Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.2%

Time: 13.8s

Alternatives: 8

Speedup: N/A×

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: 99.2% accurate, 1.0× speedup

Math

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

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (+ (- (+ (log y) (log z)) t) (* (- a 0.5) (log t))))

C

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

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
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

assert x < y && y < z && z < t && t < a;
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

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

Julia

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

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp = code(x, y, z, t, a)
	tmp = ((log(y) + log(z)) - t) + ((a - 0.5) * log(t));
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := N[(N[(N[(N[Log[y], $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. Taylor expanded in x around 0

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

   1. +-lowering-+.f64 N/A

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

   2. log-lowering-log.f64 N/A

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

   3. log-lowering-log.f64 66.9

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

5. Simplified 66.9%

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

Alternative 2: 97.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot \log t\\ t_2 := t\_1 + \left(\left(\log z + \log \left(y + x\right)\right) - t\right)\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+23}:\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{elif}\;t\_2 \leq 2000:\\ \;\;\;\;\log y + \left(\mathsf{fma}\left(\log t, -0.5, \log z\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 - t\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) (log t)))
        (t_2 (+ t_1 (- (+ (log z) (log (+ y x))) t))))
   (if (<= t_2 -1e+23)
     (- (* a (log t)) t)
     (if (<= t_2 2000.0)
       (+ (log y) (- (fma (log t) -0.5 (log z)) t))
       (- t_1 t)))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (a - 0.5) * log(t);
	double t_2 = t_1 + ((log(z) + log((y + x))) - t);
	double tmp;
	if (t_2 <= -1e+23) {
		tmp = (a * log(t)) - t;
	} else if (t_2 <= 2000.0) {
		tmp = log(y) + (fma(log(t), -0.5, log(z)) - t);
	} else {
		tmp = t_1 - t;
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(a - 0.5) * log(t))
	t_2 = Float64(t_1 + Float64(Float64(log(z) + log(Float64(y + x))) - t))
	tmp = 0.0
	if (t_2 <= -1e+23)
		tmp = Float64(Float64(a * log(t)) - t);
	elseif (t_2 <= 2000.0)
		tmp = Float64(log(y) + Float64(fma(log(t), -0.5, log(z)) - t));
	else
		tmp = Float64(t_1 - t);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
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[(t$95$1 + N[(N[(N[Log[z], $MachinePrecision] + N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+23], N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[t$95$2, 2000.0], N[(N[Log[y], $MachinePrecision] + N[(N[(N[Log[t], $MachinePrecision] * -0.5 + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], N[(t$95$1 - t), $MachinePrecision]]]]]

Derivation

1. Split input into 3 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))) < -9.9999999999999992e22

   1. Initial program 99.9%

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

      1. +-commutative N/A

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

      2. flip3-- N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 55.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(a, a \cdot a, -0.125\right) \cdot \log t, \frac{1}{\mathsf{fma}\left(a, a, \mathsf{fma}\left(a, 0.5, 0.25\right)\right)}, \log \left(\left(x + y\right) \cdot z\right) - t\right)} \]

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(\log \left(x \cdot z\right) + \frac{\log t \cdot \left({a}^{3} - \frac{1}{8}\right)}{\frac{1}{4} + \left(\frac{1}{2} \cdot a + {a}^{2}\right)}\right) - t} \]
   6. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\log \left(x \cdot z\right) + \frac{\log t \cdot \left({a}^{3} - \frac{1}{8}\right)}{\frac{1}{4} + \left(\frac{1}{2} \cdot a + {a}^{2}\right)}\right) - t} \]

   7. Simplified 40.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log t, \frac{\mathsf{fma}\left(a, a \cdot a, -0.125\right)}{\mathsf{fma}\left(a, a + 0.5, 0.25\right)}, \log \left(z \cdot x\right)\right) - t} \]

   8. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. log-lowering-log.f64 99.9

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

   10. Simplified 99.9%

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

   if -9.9999999999999992e22 < (+.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 x around 0

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

      1. +-lowering-+.f64 N/A

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

      2. log-lowering-log.f64 N/A

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

      3. log-lowering-log.f64 54.1

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

   5. Simplified 54.1%

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

   6. Taylor expanded in a around 0

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

      1. associate--l+ N/A

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

      2. +-lowering-+.f64 N/A

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

      3. log-lowering-log.f64 N/A

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

      4. --lowering--.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. log-lowering-log.f64 N/A

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

      9. log-lowering-log.f64 54.1

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

   8. Simplified 54.1%

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

   if 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 99.6%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{\left(0 - t\right)} + \left(a - \frac{1}{2}\right) \cdot \log t \]

      3. --lowering--.f64 98.8

         \[\leadsto \color{blue}{\left(0 - t\right)} + \left(a - 0.5\right) \cdot \log t \]

   5. Simplified 98.8%

      \[\leadsto \color{blue}{\left(0 - t\right)} + \left(a - 0.5\right) \cdot \log t \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a - 0.5\right) \cdot \log t + \left(\left(\log z + \log \left(y + x\right)\right) - t\right) \leq -1 \cdot 10^{+23}:\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{elif}\;\left(a - 0.5\right) \cdot \log t + \left(\left(\log z + \log \left(y + x\right)\right) - t\right) \leq 2000:\\ \;\;\;\;\log y + \left(\mathsf{fma}\left(\log t, -0.5, \log z\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a - 0.5\right) \cdot \log t - t\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.4% accurate, 0.5× speedup

Math

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

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log z) (log (+ y x)))))
   (if (<= t_1 -700.0)
     (- (* a (log t)) t)
     (if (<= t_1 700.0)
       (fma (+ a -0.5) (log t) (- (log (* y z)) t))
       (- (* (- a 0.5) (log t)) t)))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = log(z) + log((y + x));
	double tmp;
	if (t_1 <= -700.0) {
		tmp = (a * log(t)) - t;
	} else if (t_1 <= 700.0) {
		tmp = fma((a + -0.5), log(t), (log((y * z)) - t));
	} else {
		tmp = ((a - 0.5) * log(t)) - t;
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(log(z) + log(Float64(y + x)))
	tmp = 0.0
	if (t_1 <= -700.0)
		tmp = Float64(Float64(a * log(t)) - t);
	elseif (t_1 <= 700.0)
		tmp = fma(Float64(a + -0.5), log(t), Float64(log(Float64(y * z)) - t));
	else
		tmp = Float64(Float64(Float64(a - 0.5) * log(t)) - t);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[z], $MachinePrecision] + N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -700.0], N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[t$95$1, 700.0], N[(N[(a + -0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[(N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -700

   1. Initial program 99.9%

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

      1. +-commutative N/A

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

      2. flip3-- N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 2.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(a, a \cdot a, -0.125\right) \cdot \log t, \frac{1}{\mathsf{fma}\left(a, a, \mathsf{fma}\left(a, 0.5, 0.25\right)\right)}, \log \left(\left(x + y\right) \cdot z\right) - t\right)} \]

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(\log \left(x \cdot z\right) + \frac{\log t \cdot \left({a}^{3} - \frac{1}{8}\right)}{\frac{1}{4} + \left(\frac{1}{2} \cdot a + {a}^{2}\right)}\right) - t} \]
   6. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\log \left(x \cdot z\right) + \frac{\log t \cdot \left({a}^{3} - \frac{1}{8}\right)}{\frac{1}{4} + \left(\frac{1}{2} \cdot a + {a}^{2}\right)}\right) - t} \]

   7. Simplified 2.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log t, \frac{\mathsf{fma}\left(a, a \cdot a, -0.125\right)}{\mathsf{fma}\left(a, a + 0.5, 0.25\right)}, \log \left(z \cdot x\right)\right) - t} \]

   8. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. log-lowering-log.f64 84.0

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

   10. Simplified 84.0%

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

   if -700 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 700

   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

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

      1. +-lowering-+.f64 N/A

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

      2. log-lowering-log.f64 N/A

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

      3. log-lowering-log.f64 68.0

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

   5. Simplified 68.0%

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-lowering-+.f64 N/A

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

      6. log-lowering-log.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. sum-log N/A

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

      9. log-lowering-log.f64 N/A

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

      10. *-lowering-*.f64 64.0

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

   7. Applied egg-rr 64.0%

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

   if 700 < (+.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. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{\left(0 - t\right)} + \left(a - \frac{1}{2}\right) \cdot \log t \]

      3. --lowering--.f64 92.3

         \[\leadsto \color{blue}{\left(0 - t\right)} + \left(a - 0.5\right) \cdot \log t \]

   5. Simplified 92.3%

      \[\leadsto \color{blue}{\left(0 - t\right)} + \left(a - 0.5\right) \cdot \log t \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log z + \log \left(y + x\right) \leq -700:\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{elif}\;\log z + \log \left(y + x\right) \leq 700:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, \log t, \log \left(y \cdot z\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a - 0.5\right) \cdot \log t - t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 380:\\ \;\;\;\;\mathsf{fma}\left(\log t, a + -0.5, \log z + \log \left(y + x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= t 380.0)
   (fma (log t) (+ a -0.5) (+ (log z) (log (+ y x))))
   (- (* a (log t)) t)))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 380.0) {
		tmp = fma(log(t), (a + -0.5), (log(z) + log((y + x))));
	} else {
		tmp = (a * log(t)) - t;
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 380.0)
		tmp = fma(log(t), Float64(a + -0.5), Float64(log(z) + log(Float64(y + x))));
	else
		tmp = Float64(Float64(a * log(t)) - t);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[t, 380.0], N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision] + N[(N[Log[z], $MachinePrecision] + N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 380

   1. Initial program 99.3%

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

   3. 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. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. log-lowering-log.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-lowering-+.f64 N/A

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

      8. +-lowering-+.f64 N/A

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

      9. log-lowering-log.f64 N/A

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

      10. log-lowering-log.f64 N/A

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

      11. +-commutative N/A

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

      12. +-lowering-+.f64 97.2

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

   5. Simplified 97.2%

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

   if 380 < t

   1. Initial program 99.9%

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

      1. +-commutative N/A

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

      2. flip3-- N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 56.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(a, a \cdot a, -0.125\right) \cdot \log t, \frac{1}{\mathsf{fma}\left(a, a, \mathsf{fma}\left(a, 0.5, 0.25\right)\right)}, \log \left(\left(x + y\right) \cdot z\right) - t\right)} \]

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(\log \left(x \cdot z\right) + \frac{\log t \cdot \left({a}^{3} - \frac{1}{8}\right)}{\frac{1}{4} + \left(\frac{1}{2} \cdot a + {a}^{2}\right)}\right) - t} \]
   6. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\left(\log \left(x \cdot z\right) + \frac{\log t \cdot \left({a}^{3} - \frac{1}{8}\right)}{\frac{1}{4} + \left(\frac{1}{2} \cdot a + {a}^{2}\right)}\right) - t} \]

   7. Simplified 40.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log t, \frac{\mathsf{fma}\left(a, a \cdot a, -0.125\right)}{\mathsf{fma}\left(a, a + 0.5, 0.25\right)}, \log \left(z \cdot x\right)\right) - t} \]

   8. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. log-lowering-log.f64 99.2

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

   10. Simplified 99.2%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 380:\\ \;\;\;\;\mathsf{fma}\left(\log t, a + -0.5, \log z + \log \left(y + x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 62.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := a \cdot \log t\\ \mathbf{if}\;a - 0.5 \leq -5 \cdot 10^{+39}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a - 0.5 \leq 5000000000000:\\ \;\;\;\;0 - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* a (log t))))
   (if (<= (- a 0.5) -5e+39)
     t_1
     (if (<= (- a 0.5) 5000000000000.0) (- 0.0 t) t_1))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = a * log(t);
	double tmp;
	if ((a - 0.5) <= -5e+39) {
		tmp = t_1;
	} else if ((a - 0.5) <= 5000000000000.0) {
		tmp = 0.0 - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
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 = a * log(t)
    if ((a - 0.5d0) <= (-5d+39)) then
        tmp = t_1
    else if ((a - 0.5d0) <= 5000000000000.0d0) then
        tmp = 0.0d0 - t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = a * Math.log(t);
	double tmp;
	if ((a - 0.5) <= -5e+39) {
		tmp = t_1;
	} else if ((a - 0.5) <= 5000000000000.0) {
		tmp = 0.0 - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	t_1 = a * math.log(t)
	tmp = 0
	if (a - 0.5) <= -5e+39:
		tmp = t_1
	elif (a - 0.5) <= 5000000000000.0:
		tmp = 0.0 - t
	else:
		tmp = t_1
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(a * log(t))
	tmp = 0.0
	if (Float64(a - 0.5) <= -5e+39)
		tmp = t_1;
	elseif (Float64(a - 0.5) <= 5000000000000.0)
		tmp = Float64(0.0 - t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = a * log(t);
	tmp = 0.0;
	if ((a - 0.5) <= -5e+39)
		tmp = t_1;
	elseif ((a - 0.5) <= 5000000000000.0)
		tmp = 0.0 - t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a - 0.5), $MachinePrecision], -5e+39], t$95$1, If[LessEqual[N[(a - 0.5), $MachinePrecision], 5000000000000.0], N[(0.0 - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 a #s(literal 1/2 binary64)) < -5.00000000000000015e39 or 5e12 < (-.f64 a #s(literal 1/2 binary64))

   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. *-lowering-*.f64 N/A

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

      3. log-lowering-log.f64 80.5

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

   5. Simplified 80.5%

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

   if -5.00000000000000015e39 < (-.f64 a #s(literal 1/2 binary64)) < 5e12

   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. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - t} \]

      3. --lowering--.f64 57.4

         \[\leadsto \color{blue}{0 - t} \]

   5. Simplified 57.4%

      \[\leadsto \color{blue}{0 - t} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 57.4

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

   7. Applied egg-rr 57.4%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -5 \cdot 10^{+39}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{elif}\;a - 0.5 \leq 5000000000000:\\ \;\;\;\;0 - t\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.5% accurate, 2.9× speedup

Math

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

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a) :precision binary64 (- (* (- a 0.5) (log t)) t))

C

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

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = ((a - 0.5d0) * log(t)) - t
end function

Java

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

Python

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

Julia

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

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp = code(x, y, z, t, a)
	tmp = ((a - 0.5) * log(t)) - t;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := N[(N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

      \[\leadsto \color{blue}{\left(0 - t\right)} + \left(a - \frac{1}{2}\right) \cdot \log t \]

   3. --lowering--.f64 76.5

      \[\leadsto \color{blue}{\left(0 - t\right)} + \left(a - 0.5\right) \cdot \log t \]

5. Simplified 76.5%

   \[\leadsto \color{blue}{\left(0 - t\right)} + \left(a - 0.5\right) \cdot \log t \]

6. Final simplification 76.5%

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

Alternative 7: 73.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ a \cdot \log t - t \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a) :precision binary64 (- (* a (log t)) t))

C

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

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = (a * log(t)) - t
end function

Java

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

Python

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

Julia

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

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp = code(x, y, z, t, a)
	tmp = (a * log(t)) - t;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. +-commutative N/A

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

   2. flip3-- N/A

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

   3. associate-*l/ N/A

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

   4. div-inv N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

4. Applied egg-rr 58.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(a, a \cdot a, -0.125\right) \cdot \log t, \frac{1}{\mathsf{fma}\left(a, a, \mathsf{fma}\left(a, 0.5, 0.25\right)\right)}, \log \left(\left(x + y\right) \cdot z\right) - t\right)} \]

5. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\left(\log \left(x \cdot z\right) + \frac{\log t \cdot \left({a}^{3} - \frac{1}{8}\right)}{\frac{1}{4} + \left(\frac{1}{2} \cdot a + {a}^{2}\right)}\right) - t} \]
6. Step-by-step derivation

   1. --lowering--.f64 N/A

      \[\leadsto \color{blue}{\left(\log \left(x \cdot z\right) + \frac{\log t \cdot \left({a}^{3} - \frac{1}{8}\right)}{\frac{1}{4} + \left(\frac{1}{2} \cdot a + {a}^{2}\right)}\right) - t} \]

7. Simplified 38.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\log t, \frac{\mathsf{fma}\left(a, a \cdot a, -0.125\right)}{\mathsf{fma}\left(a, a + 0.5, 0.25\right)}, \log \left(z \cdot x\right)\right) - t} \]

8. Taylor expanded in a around inf

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

   1. *-commutative N/A

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

   2. *-lowering-*.f64 N/A

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

   3. log-lowering-log.f64 73.7

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

10. Simplified 73.7%

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

11. Final simplification 73.7%

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

Alternative 8: 37.4% accurate, 80.3× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ 0 - t \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a) :precision binary64 (- 0.0 t))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	return 0.0 - t;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
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 = 0.0d0 - t
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	return 0.0 - t;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	return 0.0 - t

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	return Float64(0.0 - t)
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp = code(x, y, z, t, a)
	tmp = 0.0 - t;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := N[(0.0 - t), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

      \[\leadsto \color{blue}{0 - t} \]

   3. --lowering--.f64 43.2

      \[\leadsto \color{blue}{0 - t} \]

5. Simplified 43.2%

   \[\leadsto \color{blue}{0 - t} \]
6. Step-by-step derivation

   1. sub0-neg N/A

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

   2. neg-lowering-neg.f64 43.2

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

7. Applied egg-rr 43.2%

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

8. Final simplification 43.2%

   \[\leadsto 0 - t \]
9. 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 2024195 
(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))))