Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 36.7s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a)
    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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

Alternative 2: 94.3% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log (+ x y)) (log z)))
        (t_2 (+ (* (- a 0.5) (log t)) (- (log z) t))))
   (if (<= t_1 -720.0)
     t_2
     (if (<= t_1 720.0)
       (-
        (+ (/ (log (* (/ 1.0 (+ x y)) (/ 1.0 z))) -1.0) (* (log t) (+ a -0.5)))
        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 = ((a - 0.5) * log(t)) + (log(z) - t);
	double tmp;
	if (t_1 <= -720.0) {
		tmp = t_2;
	} else if (t_1 <= 720.0) {
		tmp = ((log(((1.0 / (x + y)) * (1.0 / z))) / -1.0) + (log(t) * (a + -0.5))) - t;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    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 = ((a - 0.5d0) * log(t)) + (log(z) - t)
    if (t_1 <= (-720.0d0)) then
        tmp = t_2
    else if (t_1 <= 720.0d0) then
        tmp = ((log(((1.0d0 / (x + y)) * (1.0d0 / z))) / (-1.0d0)) + (log(t) * (a + (-0.5d0)))) - 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 = ((a - 0.5) * Math.log(t)) + (Math.log(z) - t);
	double tmp;
	if (t_1 <= -720.0) {
		tmp = t_2;
	} else if (t_1 <= 720.0) {
		tmp = ((Math.log(((1.0 / (x + y)) * (1.0 / z))) / -1.0) + (Math.log(t) * (a + -0.5))) - 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 = ((a - 0.5) * math.log(t)) + (math.log(z) - t)
	tmp = 0
	if t_1 <= -720.0:
		tmp = t_2
	elif t_1 <= 720.0:
		tmp = ((math.log(((1.0 / (x + y)) * (1.0 / z))) / -1.0) + (math.log(t) * (a + -0.5))) - 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(a - 0.5) * log(t)) + Float64(log(z) - t))
	tmp = 0.0
	if (t_1 <= -720.0)
		tmp = t_2;
	elseif (t_1 <= 720.0)
		tmp = Float64(Float64(Float64(log(Float64(Float64(1.0 / Float64(x + y)) * Float64(1.0 / z))) / -1.0) + Float64(log(t) * Float64(a + -0.5))) - 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 = ((a - 0.5) * log(t)) + (log(z) - t);
	tmp = 0.0;
	if (t_1 <= -720.0)
		tmp = t_2;
	elseif (t_1 <= 720.0)
		tmp = ((log(((1.0 / (x + y)) * (1.0 / z))) / -1.0) + (log(t) * (a + -0.5))) - 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[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision] + N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -720.0], t$95$2, If[LessEqual[t$95$1, 720.0], N[(N[(N[(N[Log[N[(N[(1.0 / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(1.0 / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / -1.0), $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. associate--l+ N/A

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

      5. +-commutative N/A

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

      6. sub-neg N/A

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

      7. associate-+l+ N/A

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

      8. lower-+.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-neg.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 84.2

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

   7. Applied rewrites 84.2%

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

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

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. associate--l+ N/A

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

      5. +-commutative N/A

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

      6. sub-neg N/A

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

      7. associate-+l+ N/A

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

      8. lower-+.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-neg.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. associate-+r+ N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. distribute-rgt-out N/A

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

      9. lower-*.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-+.f64 99.6

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in t around inf

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. log-rec N/A

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. sub-neg N/A

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

      16. metadata-eval N/A

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

      17. lower-+.f64 99.6

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

   10. Applied rewrites 99.6%

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

       1. lift-+.f64 N/A

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

       2. lift-log.f64 N/A

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

       3. lift-log.f64 N/A

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

       4. +-commutative N/A

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

       5. /-rgt-identity N/A

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

       6. frac-2neg N/A

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

       7. metadata-eval N/A

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

       8. /-rgt-identity N/A

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

       9. frac-add N/A

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

       10. metadata-eval N/A

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

       11. lower-/.f64 N/A

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

   12. Applied rewrites 99.6%

       \[\leadsto \left(\color{blue}{\frac{\log \left(\frac{1}{x + y} \cdot \frac{1}{z}\right)}{-1}} + \left(-\left(-\log t\right) \cdot \left(a + -0.5\right)\right)\right) - t \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(x + y\right) + \log z \leq -720:\\ \;\;\;\;\left(a - 0.5\right) \cdot \log t + \left(\log z - t\right)\\ \mathbf{elif}\;\log \left(x + y\right) + \log z \leq 720:\\ \;\;\;\;\left(\frac{\log \left(\frac{1}{x + y} \cdot \frac{1}{z}\right)}{-1} + \log t \cdot \left(a + -0.5\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(a - 0.5\right) \cdot \log t + \left(\log z - t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(x + y\right) + \log z\\ t_2 := \left(a - 0.5\right) \cdot \log t + \left(\log z - t\right)\\ \mathbf{if}\;t\_1 \leq -750:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 690:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, \log t, \log \left(\left(x + y\right) \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log (+ x y)) (log z)))
        (t_2 (+ (* (- a 0.5) (log t)) (- (log z) t))))
   (if (<= t_1 -750.0)
     t_2
     (if (<= t_1 690.0)
       (- (fma (+ a -0.5) (log t) (log (* (+ x y) z))) 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 = ((a - 0.5) * log(t)) + (log(z) - t);
	double tmp;
	if (t_1 <= -750.0) {
		tmp = t_2;
	} else if (t_1 <= 690.0) {
		tmp = fma((a + -0.5), log(t), log(((x + y) * z))) - 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(a - 0.5) * log(t)) + Float64(log(z) - t))
	tmp = 0.0
	if (t_1 <= -750.0)
		tmp = t_2;
	elseif (t_1 <= 690.0)
		tmp = Float64(fma(Float64(a + -0.5), log(t), log(Float64(Float64(x + y) * z))) - t);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision] + N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -750.0], t$95$2, If[LessEqual[t$95$1, 690.0], N[(N[(N[(a + -0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[Log[N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -750 or 690 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z))

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. associate--l+ N/A

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

      5. +-commutative N/A

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

      6. sub-neg N/A

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

      7. associate-+l+ N/A

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

      8. lower-+.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-neg.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 83.2

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

   7. Applied rewrites 83.2%

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

   if -750 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 690

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lift--.f64 N/A

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

      11. associate-+r- N/A

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

      12. lower--.f64 N/A

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

   4. Applied rewrites 99.7%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(x + y\right) + \log z \leq -750:\\ \;\;\;\;\left(a - 0.5\right) \cdot \log t + \left(\log z - t\right)\\ \mathbf{elif}\;\log \left(x + y\right) + \log z \leq 690:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, \log t, \log \left(\left(x + y\right) \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(a - 0.5\right) \cdot \log t + \left(\log z - t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(x + y\right) + \log z\\ t_2 := \left(a - 0.5\right) \cdot \log t + \left(\log z - t\right)\\ \mathbf{if}\;t\_1 \leq -750:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 690:\\ \;\;\;\;\log \left(y \cdot z\right) + \left(\log t \cdot \left(a + -0.5\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log (+ x y)) (log z)))
        (t_2 (+ (* (- a 0.5) (log t)) (- (log z) t))))
   (if (<= t_1 -750.0)
     t_2
     (if (<= t_1 690.0) (+ (log (* y z)) (- (* (log t) (+ a -0.5)) 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 = ((a - 0.5) * log(t)) + (log(z) - t);
	double tmp;
	if (t_1 <= -750.0) {
		tmp = t_2;
	} else if (t_1 <= 690.0) {
		tmp = log((y * z)) + ((log(t) * (a + -0.5)) - t);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    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 = ((a - 0.5d0) * log(t)) + (log(z) - t)
    if (t_1 <= (-750.0d0)) then
        tmp = t_2
    else if (t_1 <= 690.0d0) then
        tmp = log((y * z)) + ((log(t) * (a + (-0.5d0))) - 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 = ((a - 0.5) * Math.log(t)) + (Math.log(z) - t);
	double tmp;
	if (t_1 <= -750.0) {
		tmp = t_2;
	} else if (t_1 <= 690.0) {
		tmp = Math.log((y * z)) + ((Math.log(t) * (a + -0.5)) - 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 = ((a - 0.5) * math.log(t)) + (math.log(z) - t)
	tmp = 0
	if t_1 <= -750.0:
		tmp = t_2
	elif t_1 <= 690.0:
		tmp = math.log((y * z)) + ((math.log(t) * (a + -0.5)) - 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(a - 0.5) * log(t)) + Float64(log(z) - t))
	tmp = 0.0
	if (t_1 <= -750.0)
		tmp = t_2;
	elseif (t_1 <= 690.0)
		tmp = Float64(log(Float64(y * z)) + Float64(Float64(log(t) * Float64(a + -0.5)) - 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 = ((a - 0.5) * log(t)) + (log(z) - t);
	tmp = 0.0;
	if (t_1 <= -750.0)
		tmp = t_2;
	elseif (t_1 <= 690.0)
		tmp = log((y * z)) + ((log(t) * (a + -0.5)) - 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[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision] + N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -750.0], t$95$2, If[LessEqual[t$95$1, 690.0], N[(N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision] + N[(N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -750 or 690 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z))

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. associate--l+ N/A

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

      5. +-commutative N/A

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

      6. sub-neg N/A

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

      7. associate-+l+ N/A

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

      8. lower-+.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-neg.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 83.2

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

   7. Applied rewrites 83.2%

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

   if -750 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 690

   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

   4. Applied rewrites 65.9%

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

   5. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-+.f64 71.3

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

   7. Applied rewrites 71.3%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(x + y\right) + \log z \leq -750:\\ \;\;\;\;\left(a - 0.5\right) \cdot \log t + \left(\log z - t\right)\\ \mathbf{elif}\;\log \left(x + y\right) + \log z \leq 690:\\ \;\;\;\;\log \left(y \cdot z\right) + \left(\log t \cdot \left(a + -0.5\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a - 0.5\right) \cdot \log t + \left(\log z - t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. lift-+.f64 N/A

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

   2. lift-log.f64 N/A

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

   3. lift-log.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift--.f64 N/A

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

   7. lift-log.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. +-commutative N/A

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

   10. lift--.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. associate--l+ N/A

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

   13. associate-+r+ N/A

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

   14. lower-+.f64 N/A

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

4. Applied rewrites 99.6%

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

Alternative 6: 68.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in x around 0

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

   1. associate--l+ N/A

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

   2. lower-+.f64 N/A

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

   3. lower-log.f64 N/A

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

   4. +-commutative N/A

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

   5. associate--l+ N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-log.f64 N/A

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

   8. sub-neg N/A

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

   9. metadata-eval N/A

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

   10. lower-+.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lower-log.f64 74.3

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

5. Applied rewrites 74.3%

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

Alternative 7: 77.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a)
    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 - 0.5d0) * log(t)) + (log(z) - t)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. lift-+.f64 N/A

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

   2. lift-log.f64 N/A

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

   3. lift-log.f64 N/A

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

   4. associate--l+ N/A

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

   5. +-commutative N/A

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

   6. sub-neg N/A

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

   7. associate-+l+ N/A

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

   8. lower-+.f64 N/A

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

   9. lower-+.f64 N/A

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

   10. lower-neg.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 78.8

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

7. Applied rewrites 78.8%

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

8. Final simplification 78.8%

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

Alternative 8: 61.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 9.8 \cdot 10^{+58}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 9.8e+58) (* a (log t)) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 9.8e+58) {
		tmp = a * log(t);
	} else {
		tmp = -t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    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 <= 9.8d+58) then
        tmp = a * log(t)
    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 <= 9.8e+58) {
		tmp = a * Math.log(t);
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 9.8e+58:
		tmp = a * math.log(t)
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 9.8e+58)
		tmp = Float64(a * log(t));
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 9.8e+58)
		tmp = a * log(t);
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 9.8e+58], N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if t < 9.80000000000000037e58

   1. Initial program 99.4%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 54.0

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

   5. Applied rewrites 54.0%

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

   if 9.80000000000000037e58 < t

   1. Initial program 99.9%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 81.5

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

   5. Applied rewrites 81.5%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9.8 \cdot 10^{+58}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.6% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a)
    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 - 0.5d0) * log(t)) - t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[(a - 0.5), $MachinePrecision] * 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. Add Preprocessing

3. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 78.4

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

5. Applied rewrites 78.4%

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

6. Final simplification 78.4%

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

Alternative 10: 74.9% accurate, 2.9× 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

real(8) function code(x, y, z, t, a)
    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. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-log.f64 N/A

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

   3. lift-log.f64 N/A

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

   4. associate--l+ N/A

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

   5. +-commutative N/A

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

   6. sub-neg N/A

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

   7. associate-+l+ N/A

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

   8. lower-+.f64 N/A

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

   9. lower-+.f64 N/A

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

   10. lower-neg.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in a around 0

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

   1. lower--.f64 N/A

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

   2. associate-+r+ N/A

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

   3. lower-+.f64 N/A

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

   4. lower-+.f64 N/A

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

   5. lower-log.f64 N/A

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

   6. lower-log.f64 N/A

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

   7. lower-+.f64 N/A

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

   8. distribute-rgt-out N/A

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

   9. lower-*.f64 N/A

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

   10. lower-log.f64 N/A

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

   11. lower-+.f64 99.7

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

7. Applied rewrites 99.7%

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

8. Taylor expanded in a around inf

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

   1. lower-*.f64 N/A

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

   2. lower-log.f64 75.8

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

10. Applied rewrites 75.8%

    \[\leadsto \color{blue}{a \cdot \log t} - t \]
11. Add Preprocessing

Alternative 11: 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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := (-t)

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 38.7

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

5. Applied rewrites 38.7%

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

Developer Target 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a)
    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 2024220 
(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))))