Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 10.9s

Alternatives: 12

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: 97.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot \log t\\ t_2 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + t\_1\\ \mathbf{if}\;t\_2 \leq -5000 \lor \neg \left(t\_2 \leq 2000\right):\\ \;\;\;\;{\left(\frac{-1}{t}\right)}^{-1} + t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log t, \log \left(y + x\right)\right) + \log z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) (log t)))
        (t_2 (+ (- (+ (log (+ x y)) (log z)) t) t_1)))
   (if (or (<= t_2 -5000.0) (not (<= t_2 2000.0)))
     (+ (pow (/ -1.0 t) -1.0) t_1)
     (+ (fma -0.5 (log t) (log (+ y x))) (log z)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (a - 0.5) * log(t);
	double t_2 = ((log((x + y)) + log(z)) - t) + t_1;
	double tmp;
	if ((t_2 <= -5000.0) || !(t_2 <= 2000.0)) {
		tmp = pow((-1.0 / t), -1.0) + t_1;
	} else {
		tmp = fma(-0.5, log(t), log((y + x))) + log(z);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(a - 0.5) * log(t))
	t_2 = Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + t_1)
	tmp = 0.0
	if ((t_2 <= -5000.0) || !(t_2 <= 2000.0))
		tmp = Float64((Float64(-1.0 / t) ^ -1.0) + t_1);
	else
		tmp = Float64(fma(-0.5, log(t), log(Float64(y + x))) + log(z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + t$95$1), $MachinePrecision]}, If[Or[LessEqual[t$95$2, -5000.0], N[Not[LessEqual[t$95$2, 2000.0]], $MachinePrecision]], N[(N[Power[N[(-1.0 / t), $MachinePrecision], -1.0], $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(-0.5 * N[Log[t], $MachinePrecision] + N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < -5e3 or 2e3 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t)))

   1. Initial program 99.8%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   2. 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 99.7

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

   4. Applied rewrites 72.5%

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

   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 97.1

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

   7. Applied rewrites 97.1%

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

   if -5e3 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 2e3

   1. Initial program 99.2%

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

   3. Taylor expanded in a around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-log.f64 98.7

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 97.2%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \leq -5000 \lor \neg \left(\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \leq 2000\right):\\ \;\;\;\;{\left(\frac{-1}{t}\right)}^{-1} + \left(a - 0.5\right) \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log t, \log \left(y + x\right)\right) + \log z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot \log t\\ t_2 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + t\_1\\ \mathbf{if}\;t\_2 \leq -20000000000000 \lor \neg \left(t\_2 \leq 1000\right):\\ \;\;\;\;{\left(\frac{-1}{t}\right)}^{-1} + t\_1\\ \mathbf{else}:\\ \;\;\;\;\log \left(z \cdot \left(y + x\right)\right) - \left(t - -0.5 \cdot \log t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) (log t)))
        (t_2 (+ (- (+ (log (+ x y)) (log z)) t) t_1)))
   (if (or (<= t_2 -20000000000000.0) (not (<= t_2 1000.0)))
     (+ (pow (/ -1.0 t) -1.0) t_1)
     (- (log (* z (+ y x))) (- t (* -0.5 (log t)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (a - 0.5) * log(t);
	double t_2 = ((log((x + y)) + log(z)) - t) + t_1;
	double tmp;
	if ((t_2 <= -20000000000000.0) || !(t_2 <= 1000.0)) {
		tmp = pow((-1.0 / t), -1.0) + t_1;
	} else {
		tmp = log((z * (y + x))) - (t - (-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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (a - 0.5d0) * log(t)
    t_2 = ((log((x + y)) + log(z)) - t) + t_1
    if ((t_2 <= (-20000000000000.0d0)) .or. (.not. (t_2 <= 1000.0d0))) then
        tmp = (((-1.0d0) / t) ** (-1.0d0)) + t_1
    else
        tmp = log((z * (y + x))) - (t - ((-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 t_1 = (a - 0.5) * Math.log(t);
	double t_2 = ((Math.log((x + y)) + Math.log(z)) - t) + t_1;
	double tmp;
	if ((t_2 <= -20000000000000.0) || !(t_2 <= 1000.0)) {
		tmp = Math.pow((-1.0 / t), -1.0) + t_1;
	} else {
		tmp = Math.log((z * (y + x))) - (t - (-0.5 * Math.log(t)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (a - 0.5) * math.log(t)
	t_2 = ((math.log((x + y)) + math.log(z)) - t) + t_1
	tmp = 0
	if (t_2 <= -20000000000000.0) or not (t_2 <= 1000.0):
		tmp = math.pow((-1.0 / t), -1.0) + t_1
	else:
		tmp = math.log((z * (y + x))) - (t - (-0.5 * math.log(t)))
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(a - 0.5) * log(t))
	t_2 = Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + t_1)
	tmp = 0.0
	if ((t_2 <= -20000000000000.0) || !(t_2 <= 1000.0))
		tmp = Float64((Float64(-1.0 / t) ^ -1.0) + t_1);
	else
		tmp = Float64(log(Float64(z * Float64(y + x))) - Float64(t - Float64(-0.5 * log(t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (a - 0.5) * log(t);
	t_2 = ((log((x + y)) + log(z)) - t) + t_1;
	tmp = 0.0;
	if ((t_2 <= -20000000000000.0) || ~((t_2 <= 1000.0)))
		tmp = ((-1.0 / t) ^ -1.0) + t_1;
	else
		tmp = log((z * (y + x))) - (t - (-0.5 * log(t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + t$95$1), $MachinePrecision]}, If[Or[LessEqual[t$95$2, -20000000000000.0], N[Not[LessEqual[t$95$2, 1000.0]], $MachinePrecision]], N[(N[Power[N[(-1.0 / t), $MachinePrecision], -1.0], $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[Log[N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(t - N[(-0.5 * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < -2e13 or 1e3 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t)))

   1. Initial program 99.8%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   2. 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 99.7

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

   4. Applied rewrites 68.5%

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

   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 94.1

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

   7. Applied rewrites 94.1%

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

   if -2e13 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 1e3

   1. Initial program 99.1%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   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 99.1

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

   4. Applied rewrites 99.1%

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \color{blue}{\frac{\log t}{\frac{1}{a - 0.5}}} \]
   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. lower--.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-log.f64 N/A

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

      14. lift-log.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-+.f64 N/A

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

      18. log-prod N/A

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

      19. lift-*.f64 N/A

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

      20. lift-log.f64 N/A

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

      21. lower--.f64 87.9

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

   6. Applied rewrites 87.9%

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

   7. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 84.6

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

   9. Applied rewrites 84.6%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \leq -20000000000000 \lor \neg \left(\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \leq 1000\right):\\ \;\;\;\;{\left(\frac{-1}{t}\right)}^{-1} + \left(a - 0.5\right) \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\log \left(z \cdot \left(y + x\right)\right) - \left(t - -0.5 \cdot \log t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot \log t\\ t_2 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + t\_1\\ \mathbf{if}\;t\_2 \leq -20000000000000 \lor \neg \left(t\_2 \leq 1000\right):\\ \;\;\;\;{\left(\frac{-1}{t}\right)}^{-1} + t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log t, \log \left(\left(y + x\right) \cdot z\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) (log t)))
        (t_2 (+ (- (+ (log (+ x y)) (log z)) t) t_1)))
   (if (or (<= t_2 -20000000000000.0) (not (<= t_2 1000.0)))
     (+ (pow (/ -1.0 t) -1.0) t_1)
     (- (fma -0.5 (log t) (log (* (+ y x) z))) t))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (a - 0.5) * log(t);
	double t_2 = ((log((x + y)) + log(z)) - t) + t_1;
	double tmp;
	if ((t_2 <= -20000000000000.0) || !(t_2 <= 1000.0)) {
		tmp = pow((-1.0 / t), -1.0) + t_1;
	} else {
		tmp = fma(-0.5, log(t), log(((y + x) * z))) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(a - 0.5) * log(t))
	t_2 = Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + t_1)
	tmp = 0.0
	if ((t_2 <= -20000000000000.0) || !(t_2 <= 1000.0))
		tmp = Float64((Float64(-1.0 / t) ^ -1.0) + t_1);
	else
		tmp = Float64(fma(-0.5, log(t), log(Float64(Float64(y + x) * z))) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + t$95$1), $MachinePrecision]}, If[Or[LessEqual[t$95$2, -20000000000000.0], N[Not[LessEqual[t$95$2, 1000.0]], $MachinePrecision]], N[(N[Power[N[(-1.0 / t), $MachinePrecision], -1.0], $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(-0.5 * N[Log[t], $MachinePrecision] + N[Log[N[(N[(y + x), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < -2e13 or 1e3 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t)))

   1. Initial program 99.8%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   2. 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 99.7

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

   4. Applied rewrites 68.5%

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

   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 94.1

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

   7. Applied rewrites 94.1%

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

   if -2e13 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 1e3

   1. Initial program 99.1%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   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 99.1

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

   4. Applied rewrites 87.9%

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

   5. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 84.6

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

   7. Applied rewrites 84.6%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \leq -20000000000000 \lor \neg \left(\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \leq 1000\right):\\ \;\;\;\;{\left(\frac{-1}{t}\right)}^{-1} + \left(a - 0.5\right) \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log t, \log \left(\left(y + x\right) \cdot z\right)\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(x + y\right) + \log z\\ \mathbf{if}\;t\_1 \leq -800 \lor \neg \left(t\_1 \leq 705\right):\\ \;\;\;\;{\left(\frac{-1}{t}\right)}^{-1} + \left(a - 0.5\right) \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log \left(z \cdot \left(y + x\right)\right) - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log (+ x y)) (log z))))
   (if (or (<= t_1 -800.0) (not (<= t_1 705.0)))
     (+ (pow (/ -1.0 t) -1.0) (* (- a 0.5) (log t)))
     (fma (- a 0.5) (log t) (- (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 <= -800.0) || !(t_1 <= 705.0)) {
		tmp = pow((-1.0 / t), -1.0) + ((a - 0.5) * log(t));
	} else {
		tmp = fma((a - 0.5), log(t), (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 <= -800.0) || !(t_1 <= 705.0))
		tmp = Float64((Float64(-1.0 / t) ^ -1.0) + Float64(Float64(a - 0.5) * log(t)));
	else
		tmp = fma(Float64(a - 0.5), log(t), Float64(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, -800.0], N[Not[LessEqual[t$95$1, 705.0]], $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[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[(N[Log[N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -800 or 705 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z))

   1. Initial program 99.7%

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

      1. lift--.f64 N/A

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

      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 4.9%

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

   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 78.7

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

   7. Applied rewrites 78.7%

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

   if -800 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 705

   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 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(x + y\right) + \log z \leq -800 \lor \neg \left(\log \left(x + y\right) + \log z \leq 705\right):\\ \;\;\;\;{\left(\frac{-1}{t}\right)}^{-1} + \left(a - 0.5\right) \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log \left(z \cdot \left(y + x\right)\right) - t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -100 \lor \neg \left(a - 0.5 \leq -0.48\right):\\ \;\;\;\;{\left(\frac{-1}{t}\right)}^{-1} + \left(a - 0.5\right) \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log t, \log \left(y + x\right)\right) - \left(t - \log z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= (- a 0.5) -100.0) (not (<= (- a 0.5) -0.48)))
   (+ (pow (/ -1.0 t) -1.0) (* (- a 0.5) (log t)))
   (- (fma -0.5 (log t) (log (+ y x))) (- t (log z)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((a - 0.5) <= -100.0) || !((a - 0.5) <= -0.48)) {
		tmp = pow((-1.0 / t), -1.0) + ((a - 0.5) * log(t));
	} else {
		tmp = fma(-0.5, log(t), log((y + x))) - (t - log(z));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((Float64(a - 0.5) <= -100.0) || !(Float64(a - 0.5) <= -0.48))
		tmp = Float64((Float64(-1.0 / t) ^ -1.0) + Float64(Float64(a - 0.5) * log(t)));
	else
		tmp = Float64(fma(-0.5, log(t), log(Float64(y + x))) - Float64(t - log(z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[N[(a - 0.5), $MachinePrecision], -100.0], N[Not[LessEqual[N[(a - 0.5), $MachinePrecision], -0.48]], $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[(-0.5 * N[Log[t], $MachinePrecision] + N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(t - N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 a #s(literal 1/2 binary64)) < -100 or -0.47999999999999998 < (-.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. 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 99.7

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

   4. Applied rewrites 73.3%

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

   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 97.5

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

   7. Applied rewrites 97.5%

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

   if -100 < (-.f64 a #s(literal 1/2 binary64)) < -0.47999999999999998

   1. Initial program 99.5%

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

   3. Taylor expanded in a around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-log.f64 99.3

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

   5. Applied rewrites 99.3%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -100 \lor \neg \left(a - 0.5 \leq -0.48\right):\\ \;\;\;\;{\left(\frac{-1}{t}\right)}^{-1} + \left(a - 0.5\right) \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log t, \log \left(y + x\right)\right) - \left(t - \log z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -100 \lor \neg \left(a - 0.5 \leq -0.48\right):\\ \;\;\;\;{\left(\frac{-1}{t}\right)}^{-1} + \left(a - 0.5\right) \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log t, \log y\right) - \left(t - \log z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= (- a 0.5) -100.0) (not (<= (- a 0.5) -0.48)))
   (+ (pow (/ -1.0 t) -1.0) (* (- a 0.5) (log t)))
   (- (fma -0.5 (log t) (log y)) (- t (log z)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((a - 0.5) <= -100.0) || !((a - 0.5) <= -0.48)) {
		tmp = pow((-1.0 / t), -1.0) + ((a - 0.5) * log(t));
	} else {
		tmp = fma(-0.5, log(t), log(y)) - (t - log(z));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((Float64(a - 0.5) <= -100.0) || !(Float64(a - 0.5) <= -0.48))
		tmp = Float64((Float64(-1.0 / t) ^ -1.0) + Float64(Float64(a - 0.5) * log(t)));
	else
		tmp = Float64(fma(-0.5, log(t), log(y)) - Float64(t - log(z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[N[(a - 0.5), $MachinePrecision], -100.0], N[Not[LessEqual[N[(a - 0.5), $MachinePrecision], -0.48]], $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[(-0.5 * N[Log[t], $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] - N[(t - N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 a #s(literal 1/2 binary64)) < -100 or -0.47999999999999998 < (-.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. 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 99.7

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

   4. Applied rewrites 73.3%

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

   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 97.5

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

   7. Applied rewrites 97.5%

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

   if -100 < (-.f64 a #s(literal 1/2 binary64)) < -0.47999999999999998

   1. Initial program 99.5%

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

   3. Taylor expanded in a around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-log.f64 99.3

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 58.2%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -100 \lor \neg \left(a - 0.5 \leq -0.48\right):\\ \;\;\;\;{\left(\frac{-1}{t}\right)}^{-1} + \left(a - 0.5\right) \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log t, \log y\right) - \left(t - \log z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.4% 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 67.0

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

5. Applied rewrites 67.0%

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

Alternative 9: 77.4% 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 72.9%

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

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 76.0

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

7. Applied rewrites 76.0%

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

8. Final simplification 76.0%

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

Alternative 10: 74.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.85 \cdot 10^{-12}:\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot \frac{\log t}{t}, t, -t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 1.85e-12) (* (log t) a) (fma (* a (/ (log t) t)) t (- t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 1.85e-12) {
		tmp = log(t) * a;
	} else {
		tmp = fma((a * (log(t) / t)), t, -t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 1.85e-12)
		tmp = Float64(log(t) * a);
	else
		tmp = fma(Float64(a * Float64(log(t) / t)), t, Float64(-t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 1.85e-12], N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision], N[(N[(a * N[(N[Log[t], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * t + (-t)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.84999999999999999e-12

   1. Initial program 99.4%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 46.8

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

   5. Applied rewrites 46.8%

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

   if 1.84999999999999999e-12 < 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

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

      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 t around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-rgt-in N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in a around inf

      \[\leadsto \mathsf{fma}\left(\frac{a \cdot \log t}{t}, t, -t\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 96.5%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.85 \cdot 10^{-12}:\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot \frac{\log t}{t}, t, -t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 62.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -8200000 \lor \neg \left(a \leq 2.25 \cdot 10^{+62}\right):\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -8200000.0) (not (<= a 2.25e+62))) (* (log t) a) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -8200000.0) || !(a <= 2.25e+62)) {
		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 <= (-8200000.0d0)) .or. (.not. (a <= 2.25d+62))) 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 <= -8200000.0) || !(a <= 2.25e+62)) {
		tmp = Math.log(t) * a;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -8200000.0) or not (a <= 2.25e+62):
		tmp = math.log(t) * a
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -8200000.0) || !(a <= 2.25e+62))
		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 <= -8200000.0) || ~((a <= 2.25e+62)))
		tmp = log(t) * a;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -8200000.0], N[Not[LessEqual[a, 2.25e+62]], $MachinePrecision]], N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if a < -8.2e6 or 2.24999999999999999e62 < a

   1. Initial program 99.7%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   2. 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 74.1

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

   5. Applied rewrites 74.1%

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

   if -8.2e6 < a < 2.24999999999999999e62

   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 47.6

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

   5. Applied rewrites 47.6%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -8200000 \lor \neg \left(a \leq 2.25 \cdot 10^{+62}\right):\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 37.3% accurate, 107.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 36.9

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

5. Applied rewrites 36.9%

   \[\leadsto \color{blue}{-t} \]
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 2024324 
(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))))