Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.3%

Time: 12.2s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   1. associate-*r* N/A

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

   2. mul-1-neg N/A

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

   3. +-commutative N/A

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

   4. distribute-lft-in N/A

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

   5. *-rgt-identity N/A

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

   6. distribute-lft-neg-in N/A

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

   7. *-commutative N/A

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

   8. mul-1-neg N/A

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

   9. distribute-lft-neg-out N/A

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

   10. remove-double-neg N/A

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

   11. lower-fma.f64 N/A

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

5. Applied rewrites 99.6%

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

6. Final simplification 99.6%

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

Alternative 2: 90.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log z + \log \left(y + x\right)\\ t_2 := \mathsf{fma}\left(\frac{\log t}{t} \cdot a, t, -t\right)\\ \mathbf{if}\;t\_1 \leq -750:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 720:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log \left(\left(y + x\right) \cdot z\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log z) (log (+ y x))))
        (t_2 (fma (* (/ (log t) t) a) t (- t))))
   (if (<= t_1 -750.0)
     t_2
     (if (<= t_1 720.0)
       (fma (- a 0.5) (log t) (- (log (* (+ y x) z)) t))
       t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log(z) + log((y + x));
	double t_2 = fma(((log(t) / t) * a), t, -t);
	double tmp;
	if (t_1 <= -750.0) {
		tmp = t_2;
	} else if (t_1 <= 720.0) {
		tmp = fma((a - 0.5), log(t), (log(((y + x) * z)) - t));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(z) + log(Float64(y + x)))
	t_2 = fma(Float64(Float64(log(t) / t) * a), t, Float64(-t))
	tmp = 0.0
	if (t_1 <= -750.0)
		tmp = t_2;
	elseif (t_1 <= 720.0)
		tmp = fma(Float64(a - 0.5), log(t), Float64(log(Float64(Float64(y + x) * z)) - t));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[z], $MachinePrecision] + N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Log[t], $MachinePrecision] / t), $MachinePrecision] * a), $MachinePrecision] * t + (-t)), $MachinePrecision]}, If[LessEqual[t$95$1, -750.0], t$95$2, If[LessEqual[t$95$1, 720.0], N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[(N[Log[N[(N[(y + x), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -750 or 720 < (+.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

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

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

   5. Applied rewrites 95.8%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 72.1%

      \[\leadsto \mathsf{fma}\left(\frac{\log t}{t} \cdot a, t, -t\right) \]

   if -750 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 720

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      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 99.2

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

      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 99.2

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

   4. Applied rewrites 99.2%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log z + \log \left(y + x\right) \leq -750:\\ \;\;\;\;\mathsf{fma}\left(\frac{\log t}{t} \cdot a, t, -t\right)\\ \mathbf{elif}\;\log z + \log \left(y + x\right) \leq 720:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log \left(\left(y + x\right) \cdot z\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\log t}{t} \cdot a, t, -t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 64.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log z + \log \left(y + x\right)\\ t_2 := \mathsf{fma}\left(\frac{\log t}{t} \cdot a, t, -t\right)\\ \mathbf{if}\;t\_1 \leq -750:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 720:\\ \;\;\;\;\mathsf{fma}\left(\log t, a - 0.5, \log \left(z \cdot y\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log z) (log (+ y x))))
        (t_2 (fma (* (/ (log t) t) a) t (- t))))
   (if (<= t_1 -750.0)
     t_2
     (if (<= t_1 720.0) (- (fma (log t) (- a 0.5) (log (* z y))) t) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log(z) + log((y + x));
	double t_2 = fma(((log(t) / t) * a), t, -t);
	double tmp;
	if (t_1 <= -750.0) {
		tmp = t_2;
	} else if (t_1 <= 720.0) {
		tmp = fma(log(t), (a - 0.5), log((z * y))) - t;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(z) + log(Float64(y + x)))
	t_2 = fma(Float64(Float64(log(t) / t) * a), t, Float64(-t))
	tmp = 0.0
	if (t_1 <= -750.0)
		tmp = t_2;
	elseif (t_1 <= 720.0)
		tmp = Float64(fma(log(t), Float64(a - 0.5), log(Float64(z * y))) - t);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[z], $MachinePrecision] + N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Log[t], $MachinePrecision] / t), $MachinePrecision] * a), $MachinePrecision] * t + (-t)), $MachinePrecision]}, If[LessEqual[t$95$1, -750.0], t$95$2, If[LessEqual[t$95$1, 720.0], N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision] + N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -750 or 720 < (+.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

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

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

   5. Applied rewrites 95.8%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 72.1%

      \[\leadsto \mathsf{fma}\left(\frac{\log t}{t} \cdot a, t, -t\right) \]

   if -750 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 720

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 61.0

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

   7. Applied rewrites 61.0%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log z + \log \left(y + x\right) \leq -750:\\ \;\;\;\;\mathsf{fma}\left(\frac{\log t}{t} \cdot a, t, -t\right)\\ \mathbf{elif}\;\log z + \log \left(y + x\right) \leq 720:\\ \;\;\;\;\mathsf{fma}\left(\log t, a - 0.5, \log \left(z \cdot y\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\log t}{t} \cdot a, t, -t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Final simplification 99.6%

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

Alternative 5: 80.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 10.5:\\ \;\;\;\;\mathsf{fma}\left(-0.5 + a, \log t, \log z\right) + \log y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\log t}{t} \cdot a, t, -t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 10.5)
   (+ (fma (+ -0.5 a) (log t) (log z)) (log y))
   (fma (* (/ (log t) t) a) t (- t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 10.5) {
		tmp = fma((-0.5 + a), log(t), log(z)) + log(y);
	} else {
		tmp = fma(((log(t) / t) * a), t, -t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 10.5)
		tmp = Float64(fma(Float64(-0.5 + a), log(t), log(z)) + log(y));
	else
		tmp = fma(Float64(Float64(log(t) / t) * a), t, Float64(-t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 10.5], N[(N[(N[(-0.5 + a), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Log[t], $MachinePrecision] / t), $MachinePrecision] * a), $MachinePrecision] * t + (-t)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 10.5

   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 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 53.0

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

   5. Applied rewrites 53.0%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 52.5%

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

   if 10.5 < t

   1. Initial program 99.9%

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

   3. Taylor expanded in t around inf

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

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 98.8%

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

Alternative 6: 69.0% 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 63.4

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

5. Applied rewrites 63.4%

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

Alternative 7: 73.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.35 \cdot 10^{-18}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log \left(z \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\log t}{t} \cdot a, t, -t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 1.35e-18)
   (fma (- a 0.5) (log t) (log (* z y)))
   (fma (* (/ (log t) t) a) t (- t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 1.35e-18) {
		tmp = fma((a - 0.5), log(t), log((z * y)));
	} else {
		tmp = fma(((log(t) / t) * a), t, -t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 1.35e-18)
		tmp = fma(Float64(a - 0.5), log(t), log(Float64(z * y)));
	else
		tmp = fma(Float64(Float64(log(t) / t) * a), t, Float64(-t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 1.35e-18], N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Log[t], $MachinePrecision] / t), $MachinePrecision] * a), $MachinePrecision] * t + (-t)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.34999999999999994e-18

   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 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 52.1

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

   5. Applied rewrites 52.1%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 52.1%

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

   10. Applied rewrites 43.4%

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

   if 1.34999999999999994e-18 < t

   1. Initial program 99.9%

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

   3. Taylor expanded in t around inf

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

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 97.6%

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

Alternative 8: 73.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 5.9 \cdot 10^{-52}:\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\log t}{t} \cdot a, t, -t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 5.9e-52) (* (log t) a) (fma (* (/ (log t) t) a) t (- t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 5.9e-52) {
		tmp = log(t) * a;
	} else {
		tmp = fma(((log(t) / t) * a), t, -t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 5.9e-52)
		tmp = Float64(log(t) * a);
	else
		tmp = fma(Float64(Float64(log(t) / t) * a), t, Float64(-t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 5.9e-52], N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision], N[(N[(N[(N[Log[t], $MachinePrecision] / t), $MachinePrecision] * a), $MachinePrecision] * t + (-t)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 5.90000000000000019e-52

   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 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 47.9

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

   5. Applied rewrites 47.9%

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

   if 5.90000000000000019e-52 < 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. Taylor expanded in t around inf

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

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 94.0%

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

Alternative 9: 62.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.25 \cdot 10^{+40}:\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 1.25e+40) (* (log t) a) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 1.25e+40) {
		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 (t <= 1.25d+40) 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 (t <= 1.25e+40) {
		tmp = Math.log(t) * a;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 1.25e+40:
		tmp = math.log(t) * a
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 1.25e+40)
		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 (t <= 1.25e+40)
		tmp = log(t) * a;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 1.25e+40], N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if t < 1.25000000000000001e40

   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 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 52.3

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

   5. Applied rewrites 52.3%

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

   if 1.25000000000000001e40 < t

   1. Initial program 99.9%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 79.9

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

   5. Applied rewrites 79.9%

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

Alternative 10: 37.5% accurate, 107.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 39.3

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

5. Applied rewrites 39.3%

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

Alternative 11: 2.4% accurate, 321.0× 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 t
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 39.3

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

5. Applied rewrites 39.3%

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

7. Applied rewrites 17.4%

   \[\leadsto \frac{0 - t \cdot t}{\color{blue}{0 + t}} \]

9. Applied rewrites 2.3%

   \[\leadsto \color{blue}{t} \]
10. 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 2024296 
(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))))