Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.1%

Time: 13.9s

Alternatives: 10

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: 99.1% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in t around inf

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. distribute-rgt-in N/A

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

   4. mul-1-neg N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-+.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-log.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-log.f64 N/A

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

   14. lower-neg.f64 99.5

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

5. Applied rewrites 99.5%

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

6. Final simplification 99.5%

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

Alternative 2: 92.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\log z + \log \left(y + x\right)\right) - t\right) + \log t \cdot \left(a - 0.5\right)\\ t_2 := \mathsf{fma}\left(a - 0.5, \log t, -t\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+19}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 1000:\\ \;\;\;\;\mathsf{fma}\left(\log t, -0.5, \log \left(\left(y + x\right) \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- (+ (log z) (log (+ y x))) t) (* (log t) (- a 0.5))))
        (t_2 (fma (- a 0.5) (log t) (- t))))
   (if (<= t_1 -2e+19)
     t_2
     (if (<= t_1 1000.0) (- (fma (log t) -0.5 (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))) - t) + (log(t) * (a - 0.5));
	double t_2 = fma((a - 0.5), log(t), -t);
	double tmp;
	if (t_1 <= -2e+19) {
		tmp = t_2;
	} else if (t_1 <= 1000.0) {
		tmp = fma(log(t), -0.5, log(((y + x) * z))) - t;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(log(z) + log(Float64(y + x))) - t) + Float64(log(t) * Float64(a - 0.5)))
	t_2 = fma(Float64(a - 0.5), log(t), Float64(-t))
	tmp = 0.0
	if (t_1 <= -2e+19)
		tmp = t_2;
	elseif (t_1 <= 1000.0)
		tmp = Float64(fma(log(t), -0.5, 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[(N[(N[Log[z], $MachinePrecision] + N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + (-t)), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+19], t$95$2, If[LessEqual[t$95$1, 1000.0], N[(N[(N[Log[t], $MachinePrecision] * -0.5 + N[Log[N[(N[(y + x), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < -2e19 or 1e3 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t)))

   1. Initial program 99.7%

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

   3. Taylor expanded in t around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-rgt-in N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-log.f64 N/A

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

      14. lower-neg.f64 99.7

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

   5. Applied rewrites 99.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.7

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

   7. Applied rewrites 67.2%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 94.3%

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

   if -2e19 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 1e3

   1. Initial program 98.8%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      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 96.2%

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

   5. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 95.3

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

   7. Applied rewrites 95.3%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\log z + \log \left(y + x\right)\right) - t\right) + \log t \cdot \left(a - 0.5\right) \leq -2 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, -t\right)\\ \mathbf{elif}\;\left(\left(\log z + \log \left(y + x\right)\right) - t\right) + \log t \cdot \left(a - 0.5\right) \leq 1000:\\ \;\;\;\;\mathsf{fma}\left(\log t, -0.5, \log \left(\left(y + x\right) \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, -t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log z + \log \left(y + x\right)\\ \mathbf{if}\;t\_1 \leq -578:\\ \;\;\;\;\frac{\log t}{\frac{1}{a - 0.5}} + \left(-t\right)\\ \mathbf{elif}\;t\_1 \leq 710:\\ \;\;\;\;\mathsf{fma}\left(\log t, a - 0.5, \log \left(\left(y + x\right) \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, -t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log z) (log (+ y x)))))
   (if (<= t_1 -578.0)
     (+ (/ (log t) (/ 1.0 (- a 0.5))) (- t))
     (if (<= t_1 710.0)
       (- (fma (log t) (- a 0.5) (log (* (+ y x) z))) t)
       (fma (- a 0.5) (log t) (- t))))))

C

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 <= -578.0) {
		tmp = (log(t) / (1.0 / (a - 0.5))) + -t;
	} else if (t_1 <= 710.0) {
		tmp = fma(log(t), (a - 0.5), log(((y + x) * z))) - t;
	} else {
		tmp = fma((a - 0.5), log(t), -t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(z) + log(Float64(y + x)))
	tmp = 0.0
	if (t_1 <= -578.0)
		tmp = Float64(Float64(log(t) / Float64(1.0 / Float64(a - 0.5))) + Float64(-t));
	elseif (t_1 <= 710.0)
		tmp = Float64(fma(log(t), Float64(a - 0.5), log(Float64(Float64(y + x) * z))) - t);
	else
		tmp = fma(Float64(a - 0.5), log(t), Float64(-t));
	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]}, If[LessEqual[t$95$1, -578.0], N[(N[(N[Log[t], $MachinePrecision] / N[(1.0 / N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + (-t)), $MachinePrecision], If[LessEqual[t$95$1, 710.0], N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision] + N[Log[N[(N[(y + x), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + (-t)), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -578

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. flip3-- N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. clear-num N/A

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

      9. flip3-- N/A

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

      10. lift--.f64 N/A

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

      11. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 92.4

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

   7. Applied rewrites 92.4%

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

   if -578 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 710

   1. Initial program 99.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+r- N/A

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

      5. lower--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 99.5

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

      9. lift-+.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. sum-log N/A

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

      13. lower-log.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 99.7

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 99.7

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

   4. Applied rewrites 99.7%

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

   if 710 < (+.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

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-rgt-in N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-log.f64 N/A

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

      14. lower-neg.f64 99.7

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

   5. Applied rewrites 99.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.7

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

   7. Applied rewrites 1.8%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 81.4%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log z + \log \left(y + x\right) \leq -578:\\ \;\;\;\;\frac{\log t}{\frac{1}{a - 0.5}} + \left(-t\right)\\ \mathbf{elif}\;\log z + \log \left(y + x\right) \leq 710:\\ \;\;\;\;\mathsf{fma}\left(\log t, a - 0.5, \log \left(\left(y + x\right) \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, -t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log z + \log \left(y + x\right)\\ \mathbf{if}\;t\_1 \leq -578:\\ \;\;\;\;\frac{\log t}{\frac{1}{a - 0.5}} + \left(-t\right)\\ \mathbf{elif}\;t\_1 \leq 710:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log \left(z \cdot y\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, -t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log z) (log (+ y x)))))
   (if (<= t_1 -578.0)
     (+ (/ (log t) (/ 1.0 (- a 0.5))) (- t))
     (if (<= t_1 710.0)
       (- (fma (- a 0.5) (log t) (log (* z y))) t)
       (fma (- a 0.5) (log t) (- t))))))

C

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 <= -578.0) {
		tmp = (log(t) / (1.0 / (a - 0.5))) + -t;
	} else if (t_1 <= 710.0) {
		tmp = fma((a - 0.5), log(t), log((z * y))) - t;
	} else {
		tmp = fma((a - 0.5), log(t), -t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(z) + log(Float64(y + x)))
	tmp = 0.0
	if (t_1 <= -578.0)
		tmp = Float64(Float64(log(t) / Float64(1.0 / Float64(a - 0.5))) + Float64(-t));
	elseif (t_1 <= 710.0)
		tmp = Float64(fma(Float64(a - 0.5), log(t), log(Float64(z * y))) - t);
	else
		tmp = fma(Float64(a - 0.5), log(t), Float64(-t));
	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]}, If[LessEqual[t$95$1, -578.0], N[(N[(N[Log[t], $MachinePrecision] / N[(1.0 / N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + (-t)), $MachinePrecision], If[LessEqual[t$95$1, 710.0], N[(N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + (-t)), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -578

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. flip3-- N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. clear-num N/A

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

      9. flip3-- N/A

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

      10. lift--.f64 N/A

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

      11. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 92.4

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

   7. Applied rewrites 92.4%

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

   if -578 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 710

   1. Initial program 99.5%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower-*.f64 60.3

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

   7. Applied rewrites 60.3%

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

   if 710 < (+.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

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-rgt-in N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-log.f64 N/A

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

      14. lower-neg.f64 99.7

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

   5. Applied rewrites 99.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.7

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

   7. Applied rewrites 1.8%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 81.4%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log z + \log \left(y + x\right) \leq -578:\\ \;\;\;\;\frac{\log t}{\frac{1}{a - 0.5}} + \left(-t\right)\\ \mathbf{elif}\;\log z + \log \left(y + x\right) \leq 710:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log \left(z \cdot y\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, -t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 140:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log \left(y + x\right)\right) + \log z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, -t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 140.0)
   (+ (fma (- a 0.5) (log t) (log (+ y x))) (log z))
   (fma (- a 0.5) (log t) (- t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 140.0) {
		tmp = fma((a - 0.5), log(t), log((y + x))) + log(z);
	} else {
		tmp = fma((a - 0.5), log(t), -t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 140.0)
		tmp = Float64(fma(Float64(a - 0.5), log(t), log(Float64(y + x))) + log(z));
	else
		tmp = fma(Float64(a - 0.5), log(t), Float64(-t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 140.0], N[(N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision], N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + (-t)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 140

   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 t around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-log.f64 98.2

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

   5. Applied rewrites 98.2%

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

   if 140 < 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

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-rgt-in N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-log.f64 N/A

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

      14. lower-neg.f64 99.8

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

   5. Applied rewrites 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.8

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

   7. Applied rewrites 68.8%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 99.4%

       \[\leadsto \mathsf{fma}\left(a - 0.5, \log t, -t\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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) + (log(t) * (a - 0.5d0))
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) + (Math.log(t) * (a - 0.5));
}

Python

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

Julia

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

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = ((log(z) + log((y + x))) - t) + (log(t) * (a - 0.5));
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[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

3. Final simplification 99.5%

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

Alternative 7: 68.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in y around inf

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

   1. associate-+r+ N/A

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

   2. associate--l+ N/A

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

   3. +-commutative N/A

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

   4. mul-1-neg N/A

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

   5. log-rec N/A

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

   6. remove-double-neg N/A

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

   7. associate--l+ N/A

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

   8. associate-+r+ N/A

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

   9. lower--.f64 N/A

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

5. Applied rewrites 67.7%

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

6. Final simplification 67.7%

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

Alternative 8: 60.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log t \cdot a\\ \mathbf{if}\;a - 0.5 \leq -5 \cdot 10^{+79}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a - 0.5 \leq 2 \cdot 10^{+73}:\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (log t) a)))
   (if (<= (- a 0.5) -5e+79) t_1 (if (<= (- a 0.5) 2e+73) (- t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log(t) * a;
	double tmp;
	if ((a - 0.5) <= -5e+79) {
		tmp = t_1;
	} else if ((a - 0.5) <= 2e+73) {
		tmp = -t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = log(t) * a
    if ((a - 0.5d0) <= (-5d+79)) then
        tmp = t_1
    else if ((a - 0.5d0) <= 2d+73) then
        tmp = -t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = Math.log(t) * a;
	double tmp;
	if ((a - 0.5) <= -5e+79) {
		tmp = t_1;
	} else if ((a - 0.5) <= 2e+73) {
		tmp = -t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = math.log(t) * a
	tmp = 0
	if (a - 0.5) <= -5e+79:
		tmp = t_1
	elif (a - 0.5) <= 2e+73:
		tmp = -t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(t) * a)
	tmp = 0.0
	if (Float64(a - 0.5) <= -5e+79)
		tmp = t_1;
	elseif (Float64(a - 0.5) <= 2e+73)
		tmp = Float64(-t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = log(t) * a;
	tmp = 0.0;
	if ((a - 0.5) <= -5e+79)
		tmp = t_1;
	elseif ((a - 0.5) <= 2e+73)
		tmp = -t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[N[(a - 0.5), $MachinePrecision], -5e+79], t$95$1, If[LessEqual[N[(a - 0.5), $MachinePrecision], 2e+73], (-t), t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 a #s(literal 1/2 binary64)) < -5e79 or 1.99999999999999997e73 < (-.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. lower-*.f64 N/A

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

      2. lower-log.f64 81.7

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

   5. Applied rewrites 81.7%

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

   if -5e79 < (-.f64 a #s(literal 1/2 binary64)) < 1.99999999999999997e73

   1. Initial program 99.5%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 50.7

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

   5. Applied rewrites 50.7%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -5 \cdot 10^{+79}:\\ \;\;\;\;\log t \cdot a\\ \mathbf{elif}\;a - 0.5 \leq 2 \cdot 10^{+73}:\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.5% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in t around inf

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. distribute-rgt-in N/A

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

   4. mul-1-neg N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-+.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-log.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-log.f64 N/A

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

   14. lower-neg.f64 99.5

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

5. Applied rewrites 99.5%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 99.5

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

7. Applied rewrites 74.2%

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

8. Taylor expanded in t around inf

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

10. Applied rewrites 75.2%

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

Alternative 10: 37.1% accurate, 107.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in 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 38.6

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

5. Applied rewrites 38.6%

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

Developer Target 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024255 
(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))))