Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 15.7s

Alternatives: 13

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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Final simplification 99.7%

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

Alternative 2: 89.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log t \cdot a\\ t_2 := \log z + \log \left(y + x\right)\\ \mathbf{if}\;t\_2 \leq -750:\\ \;\;\;\;\frac{1}{\frac{1}{\left(\log z - t\right) + t\_1}}\\ \mathbf{elif}\;t\_2 \leq 655:\\ \;\;\;\;\log \left(z \cdot \left(y + x\right)\right) - \left(t - \log t \cdot \left(a - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 + \log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (log t) a)) (t_2 (+ (log z) (log (+ y x)))))
   (if (<= t_2 -750.0)
     (/ 1.0 (/ 1.0 (+ (- (log z) t) t_1)))
     (if (<= t_2 655.0)
       (- (log (* z (+ y x))) (- t (* (log t) (- a 0.5))))
       (- (+ t_1 (log y)) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log(t) * a;
	double t_2 = log(z) + log((y + x));
	double tmp;
	if (t_2 <= -750.0) {
		tmp = 1.0 / (1.0 / ((log(z) - t) + t_1));
	} else if (t_2 <= 655.0) {
		tmp = log((z * (y + x))) - (t - (log(t) * (a - 0.5)));
	} else {
		tmp = (t_1 + log(y)) - 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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = log(t) * a
    t_2 = log(z) + log((y + x))
    if (t_2 <= (-750.0d0)) then
        tmp = 1.0d0 / (1.0d0 / ((log(z) - t) + t_1))
    else if (t_2 <= 655.0d0) then
        tmp = log((z * (y + x))) - (t - (log(t) * (a - 0.5d0)))
    else
        tmp = (t_1 + log(y)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = Math.log(t) * a;
	double t_2 = Math.log(z) + Math.log((y + x));
	double tmp;
	if (t_2 <= -750.0) {
		tmp = 1.0 / (1.0 / ((Math.log(z) - t) + t_1));
	} else if (t_2 <= 655.0) {
		tmp = Math.log((z * (y + x))) - (t - (Math.log(t) * (a - 0.5)));
	} else {
		tmp = (t_1 + Math.log(y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = math.log(t) * a
	t_2 = math.log(z) + math.log((y + x))
	tmp = 0
	if t_2 <= -750.0:
		tmp = 1.0 / (1.0 / ((math.log(z) - t) + t_1))
	elif t_2 <= 655.0:
		tmp = math.log((z * (y + x))) - (t - (math.log(t) * (a - 0.5)))
	else:
		tmp = (t_1 + math.log(y)) - t
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(t) * a)
	t_2 = Float64(log(z) + log(Float64(y + x)))
	tmp = 0.0
	if (t_2 <= -750.0)
		tmp = Float64(1.0 / Float64(1.0 / Float64(Float64(log(z) - t) + t_1)));
	elseif (t_2 <= 655.0)
		tmp = Float64(log(Float64(z * Float64(y + x))) - Float64(t - Float64(log(t) * Float64(a - 0.5))));
	else
		tmp = Float64(Float64(t_1 + log(y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = log(t) * a;
	t_2 = log(z) + log((y + x));
	tmp = 0.0;
	if (t_2 <= -750.0)
		tmp = 1.0 / (1.0 / ((log(z) - t) + t_1));
	elseif (t_2 <= 655.0)
		tmp = log((z * (y + x))) - (t - (log(t) * (a - 0.5)));
	else
		tmp = (t_1 + log(y)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]}, Block[{t$95$2 = N[(N[Log[z], $MachinePrecision] + N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -750.0], N[(1.0 / N[(1.0 / N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 655.0], N[(N[Log[N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(t - N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 + N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   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

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

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

   4. Applied rewrites 2.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. log-prod N/A

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

      7. lift-log.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. +-commutative N/A

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

      13. associate--l+ N/A

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

      14. associate-+r+ N/A

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

      15. lower-+.f64 N/A

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 86.1

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

   9. Applied rewrites 86.1%

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

   if -750 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 655

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. sum-log N/A

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

      9. lower-log.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 99.7

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 99.7

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

   4. Applied rewrites 99.7%

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

   if 655 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z))

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. flip3-- N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. clear-num N/A

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

      9. flip3-- N/A

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

      10. lift--.f64 N/A

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

      11. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-log.f64 62.9

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

   7. Applied rewrites 62.9%

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

   8. Taylor expanded in a around inf

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

   10. Applied rewrites 56.9%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log z + \log \left(y + x\right) \leq -750:\\ \;\;\;\;\frac{1}{\frac{1}{\left(\log z - t\right) + \log t \cdot a}}\\ \mathbf{elif}\;\log z + \log \left(y + x\right) \leq 655:\\ \;\;\;\;\log \left(z \cdot \left(y + x\right)\right) - \left(t - \log t \cdot \left(a - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log t \cdot a + \log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log z + \log \left(y + x\right)\\ t_2 := \left(\log t \cdot a + \log y\right) - t\\ \mathbf{if}\;t\_1 \leq -750:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 655:\\ \;\;\;\;\log \left(z \cdot \left(y + x\right)\right) - \left(t - \log t \cdot \left(a - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log z) (log (+ y x)))) (t_2 (- (+ (* (log t) a) (log y)) t)))
   (if (<= t_1 -750.0)
     t_2
     (if (<= t_1 655.0)
       (- (log (* z (+ y x))) (- t (* (log t) (- a 0.5))))
       t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log(z) + log((y + x));
	double t_2 = ((log(t) * a) + log(y)) - t;
	double tmp;
	if (t_1 <= -750.0) {
		tmp = t_2;
	} else if (t_1 <= 655.0) {
		tmp = log((z * (y + x))) - (t - (log(t) * (a - 0.5)));
	} 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(z) + log((y + x))
    t_2 = ((log(t) * a) + log(y)) - t
    if (t_1 <= (-750.0d0)) then
        tmp = t_2
    else if (t_1 <= 655.0d0) then
        tmp = log((z * (y + x))) - (t - (log(t) * (a - 0.5d0)))
    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(z) + Math.log((y + x));
	double t_2 = ((Math.log(t) * a) + Math.log(y)) - t;
	double tmp;
	if (t_1 <= -750.0) {
		tmp = t_2;
	} else if (t_1 <= 655.0) {
		tmp = Math.log((z * (y + x))) - (t - (Math.log(t) * (a - 0.5)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = math.log(z) + math.log((y + x))
	t_2 = ((math.log(t) * a) + math.log(y)) - t
	tmp = 0
	if t_1 <= -750.0:
		tmp = t_2
	elif t_1 <= 655.0:
		tmp = math.log((z * (y + x))) - (t - (math.log(t) * (a - 0.5)))
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(z) + log(Float64(y + x)))
	t_2 = Float64(Float64(Float64(log(t) * a) + log(y)) - t)
	tmp = 0.0
	if (t_1 <= -750.0)
		tmp = t_2;
	elseif (t_1 <= 655.0)
		tmp = Float64(log(Float64(z * Float64(y + x))) - Float64(t - Float64(log(t) * Float64(a - 0.5))));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = log(z) + log((y + x));
	t_2 = ((log(t) * a) + log(y)) - t;
	tmp = 0.0;
	if (t_1 <= -750.0)
		tmp = t_2;
	elseif (t_1 <= 655.0)
		tmp = log((z * (y + x))) - (t - (log(t) * (a - 0.5)));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[z], $MachinePrecision] + N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[t$95$1, -750.0], t$95$2, If[LessEqual[t$95$1, 655.0], N[(N[Log[N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(t - N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $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 655 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z))

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. flip3-- N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. clear-num N/A

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

      9. flip3-- N/A

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

      10. lift--.f64 N/A

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

      11. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-log.f64 63.0

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

   7. Applied rewrites 63.0%

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

   8. Taylor expanded in a around inf

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

   10. Applied rewrites 56.8%

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

   if -750 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 655

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. sum-log N/A

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

      9. lower-log.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 99.7

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 99.7

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

   4. Applied rewrites 99.7%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log z + \log \left(y + x\right) \leq -750:\\ \;\;\;\;\left(\log t \cdot a + \log y\right) - t\\ \mathbf{elif}\;\log z + \log \left(y + x\right) \leq 655:\\ \;\;\;\;\log \left(z \cdot \left(y + x\right)\right) - \left(t - \log t \cdot \left(a - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log t \cdot a + \log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log z + \log \left(y + x\right)\\ t_2 := \left(\log t \cdot a + \log y\right) - t\\ \mathbf{if}\;t\_1 \leq -750:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 655:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log \left(z \cdot \left(y + x\right)\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log z) (log (+ y x)))) (t_2 (- (+ (* (log t) a) (log y)) t)))
   (if (<= t_1 -750.0)
     t_2
     (if (<= t_1 655.0)
       (fma (- a 0.5) (log t) (- (log (* z (+ y x))) t))
       t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log(z) + log((y + x));
	double t_2 = ((log(t) * a) + log(y)) - t;
	double tmp;
	if (t_1 <= -750.0) {
		tmp = t_2;
	} else if (t_1 <= 655.0) {
		tmp = fma((a - 0.5), log(t), (log((z * (y + x))) - t));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(z) + log(Float64(y + x)))
	t_2 = Float64(Float64(Float64(log(t) * a) + log(y)) - t)
	tmp = 0.0
	if (t_1 <= -750.0)
		tmp = t_2;
	elseif (t_1 <= 655.0)
		tmp = fma(Float64(a - 0.5), log(t), Float64(log(Float64(z * Float64(y + x))) - t));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[z], $MachinePrecision] + N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[t$95$1, -750.0], t$95$2, If[LessEqual[t$95$1, 655.0], N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[(N[Log[N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -750 or 655 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z))

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. flip3-- N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. clear-num N/A

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

      9. flip3-- N/A

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

      10. lift--.f64 N/A

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

      11. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-log.f64 63.0

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

   7. Applied rewrites 63.0%

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

   8. Taylor expanded in a around inf

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

   10. Applied rewrites 56.8%

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

   if -750 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 655

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.6

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

      5. lift-+.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. sum-log N/A

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

      9. lower-log.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.7

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 99.7

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

   4. Applied rewrites 99.7%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log z + \log \left(y + x\right) \leq -750:\\ \;\;\;\;\left(\log t \cdot a + \log y\right) - t\\ \mathbf{elif}\;\log z + \log \left(y + x\right) \leq 655:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log \left(z \cdot \left(y + x\right)\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log t \cdot a + \log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log z + \log \left(y + x\right)\\ t_2 := \left(\log t \cdot a + \log y\right) - t\\ \mathbf{if}\;t\_1 \leq -750:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 655:\\ \;\;\;\;\log \left(z \cdot y\right) - \left(t - \log t \cdot \left(a - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log z) (log (+ y x)))) (t_2 (- (+ (* (log t) a) (log y)) t)))
   (if (<= t_1 -750.0)
     t_2
     (if (<= t_1 655.0) (- (log (* z y)) (- t (* (log t) (- a 0.5)))) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log(z) + log((y + x));
	double t_2 = ((log(t) * a) + log(y)) - t;
	double tmp;
	if (t_1 <= -750.0) {
		tmp = t_2;
	} else if (t_1 <= 655.0) {
		tmp = log((z * y)) - (t - (log(t) * (a - 0.5)));
	} 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(z) + log((y + x))
    t_2 = ((log(t) * a) + log(y)) - t
    if (t_1 <= (-750.0d0)) then
        tmp = t_2
    else if (t_1 <= 655.0d0) then
        tmp = log((z * y)) - (t - (log(t) * (a - 0.5d0)))
    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(z) + Math.log((y + x));
	double t_2 = ((Math.log(t) * a) + Math.log(y)) - t;
	double tmp;
	if (t_1 <= -750.0) {
		tmp = t_2;
	} else if (t_1 <= 655.0) {
		tmp = Math.log((z * y)) - (t - (Math.log(t) * (a - 0.5)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = math.log(z) + math.log((y + x))
	t_2 = ((math.log(t) * a) + math.log(y)) - t
	tmp = 0
	if t_1 <= -750.0:
		tmp = t_2
	elif t_1 <= 655.0:
		tmp = math.log((z * y)) - (t - (math.log(t) * (a - 0.5)))
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(z) + log(Float64(y + x)))
	t_2 = Float64(Float64(Float64(log(t) * a) + log(y)) - t)
	tmp = 0.0
	if (t_1 <= -750.0)
		tmp = t_2;
	elseif (t_1 <= 655.0)
		tmp = Float64(log(Float64(z * y)) - Float64(t - Float64(log(t) * Float64(a - 0.5))));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = log(z) + log((y + x));
	t_2 = ((log(t) * a) + log(y)) - t;
	tmp = 0.0;
	if (t_1 <= -750.0)
		tmp = t_2;
	elseif (t_1 <= 655.0)
		tmp = log((z * y)) - (t - (log(t) * (a - 0.5)));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[z], $MachinePrecision] + N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[t$95$1, -750.0], t$95$2, If[LessEqual[t$95$1, 655.0], N[(N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision] - N[(t - N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $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 655 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z))

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. flip3-- N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. clear-num N/A

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

      9. flip3-- N/A

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

      10. lift--.f64 N/A

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

      11. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-log.f64 63.0

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

   7. Applied rewrites 63.0%

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

   8. Taylor expanded in a around inf

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

   10. Applied rewrites 56.8%

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

   if -750 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 655

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. log-rec N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-log.f64 68.7

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

   7. Applied rewrites 68.7%

      \[\leadsto \frac{1}{\frac{1}{\mathsf{fma}\left(\log t, a - 0.5, \color{blue}{\left(\log z - \left(-\log y\right)\right)} - t\right)}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. remove-double-div 68.8

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lift--.f64 N/A

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

      7. associate-+l- N/A

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

      8. *-commutative N/A

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

      9. lower--.f64 N/A

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

   9. Applied rewrites 67.0%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log z + \log \left(y + x\right) \leq -750:\\ \;\;\;\;\left(\log t \cdot a + \log y\right) - t\\ \mathbf{elif}\;\log z + \log \left(y + x\right) \leq 655:\\ \;\;\;\;\log \left(z \cdot y\right) - \left(t - \log t \cdot \left(a - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log t \cdot a + \log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log z + \log \left(y + x\right)\\ t_2 := \left(\log t \cdot a + \log y\right) - t\\ \mathbf{if}\;t\_1 \leq -750:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 655:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log \left(z \cdot y\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log z) (log (+ y x)))) (t_2 (- (+ (* (log t) a) (log y)) t)))
   (if (<= t_1 -750.0)
     t_2
     (if (<= t_1 655.0) (- (fma (- a 0.5) (log t) (log (* z y))) t) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log(z) + log((y + x));
	double t_2 = ((log(t) * a) + log(y)) - t;
	double tmp;
	if (t_1 <= -750.0) {
		tmp = t_2;
	} else if (t_1 <= 655.0) {
		tmp = fma((a - 0.5), log(t), log((z * y))) - t;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(z) + log(Float64(y + x)))
	t_2 = Float64(Float64(Float64(log(t) * a) + log(y)) - t)
	tmp = 0.0
	if (t_1 <= -750.0)
		tmp = t_2;
	elseif (t_1 <= 655.0)
		tmp = Float64(fma(Float64(a - 0.5), log(t), log(Float64(z * y))) - t);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[z], $MachinePrecision] + N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[t$95$1, -750.0], t$95$2, If[LessEqual[t$95$1, 655.0], N[(N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[Log[N[(z * y), $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 655 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z))

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. flip3-- N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. clear-num N/A

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

      9. flip3-- N/A

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

      10. lift--.f64 N/A

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

      11. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-log.f64 63.0

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

   7. Applied rewrites 63.0%

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

   8. Taylor expanded in a around inf

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

   10. Applied rewrites 56.8%

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

   if -750 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 655

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

   4. Applied rewrites 99.5%

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

   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. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower-*.f64 67.0

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

   7. Applied rewrites 67.0%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log z + \log \left(y + x\right) \leq -750:\\ \;\;\;\;\left(\log t \cdot a + \log y\right) - t\\ \mathbf{elif}\;\log z + \log \left(y + x\right) \leq 655:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log \left(z \cdot y\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log t \cdot a + \log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 520:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log z\right) + \log y\\ \mathbf{else}:\\ \;\;\;\;\left(\log t \cdot a + \log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 520.0)
   (+ (fma (- a 0.5) (log t) (log z)) (log y))
   (- (+ (* (log t) a) (log y)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 520.0) {
		tmp = fma((a - 0.5), log(t), log(z)) + log(y);
	} else {
		tmp = ((log(t) * a) + log(y)) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 520.0)
		tmp = Float64(fma(Float64(a - 0.5), log(t), log(z)) + log(y));
	else
		tmp = Float64(Float64(Float64(log(t) * a) + log(y)) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 520.0], N[(N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 520

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. flip3-- N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. clear-num N/A

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

      9. flip3-- N/A

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

      10. lift--.f64 N/A

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

      11. lower-/.f64 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-log.f64 61.5

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

   7. Applied rewrites 61.5%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 61.3%

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

   if 520 < 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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. flip3-- N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. clear-num N/A

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

      9. flip3-- N/A

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

      10. lift--.f64 N/A

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

      11. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-log.f64 72.2

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

   7. Applied rewrites 72.2%

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

   8. Taylor expanded in a around inf

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

   10. Applied rewrites 72.0%

       \[\leadsto \left(\log y + a \cdot \log t\right) - t \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 520:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log z\right) + \log y\\ \mathbf{else}:\\ \;\;\;\;\left(\log t \cdot a + \log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in y around inf

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

   1. associate-+r+ N/A

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

   2. associate--l+ N/A

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

   3. +-commutative N/A

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

   4. mul-1-neg N/A

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

   5. log-rec N/A

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

   6. remove-double-neg N/A

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

   7. associate--l+ N/A

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

   8. associate-+r+ N/A

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

   9. lower--.f64 N/A

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

5. Applied rewrites 67.1%

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

6. Final simplification 67.1%

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

Alternative 9: 58.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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

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

   2. *-commutative N/A

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

   3. lift--.f64 N/A

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

   4. flip3-- N/A

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

   5. clear-num N/A

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

   6. un-div-inv N/A

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

   7. lower-/.f64 N/A

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

   8. clear-num N/A

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

   9. flip3-- N/A

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

   10. lift--.f64 N/A

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

   11. lower-/.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in x around 0

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

   1. lower--.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-log.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower--.f64 N/A

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

   8. lower-log.f64 N/A

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

   9. lower-log.f64 67.1

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

7. Applied rewrites 67.1%

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

8. Taylor expanded in a around inf

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

10. Applied rewrites 57.1%

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

11. Final simplification 57.1%

    \[\leadsto \left(\log t \cdot a + \log y\right) - t \]
12. Add Preprocessing

Alternative 10: 61.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log t \cdot a\\ \mathbf{if}\;a - 0.5 \leq -2 \cdot 10^{+23}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a - 0.5 \leq 10^{+55}:\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (log t) a)))
   (if (<= (- a 0.5) -2e+23) t_1 (if (<= (- a 0.5) 1e+55) (- t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log(t) * a;
	double tmp;
	if ((a - 0.5) <= -2e+23) {
		tmp = t_1;
	} else if ((a - 0.5) <= 1e+55) {
		tmp = -t;
	} else {
		tmp = t_1;
	}
	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) :: tmp
    t_1 = log(t) * a
    if ((a - 0.5d0) <= (-2d+23)) then
        tmp = t_1
    else if ((a - 0.5d0) <= 1d+55) then
        tmp = -t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = Math.log(t) * a;
	double tmp;
	if ((a - 0.5) <= -2e+23) {
		tmp = t_1;
	} else if ((a - 0.5) <= 1e+55) {
		tmp = -t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = math.log(t) * a
	tmp = 0
	if (a - 0.5) <= -2e+23:
		tmp = t_1
	elif (a - 0.5) <= 1e+55:
		tmp = -t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(t) * a)
	tmp = 0.0
	if (Float64(a - 0.5) <= -2e+23)
		tmp = t_1;
	elseif (Float64(a - 0.5) <= 1e+55)
		tmp = Float64(-t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = log(t) * a;
	tmp = 0.0;
	if ((a - 0.5) <= -2e+23)
		tmp = t_1;
	elseif ((a - 0.5) <= 1e+55)
		tmp = -t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[N[(a - 0.5), $MachinePrecision], -2e+23], t$95$1, If[LessEqual[N[(a - 0.5), $MachinePrecision], 1e+55], (-t), t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 a #s(literal 1/2 binary64)) < -1.9999999999999998e23 or 1.00000000000000001e55 < (-.f64 a #s(literal 1/2 binary64))

   1. Initial program 99.7%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 81.6

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

   5. Applied rewrites 81.6%

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

   if -1.9999999999999998e23 < (-.f64 a #s(literal 1/2 binary64)) < 1.00000000000000001e55

   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 56.0

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

   5. Applied rewrites 56.0%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -2 \cdot 10^{+23}:\\ \;\;\;\;\log t \cdot a\\ \mathbf{elif}\;a - 0.5 \leq 10^{+55}:\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 76.7% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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

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

   2. *-commutative N/A

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

   3. lift--.f64 N/A

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

   4. flip3-- N/A

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

   5. clear-num N/A

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

   6. un-div-inv N/A

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

   7. lower-/.f64 N/A

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

   8. clear-num N/A

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

   9. flip3-- N/A

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

   10. lift--.f64 N/A

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

   11. lower-/.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 78.1

      \[\leadsto \color{blue}{\left(-t\right)} + \frac{\log t}{\frac{1}{a - 0.5}} \]

7. Applied rewrites 78.1%

   \[\leadsto \color{blue}{\left(-t\right)} + \frac{\log t}{\frac{1}{a - 0.5}} \]
8. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. associate-/r/ N/A

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

   6. /-rgt-identity N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 78.2

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

9. Applied rewrites 78.2%

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

Alternative 12: 74.1% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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(t) * a) - t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift--.f64 N/A

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

   4. flip3-- N/A

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

   5. clear-num N/A

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

   6. un-div-inv N/A

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

   7. lower-/.f64 N/A

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

   8. clear-num N/A

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

   9. flip3-- N/A

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

   10. lift--.f64 N/A

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

   11. lower-/.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in x around 0

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

   1. lower--.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-log.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower--.f64 N/A

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

   8. lower-log.f64 N/A

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

   9. lower-log.f64 67.1

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

7. Applied rewrites 67.1%

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

8. Taylor expanded in a around inf

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

10. Applied rewrites 75.8%

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

11. Final simplification 75.8%

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

Alternative 13: 37.2% 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 99.7%

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

3. Taylor expanded in 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 40.2

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

5. Applied rewrites 40.2%

   \[\leadsto \color{blue}{-t} \]
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 2024235 
(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))))