(z \cdot y\right) + \mathsf{fma}\left(\log t, a - \frac{1}{2}, \frac{x}{y}\right)\right) - t \]

   11. lower-/.f64 44.9

       \[\leadsto \left(\log \left(z \cdot y\right) + \mathsf{fma}\left(\log t, a - 0.5, \frac{x}{y}\right)\right) - t \]

5. Applied rewrites 44.9%

   \[\leadsto \color{blue}{\left(\log \left(z \cdot y\right) + \mathsf{fma}\left(\log t, a - 0.5, \frac{x}{y}\right)\right) - t} \]

6. Taylor expanded in x around inf

   \[\leadsto \frac{x}{y} - t \]
7. Step-by-step derivation

   1. lift-/.f64 25.5

      \[\leadsto \frac{x}{y} - t \]

8. Applied rewrites 25.5%

   \[\leadsto \frac{x}{y} - t \]
9. Add Preprocessing

Alternative 20: 37.5% 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 99.6%

   \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
2. Add Preprocessing

3. Taylor expanded in 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 38.4

      \[\leadsto -t \]

5. Applied rewrites 38.4%

   \[\leadsto \color{blue}{-t} \]
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 2025051 
(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))))