Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 8.6s

Alternatives: 20

Speedup: N/A×

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

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

FPCore

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

C

double code(double x, double y, double z, double t, double a) {
	return ((((-0.5 + a) * log(t)) + log((y + x))) + log(z)) - 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 = (((((-0.5d0) + a) * log(t)) + log((y + x))) + log(z)) - t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in a around 0

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

   1. lower--.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 N/A

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

   4. +-commutative N/A

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

   5. distribute-rgt-out N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-log.f64 N/A

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

   8. lower-+.f64 N/A

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

   9. lift-log.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 N/A

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

   12. lift-log.f64 99.6

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

4. Applied rewrites 99.6%

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

   1. lift-log.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. distribute-rgt-in N/A

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

   5. lift-+.f64 N/A

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

   6. lift-log.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-+.f64 N/A

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

   9. distribute-rgt-in N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. lift-log.f64 N/A

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

   14. +-commutative N/A

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

   15. lift-log.f64 N/A

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

   16. lift-+.f64 99.6

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

6. Applied rewrites 99.6%

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

Alternative 2: 80.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+14}:\\ \;\;\;\;\log t \cdot a - t\\ \mathbf{elif}\;t\_1 \leq 680:\\ \;\;\;\;\mathsf{fma}\left(\frac{\log \left(\left({t}^{\left(a - 0.5\right)} \cdot z\right) \cdot y\right)}{t} - 1, t, \frac{x}{y}\right)\\ \mathbf{elif}\;t\_1 \leq 2000:\\ \;\;\;\;\left(\log z - 0.5 \cdot \log t\right) + \log y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log t, a, \log z\right) + \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t)))))
   (if (<= t_1 -1e+14)
     (- (* (log t) a) t)
     (if (<= t_1 680.0)
       (fma (- (/ (log (* (* (pow t (- a 0.5)) z) y)) t) 1.0) t (/ x y))
       (if (<= t_1 2000.0)
         (+ (- (log z) (* 0.5 (log t))) (log y))
         (+ (fma (log t) a (log z)) (log y)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
	double tmp;
	if (t_1 <= -1e+14) {
		tmp = (log(t) * a) - t;
	} else if (t_1 <= 680.0) {
		tmp = fma(((log(((pow(t, (a - 0.5)) * z) * y)) / t) - 1.0), t, (x / y));
	} else if (t_1 <= 2000.0) {
		tmp = (log(z) - (0.5 * log(t))) + log(y);
	} else {
		tmp = fma(log(t), a, log(z)) + log(y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + Float64(Float64(a - 0.5) * log(t)))
	tmp = 0.0
	if (t_1 <= -1e+14)
		tmp = Float64(Float64(log(t) * a) - t);
	elseif (t_1 <= 680.0)
		tmp = fma(Float64(Float64(log(Float64(Float64((t ^ Float64(a - 0.5)) * z) * y)) / t) - 1.0), t, Float64(x / y));
	elseif (t_1 <= 2000.0)
		tmp = Float64(Float64(log(z) - Float64(0.5 * log(t))) + log(y));
	else
		tmp = Float64(fma(log(t), a, log(z)) + log(y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+14], N[(N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[t$95$1, 680.0], N[(N[(N[(N[Log[N[(N[(N[Power[t, N[(a - 0.5), $MachinePrecision]], $MachinePrecision] * z), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision] - 1.0), $MachinePrecision] * t + N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2000.0], N[(N[(N[Log[z], $MachinePrecision] - N[(0.5 * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[t], $MachinePrecision] * a + N[Log[z], $MachinePrecision]), $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < -1e14

   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. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 99.7

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

   7. Applied rewrites 99.7%

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

   if -1e14 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 680

   1. Initial program 98.9%

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

   2. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 91.1%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   7. Applied rewrites 45.4%

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

   if 680 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 2e3

   1. Initial program 99.5%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. associate-+r+ N/A

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

      3. sum-log N/A

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

      4. *-commutative N/A

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

      5. log-pow-rev N/A

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

      6. sum-log N/A

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

      7. lower-log.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lift--.f64 7.2

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

   4. Applied rewrites 7.2%

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

   5. Taylor expanded in t around 0

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

      1. pow-to-exp N/A

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

      2. associate-*r* N/A

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

      3. lower-log.f64 N/A

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

      4. associate-*r* N/A

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

      5. pow-to-exp N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. pow-to-exp N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift--.f64 6.1

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

   7. Applied rewrites 6.1%

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

      1. lift-log.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. log-prod N/A

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

      7. *-commutative N/A

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

      8. pow-to-exp N/A

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

      9. lower-+.f64 N/A

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

      10. log-prod N/A

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

      11. pow-to-exp N/A

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

      12. log-pow-rev N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lift-log.f64 N/A

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

      17. lift--.f64 N/A

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

      18. lift-log.f64 N/A

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

      19. lower-log.f64 52.9

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

   9. Applied rewrites 52.9%

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

   10. Taylor expanded in a around 0

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

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

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

       2. lower--.f64 N/A

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

       3. lift-log.f64 N/A

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

       4. metadata-eval N/A

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

       5. lower-*.f64 N/A

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

       6. lift-log.f64 52.1

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

   12. Applied rewrites 52.1%

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

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

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. associate-+r+ N/A

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

      3. sum-log N/A

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

      4. *-commutative N/A

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

      5. log-pow-rev N/A

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

      6. sum-log N/A

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

      7. lower-log.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lift--.f64 4.5

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

   4. Applied rewrites 4.5%

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

   5. Taylor expanded in t around 0

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

      1. pow-to-exp N/A

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

      2. associate-*r* N/A

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

      3. lower-log.f64 N/A

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

      4. associate-*r* N/A

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

      5. pow-to-exp N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. pow-to-exp N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift--.f64 5.0

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

   7. Applied rewrites 5.0%

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

      1. lift-log.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. log-prod N/A

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

      7. *-commutative N/A

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

      8. pow-to-exp N/A

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

      9. lower-+.f64 N/A

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

      10. log-prod N/A

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

      11. pow-to-exp N/A

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

      12. log-pow-rev N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lift-log.f64 N/A

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

      17. lift--.f64 N/A

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

      18. lift-log.f64 N/A

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

      19. lower-log.f64 71.9

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

   9. Applied rewrites 71.9%

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

   10. Taylor expanded in a around inf

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

   12. Applied rewrites 71.3%

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

Alternative 3: 92.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ t_2 := \left(\log t \cdot a + \log z\right) - t\\ \mathbf{if}\;t\_1 \leq -200000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 920:\\ \;\;\;\;\mathsf{fma}\left(\log t, a - 0.5, \log \left(\left(y + x\right) \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))
        (t_2 (- (+ (* (log t) a) (log z)) t)))
   (if (<= t_1 -200000.0)
     t_2
     (if (<= t_1 920.0) (fma (log t) (- a 0.5) (log (* (+ y x) z))) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
	double t_2 = ((log(t) * a) + log(z)) - t;
	double tmp;
	if (t_1 <= -200000.0) {
		tmp = t_2;
	} else if (t_1 <= 920.0) {
		tmp = fma(log(t), (a - 0.5), log(((y + x) * z)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + Float64(Float64(a - 0.5) * log(t)))
	t_2 = Float64(Float64(Float64(log(t) * a) + log(z)) - t)
	tmp = 0.0
	if (t_1 <= -200000.0)
		tmp = t_2;
	elseif (t_1 <= 920.0)
		tmp = fma(log(t), Float64(a - 0.5), log(Float64(Float64(y + x) * z)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[t$95$1, -200000.0], t$95$2, If[LessEqual[t$95$1, 920.0], N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision] + N[Log[N[(N[(y + x), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < -2e5 or 920 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t)))

   1. Initial program 99.8%

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

   2. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lift-log.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 93.4

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

   7. Applied rewrites 93.4%

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

   if -2e5 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 920

   1. Initial program 98.9%

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

   2. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lift-log.f64 98.9

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in t around 0

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

      1. associate-+r+ N/A

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

      2. sum-log N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift--.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-log.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-+.f64 87.9

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

   7. Applied rewrites 87.9%

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

Alternative 4: 82.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ t_2 := \log t \cdot a\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+14}:\\ \;\;\;\;t\_2 - t\\ \mathbf{elif}\;t\_1 \leq 720:\\ \;\;\;\;\log \left(\left(y \cdot z\right) \cdot {t}^{\left(a - 0.5\right)}\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(t\_2 + \log z\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))
        (t_2 (* (log t) a)))
   (if (<= t_1 -1e+14)
     (- t_2 t)
     (if (<= t_1 720.0)
       (- (log (* (* y z) (pow t (- a 0.5)))) t)
       (- (+ t_2 (log z)) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
	double t_2 = log(t) * a;
	double tmp;
	if (t_1 <= -1e+14) {
		tmp = t_2 - t;
	} else if (t_1 <= 720.0) {
		tmp = log(((y * z) * pow(t, (a - 0.5)))) - t;
	} else {
		tmp = (t_2 + log(z)) - t;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = ((log((x + y)) + log(z)) - t) + ((a - 0.5d0) * log(t))
    t_2 = log(t) * a
    if (t_1 <= (-1d+14)) then
        tmp = t_2 - t
    else if (t_1 <= 720.0d0) then
        tmp = log(((y * z) * (t ** (a - 0.5d0)))) - t
    else
        tmp = (t_2 + log(z)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = ((Math.log((x + y)) + Math.log(z)) - t) + ((a - 0.5) * Math.log(t));
	double t_2 = Math.log(t) * a;
	double tmp;
	if (t_1 <= -1e+14) {
		tmp = t_2 - t;
	} else if (t_1 <= 720.0) {
		tmp = Math.log(((y * z) * Math.pow(t, (a - 0.5)))) - t;
	} else {
		tmp = (t_2 + Math.log(z)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = ((math.log((x + y)) + math.log(z)) - t) + ((a - 0.5) * math.log(t))
	t_2 = math.log(t) * a
	tmp = 0
	if t_1 <= -1e+14:
		tmp = t_2 - t
	elif t_1 <= 720.0:
		tmp = math.log(((y * z) * math.pow(t, (a - 0.5)))) - t
	else:
		tmp = (t_2 + math.log(z)) - t
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + Float64(Float64(a - 0.5) * log(t)))
	t_2 = Float64(log(t) * a)
	tmp = 0.0
	if (t_1 <= -1e+14)
		tmp = Float64(t_2 - t);
	elseif (t_1 <= 720.0)
		tmp = Float64(log(Float64(Float64(y * z) * (t ^ Float64(a - 0.5)))) - t);
	else
		tmp = Float64(Float64(t_2 + log(z)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
	t_2 = log(t) * a;
	tmp = 0.0;
	if (t_1 <= -1e+14)
		tmp = t_2 - t;
	elseif (t_1 <= 720.0)
		tmp = log(((y * z) * (t ^ (a - 0.5)))) - t;
	else
		tmp = (t_2 + log(z)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+14], N[(t$95$2 - t), $MachinePrecision], If[LessEqual[t$95$1, 720.0], N[(N[Log[N[(N[(y * z), $MachinePrecision] * N[Power[t, N[(a - 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision], N[(N[(t$95$2 + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < -1e14

   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. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 99.7

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

   7. Applied rewrites 99.7%

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

   if -1e14 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 720

   1. Initial program 98.9%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. associate-+r+ N/A

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

      3. sum-log N/A

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

      4. *-commutative N/A

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

      5. log-pow-rev N/A

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

      6. sum-log N/A

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

      7. lower-log.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lift--.f64 45.9

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

   4. Applied rewrites 45.9%

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

   if 720 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t)))

   1. Initial program 99.6%

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

   2. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lift-log.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 74.0

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

   7. Applied rewrites 74.0%

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

Alternative 5: 83.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ t_2 := \log t \cdot a\\ \mathbf{if}\;t\_1 \leq -550:\\ \;\;\;\;\left(t\_2 + \log \left(y + x\right)\right) - t\\ \mathbf{elif}\;t\_1 \leq 920:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log \left(z \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_2 + \log z\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))
        (t_2 (* (log t) a)))
   (if (<= t_1 -550.0)
     (- (+ t_2 (log (+ y x))) t)
     (if (<= t_1 920.0)
       (fma (- a 0.5) (log t) (log (* z y)))
       (- (+ t_2 (log z)) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
	double t_2 = log(t) * a;
	double tmp;
	if (t_1 <= -550.0) {
		tmp = (t_2 + log((y + x))) - t;
	} else if (t_1 <= 920.0) {
		tmp = fma((a - 0.5), log(t), log((z * y)));
	} else {
		tmp = (t_2 + log(z)) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + Float64(Float64(a - 0.5) * log(t)))
	t_2 = Float64(log(t) * a)
	tmp = 0.0
	if (t_1 <= -550.0)
		tmp = Float64(Float64(t_2 + log(Float64(y + x))) - t);
	elseif (t_1 <= 920.0)
		tmp = fma(Float64(a - 0.5), log(t), log(Float64(z * y)));
	else
		tmp = Float64(Float64(t_2 + log(z)) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[t$95$1, -550.0], N[(N[(t$95$2 + N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[t$95$1, 920.0], N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < -550

   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. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 96.7

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

   7. Applied rewrites 96.7%

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

   if -550 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 920

   1. Initial program 98.9%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. associate-+r+ N/A

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

      3. sum-log N/A

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

      4. *-commutative N/A

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

      5. log-pow-rev N/A

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

      6. sum-log N/A

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

      7. lower-log.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lift--.f64 43.1

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

   4. Applied rewrites 43.1%

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

   5. Taylor expanded in t around 0

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

      1. pow-to-exp N/A

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

      2. associate-*r* N/A

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

      3. lower-log.f64 N/A

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

      4. associate-*r* N/A

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

      5. pow-to-exp N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. pow-to-exp N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift--.f64 41.8

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

   7. Applied rewrites 41.8%

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

      1. lift-log.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. log-prod N/A

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

      9. log-pow-rev N/A

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

      10. sum-log N/A

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

      11. lower-fma.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lift-log.f64 N/A

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

      14. +-commutative N/A

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

      15. log-prod N/A

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

      16. lift-log.f64 N/A

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

      17. lift-*.f64 47.5

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

   9. Applied rewrites 47.5%

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

   if 920 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t)))

   1. Initial program 99.6%

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

   2. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lift-log.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 81.2

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

   7. Applied rewrites 81.2%

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

Alternative 6: 83.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ t_2 := \left(\log t \cdot a + \log z\right) - t\\ \mathbf{if}\;t\_1 \leq -550:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 920:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log \left(z \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))
        (t_2 (- (+ (* (log t) a) (log z)) t)))
   (if (<= t_1 -550.0)
     t_2
     (if (<= t_1 920.0) (fma (- a 0.5) (log t) (log (* z y))) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
	double t_2 = ((log(t) * a) + log(z)) - t;
	double tmp;
	if (t_1 <= -550.0) {
		tmp = t_2;
	} else if (t_1 <= 920.0) {
		tmp = fma((a - 0.5), log(t), log((z * y)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + Float64(Float64(a - 0.5) * log(t)))
	t_2 = Float64(Float64(Float64(log(t) * a) + log(z)) - t)
	tmp = 0.0
	if (t_1 <= -550.0)
		tmp = t_2;
	elseif (t_1 <= 920.0)
		tmp = fma(Float64(a - 0.5), log(t), log(Float64(z * y)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[t$95$1, -550.0], t$95$2, If[LessEqual[t$95$1, 920.0], N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < -550 or 920 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t)))

   1. Initial program 99.8%

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

   2. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lift-log.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 92.2

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

   7. Applied rewrites 92.2%

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

   if -550 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 920

   1. Initial program 98.9%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. associate-+r+ N/A

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

      3. sum-log N/A

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

      4. *-commutative N/A

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

      5. log-pow-rev N/A

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

      6. sum-log N/A

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

      7. lower-log.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lift--.f64 43.1

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

   4. Applied rewrites 43.1%

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

   5. Taylor expanded in t around 0

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

      1. pow-to-exp N/A

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

      2. associate-*r* N/A

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

      3. lower-log.f64 N/A

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

      4. associate-*r* N/A

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

      5. pow-to-exp N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. pow-to-exp N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift--.f64 41.8

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

   7. Applied rewrites 41.8%

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

      1. lift-log.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. log-prod N/A

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

      9. log-pow-rev N/A

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

      10. sum-log N/A

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

      11. lower-fma.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lift-log.f64 N/A

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

      14. +-commutative N/A

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

      15. log-prod N/A

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

      16. lift-log.f64 N/A

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

      17. lift-*.f64 47.5

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

   9. Applied rewrites 47.5%

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

Alternative 7: 83.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ t_2 := \left(\log t \cdot a + \log z\right) - t\\ \mathbf{if}\;t\_1 \leq -550:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 920:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log t, \log \left(z \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))
        (t_2 (- (+ (* (log t) a) (log z)) t)))
   (if (<= t_1 -550.0)
     t_2
     (if (<= t_1 920.0) (fma -0.5 (log t) (log (* z y))) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
	double t_2 = ((log(t) * a) + log(z)) - t;
	double tmp;
	if (t_1 <= -550.0) {
		tmp = t_2;
	} else if (t_1 <= 920.0) {
		tmp = fma(-0.5, log(t), log((z * y)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + Float64(Float64(a - 0.5) * log(t)))
	t_2 = Float64(Float64(Float64(log(t) * a) + log(z)) - t)
	tmp = 0.0
	if (t_1 <= -550.0)
		tmp = t_2;
	elseif (t_1 <= 920.0)
		tmp = fma(-0.5, log(t), log(Float64(z * y)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[t$95$1, -550.0], t$95$2, If[LessEqual[t$95$1, 920.0], N[(-0.5 * N[Log[t], $MachinePrecision] + N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < -550 or 920 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t)))

   1. Initial program 99.8%

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

   2. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lift-log.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 92.2

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

   7. Applied rewrites 92.2%

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

   if -550 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 920

   1. Initial program 98.9%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. associate-+r+ N/A

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

      3. sum-log N/A

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

      4. *-commutative N/A

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

      5. log-pow-rev N/A

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

      6. sum-log N/A

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

      7. lower-log.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lift--.f64 43.1

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

   4. Applied rewrites 43.1%

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

   5. Taylor expanded in t around 0

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

      1. pow-to-exp N/A

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

      2. associate-*r* N/A

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

      3. lower-log.f64 N/A

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

      4. associate-*r* N/A

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

      5. pow-to-exp N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. pow-to-exp N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift--.f64 41.8

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

   7. Applied rewrites 41.8%

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

   8. Taylor expanded in a around 0

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

      1. log-prod N/A

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

      2. pow1/2 N/A

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

      3. inv-pow N/A

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

      4. pow-pow N/A

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

      5. metadata-eval N/A

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

      6. log-pow-rev N/A

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

      7. sum-log N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. +-commutative N/A

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

      11. log-prod N/A

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

      12. lift-log.f64 N/A

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

      13. lift-*.f64 46.3

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

   10. Applied rewrites 46.3%

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

Alternative 8: 77.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot \log t\\ t_2 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + t\_1\\ \mathbf{if}\;t\_2 \leq -200000:\\ \;\;\;\;\left(-t\right) + t\_1\\ \mathbf{elif}\;t\_2 \leq 920:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log t, \log \left(z \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t + \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) (log t)))
        (t_2 (+ (- (+ (log (+ x y)) (log z)) t) t_1)))
   (if (<= t_2 -200000.0)
     (+ (- t) t_1)
     (if (<= t_2 920.0)
       (fma -0.5 (log t) (log (* z y)))
       (+ (* a (log t)) (log y))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (a - 0.5) * log(t);
	double t_2 = ((log((x + y)) + log(z)) - t) + t_1;
	double tmp;
	if (t_2 <= -200000.0) {
		tmp = -t + t_1;
	} else if (t_2 <= 920.0) {
		tmp = fma(-0.5, log(t), log((z * y)));
	} else {
		tmp = (a * log(t)) + log(y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(a - 0.5) * log(t))
	t_2 = Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + t_1)
	tmp = 0.0
	if (t_2 <= -200000.0)
		tmp = Float64(Float64(-t) + t_1);
	elseif (t_2 <= 920.0)
		tmp = fma(-0.5, log(t), log(Float64(z * y)));
	else
		tmp = Float64(Float64(a * log(t)) + log(y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -200000.0], N[((-t) + t$95$1), $MachinePrecision], If[LessEqual[t$95$2, 920.0], N[(-0.5 * N[Log[t], $MachinePrecision] + N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < -2e5

   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. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 98.7

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

   4. Applied rewrites 98.7%

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

   if -2e5 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 920

   1. Initial program 98.9%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. associate-+r+ N/A

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

      3. sum-log N/A

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

      4. *-commutative N/A

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

      5. log-pow-rev N/A

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

      6. sum-log N/A

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

      7. lower-log.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lift--.f64 41.2

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

   4. Applied rewrites 41.2%

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

   5. Taylor expanded in t around 0

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

      1. pow-to-exp N/A

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

      2. associate-*r* N/A

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

      3. lower-log.f64 N/A

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

      4. associate-*r* N/A

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

      5. pow-to-exp N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. pow-to-exp N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift--.f64 40.8

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

   7. Applied rewrites 40.8%

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

   8. Taylor expanded in a around 0

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

      1. log-prod N/A

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

      2. pow1/2 N/A

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

      3. inv-pow N/A

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

      4. pow-pow N/A

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

      5. metadata-eval N/A

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

      6. log-pow-rev N/A

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

      7. sum-log N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. +-commutative N/A

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

      11. log-prod N/A

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

      12. lift-log.f64 N/A

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

      13. lift-*.f64 43.9

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

   10. Applied rewrites 43.9%

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

   if 920 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t)))

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. associate-+r+ N/A

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

      3. sum-log N/A

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

      4. *-commutative N/A

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

      5. log-pow-rev N/A

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

      6. sum-log N/A

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

      7. lower-log.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lift--.f64 4.5

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

   4. Applied rewrites 4.5%

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

   5. Taylor expanded in t around 0

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

      1. pow-to-exp N/A

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

      2. associate-*r* N/A

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

      3. lower-log.f64 N/A

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

      4. associate-*r* N/A

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

      5. pow-to-exp N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. pow-to-exp N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift--.f64 4.7

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

   7. Applied rewrites 4.7%

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

      1. lift-log.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. log-prod N/A

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

      7. *-commutative N/A

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

      8. pow-to-exp N/A

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

      9. lower-+.f64 N/A

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

      10. log-prod N/A

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

      11. pow-to-exp N/A

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

      12. log-pow-rev N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lift-log.f64 N/A

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

      17. lift--.f64 N/A

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

      18. lift-log.f64 N/A

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

      19. lower-log.f64 68.2

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

   9. Applied rewrites 68.2%

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

   10. Taylor expanded in a around inf

       \[\leadsto a \cdot \log t + \log y \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

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

       2. lift-log.f64 58.7

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

   12. Applied rewrites 58.7%

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

Alternative 9: 76.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+14}:\\ \;\;\;\;\log t \cdot a - t\\ \mathbf{elif}\;t\_1 \leq 720:\\ \;\;\;\;\log \left(\left(y \cdot z\right) \cdot \frac{1}{\sqrt{t}}\right) - t\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t + \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t)))))
   (if (<= t_1 -1e+14)
     (- (* (log t) a) t)
     (if (<= t_1 720.0)
       (- (log (* (* y z) (/ 1.0 (sqrt t)))) t)
       (+ (* a (log t)) (log y))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
	double tmp;
	if (t_1 <= -1e+14) {
		tmp = (log(t) * a) - t;
	} else if (t_1 <= 720.0) {
		tmp = log(((y * z) * (1.0 / sqrt(t)))) - t;
	} else {
		tmp = (a * log(t)) + log(y);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((log((x + y)) + log(z)) - t) + ((a - 0.5d0) * log(t))
    if (t_1 <= (-1d+14)) then
        tmp = (log(t) * a) - t
    else if (t_1 <= 720.0d0) then
        tmp = log(((y * z) * (1.0d0 / sqrt(t)))) - t
    else
        tmp = (a * log(t)) + log(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = ((Math.log((x + y)) + Math.log(z)) - t) + ((a - 0.5) * Math.log(t));
	double tmp;
	if (t_1 <= -1e+14) {
		tmp = (Math.log(t) * a) - t;
	} else if (t_1 <= 720.0) {
		tmp = Math.log(((y * z) * (1.0 / Math.sqrt(t)))) - t;
	} else {
		tmp = (a * Math.log(t)) + Math.log(y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = ((math.log((x + y)) + math.log(z)) - t) + ((a - 0.5) * math.log(t))
	tmp = 0
	if t_1 <= -1e+14:
		tmp = (math.log(t) * a) - t
	elif t_1 <= 720.0:
		tmp = math.log(((y * z) * (1.0 / math.sqrt(t)))) - t
	else:
		tmp = (a * math.log(t)) + math.log(y)
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + Float64(Float64(a - 0.5) * log(t)))
	tmp = 0.0
	if (t_1 <= -1e+14)
		tmp = Float64(Float64(log(t) * a) - t);
	elseif (t_1 <= 720.0)
		tmp = Float64(log(Float64(Float64(y * z) * Float64(1.0 / sqrt(t)))) - t);
	else
		tmp = Float64(Float64(a * log(t)) + log(y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
	tmp = 0.0;
	if (t_1 <= -1e+14)
		tmp = (log(t) * a) - t;
	elseif (t_1 <= 720.0)
		tmp = log(((y * z) * (1.0 / sqrt(t)))) - t;
	else
		tmp = (a * log(t)) + log(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+14], N[(N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[t$95$1, 720.0], N[(N[Log[N[(N[(y * z), $MachinePrecision] * N[(1.0 / N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision], N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < -1e14

   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. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 99.7

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

   7. Applied rewrites 99.7%

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

   if -1e14 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 720

   1. Initial program 98.9%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. associate-+r+ N/A

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

      3. sum-log N/A

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

      4. *-commutative N/A

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

      5. log-pow-rev N/A

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

      6. sum-log N/A

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

      7. lower-log.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lift--.f64 45.9

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

   4. Applied rewrites 45.9%

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

   5. Taylor expanded in a around 0

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

      1. sqrt-div N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 44.9

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

   7. Applied rewrites 44.9%

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

   if 720 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t)))

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. associate-+r+ N/A

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

      3. sum-log N/A

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

      4. *-commutative N/A

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

      5. log-pow-rev N/A

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

      6. sum-log N/A

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

      7. lower-log.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lift--.f64 4.6

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

   4. Applied rewrites 4.6%

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

   5. Taylor expanded in t around 0

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

      1. pow-to-exp N/A

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

      2. associate-*r* N/A

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

      3. lower-log.f64 N/A

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

      4. associate-*r* N/A

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

      5. pow-to-exp N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. pow-to-exp N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift--.f64 4.6

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

   7. Applied rewrites 4.6%

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

      1. lift-log.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. log-prod N/A

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

      7. *-commutative N/A

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

      8. pow-to-exp N/A

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

      9. lower-+.f64 N/A

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

      10. log-prod N/A

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

      11. pow-to-exp N/A

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

      12. log-pow-rev N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lift-log.f64 N/A

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

      17. lift--.f64 N/A

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

      18. lift-log.f64 N/A

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

      19. lower-log.f64 66.3

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

   9. Applied rewrites 66.3%

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

   10. Taylor expanded in a around inf

       \[\leadsto a \cdot \log t + \log y \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

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

       2. lift-log.f64 53.4

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

   12. Applied rewrites 53.4%

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

Alternative 10: 82.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot \log t\\ t_2 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + t\_1\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+14}:\\ \;\;\;\;\log t \cdot a - t\\ \mathbf{elif}\;t\_2 \leq 720:\\ \;\;\;\;\log \left(\left(y \cdot z\right) \cdot \frac{1}{\sqrt{t}}\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) + t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) (log t)))
        (t_2 (+ (- (+ (log (+ x y)) (log z)) t) t_1)))
   (if (<= t_2 -1e+14)
     (- (* (log t) a) t)
     (if (<= t_2 720.0)
       (- (log (* (* y z) (/ 1.0 (sqrt t)))) t)
       (+ (- t) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (a - 0.5) * log(t);
	double t_2 = ((log((x + y)) + log(z)) - t) + t_1;
	double tmp;
	if (t_2 <= -1e+14) {
		tmp = (log(t) * a) - t;
	} else if (t_2 <= 720.0) {
		tmp = log(((y * z) * (1.0 / sqrt(t)))) - t;
	} else {
		tmp = -t + t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (a - 0.5d0) * log(t)
    t_2 = ((log((x + y)) + log(z)) - t) + t_1
    if (t_2 <= (-1d+14)) then
        tmp = (log(t) * a) - t
    else if (t_2 <= 720.0d0) then
        tmp = log(((y * z) * (1.0d0 / sqrt(t)))) - t
    else
        tmp = -t + t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (a - 0.5) * Math.log(t);
	double t_2 = ((Math.log((x + y)) + Math.log(z)) - t) + t_1;
	double tmp;
	if (t_2 <= -1e+14) {
		tmp = (Math.log(t) * a) - t;
	} else if (t_2 <= 720.0) {
		tmp = Math.log(((y * z) * (1.0 / Math.sqrt(t)))) - t;
	} else {
		tmp = -t + t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (a - 0.5) * math.log(t)
	t_2 = ((math.log((x + y)) + math.log(z)) - t) + t_1
	tmp = 0
	if t_2 <= -1e+14:
		tmp = (math.log(t) * a) - t
	elif t_2 <= 720.0:
		tmp = math.log(((y * z) * (1.0 / math.sqrt(t)))) - t
	else:
		tmp = -t + t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(a - 0.5) * log(t))
	t_2 = Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + t_1)
	tmp = 0.0
	if (t_2 <= -1e+14)
		tmp = Float64(Float64(log(t) * a) - t);
	elseif (t_2 <= 720.0)
		tmp = Float64(log(Float64(Float64(y * z) * Float64(1.0 / sqrt(t)))) - t);
	else
		tmp = Float64(Float64(-t) + t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (a - 0.5) * log(t);
	t_2 = ((log((x + y)) + log(z)) - t) + t_1;
	tmp = 0.0;
	if (t_2 <= -1e+14)
		tmp = (log(t) * a) - t;
	elseif (t_2 <= 720.0)
		tmp = log(((y * z) * (1.0 / sqrt(t)))) - t;
	else
		tmp = -t + t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+14], N[(N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[t$95$2, 720.0], N[(N[Log[N[(N[(y * z), $MachinePrecision] * N[(1.0 / N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision], N[((-t) + t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < -1e14

   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. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 99.7

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

   7. Applied rewrites 99.7%

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

   if -1e14 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 720

   1. Initial program 98.9%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. associate-+r+ N/A

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

      3. sum-log N/A

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

      4. *-commutative N/A

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

      5. log-pow-rev N/A

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

      6. sum-log N/A

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

      7. lower-log.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lift--.f64 45.9

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

   4. Applied rewrites 45.9%

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

   5. Taylor expanded in a around 0

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

      1. sqrt-div N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 44.9

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

   7. Applied rewrites 44.9%

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

   if 720 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t)))

   1. Initial program 99.6%

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

   2. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 73.8

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

   4. Applied rewrites 73.8%

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

Alternative 11: 82.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot \log t\\ t_2 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + t\_1\\ t_3 := \left(-t\right) + t\_1\\ \mathbf{if}\;t\_2 \leq -200000:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 720:\\ \;\;\;\;\log \left(\left(\frac{1}{\sqrt{t}} \cdot z\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) (log t)))
        (t_2 (+ (- (+ (log (+ x y)) (log z)) t) t_1))
        (t_3 (+ (- t) t_1)))
   (if (<= t_2 -200000.0)
     t_3
     (if (<= t_2 720.0) (log (* (* (/ 1.0 (sqrt t)) z) y)) t_3))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (a - 0.5) * log(t);
	double t_2 = ((log((x + y)) + log(z)) - t) + t_1;
	double t_3 = -t + t_1;
	double tmp;
	if (t_2 <= -200000.0) {
		tmp = t_3;
	} else if (t_2 <= 720.0) {
		tmp = log((((1.0 / sqrt(t)) * z) * y));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = (a - 0.5d0) * log(t)
    t_2 = ((log((x + y)) + log(z)) - t) + t_1
    t_3 = -t + t_1
    if (t_2 <= (-200000.0d0)) then
        tmp = t_3
    else if (t_2 <= 720.0d0) then
        tmp = log((((1.0d0 / sqrt(t)) * z) * y))
    else
        tmp = t_3
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (a - 0.5) * Math.log(t);
	double t_2 = ((Math.log((x + y)) + Math.log(z)) - t) + t_1;
	double t_3 = -t + t_1;
	double tmp;
	if (t_2 <= -200000.0) {
		tmp = t_3;
	} else if (t_2 <= 720.0) {
		tmp = Math.log((((1.0 / Math.sqrt(t)) * z) * y));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (a - 0.5) * math.log(t)
	t_2 = ((math.log((x + y)) + math.log(z)) - t) + t_1
	t_3 = -t + t_1
	tmp = 0
	if t_2 <= -200000.0:
		tmp = t_3
	elif t_2 <= 720.0:
		tmp = math.log((((1.0 / math.sqrt(t)) * z) * y))
	else:
		tmp = t_3
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(a - 0.5) * log(t))
	t_2 = Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + t_1)
	t_3 = Float64(Float64(-t) + t_1)
	tmp = 0.0
	if (t_2 <= -200000.0)
		tmp = t_3;
	elseif (t_2 <= 720.0)
		tmp = log(Float64(Float64(Float64(1.0 / sqrt(t)) * z) * y));
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (a - 0.5) * log(t);
	t_2 = ((log((x + y)) + log(z)) - t) + t_1;
	t_3 = -t + t_1;
	tmp = 0.0;
	if (t_2 <= -200000.0)
		tmp = t_3;
	elseif (t_2 <= 720.0)
		tmp = log((((1.0 / sqrt(t)) * z) * y));
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[((-t) + t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -200000.0], t$95$3, If[LessEqual[t$95$2, 720.0], N[Log[N[(N[(N[(1.0 / N[Sqrt[t], $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision], t$95$3]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < -2e5 or 720 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t)))

   1. Initial program 99.8%

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

   2. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 90.6

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

   4. Applied rewrites 90.6%

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

   if -2e5 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 720

   1. Initial program 98.8%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. associate-+r+ N/A

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

      3. sum-log N/A

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

      4. *-commutative N/A

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

      5. log-pow-rev N/A

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

      6. sum-log N/A

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

      7. lower-log.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lift--.f64 47.2

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

   4. Applied rewrites 47.2%

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

   5. Taylor expanded in t around 0

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

      1. pow-to-exp N/A

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

      2. associate-*r* N/A

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

      3. lower-log.f64 N/A

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

      4. associate-*r* N/A

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

      5. pow-to-exp N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. pow-to-exp N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift--.f64 46.7

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

   7. Applied rewrites 46.7%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 45.6%

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

   11. Taylor expanded in a around 0

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

       1. sqrt-div N/A

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

       2. metadata-eval N/A

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

       3. lower-/.f64 N/A

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

       4. lower-sqrt.f64 45.6

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

   13. Applied rewrites 45.6%

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

Alternative 12: 82.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot \log t\\ t_2 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + t\_1\\ t_3 := \left(-t\right) + t\_1\\ \mathbf{if}\;t\_2 \leq -200000:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 720:\\ \;\;\;\;\log \left(\frac{1}{\sqrt{t}} \cdot \left(y \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) (log t)))
        (t_2 (+ (- (+ (log (+ x y)) (log z)) t) t_1))
        (t_3 (+ (- t) t_1)))
   (if (<= t_2 -200000.0)
     t_3
     (if (<= t_2 720.0) (log (* (/ 1.0 (sqrt t)) (* y z))) t_3))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (a - 0.5) * log(t);
	double t_2 = ((log((x + y)) + log(z)) - t) + t_1;
	double t_3 = -t + t_1;
	double tmp;
	if (t_2 <= -200000.0) {
		tmp = t_3;
	} else if (t_2 <= 720.0) {
		tmp = log(((1.0 / sqrt(t)) * (y * z)));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = (a - 0.5d0) * log(t)
    t_2 = ((log((x + y)) + log(z)) - t) + t_1
    t_3 = -t + t_1
    if (t_2 <= (-200000.0d0)) then
        tmp = t_3
    else if (t_2 <= 720.0d0) then
        tmp = log(((1.0d0 / sqrt(t)) * (y * z)))
    else
        tmp = t_3
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (a - 0.5) * Math.log(t);
	double t_2 = ((Math.log((x + y)) + Math.log(z)) - t) + t_1;
	double t_3 = -t + t_1;
	double tmp;
	if (t_2 <= -200000.0) {
		tmp = t_3;
	} else if (t_2 <= 720.0) {
		tmp = Math.log(((1.0 / Math.sqrt(t)) * (y * z)));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (a - 0.5) * math.log(t)
	t_2 = ((math.log((x + y)) + math.log(z)) - t) + t_1
	t_3 = -t + t_1
	tmp = 0
	if t_2 <= -200000.0:
		tmp = t_3
	elif t_2 <= 720.0:
		tmp = math.log(((1.0 / math.sqrt(t)) * (y * z)))
	else:
		tmp = t_3
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(a - 0.5) * log(t))
	t_2 = Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + t_1)
	t_3 = Float64(Float64(-t) + t_1)
	tmp = 0.0
	if (t_2 <= -200000.0)
		tmp = t_3;
	elseif (t_2 <= 720.0)
		tmp = log(Float64(Float64(1.0 / sqrt(t)) * Float64(y * z)));
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (a - 0.5) * log(t);
	t_2 = ((log((x + y)) + log(z)) - t) + t_1;
	t_3 = -t + t_1;
	tmp = 0.0;
	if (t_2 <= -200000.0)
		tmp = t_3;
	elseif (t_2 <= 720.0)
		tmp = log(((1.0 / sqrt(t)) * (y * z)));
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[((-t) + t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -200000.0], t$95$3, If[LessEqual[t$95$2, 720.0], N[Log[N[(N[(1.0 / N[Sqrt[t], $MachinePrecision]), $MachinePrecision] * N[(y * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$3]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < -2e5 or 720 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t)))

   1. Initial program 99.8%

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

   2. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 90.6

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

   4. Applied rewrites 90.6%

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

   if -2e5 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 720

   1. Initial program 98.8%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. associate-+r+ N/A

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

      3. sum-log N/A

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

      4. *-commutative N/A

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

      5. log-pow-rev N/A

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

      6. sum-log N/A

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

      7. lower-log.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lift--.f64 47.2

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

   4. Applied rewrites 47.2%

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

   5. Taylor expanded in t around 0

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

      1. pow-to-exp N/A

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

      2. associate-*r* N/A

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

      3. lower-log.f64 N/A

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

      4. associate-*r* N/A

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

      5. pow-to-exp N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. pow-to-exp N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift--.f64 46.7

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

   7. Applied rewrites 46.7%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 45.6%

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

   11. Taylor expanded in a around 0

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

       1. lower-*.f64 N/A

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

       2. sqrt-div N/A

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

       3. metadata-eval N/A

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

       4. lower-/.f64 N/A

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

       5. lower-sqrt.f64 N/A

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

       6. lower-*.f64 44.8

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

   13. Applied rewrites 44.8%

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

Alternative 13: 68.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(x + y\right) + \log z\\ t_2 := \left(\log t \cdot a + \log z\right) - t\\ \mathbf{if}\;t\_1 \leq -800:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 672:\\ \;\;\;\;\left(\log \left(z \cdot y\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log (+ x y)) (log z))) (t_2 (- (+ (* (log t) a) (log z)) t)))
   (if (<= t_1 -800.0)
     t_2
     (if (<= t_1 672.0) (+ (- (log (* z y)) t) (* (- a 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 = ((log(t) * a) + log(z)) - t;
	double tmp;
	if (t_1 <= -800.0) {
		tmp = t_2;
	} else if (t_1 <= 672.0) {
		tmp = (log((z * y)) - t) + ((a - 0.5) * log(t));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = log((x + y)) + log(z)
    t_2 = ((log(t) * a) + log(z)) - t
    if (t_1 <= (-800.0d0)) then
        tmp = t_2
    else if (t_1 <= 672.0d0) then
        tmp = (log((z * y)) - t) + ((a - 0.5d0) * log(t))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = Math.log((x + y)) + Math.log(z);
	double t_2 = ((Math.log(t) * a) + Math.log(z)) - t;
	double tmp;
	if (t_1 <= -800.0) {
		tmp = t_2;
	} else if (t_1 <= 672.0) {
		tmp = (Math.log((z * y)) - t) + ((a - 0.5) * Math.log(t));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = math.log((x + y)) + math.log(z)
	t_2 = ((math.log(t) * a) + math.log(z)) - t
	tmp = 0
	if t_1 <= -800.0:
		tmp = t_2
	elif t_1 <= 672.0:
		tmp = (math.log((z * y)) - t) + ((a - 0.5) * math.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(Float64(Float64(log(t) * a) + log(z)) - t)
	tmp = 0.0
	if (t_1 <= -800.0)
		tmp = t_2;
	elseif (t_1 <= 672.0)
		tmp = Float64(Float64(log(Float64(z * y)) - t) + Float64(Float64(a - 0.5) * log(t)));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = log((x + y)) + log(z);
	t_2 = ((log(t) * a) + log(z)) - t;
	tmp = 0.0;
	if (t_1 <= -800.0)
		tmp = t_2;
	elseif (t_1 <= 672.0)
		tmp = (log((z * y)) - t) + ((a - 0.5) * log(t));
	else
		tmp = t_2;
	end
	tmp_2 = 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[(N[Log[t], $MachinePrecision] * a), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[t$95$1, -800.0], t$95$2, If[LessEqual[t$95$1, 672.0], N[(N[(N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -800 or 672 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z))

   1. Initial program 99.7%

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

   2. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lift-log.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 77.5

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

   7. Applied rewrites 77.5%

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

   if -800 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 672

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. sum-log N/A

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

      3. lower-log.f64 N/A

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

      4. lower-*.f64 64.9

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

   4. Applied rewrites 64.9%

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

Alternative 14: 80.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 300:\\ \;\;\;\;\mathsf{fma}\left(\log t, a - 0.5, \log z\right) + \log y\\ \mathbf{else}:\\ \;\;\;\;\left(\log t \cdot a + \log \left(y + x\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 300.0)
   (+ (fma (log t) (- a 0.5) (log z)) (log y))
   (- (+ (* (log t) a) (log (+ y x))) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 300.0) {
		tmp = fma(log(t), (a - 0.5), log(z)) + log(y);
	} else {
		tmp = ((log(t) * a) + log((y + x))) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 300.0)
		tmp = Float64(fma(log(t), Float64(a - 0.5), log(z)) + log(y));
	else
		tmp = Float64(Float64(Float64(log(t) * a) + log(Float64(y + x))) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 300.0], N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision] + N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 300

   1. Initial program 99.3%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. associate-+r+ N/A

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

      3. sum-log N/A

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

      4. *-commutative N/A

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

      5. log-pow-rev N/A

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

      6. sum-log N/A

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

      7. lower-log.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lift--.f64 19.5

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

   4. Applied rewrites 19.5%

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

   5. Taylor expanded in t around 0

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

      1. pow-to-exp N/A

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

      2. associate-*r* N/A

         \[\leadsto \log \left(\left(y \cdot z\right) \cdot {t}^{\left(a - \frac{1}{2}\right)}\right) \]

      3. lower-log.f64 N/A

         \[\leadsto \log \left(\left(y \cdot z\right) \cdot {t}^{\left(a - \frac{1}{2}\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \log \left(y \cdot \left(z \cdot {t}^{\left(a - \frac{1}{2}\right)}\right)\right) \]

      5. pow-to-exp N/A

         \[\leadsto \log \left(y \cdot \left(z \cdot e^{\log t \cdot \left(a - \frac{1}{2}\right)}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \log \left(\left(z \cdot e^{\log t \cdot \left(a - \frac{1}{2}\right)}\right) \cdot y\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \log \left(\left(z \cdot e^{\log t \cdot \left(a - \frac{1}{2}\right)}\right) \cdot y\right) \]

      8. pow-to-exp N/A

         \[\leadsto \log \left(\left(z \cdot {t}^{\left(a - \frac{1}{2}\right)}\right) \cdot y\right) \]

      9. *-commutative N/A

         \[\leadsto \log \left(\left({t}^{\left(a - \frac{1}{2}\right)} \cdot z\right) \cdot y\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \log \left(\left({t}^{\left(a - \frac{1}{2}\right)} \cdot z\right) \cdot y\right) \]

      11. lift-pow.f64 N/A

          \[\leadsto \log \left(\left({t}^{\left(a - \frac{1}{2}\right)} \cdot z\right) \cdot y\right) \]

      12. lift--.f64 19.7

          \[\leadsto \log \left(\left({t}^{\left(a - 0.5\right)} \cdot z\right) \cdot y\right) \]

   7. Applied rewrites 19.7%

      \[\leadsto \log \left(\left({t}^{\left(a - 0.5\right)} \cdot z\right) \cdot y\right) \]
   8. Step-by-step derivation

      1. lift-log.f64 N/A

         \[\leadsto \log \left(\left({t}^{\left(a - \frac{1}{2}\right)} \cdot z\right) \cdot y\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \log \left(\left({t}^{\left(a - \frac{1}{2}\right)} \cdot z\right) \cdot y\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \log \left(\left({t}^{\left(a - \frac{1}{2}\right)} \cdot z\right) \cdot y\right) \]

      4. lift--.f64 N/A

         \[\leadsto \log \left(\left({t}^{\left(a - \frac{1}{2}\right)} \cdot z\right) \cdot y\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \log \left(\left({t}^{\left(a - \frac{1}{2}\right)} \cdot z\right) \cdot y\right) \]

      6. log-prod N/A

         \[\leadsto \log \left({t}^{\left(a - \frac{1}{2}\right)} \cdot z\right) + \log y \]

      7. *-commutative N/A

         \[\leadsto \log \left(z \cdot {t}^{\left(a - \frac{1}{2}\right)}\right) + \log y \]

      8. pow-to-exp N/A

         \[\leadsto \log \left(z \cdot e^{\log t \cdot \left(a - \frac{1}{2}\right)}\right) + \log y \]

      9. lower-+.f64 N/A

         \[\leadsto \log \left(z \cdot e^{\log t \cdot \left(a - \frac{1}{2}\right)}\right) + \log y \]

      10. log-prod N/A

          \[\leadsto \left(\log z + \log \left(e^{\log t \cdot \left(a - \frac{1}{2}\right)}\right)\right) + \log y \]

      11. pow-to-exp N/A

          \[\leadsto \left(\log z + \log \left({t}^{\left(a - \frac{1}{2}\right)}\right)\right) + \log y \]

      12. log-pow-rev N/A

          \[\leadsto \left(\log z + \left(a - \frac{1}{2}\right) \cdot \log t\right) + \log y \]

      13. *-commutative N/A

          \[\leadsto \left(\log z + \log t \cdot \left(a - \frac{1}{2}\right)\right) + \log y \]

      14. +-commutative N/A

          \[\leadsto \left(\log t \cdot \left(a - \frac{1}{2}\right) + \log z\right) + \log y \]

      15. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\log t, a - \frac{1}{2}, \log z\right) + \log y \]

      16. lift-log.f64 N/A

          \[\leadsto \mathsf{fma}\left(\log t, a - \frac{1}{2}, \log z\right) + \log y \]

      17. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\log t, a - \frac{1}{2}, \log z\right) + \log y \]

      18. lift-log.f64 N/A

          \[\leadsto \mathsf{fma}\left(\log t, a - \frac{1}{2}, \log z\right) + \log y \]

      19. lower-log.f64 62.2

          \[\leadsto \mathsf{fma}\left(\log t, a - 0.5, \log z\right) + \log y \]

   9. Applied rewrites 62.2%

      \[\leadsto \mathsf{fma}\left(\log t, a - 0.5, \log z\right) + \log y \]

   if 300 < t

   1. Initial program 99.9%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\log \left(x + y\right) + \left(-1 \cdot \log \left(\frac{1}{z}\right) + \log t \cdot \left(a - \frac{1}{2}\right)\right)\right) - t} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(\log \left(x + y\right) + \left(-1 \cdot \log \left(\frac{1}{z}\right) + \log t \cdot \left(a - \frac{1}{2}\right)\right)\right) - \color{blue}{t} \]

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\log t, a - 0.5, -\left(-\log z\right)\right) + \log \left(y + x\right)\right) - t} \]

   5. Taylor expanded in a around inf

      \[\leadsto \left(a \cdot \log t + \log \left(y + x\right)\right) - t \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\log t \cdot a + \log \left(y + x\right)\right) - t \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\log t \cdot a + \log \left(y + x\right)\right) - t \]

      3. lift-log.f64 98.7

         \[\leadsto \left(\log t \cdot a + \log \left(y + x\right)\right) - t \]

   7. Applied rewrites 98.7%

      \[\leadsto \left(\log t \cdot a + \log \left(y + x\right)\right) - t \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(\log t, -0.5 + a, \log \left(y + x\right)\right) + \log z\right) - t \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (- (+ (fma (log t) (+ -0.5 a) (log (+ y x))) (log z)) t))

C

double code(double x, double y, double z, double t, double a) {
	return (fma(log(t), (-0.5 + a), log((y + x))) + log(z)) - t;
}

Julia

function code(x, y, z, t, a)
	return Float64(Float64(fma(log(t), Float64(-0.5 + a), log(Float64(y + x))) + log(z)) - t)
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[(N[Log[t], $MachinePrecision] * N[(-0.5 + a), $MachinePrecision] + N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]

2. Taylor expanded in a around 0

   \[\leadsto \color{blue}{\left(\log z + \left(\log \left(x + y\right) + \left(\frac{-1}{2} \cdot \log t + a \cdot \log t\right)\right)\right) - t} \]
3. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto \left(\log z + \left(\log \left(x + y\right) + \left(\frac{-1}{2} \cdot \log t + a \cdot \log t\right)\right)\right) - \color{blue}{t} \]

   2. +-commutative N/A

      \[\leadsto \left(\left(\log \left(x + y\right) + \left(\frac{-1}{2} \cdot \log t + a \cdot \log t\right)\right) + \log z\right) - t \]

   3. lower-+.f64 N/A

      \[\leadsto \left(\left(\log \left(x + y\right) + \left(\frac{-1}{2} \cdot \log t + a \cdot \log t\right)\right) + \log z\right) - t \]

   4. +-commutative N/A

      \[\leadsto \left(\left(\left(\frac{-1}{2} \cdot \log t + a \cdot \log t\right) + \log \left(x + y\right)\right) + \log z\right) - t \]

   5. distribute-rgt-out N/A

      \[\leadsto \left(\left(\log t \cdot \left(\frac{-1}{2} + a\right) + \log \left(x + y\right)\right) + \log z\right) - t \]

   6. lower-fma.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\log t, \frac{-1}{2} + a, \log \left(x + y\right)\right) + \log z\right) - t \]

   7. lift-log.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\log t, \frac{-1}{2} + a, \log \left(x + y\right)\right) + \log z\right) - t \]

   8. lower-+.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\log t, \frac{-1}{2} + a, \log \left(x + y\right)\right) + \log z\right) - t \]

   9. lift-log.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\log t, \frac{-1}{2} + a, \log \left(x + y\right)\right) + \log z\right) - t \]

   10. +-commutative N/A

       \[\leadsto \left(\mathsf{fma}\left(\log t, \frac{-1}{2} + a, \log \left(y + x\right)\right) + \log z\right) - t \]

   11. lower-+.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\log t, \frac{-1}{2} + a, \log \left(y + x\right)\right) + \log z\right) - t \]

   12. lift-log.f64 99.6

       \[\leadsto \left(\mathsf{fma}\left(\log t, -0.5 + a, \log \left(y + x\right)\right) + \log z\right) - t \]

4. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\log t, -0.5 + a, \log \left(y + x\right)\right) + \log z\right) - t} \]
5. Add Preprocessing

Alternative 16: 69.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(\log t, -0.5 + a, \log y\right) + \log z\right) - t \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (- (+ (fma (log t) (+ -0.5 a) (log y)) (log z)) t))

C

double code(double x, double y, double z, double t, double a) {
	return (fma(log(t), (-0.5 + a), log(y)) + log(z)) - t;
}

Julia

function code(x, y, z, t, a)
	return Float64(Float64(fma(log(t), Float64(-0.5 + a), log(y)) + log(z)) - t)
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[(N[Log[t], $MachinePrecision] * N[(-0.5 + a), $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]

2. Taylor expanded in a around 0

   \[\leadsto \color{blue}{\left(\log z + \left(\log \left(x + y\right) + \left(\frac{-1}{2} \cdot \log t + a \cdot \log t\right)\right)\right) - t} \]
3. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto \left(\log z + \left(\log \left(x + y\right) + \left(\frac{-1}{2} \cdot \log t + a \cdot \log t\right)\right)\right) - \color{blue}{t} \]

   2. +-commutative N/A

      \[\leadsto \left(\left(\log \left(x + y\right) + \left(\frac{-1}{2} \cdot \log t + a \cdot \log t\right)\right) + \log z\right) - t \]

   3. lower-+.f64 N/A

      \[\leadsto \left(\left(\log \left(x + y\right) + \left(\frac{-1}{2} \cdot \log t + a \cdot \log t\right)\right) + \log z\right) - t \]

   4. +-commutative N/A

      \[\leadsto \left(\left(\left(\frac{-1}{2} \cdot \log t + a \cdot \log t\right) + \log \left(x + y\right)\right) + \log z\right) - t \]

   5. distribute-rgt-out N/A

      \[\leadsto \left(\left(\log t \cdot \left(\frac{-1}{2} + a\right) + \log \left(x + y\right)\right) + \log z\right) - t \]

   6. lower-fma.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\log t, \frac{-1}{2} + a, \log \left(x + y\right)\right) + \log z\right) - t \]

   7. lift-log.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\log t, \frac{-1}{2} + a, \log \left(x + y\right)\right) + \log z\right) - t \]

   8. lower-+.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\log t, \frac{-1}{2} + a, \log \left(x + y\right)\right) + \log z\right) - t \]

   9. lift-log.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\log t, \frac{-1}{2} + a, \log \left(x + y\right)\right) + \log z\right) - t \]

   10. +-commutative N/A

       \[\leadsto \left(\mathsf{fma}\left(\log t, \frac{-1}{2} + a, \log \left(y + x\right)\right) + \log z\right) - t \]

   11. lower-+.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\log t, \frac{-1}{2} + a, \log \left(y + x\right)\right) + \log z\right) - t \]

   12. lift-log.f64 99.6

       \[\leadsto \left(\mathsf{fma}\left(\log t, -0.5 + a, \log \left(y + x\right)\right) + \log z\right) - t \]

4. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\log t, -0.5 + a, \log \left(y + x\right)\right) + \log z\right) - t} \]

5. Taylor expanded in x around 0

   \[\leadsto \left(\mathsf{fma}\left(\log t, \frac{-1}{2} + a, \log y\right) + \log z\right) - t \]
6. Step-by-step derivation

7. Applied rewrites 69.0%

   \[\leadsto \left(\mathsf{fma}\left(\log t, -0.5 + a, \log y\right) + \log z\right) - t \]
8. Add Preprocessing

Alternative 17: 76.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \left(-t\right) + \left(a - 0.5\right) \cdot \log t \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (+ (- t) (* (- a 0.5) (log t))))

C

double code(double x, double y, double z, double t, double a) {
	return -t + ((a - 0.5) * log(t));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = -t + ((a - 0.5d0) * log(t))
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return -t + ((a - 0.5) * Math.log(t));
}

Python

def code(x, y, z, t, a):
	return -t + ((a - 0.5) * math.log(t))

Julia

function code(x, y, z, t, a)
	return Float64(Float64(-t) + Float64(Float64(a - 0.5) * log(t)))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = -t + ((a - 0.5) * log(t));
end

Wolfram

code[x_, y_, z_, t_, a_] := N[((-t) + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]

2. Taylor expanded in t around inf

   \[\leadsto \color{blue}{-1 \cdot t} + \left(a - \frac{1}{2}\right) \cdot \log t \]
3. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \left(\mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

   2. lower-neg.f64 76.6

      \[\leadsto \left(-t\right) + \left(a - 0.5\right) \cdot \log t \]

4. Applied rewrites 76.6%

   \[\leadsto \color{blue}{\left(-t\right)} + \left(a - 0.5\right) \cdot \log t \]
5. Add Preprocessing

Alternative 18: 60.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 8 \cdot 10^{+64}:\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (if (<= t 8e+64) (* (log t) a) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 8e+64) {
		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 <= 8d+64) 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 <= 8e+64) {
		tmp = Math.log(t) * a;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 8e+64:
		tmp = math.log(t) * a
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 8e+64)
		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 <= 8e+64)
		tmp = log(t) * a;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 8e+64], N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if t < 8.00000000000000017e64

   1. Initial program 99.4%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \log t} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \log t \cdot \color{blue}{a} \]

      2. lower-*.f64 N/A

         \[\leadsto \log t \cdot \color{blue}{a} \]

      3. lift-log.f64 49.3

         \[\leadsto \log t \cdot a \]

   4. Applied rewrites 49.3%

      \[\leadsto \color{blue}{\log t \cdot a} \]

   if 8.00000000000000017e64 < t

   1. Initial program 99.9%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]

   2. Taylor expanded in t around inf

      \[\leadsto \color{blue}{-1 \cdot t} \]
   3. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(t\right) \]

      2. lower-neg.f64 78.9

         \[\leadsto -t \]

   4. Applied rewrites 78.9%

      \[\leadsto \color{blue}{-t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 74.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \log t \cdot a - t \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (- (* (log t) a) t))

C

double code(double x, double y, double z, double t, double a) {
	return (log(t) * a) - t;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = (log(t) * a) - t
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return (Math.log(t) * a) - t;
}

Python

def code(x, y, z, t, a):
	return (math.log(t) * a) - t

Julia

function code(x, y, z, t, a)
	return Float64(Float64(log(t) * a) - t)
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = (log(t) * a) - t;
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]

2. Taylor expanded in z around inf

   \[\leadsto \color{blue}{\left(\log \left(x + y\right) + \left(-1 \cdot \log \left(\frac{1}{z}\right) + \log t \cdot \left(a - \frac{1}{2}\right)\right)\right) - t} \]
3. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto \left(\log \left(x + y\right) + \left(-1 \cdot \log \left(\frac{1}{z}\right) + \log t \cdot \left(a - \frac{1}{2}\right)\right)\right) - \color{blue}{t} \]

4. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\log t, a - 0.5, -\left(-\log z\right)\right) + \log \left(y + x\right)\right) - t} \]

5. Taylor expanded in a around inf

   \[\leadsto a \cdot \log t - t \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \log t \cdot a - t \]

   2. lower-*.f64 N/A

      \[\leadsto \log t \cdot a - t \]

   3. lift-log.f64 74.0

      \[\leadsto \log t \cdot a - t \]

7. Applied rewrites 74.0%

   \[\leadsto \log t \cdot a - t \]
8. Add Preprocessing

Alternative 20: 37.3% 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. Taylor expanded in t around inf

   \[\leadsto \color{blue}{-1 \cdot t} \]
3. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(t\right) \]

   2. lower-neg.f64 37.3

      \[\leadsto -t \]

4. Applied rewrites 37.3%

   \[\leadsto \color{blue}{-t} \]
5. 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 2025095 
(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))))