Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 11.8s

Alternatives: 15

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: 85.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(x + y\right) + \log z\\ t_2 := \mathsf{fma}\left(\log t, a - 0.5, \log z\right) + \log y\\ \mathbf{if}\;t\_1 \leq -750:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 710:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log \left(z \cdot \left(y + x\right)\right) - t\right)\\ \mathbf{elif}\;t\_1 \leq 950:\\ \;\;\;\;\left(\left(\log y + \log z\right) - t\right) + -0.5 \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log (+ x y)) (log z)))
        (t_2 (+ (fma (log t) (- a 0.5) (log z)) (log y))))
   (if (<= t_1 -750.0)
     t_2
     (if (<= t_1 710.0)
       (fma (- a 0.5) (log t) (- (log (* z (+ y x))) t))
       (if (<= t_1 950.0)
         (+ (- (+ (log y) (log z)) t) (* -0.5 (log t)))
         t_2)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((x + y)) + log(z);
	double t_2 = fma(log(t), (a - 0.5), log(z)) + log(y);
	double tmp;
	if (t_1 <= -750.0) {
		tmp = t_2;
	} else if (t_1 <= 710.0) {
		tmp = fma((a - 0.5), log(t), (log((z * (y + x))) - t));
	} else if (t_1 <= 950.0) {
		tmp = ((log(y) + log(z)) - t) + (-0.5 * log(t));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(Float64(x + y)) + log(z))
	t_2 = Float64(fma(log(t), Float64(a - 0.5), log(z)) + log(y))
	tmp = 0.0
	if (t_1 <= -750.0)
		tmp = t_2;
	elseif (t_1 <= 710.0)
		tmp = fma(Float64(a - 0.5), log(t), Float64(log(Float64(z * Float64(y + x))) - t));
	elseif (t_1 <= 950.0)
		tmp = Float64(Float64(Float64(log(y) + log(z)) - t) + Float64(-0.5 * log(t)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -750.0], t$95$2, If[LessEqual[t$95$1, 710.0], N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[(N[Log[N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 950.0], N[(N[(N[(N[Log[y], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + N[(-0.5 * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -750 or 950 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z))

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-out-- N/A

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

      6. metadata-eval N/A

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

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

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

      8. distribute-rgt-out N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-log.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-log.f64 74.7

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

   5. Applied rewrites 74.7%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 50.3%

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

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

   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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.6

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

      5. lift-+.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. sum-log N/A

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

      9. lower-log.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.7

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 99.7

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

   4. Applied rewrites 99.7%

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

   if 710 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 950

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. lower-log.f64 72.3

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

   5. Applied rewrites 72.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 54.4%

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

Alternative 3: 85.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(x + y\right) + \log z\\ t_2 := \mathsf{fma}\left(\log t, a - 0.5, \log z\right) + \log y\\ \mathbf{if}\;t\_1 \leq -750:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 710:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log \left(z \cdot \left(y + x\right)\right) - t\right)\\ \mathbf{elif}\;t\_1 \leq 950:\\ \;\;\;\;\log z + \mathsf{fma}\left(-0.5, \log t, \log y - t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log (+ x y)) (log z)))
        (t_2 (+ (fma (log t) (- a 0.5) (log z)) (log y))))
   (if (<= t_1 -750.0)
     t_2
     (if (<= t_1 710.0)
       (fma (- a 0.5) (log t) (- (log (* z (+ y x))) t))
       (if (<= t_1 950.0) (+ (log z) (fma -0.5 (log t) (- (log y) t))) t_2)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((x + y)) + log(z);
	double t_2 = fma(log(t), (a - 0.5), log(z)) + log(y);
	double tmp;
	if (t_1 <= -750.0) {
		tmp = t_2;
	} else if (t_1 <= 710.0) {
		tmp = fma((a - 0.5), log(t), (log((z * (y + x))) - t));
	} else if (t_1 <= 950.0) {
		tmp = log(z) + fma(-0.5, log(t), (log(y) - t));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(Float64(x + y)) + log(z))
	t_2 = Float64(fma(log(t), Float64(a - 0.5), log(z)) + log(y))
	tmp = 0.0
	if (t_1 <= -750.0)
		tmp = t_2;
	elseif (t_1 <= 710.0)
		tmp = fma(Float64(a - 0.5), log(t), Float64(log(Float64(z * Float64(y + x))) - t));
	elseif (t_1 <= 950.0)
		tmp = Float64(log(z) + fma(-0.5, log(t), Float64(log(y) - t)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -750.0], t$95$2, If[LessEqual[t$95$1, 710.0], N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[(N[Log[N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 950.0], N[(N[Log[z], $MachinePrecision] + N[(-0.5 * N[Log[t], $MachinePrecision] + N[(N[Log[y], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -750 or 950 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z))

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-out-- N/A

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

      6. metadata-eval N/A

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

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

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

      8. distribute-rgt-out N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-log.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-log.f64 74.7

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

   5. Applied rewrites 74.7%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 50.3%

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

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

   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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.6

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

      5. lift-+.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. sum-log N/A

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

      9. lower-log.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.7

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 99.7

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

   4. Applied rewrites 99.7%

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

   if 710 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 950

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-out-- N/A

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

      6. metadata-eval N/A

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

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

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

      8. distribute-rgt-out N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-log.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-log.f64 72.3

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

   5. Applied rewrites 72.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 54.4%

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

   10. Applied rewrites 54.2%

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

Alternative 4: 83.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(x + y\right) + \log z\\ \mathbf{if}\;t\_1 \leq 722:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log \left(z \cdot \left(y + x\right)\right) - t\right)\\ \mathbf{elif}\;t\_1 \leq 950:\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;\log \left(y + x\right) + \log t \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log (+ x y)) (log z))))
   (if (<= t_1 722.0)
     (fma (- a 0.5) (log t) (- (log (* z (+ y x))) t))
     (if (<= t_1 950.0) (- t) (+ (log (+ y x)) (* (log t) a))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((x + y)) + log(z);
	double tmp;
	if (t_1 <= 722.0) {
		tmp = fma((a - 0.5), log(t), (log((z * (y + x))) - t));
	} else if (t_1 <= 950.0) {
		tmp = -t;
	} else {
		tmp = log((y + x)) + (log(t) * a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(Float64(x + y)) + log(z))
	tmp = 0.0
	if (t_1 <= 722.0)
		tmp = fma(Float64(a - 0.5), log(t), Float64(log(Float64(z * Float64(y + x))) - t));
	elseif (t_1 <= 950.0)
		tmp = Float64(-t);
	else
		tmp = Float64(log(Float64(y + x)) + Float64(log(t) * a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 722.0], N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[(N[Log[N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 950.0], (-t), N[(N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 722

   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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.6

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

      5. lift-+.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. sum-log N/A

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

      9. lower-log.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 93.6

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 93.6

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

   4. Applied rewrites 93.6%

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

   if 722 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 950

   1. Initial program 99.9%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 59.5

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

   5. Applied rewrites 59.5%

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

   if 950 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z))

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lift-+.f64 N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower--.f64 99.6

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 53.1

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

   7. Applied rewrites 53.1%

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

Alternative 5: 57.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(x + y\right) + \log z\\ \mathbf{if}\;t\_1 \leq 722:\\ \;\;\;\;\mathsf{fma}\left(-0.5 + a, \log t, \log \left(z \cdot y\right) - t\right)\\ \mathbf{elif}\;t\_1 \leq 950:\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;\log \left(y + x\right) + \log t \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log (+ x y)) (log z))))
   (if (<= t_1 722.0)
     (fma (+ -0.5 a) (log t) (- (log (* z y)) t))
     (if (<= t_1 950.0) (- t) (+ (log (+ y x)) (* (log t) a))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((x + y)) + log(z);
	double tmp;
	if (t_1 <= 722.0) {
		tmp = fma((-0.5 + a), log(t), (log((z * y)) - t));
	} else if (t_1 <= 950.0) {
		tmp = -t;
	} else {
		tmp = log((y + x)) + (log(t) * a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(Float64(x + y)) + log(z))
	tmp = 0.0
	if (t_1 <= 722.0)
		tmp = fma(Float64(-0.5 + a), log(t), Float64(log(Float64(z * y)) - t));
	elseif (t_1 <= 950.0)
		tmp = Float64(-t);
	else
		tmp = Float64(log(Float64(y + x)) + Float64(log(t) * a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 722.0], N[(N[(-0.5 + a), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[(N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 950.0], (-t), N[(N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 722

   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. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-out-- N/A

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

      6. metadata-eval N/A

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

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

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

      8. distribute-rgt-out N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-log.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-log.f64 71.7

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

   5. Applied rewrites 71.7%

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

   7. Applied rewrites 63.0%

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

   if 722 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 950

   1. Initial program 99.9%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 59.5

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

   5. Applied rewrites 59.5%

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

   if 950 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z))

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lift-+.f64 N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower--.f64 99.6

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 53.1

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

   7. Applied rewrites 53.1%

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

Alternative 6: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. associate-+l- N/A

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

   4. lift-+.f64 N/A

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

   5. associate--l+ N/A

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

   6. lower-+.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lower--.f64 99.6

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 99.6

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

4. Applied rewrites 99.6%

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

Alternative 7: 67.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 43000:\\ \;\;\;\;\mathsf{fma}\left(\log t, a - 0.5, \log z\right) + \log y\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{+68}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 + a, \log t, \log \left(z \cdot y\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 43000.0)
   (+ (fma (log t) (- a 0.5) (log z)) (log y))
   (if (<= t 2.5e+68) (fma (+ -0.5 a) (log t) (- (log (* z y)) t)) (- t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 43000.0) {
		tmp = fma(log(t), (a - 0.5), log(z)) + log(y);
	} else if (t <= 2.5e+68) {
		tmp = fma((-0.5 + a), log(t), (log((z * y)) - t));
	} else {
		tmp = -t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 43000.0)
		tmp = Float64(fma(log(t), Float64(a - 0.5), log(z)) + log(y));
	elseif (t <= 2.5e+68)
		tmp = fma(Float64(-0.5 + a), log(t), Float64(log(Float64(z * y)) - t));
	else
		tmp = Float64(-t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 43000.0], N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.5e+68], N[(N[(-0.5 + a), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[(N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], (-t)]]

Derivation

1. Split input into 3 regimes

2. if t < 43000

   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. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-out-- N/A

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

      6. metadata-eval N/A

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

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

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

      8. distribute-rgt-out N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-log.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-log.f64 67.9

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

   5. Applied rewrites 67.9%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 67.0%

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

   if 43000 < t < 2.5000000000000002e68

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-out-- N/A

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

      6. metadata-eval N/A

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

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

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

      8. distribute-rgt-out N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-log.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-log.f64 85.2

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

   5. Applied rewrites 85.2%

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

   7. Applied rewrites 76.0%

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

   if 2.5000000000000002e68 < 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 t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 84.5

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

   5. Applied rewrites 84.5%

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

Alternative 8: 68.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in x around 0

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

   1. lower-log.f64 71.7

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

5. Applied rewrites 71.7%

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

Alternative 9: 68.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in x around 0

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

   1. lower-log.f64 71.7

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

5. Applied rewrites 71.7%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 71.7

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

   5. lift--.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. associate--l+ N/A

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

   8. lift--.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 71.6

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

7. Applied rewrites 71.6%

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

Alternative 10: 68.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[Log[z], $MachinePrecision] + N[(N[(-0.5 + a), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[(N[Log[y], $MachinePrecision] - 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 x around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. lower-+.f64 N/A

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

   4. +-commutative N/A

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

   5. distribute-rgt-out-- N/A

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

   6. metadata-eval N/A

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

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

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

   8. distribute-rgt-out N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. lower-+.f64 N/A

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

   13. lower-log.f64 N/A

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

   14. lower-log.f64 N/A

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

   15. lower--.f64 N/A

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

   16. lower-log.f64 71.6

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

5. Applied rewrites 71.6%

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

7. Applied rewrites 71.6%

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

Alternative 11: 61.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -165 \lor \neg \left(a \leq 4.9 \cdot 10^{+18}\right):\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log t, \log \left(z \cdot y\right) - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -165.0) (not (<= a 4.9e+18)))
   (* (log t) a)
   (fma -0.5 (log t) (- (log (* z y)) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -165.0) || !(a <= 4.9e+18)) {
		tmp = log(t) * a;
	} else {
		tmp = fma(-0.5, log(t), (log((z * y)) - t));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -165.0) || !(a <= 4.9e+18))
		tmp = Float64(log(t) * a);
	else
		tmp = fma(-0.5, log(t), Float64(log(Float64(z * y)) - t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -165.0], N[Not[LessEqual[a, 4.9e+18]], $MachinePrecision]], N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision], N[(-0.5 * N[Log[t], $MachinePrecision] + N[(N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -165 or 4.9e18 < 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

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 78.8

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

   5. Applied rewrites 78.8%

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

   if -165 < a < 4.9e18

   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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lift-+.f64 N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower--.f64 99.5

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 99.5

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

   4. Applied rewrites 99.5%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+r- N/A

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

      4. lower-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lift--.f64 N/A

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

      9. associate-+l- N/A

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

      10. lift--.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 99.6

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

   6. Applied rewrites 71.4%

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

   7. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 48.2

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

   9. Applied rewrites 48.2%

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

   10. Taylor expanded in a around 0

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

   12. Applied rewrites 47.0%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -165 \lor \neg \left(a \leq 4.9 \cdot 10^{+18}\right):\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log t, \log \left(z \cdot y\right) - t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 58.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -165 \lor \neg \left(a \leq 9 \cdot 10^{+18}\right):\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left({t}^{-0.5} \cdot z\right) \cdot y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -165.0) (not (<= a 9e+18)))
   (* (log t) a)
   (- (log (* (* (pow t -0.5) z) y)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -165.0) || !(a <= 9e+18)) {
		tmp = log(t) * a;
	} else {
		tmp = log(((pow(t, -0.5) * z) * 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 <= (-165.0d0)) .or. (.not. (a <= 9d+18))) then
        tmp = log(t) * a
    else
        tmp = log((((t ** (-0.5d0)) * z) * 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 <= -165.0) || !(a <= 9e+18)) {
		tmp = Math.log(t) * a;
	} else {
		tmp = Math.log(((Math.pow(t, -0.5) * z) * y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -165.0) or not (a <= 9e+18):
		tmp = math.log(t) * a
	else:
		tmp = math.log(((math.pow(t, -0.5) * z) * y)) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -165.0) || !(a <= 9e+18))
		tmp = Float64(log(t) * a);
	else
		tmp = Float64(log(Float64(Float64((t ^ -0.5) * z) * y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -165.0) || ~((a <= 9e+18)))
		tmp = log(t) * a;
	else
		tmp = log((((t ^ -0.5) * z) * y)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -165.0], N[Not[LessEqual[a, 9e+18]], $MachinePrecision]], N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision], N[(N[Log[N[(N[(N[Power[t, -0.5], $MachinePrecision] * z), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -165 or 9e18 < 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

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 78.8

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

   5. Applied rewrites 78.8%

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

   if -165 < a < 9e18

   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. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-out-- N/A

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

      6. metadata-eval N/A

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

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

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

      8. distribute-rgt-out N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-log.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-log.f64 67.3

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

   5. Applied rewrites 67.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 65.1%

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

   10. Applied rewrites 47.2%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -165 \lor \neg \left(a \leq 9 \cdot 10^{+18}\right):\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left({t}^{-0.5} \cdot z\right) \cdot y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 65.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 3.8 \cdot 10^{+28}:\\ \;\;\;\;\log \left(y + x\right) + \log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 3.8e+28) (+ (log (+ y x)) (* (log t) a)) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 3.8e+28) {
		tmp = log((y + x)) + (log(t) * a);
	} else {
		tmp = -t;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 3.8d+28) then
        tmp = log((y + x)) + (log(t) * a)
    else
        tmp = -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 3.8e+28) {
		tmp = Math.log((y + x)) + (Math.log(t) * a);
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 3.8e+28:
		tmp = math.log((y + x)) + (math.log(t) * a)
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 3.8e+28)
		tmp = Float64(log(Float64(y + x)) + Float64(log(t) * a));
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 3.8e+28)
		tmp = log((y + x)) + (log(t) * a);
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 3.8e+28], N[(N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if t < 3.7999999999999999e28

   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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lift-+.f64 N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower--.f64 99.4

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 61.5

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

   7. Applied rewrites 61.5%

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

   if 3.7999999999999999e28 < 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 t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 81.1

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

   5. Applied rewrites 81.1%

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

Alternative 14: 62.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 5.2 \cdot 10^{+28}:\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 5.2e+28) (* (log t) a) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 5.2e+28) {
		tmp = log(t) * a;
	} else {
		tmp = -t;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 5.2d+28) then
        tmp = log(t) * a
    else
        tmp = -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 5.2e+28) {
		tmp = Math.log(t) * a;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 5.2e+28:
		tmp = math.log(t) * a
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 5.2e+28)
		tmp = Float64(log(t) * a);
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 5.2e+28)
		tmp = log(t) * a;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 5.2e+28], N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if t < 5.2000000000000004e28

   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 a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 56.1

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

   5. Applied rewrites 56.1%

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

   if 5.2000000000000004e28 < 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 t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 81.1

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

   5. Applied rewrites 81.1%

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

Alternative 15: 37.7% 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

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

   2. lower-neg.f64 38.5

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

5. Applied rewrites 38.5%

   \[\leadsto \color{blue}{-t} \]
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 2024364 
(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))))