Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 9.5s

Alternatives: 11

Speedup: 1.1×

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, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

Alternative 3: 99.6% accurate, 1.1× 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 4: 92.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -8.2:\\ \;\;\;\;\left(-t\right) + \left(a - 0.5\right) \cdot \log t\\ \mathbf{elif}\;a \leq 1.75:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.5, \log t, \log z\right) + \log \left(y + x\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \log t + \log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -8.2)
   (+ (- t) (* (- a 0.5) (log t)))
   (if (<= a 1.75)
     (- (+ (fma -0.5 (log t) (log z)) (log (+ y x))) t)
     (- (+ (* a (log t)) (log y)) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -8.2) {
		tmp = -t + ((a - 0.5) * log(t));
	} else if (a <= 1.75) {
		tmp = (fma(-0.5, log(t), log(z)) + log((y + x))) - t;
	} else {
		tmp = ((a * log(t)) + log(y)) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -8.2)
		tmp = Float64(Float64(-t) + Float64(Float64(a - 0.5) * log(t)));
	elseif (a <= 1.75)
		tmp = Float64(Float64(fma(-0.5, log(t), log(z)) + log(Float64(y + x))) - t);
	else
		tmp = Float64(Float64(Float64(a * log(t)) + log(y)) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -8.2], N[((-t) + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.75], N[(N[(N[(-0.5 * N[Log[t], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] + N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -8.1999999999999993

   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 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 98.9

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

   4. Applied rewrites 98.9%

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

   if -8.1999999999999993 < a < 1.75

   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 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 99.5%

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

   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. lower-*.f64 N/A

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

      2. lift-log.f64 56.7

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

   7. Applied rewrites 56.7%

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

   8. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-log.f64 98.3

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

   10. Applied rewrites 98.3%

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

   if 1.75 < a

   1. Initial program 99.7%

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

   2. 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 99.7%

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

   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. lower-*.f64 N/A

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

      2. lift-log.f64 98.3

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

   7. Applied rewrites 98.3%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative 76.1

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

   10. Applied rewrites 76.1%

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

Alternative 5: 83.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -8.2:\\ \;\;\;\;\left(-t\right) + \left(a - 0.5\right) \cdot \log t\\ \mathbf{elif}\;a \leq 1.75:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.5, \log t, \log z\right) + \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \log t + \log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -8.2)
   (+ (- t) (* (- a 0.5) (log t)))
   (if (<= a 1.75)
     (- (+ (fma -0.5 (log t) (log z)) (log y)) t)
     (- (+ (* a (log t)) (log y)) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -8.2) {
		tmp = -t + ((a - 0.5) * log(t));
	} else if (a <= 1.75) {
		tmp = (fma(-0.5, log(t), log(z)) + log(y)) - t;
	} else {
		tmp = ((a * log(t)) + log(y)) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -8.2)
		tmp = Float64(Float64(-t) + Float64(Float64(a - 0.5) * log(t)));
	elseif (a <= 1.75)
		tmp = Float64(Float64(fma(-0.5, log(t), log(z)) + log(y)) - t);
	else
		tmp = Float64(Float64(Float64(a * log(t)) + log(y)) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -8.2], N[((-t) + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.75], N[(N[(N[(-0.5 * N[Log[t], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -8.1999999999999993

   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 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 98.9

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

   4. Applied rewrites 98.9%

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

   if -8.1999999999999993 < a < 1.75

   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 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 99.5%

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

   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. lower-*.f64 N/A

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

      2. lift-log.f64 56.7

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

   7. Applied rewrites 56.7%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative 41.9

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

   10. Applied rewrites 41.9%

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

   11. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. lift-log.f64 N/A

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

       4. lift-log.f64 63.1

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

   13. Applied rewrites 63.1%

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

   if 1.75 < a

   1. Initial program 99.7%

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

   2. 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 99.7%

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

   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. lower-*.f64 N/A

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

      2. lift-log.f64 98.3

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

   7. Applied rewrites 98.3%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative 76.1

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

   10. Applied rewrites 76.1%

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

Alternative 6: 75.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ t_2 := \left(a \cdot \log t + \log y\right) - t\\ \mathbf{if}\;t\_1 \leq -500:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 1020:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log \left(z \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 (- (+ (* a (log t)) (log y)) t)))
   (if (<= t_1 -500.0)
     t_2
     (if (<= t_1 1020.0) (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 = ((a * log(t)) + log(y)) - t;
	double tmp;
	if (t_1 <= -500.0) {
		tmp = t_2;
	} else if (t_1 <= 1020.0) {
		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(a * log(t)) + log(y)) - t)
	tmp = 0.0
	if (t_1 <= -500.0)
		tmp = t_2;
	elseif (t_1 <= 1020.0)
		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[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[t$95$1, -500.0], t$95$2, If[LessEqual[t$95$1, 1020.0], 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))) < -500 or 1020 < (+.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 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 99.8%

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

   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. lower-*.f64 N/A

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

      2. lift-log.f64 93.6

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

   7. Applied rewrites 93.6%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative 70.9

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

   10. Applied rewrites 70.9%

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

   if -500 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 1020

   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 44.1

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

   4. Applied rewrites 44.1%

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

   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. lift-pow.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-log.f64 42.9

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

   7. Applied rewrites 42.9%

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

      1. lift-log.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. log-prod N/A

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

      8. log-pow-rev N/A

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

      9. sum-log N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift-log.f64 N/A

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

      13. +-commutative N/A

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

      14. log-prod N/A

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

      15. lift-log.f64 N/A

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

      16. lift-*.f64 47.2

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

   9. Applied rewrites 47.2%

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

Alternative 7: 74.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ t_2 := \left(a \cdot \log t + \log y\right) - t\\ \mathbf{if}\;t\_1 \leq -500:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 1020:\\ \;\;\;\;\mathsf{fma}\left(0.5, -\log t, \log \left(z \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 (- (+ (* a (log t)) (log y)) t)))
   (if (<= t_1 -500.0)
     t_2
     (if (<= t_1 1020.0) (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 = ((a * log(t)) + log(y)) - t;
	double tmp;
	if (t_1 <= -500.0) {
		tmp = t_2;
	} else if (t_1 <= 1020.0) {
		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(a * log(t)) + log(y)) - t)
	tmp = 0.0
	if (t_1 <= -500.0)
		tmp = t_2;
	elseif (t_1 <= 1020.0)
		tmp = fma(0.5, Float64(-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[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[t$95$1, -500.0], t$95$2, If[LessEqual[t$95$1, 1020.0], 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))) < -500 or 1020 < (+.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 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 99.8%

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

   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. lower-*.f64 N/A

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

      2. lift-log.f64 93.6

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

   7. Applied rewrites 93.6%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative 70.9

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

   10. Applied rewrites 70.9%

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

   if -500 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 1020

   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 44.1

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

   4. Applied rewrites 44.1%

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

   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. lift-pow.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-log.f64 42.9

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

   7. Applied rewrites 42.9%

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

   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. log-pow N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. sum-log N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. +-commutative N/A

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

      12. log-prod N/A

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

      13. lift-log.f64 N/A

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

      14. lift-*.f64 46.1

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

   10. Applied rewrites 46.1%

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

Alternative 8: 66.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ t_2 := a \cdot \log t - t\\ \mathbf{if}\;t\_1 \leq -700:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2000:\\ \;\;\;\;\mathsf{fma}\left(0.5, -\log t, \log \left(z \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 (- (* a (log t)) t)))
   (if (<= t_1 -700.0)
     t_2
     (if (<= t_1 2000.0) (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 = (a * log(t)) - t;
	double tmp;
	if (t_1 <= -700.0) {
		tmp = t_2;
	} else if (t_1 <= 2000.0) {
		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(a * log(t)) - t)
	tmp = 0.0
	if (t_1 <= -700.0)
		tmp = t_2;
	elseif (t_1 <= 2000.0)
		tmp = fma(0.5, Float64(-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[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[t$95$1, -700.0], t$95$2, If[LessEqual[t$95$1, 2000.0], 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))) < -700 or 2e3 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t)))

   1. Initial program 99.8%

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

   2. 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 99.8%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lift-log.f64 97.7

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

   7. Applied rewrites 97.7%

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

   if -700 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 2e3

   1. Initial program 99.0%

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

   2. 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 38.5

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

   4. Applied rewrites 38.5%

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

   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. lift-pow.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-log.f64 37.4

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

   7. Applied rewrites 37.4%

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

   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. log-pow N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. sum-log N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. +-commutative N/A

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

      12. log-prod N/A

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

      13. lift-log.f64 N/A

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

      14. lift-*.f64 40.2

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

   10. Applied rewrites 40.2%

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

Alternative 9: 65.8% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(a * N[Log[t], $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 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. lower-*.f64 N/A

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

   2. lift-log.f64 74.6

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

7. Applied rewrites 74.6%

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

Alternative 10: 61.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.4 \cdot 10^{+47}:\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 1.4e+47) (* (log t) a) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 1.4e+47) {
		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 <= 1.4d+47) 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 <= 1.4e+47) {
		tmp = Math.log(t) * a;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 1.4e+47:
		tmp = math.log(t) * a
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 1.4e+47)
		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 <= 1.4e+47)
		tmp = log(t) * a;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 1.4e+47], N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if t < 1.39999999999999994e47

   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 50.1

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

   4. Applied rewrites 50.1%

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

   if 1.39999999999999994e47 < 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 78.0

         \[\leadsto -t \]

   4. Applied rewrites 78.0%

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

Alternative 11: 38.0% accurate, 8.0× 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 38.0

      \[\leadsto -t \]

4. Applied rewrites 38.0%

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

Reproduce

herbie shell --seed 2025101 
(FPCore (x y z t a)
  :name "Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2"
  :precision binary64
  (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))