Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 12.0s

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

Alternative 2: 93.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(x + y\right) + \log z\\ \mathbf{if}\;t\_1 \leq -750 \lor \neg \left(t\_1 \leq 710\right):\\ \;\;\;\;{\left(\frac{-1}{t}\right)}^{-1} + \left(a - 0.5\right) \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\log \left(z \cdot \left(y + x\right)\right) - \left(t - \log t \cdot \left(a - 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log (+ x y)) (log z))))
   (if (or (<= t_1 -750.0) (not (<= t_1 710.0)))
     (+ (pow (/ -1.0 t) -1.0) (* (- a 0.5) (log t)))
     (- (log (* z (+ y x))) (- t (* (log t) (- a 0.5)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((x + y)) + log(z);
	double tmp;
	if ((t_1 <= -750.0) || !(t_1 <= 710.0)) {
		tmp = pow((-1.0 / t), -1.0) + ((a - 0.5) * log(t));
	} else {
		tmp = log((z * (y + x))) - (t - (log(t) * (a - 0.5)));
	}
	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((x + y)) + log(z)
    if ((t_1 <= (-750.0d0)) .or. (.not. (t_1 <= 710.0d0))) then
        tmp = (((-1.0d0) / t) ** (-1.0d0)) + ((a - 0.5d0) * log(t))
    else
        tmp = log((z * (y + x))) - (t - (log(t) * (a - 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = Math.log((x + y)) + Math.log(z);
	double tmp;
	if ((t_1 <= -750.0) || !(t_1 <= 710.0)) {
		tmp = Math.pow((-1.0 / t), -1.0) + ((a - 0.5) * Math.log(t));
	} else {
		tmp = Math.log((z * (y + x))) - (t - (Math.log(t) * (a - 0.5)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = math.log((x + y)) + math.log(z)
	tmp = 0
	if (t_1 <= -750.0) or not (t_1 <= 710.0):
		tmp = math.pow((-1.0 / t), -1.0) + ((a - 0.5) * math.log(t))
	else:
		tmp = math.log((z * (y + x))) - (t - (math.log(t) * (a - 0.5)))
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(Float64(x + y)) + log(z))
	tmp = 0.0
	if ((t_1 <= -750.0) || !(t_1 <= 710.0))
		tmp = Float64((Float64(-1.0 / t) ^ -1.0) + Float64(Float64(a - 0.5) * log(t)));
	else
		tmp = Float64(log(Float64(z * Float64(y + x))) - Float64(t - Float64(log(t) * Float64(a - 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = log((x + y)) + log(z);
	tmp = 0.0;
	if ((t_1 <= -750.0) || ~((t_1 <= 710.0)))
		tmp = ((-1.0 / t) ^ -1.0) + ((a - 0.5) * log(t));
	else
		tmp = log((z * (y + x))) - (t - (log(t) * (a - 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -750.0], N[Not[LessEqual[t$95$1, 710.0]], $MachinePrecision]], N[(N[Power[N[(-1.0 / t), $MachinePrecision], -1.0], $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 99.6

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

   4. Applied rewrites 2.3%

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

   5. Taylor expanded in t around inf

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

      1. lower-/.f64 81.9

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

   7. Applied rewrites 81.9%

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

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

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. 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 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(x + y\right) + \log z \leq -750 \lor \neg \left(\log \left(x + y\right) + \log z \leq 710\right):\\ \;\;\;\;{\left(\frac{-1}{t}\right)}^{-1} + \left(a - 0.5\right) \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\log \left(z \cdot \left(y + x\right)\right) - \left(t - \log t \cdot \left(a - 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(x + y\right) + \log z\\ \mathbf{if}\;t\_1 \leq -750 \lor \neg \left(t\_1 \leq 710\right):\\ \;\;\;\;{\left(\frac{-1}{t}\right)}^{-1} + \left(a - 0.5\right) \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log t, a - 0.5, \log \left(z \cdot \left(y + x\right)\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log (+ x y)) (log z))))
   (if (or (<= t_1 -750.0) (not (<= t_1 710.0)))
     (+ (pow (/ -1.0 t) -1.0) (* (- a 0.5) (log t)))
     (- (fma (log t) (- a 0.5) (log (* z (+ y x)))) t))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((x + y)) + log(z);
	double tmp;
	if ((t_1 <= -750.0) || !(t_1 <= 710.0)) {
		tmp = pow((-1.0 / t), -1.0) + ((a - 0.5) * log(t));
	} else {
		tmp = fma(log(t), (a - 0.5), log((z * (y + x)))) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(Float64(x + y)) + log(z))
	tmp = 0.0
	if ((t_1 <= -750.0) || !(t_1 <= 710.0))
		tmp = Float64((Float64(-1.0 / t) ^ -1.0) + Float64(Float64(a - 0.5) * log(t)));
	else
		tmp = Float64(fma(log(t), Float64(a - 0.5), log(Float64(z * Float64(y + x)))) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -750.0], N[Not[LessEqual[t$95$1, 710.0]], $MachinePrecision]], N[(N[Power[N[(-1.0 / t), $MachinePrecision], -1.0], $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision] + N[Log[N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 99.6

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

   4. Applied rewrites 2.3%

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

   5. Taylor expanded in t around inf

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

      1. lower-/.f64 81.9

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

   7. Applied rewrites 81.9%

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

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

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+r- N/A

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

      5. lower--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 99.6

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

      9. lift-+.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. sum-log N/A

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

      13. lower-log.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 99.7

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 99.7

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

   4. Applied rewrites 99.7%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(x + y\right) + \log z \leq -750 \lor \neg \left(\log \left(x + y\right) + \log z \leq 710\right):\\ \;\;\;\;{\left(\frac{-1}{t}\right)}^{-1} + \left(a - 0.5\right) \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log t, a - 0.5, \log \left(z \cdot \left(y + x\right)\right)\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(x + y\right) + \log z\\ t_2 := \left(a - 0.5\right) \cdot \log t\\ \mathbf{if}\;t\_1 \leq -750 \lor \neg \left(t\_1 \leq 710\right):\\ \;\;\;\;{\left(\frac{-1}{t}\right)}^{-1} + t\_2\\ \mathbf{else}:\\ \;\;\;\;\log \left(z \cdot y\right) - \left(t - t\_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log (+ x y)) (log z))) (t_2 (* (- a 0.5) (log t))))
   (if (or (<= t_1 -750.0) (not (<= t_1 710.0)))
     (+ (pow (/ -1.0 t) -1.0) t_2)
     (- (log (* z y)) (- t t_2)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((x + y)) + log(z);
	double t_2 = (a - 0.5) * log(t);
	double tmp;
	if ((t_1 <= -750.0) || !(t_1 <= 710.0)) {
		tmp = pow((-1.0 / t), -1.0) + t_2;
	} else {
		tmp = log((z * y)) - (t - t_2);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = log((x + y)) + log(z)
    t_2 = (a - 0.5d0) * log(t)
    if ((t_1 <= (-750.0d0)) .or. (.not. (t_1 <= 710.0d0))) then
        tmp = (((-1.0d0) / t) ** (-1.0d0)) + t_2
    else
        tmp = log((z * y)) - (t - t_2)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = Math.log((x + y)) + Math.log(z);
	double t_2 = (a - 0.5) * Math.log(t);
	double tmp;
	if ((t_1 <= -750.0) || !(t_1 <= 710.0)) {
		tmp = Math.pow((-1.0 / t), -1.0) + t_2;
	} else {
		tmp = Math.log((z * y)) - (t - t_2);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = math.log((x + y)) + math.log(z)
	t_2 = (a - 0.5) * math.log(t)
	tmp = 0
	if (t_1 <= -750.0) or not (t_1 <= 710.0):
		tmp = math.pow((-1.0 / t), -1.0) + t_2
	else:
		tmp = math.log((z * y)) - (t - t_2)
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(Float64(x + y)) + log(z))
	t_2 = Float64(Float64(a - 0.5) * log(t))
	tmp = 0.0
	if ((t_1 <= -750.0) || !(t_1 <= 710.0))
		tmp = Float64((Float64(-1.0 / t) ^ -1.0) + t_2);
	else
		tmp = Float64(log(Float64(z * y)) - Float64(t - t_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = log((x + y)) + log(z);
	t_2 = (a - 0.5) * log(t);
	tmp = 0.0;
	if ((t_1 <= -750.0) || ~((t_1 <= 710.0)))
		tmp = ((-1.0 / t) ^ -1.0) + t_2;
	else
		tmp = log((z * y)) - (t - t_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -750.0], N[Not[LessEqual[t$95$1, 710.0]], $MachinePrecision]], N[(N[Power[N[(-1.0 / t), $MachinePrecision], -1.0], $MachinePrecision] + t$95$2), $MachinePrecision], N[(N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision] - N[(t - t$95$2), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 99.6

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

   4. Applied rewrites 2.3%

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

   5. Taylor expanded in t around inf

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

      1. lower-/.f64 81.9

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

   7. Applied rewrites 81.9%

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

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

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lift-/.f64 N/A

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

      6. remove-double-div N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-+l- N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-log.f64 N/A

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

      13. lift-log.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. log-prod N/A

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

      18. lift-*.f64 N/A

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

      19. lift-log.f64 N/A

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 66.2

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

   9. Applied rewrites 66.2%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(x + y\right) + \log z \leq -750 \lor \neg \left(\log \left(x + y\right) + \log z \leq 710\right):\\ \;\;\;\;{\left(\frac{-1}{t}\right)}^{-1} + \left(a - 0.5\right) \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\log \left(z \cdot y\right) - \left(t - \left(a - 0.5\right) \cdot \log t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(x + y\right) + \log z\\ \mathbf{if}\;t\_1 \leq -750 \lor \neg \left(t\_1 \leq 710\right):\\ \;\;\;\;{\left(\frac{-1}{t}\right)}^{-1} + \left(a - 0.5\right) \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log t, a - 0.5, \log \left(z \cdot y\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log (+ x y)) (log z))))
   (if (or (<= t_1 -750.0) (not (<= t_1 710.0)))
     (+ (pow (/ -1.0 t) -1.0) (* (- a 0.5) (log t)))
     (- (fma (log t) (- a 0.5) (log (* z y))) t))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((x + y)) + log(z);
	double tmp;
	if ((t_1 <= -750.0) || !(t_1 <= 710.0)) {
		tmp = pow((-1.0 / t), -1.0) + ((a - 0.5) * log(t));
	} else {
		tmp = fma(log(t), (a - 0.5), log((z * y))) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(Float64(x + y)) + log(z))
	tmp = 0.0
	if ((t_1 <= -750.0) || !(t_1 <= 710.0))
		tmp = Float64((Float64(-1.0 / t) ^ -1.0) + Float64(Float64(a - 0.5) * log(t)));
	else
		tmp = Float64(fma(log(t), Float64(a - 0.5), log(Float64(z * y))) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -750.0], N[Not[LessEqual[t$95$1, 710.0]], $MachinePrecision]], N[(N[Power[N[(-1.0 / t), $MachinePrecision], -1.0], $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision] + N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 99.6

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

   4. Applied rewrites 2.3%

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

   5. Taylor expanded in t around inf

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

      1. lower-/.f64 81.9

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

   7. Applied rewrites 81.9%

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

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

   1. Initial program 99.6%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 99.5

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

   4. Applied rewrites 99.6%

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

   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. lower-fma.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 66.2

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

   7. Applied rewrites 66.2%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(x + y\right) + \log z \leq -750 \lor \neg \left(\log \left(x + y\right) + \log z \leq 710\right):\\ \;\;\;\;{\left(\frac{-1}{t}\right)}^{-1} + \left(a - 0.5\right) \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log t, a - 0.5, \log \left(z \cdot y\right)\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 440:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log \left(y + x\right)\right) + \log z\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{-1}{t}\right)}^{-1} + \left(a - 0.5\right) \cdot \log t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 440.0)
   (+ (fma (- a 0.5) (log t) (log (+ y x))) (log z))
   (+ (pow (/ -1.0 t) -1.0) (* (- a 0.5) (log t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 440.0) {
		tmp = fma((a - 0.5), log(t), log((y + x))) + log(z);
	} else {
		tmp = pow((-1.0 / t), -1.0) + ((a - 0.5) * log(t));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 440.0)
		tmp = Float64(fma(Float64(a - 0.5), log(t), log(Float64(y + x))) + log(z));
	else
		tmp = Float64((Float64(-1.0 / t) ^ -1.0) + Float64(Float64(a - 0.5) * log(t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 440.0], N[(N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(-1.0 / t), $MachinePrecision], -1.0], $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 440

   1. Initial program 99.4%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-log.f64 99.0

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

   5. Applied rewrites 99.0%

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

   if 440 < t

   1. Initial program 99.8%

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

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 99.6

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

   4. Applied rewrites 78.5%

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

   5. Taylor expanded in t around inf

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

      1. lower-/.f64 98.5

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

   7. Applied rewrites 98.5%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 440:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log \left(y + x\right)\right) + \log z\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{-1}{t}\right)}^{-1} + \left(a - 0.5\right) \cdot \log t\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 440:\\ \;\;\;\;\mathsf{fma}\left(\log t, -0.5 + a, \log y\right) + \log z\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{-1}{t}\right)}^{-1} + \left(a - 0.5\right) \cdot \log t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 440.0)
   (+ (fma (log t) (+ -0.5 a) (log y)) (log z))
   (+ (pow (/ -1.0 t) -1.0) (* (- a 0.5) (log t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 440.0) {
		tmp = fma(log(t), (-0.5 + a), log(y)) + log(z);
	} else {
		tmp = pow((-1.0 / t), -1.0) + ((a - 0.5) * log(t));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 440.0)
		tmp = Float64(fma(log(t), Float64(-0.5 + a), log(y)) + log(z));
	else
		tmp = Float64((Float64(-1.0 / t) ^ -1.0) + Float64(Float64(a - 0.5) * log(t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 440.0], N[(N[(N[Log[t], $MachinePrecision] * N[(-0.5 + a), $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(-1.0 / t), $MachinePrecision], -1.0], $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 440

   1. Initial program 99.4%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   2. Add Preprocessing
   3. 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 t around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-log.f64 99.0

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

   7. Applied rewrites 99.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 61.4%

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

   if 440 < t

   1. Initial program 99.8%

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

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 99.6

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

   4. Applied rewrites 78.5%

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

   5. Taylor expanded in t around inf

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

      1. lower-/.f64 98.5

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

   7. Applied rewrites 98.5%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 440:\\ \;\;\;\;\mathsf{fma}\left(\log t, -0.5 + a, \log y\right) + \log z\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{-1}{t}\right)}^{-1} + \left(a - 0.5\right) \cdot \log t\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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(y) + log(z)) - (t - ((a - 0.5d0) * log(t)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. lift-*.f64 N/A

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

   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. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. div-inv N/A

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

   5. lift-/.f64 N/A

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

   6. remove-double-div N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. associate-+l- N/A

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lift-log.f64 N/A

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

   13. lift-log.f64 N/A

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. lift-+.f64 N/A

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

   17. log-prod N/A

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

   18. lift-*.f64 N/A

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

   19. lift-log.f64 N/A

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

6. Applied rewrites 80.3%

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

7. Taylor expanded in y around inf

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. log-rec N/A

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

   4. remove-double-neg N/A

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

   5. lower-+.f64 N/A

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

   6. lower-log.f64 N/A

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

   7. lower-log.f64 71.2

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

9. Applied rewrites 71.2%

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

Alternative 9: 69.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in x around 0

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

   1. associate--l+ N/A

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

   2. +-commutative N/A

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

   3. associate-+l- N/A

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

   4. lower--.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-log.f64 N/A

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

   10. lower-log.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lower-log.f64 71.2

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

5. Applied rewrites 71.2%

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

Alternative 10: 64.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.32 \cdot 10^{+51} \lor \neg \left(a \leq 3 \cdot 10^{+56}\right):\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{-1}{t}\right)}^{-1} + -0.5 \cdot \log t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -1.32e+51) (not (<= a 3e+56)))
   (* (log t) a)
   (+ (pow (/ -1.0 t) -1.0) (* -0.5 (log t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -1.32e+51) || !(a <= 3e+56)) {
		tmp = log(t) * a;
	} else {
		tmp = pow((-1.0 / t), -1.0) + (-0.5 * log(t));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((a <= (-1.32d+51)) .or. (.not. (a <= 3d+56))) then
        tmp = log(t) * a
    else
        tmp = (((-1.0d0) / t) ** (-1.0d0)) + ((-0.5d0) * log(t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -1.32e+51) || !(a <= 3e+56)) {
		tmp = Math.log(t) * a;
	} else {
		tmp = Math.pow((-1.0 / t), -1.0) + (-0.5 * Math.log(t));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -1.32e+51) or not (a <= 3e+56):
		tmp = math.log(t) * a
	else:
		tmp = math.pow((-1.0 / t), -1.0) + (-0.5 * math.log(t))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -1.32e+51) || !(a <= 3e+56))
		tmp = Float64(log(t) * a);
	else
		tmp = Float64((Float64(-1.0 / t) ^ -1.0) + Float64(-0.5 * log(t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -1.32e+51) || ~((a <= 3e+56)))
		tmp = log(t) * a;
	else
		tmp = ((-1.0 / t) ^ -1.0) + (-0.5 * log(t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -1.32e+51], N[Not[LessEqual[a, 3e+56]], $MachinePrecision]], N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision], N[(N[Power[N[(-1.0 / t), $MachinePrecision], -1.0], $MachinePrecision] + N[(-0.5 * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.32e51 or 3.00000000000000006e56 < a

   1. Initial program 99.6%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 76.8

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

   5. Applied rewrites 76.8%

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

   if -1.32e51 < a < 3.00000000000000006e56

   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

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 99.5

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

   4. Applied rewrites 76.8%

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

   5. Taylor expanded in t around inf

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

      1. lower-/.f64 61.8

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

   7. Applied rewrites 61.8%

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

   8. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{-1}{t}} + \color{blue}{\frac{-1}{2} \cdot \log t} \]

      2. lower-log.f64 56.2

         \[\leadsto \frac{1}{\frac{-1}{t}} + -0.5 \cdot \color{blue}{\log t} \]

   10. Applied rewrites 56.2%

       \[\leadsto \frac{1}{\frac{-1}{t}} + \color{blue}{-0.5 \cdot \log t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.32 \cdot 10^{+51} \lor \neg \left(a \leq 3 \cdot 10^{+56}\right):\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{-1}{t}\right)}^{-1} + -0.5 \cdot \log t\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 77.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{-1}{t}\right)}^{-1} + \left(a - 0.5\right) \cdot \log t \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (+ (pow (/ -1.0 t) -1.0) (* (- a 0.5) (log t))))

C

double code(double x, double y, double z, double t, double a) {
	return pow((-1.0 / t), -1.0) + ((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 = (((-1.0d0) / t) ** (-1.0d0)) + ((a - 0.5d0) * log(t))
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return Math.pow((-1.0 / t), -1.0) + ((a - 0.5) * Math.log(t));
}

Python

def code(x, y, z, t, a):
	return math.pow((-1.0 / t), -1.0) + ((a - 0.5) * math.log(t))

Julia

function code(x, y, z, t, a)
	return Float64((Float64(-1.0 / t) ^ -1.0) + Float64(Float64(a - 0.5) * log(t)))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = ((-1.0 / t) ^ -1.0) + ((a - 0.5) * log(t));
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[Power[N[(-1.0 / t), $MachinePrecision], -1.0], $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

   5. clear-num N/A

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

   6. flip-- N/A

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

   7. lift--.f64 N/A

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

   8. lower-/.f64 99.5

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

4. Applied rewrites 80.2%

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

5. Taylor expanded in t around inf

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

   1. lower-/.f64 76.8

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

7. Applied rewrites 76.8%

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

8. Final simplification 76.8%

   \[\leadsto {\left(\frac{-1}{t}\right)}^{-1} + \left(a - 0.5\right) \cdot \log t \]
9. Add Preprocessing

Alternative 12: 61.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.3 \cdot 10^{+43} \lor \neg \left(a \leq 3 \cdot 10^{+56}\right):\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -4.3e+43) (not (<= a 3e+56))) (* (log t) a) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -4.3e+43) || !(a <= 3e+56)) {
		tmp = log(t) * a;
	} else {
		tmp = -t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((a <= (-4.3d+43)) .or. (.not. (a <= 3d+56))) then
        tmp = log(t) * a
    else
        tmp = -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -4.3e+43) || !(a <= 3e+56)) {
		tmp = Math.log(t) * a;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -4.3e+43) or not (a <= 3e+56):
		tmp = math.log(t) * a
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -4.3e+43) || !(a <= 3e+56))
		tmp = Float64(log(t) * a);
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -4.3e+43) || ~((a <= 3e+56)))
		tmp = log(t) * a;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -4.3e+43], N[Not[LessEqual[a, 3e+56]], $MachinePrecision]], N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if a < -4.3e43 or 3.00000000000000006e56 < a

   1. Initial program 99.6%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 76.4

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

   5. Applied rewrites 76.4%

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

   if -4.3e43 < a < 3.00000000000000006e56

   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 50.4

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

   5. Applied rewrites 50.4%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.3 \cdot 10^{+43} \lor \neg \left(a \leq 3 \cdot 10^{+56}\right):\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 38.1% accurate, 107.0× speedup

Math

\[\begin{array}{l} \\ -t \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 39.5

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

5. Applied rewrites 39.5%

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

Developer Target 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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 2024315 
(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))))