Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.3%

Time: 11.9s

Alternatives: 10

Speedup: N/A×

• Could not converge on a ground truth

  1. x = 2.699162526849881e-273
  2. y = 2.248521185053621e+71
  3. z = 2.2113130524210697e-251
  4. t = 29784896915.54911
  5. a = 1.8077626701764913e+219

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in x around 0

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

   1. lower-log.f64 66.5

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

5. Applied rewrites 66.5%

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

Alternative 2: 86.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ \mathbf{if}\;t\_1 \leq -200000:\\ \;\;\;\;\mathsf{fma}\left(\left(-\log t\right) \cdot \frac{a - 0.5}{t}, -1, -1\right) \cdot t\\ \mathbf{elif}\;t\_1 \leq 1100:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log t, \log \left(y \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot a\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t)))))
   (if (<= t_1 -200000.0)
     (* (fma (* (- (log t)) (/ (- a 0.5) t)) -1.0 -1.0) t)
     (if (<= t_1 1100.0)
       (- (fma -0.5 (log t) (log (* y z))) t)
       (* (log t) a)))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
	double tmp;
	if (t_1 <= -200000.0) {
		tmp = fma((-log(t) * ((a - 0.5) / t)), -1.0, -1.0) * t;
	} else if (t_1 <= 1100.0) {
		tmp = fma(-0.5, log(t), log((y * z))) - t;
	} else {
		tmp = log(t) * a;
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + Float64(Float64(a - 0.5) * log(t)))
	tmp = 0.0
	if (t_1 <= -200000.0)
		tmp = Float64(fma(Float64(Float64(-log(t)) * Float64(Float64(a - 0.5) / t)), -1.0, -1.0) * t);
	elseif (t_1 <= 1100.0)
		tmp = Float64(fma(-0.5, log(t), log(Float64(y * z))) - t);
	else
		tmp = Float64(log(t) * a);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(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]}, If[LessEqual[t$95$1, -200000.0], N[(N[(N[((-N[Log[t], $MachinePrecision]) * N[(N[(a - 0.5), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * -1.0 + -1.0), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[t$95$1, 1100.0], N[(N[(-0.5 * N[Log[t], $MachinePrecision] + N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < -2e5

   1. Initial program 99.8%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. flip-+ N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. pow2 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower--.f64 N/A

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

      11. pow2 N/A

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

      12. lower-pow.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower--.f64 67.0

          \[\leadsto \frac{{\left(\log z - t\right)}^{2} - {\log \left(y + x\right)}^{2}}{\color{blue}{\left(\log z - t\right)} - \log \left(x + y\right)} + \left(a - 0.5\right) \cdot \log t \]

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 67.0

          \[\leadsto \frac{{\left(\log z - t\right)}^{2} - {\log \left(y + x\right)}^{2}}{\left(\log z - t\right) - \log \color{blue}{\left(y + x\right)}} + \left(a - 0.5\right) \cdot \log t \]

   4. Applied rewrites 67.0%

      \[\leadsto \color{blue}{\frac{{\left(\log z - t\right)}^{2} - {\log \left(y + x\right)}^{2}}{\left(\log z - t\right) - \log \left(y + x\right)}} + \left(a - 0.5\right) \cdot \log t \]

   5. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 91.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(-\log t\right) \cdot \frac{a - 0.5}{t}, -1, \frac{\mathsf{fma}\left(2, \log z, \log \left(y + x\right)\right)}{t} - \left(\frac{\log z}{t} + 1\right)\right) \cdot t} \]

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 90.3%

       \[\leadsto \mathsf{fma}\left(\left(-\log t\right) \cdot \frac{a - 0.5}{t}, -1, -1\right) \cdot t \]

   if -2e5 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 1100

   1. Initial program 99.1%

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

   3. Taylor expanded in x around 0

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

      1. lower-log.f64 50.1

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

   5. Applied rewrites 50.1%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-log.f64 50.0

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

   8. Applied rewrites 50.0%

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

   9. Taylor expanded in a around 0

      \[\leadsto \left(\mathsf{fma}\left(\log t, \frac{-1}{2}, \log z\right) + \log y\right) - t \]
   10. Step-by-step derivation

   11. Applied rewrites 49.9%

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

   13. Applied rewrites 44.5%

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

   if 1100 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t)))

   1. Initial program 99.6%

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

   3. Taylor expanded in 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 84.2

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

   5. Applied rewrites 84.2%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \leq -200000:\\ \;\;\;\;\mathsf{fma}\left(\left(-\log t\right) \cdot \frac{a - 0.5}{t}, -1, -1\right) \cdot t\\ \mathbf{elif}\;\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \leq 1100:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log t, \log \left(y \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ \mathbf{if}\;t\_1 \leq -200000:\\ \;\;\;\;\mathsf{fma}\left(\left(-\log t\right) \cdot \frac{a - 0.5}{t}, -1, -1\right) \cdot t\\ \mathbf{elif}\;t\_1 \leq 700:\\ \;\;\;\;\log \left({t}^{-0.5} \cdot \left(y \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot a\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t)))))
   (if (<= t_1 -200000.0)
     (* (fma (* (- (log t)) (/ (- a 0.5) t)) -1.0 -1.0) t)
     (if (<= t_1 700.0) (- (log (* (pow t -0.5) (* y z))) t) (* (log t) a)))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
	double tmp;
	if (t_1 <= -200000.0) {
		tmp = fma((-log(t) * ((a - 0.5) / t)), -1.0, -1.0) * t;
	} else if (t_1 <= 700.0) {
		tmp = log((pow(t, -0.5) * (y * z))) - t;
	} else {
		tmp = log(t) * a;
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + Float64(Float64(a - 0.5) * log(t)))
	tmp = 0.0
	if (t_1 <= -200000.0)
		tmp = Float64(fma(Float64(Float64(-log(t)) * Float64(Float64(a - 0.5) / t)), -1.0, -1.0) * t);
	elseif (t_1 <= 700.0)
		tmp = Float64(log(Float64((t ^ -0.5) * Float64(y * z))) - t);
	else
		tmp = Float64(log(t) * a);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(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]}, If[LessEqual[t$95$1, -200000.0], N[(N[(N[((-N[Log[t], $MachinePrecision]) * N[(N[(a - 0.5), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * -1.0 + -1.0), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[t$95$1, 700.0], N[(N[Log[N[(N[Power[t, -0.5], $MachinePrecision] * N[(y * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision], N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < -2e5

   1. Initial program 99.8%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. flip-+ N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. pow2 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower--.f64 N/A

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

      11. pow2 N/A

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

      12. lower-pow.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower--.f64 67.0

          \[\leadsto \frac{{\left(\log z - t\right)}^{2} - {\log \left(y + x\right)}^{2}}{\color{blue}{\left(\log z - t\right)} - \log \left(x + y\right)} + \left(a - 0.5\right) \cdot \log t \]

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 67.0

          \[\leadsto \frac{{\left(\log z - t\right)}^{2} - {\log \left(y + x\right)}^{2}}{\left(\log z - t\right) - \log \color{blue}{\left(y + x\right)}} + \left(a - 0.5\right) \cdot \log t \]

   4. Applied rewrites 67.0%

      \[\leadsto \color{blue}{\frac{{\left(\log z - t\right)}^{2} - {\log \left(y + x\right)}^{2}}{\left(\log z - t\right) - \log \left(y + x\right)}} + \left(a - 0.5\right) \cdot \log t \]

   5. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 91.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(-\log t\right) \cdot \frac{a - 0.5}{t}, -1, \frac{\mathsf{fma}\left(2, \log z, \log \left(y + x\right)\right)}{t} - \left(\frac{\log z}{t} + 1\right)\right) \cdot t} \]

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 90.3%

       \[\leadsto \mathsf{fma}\left(\left(-\log t\right) \cdot \frac{a - 0.5}{t}, -1, -1\right) \cdot t \]

   if -2e5 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 700

   1. Initial program 99.1%

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

   3. Taylor expanded in x around 0

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

      1. lower-log.f64 52.0

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

   5. Applied rewrites 52.0%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-log.f64 51.7

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

   8. Applied rewrites 51.7%

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

   9. Taylor expanded in a around 0

      \[\leadsto \left(\mathsf{fma}\left(\log t, \frac{-1}{2}, \log z\right) + \log y\right) - t \]
   10. Step-by-step derivation

   11. Applied rewrites 51.7%

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

   13. Applied rewrites 51.6%

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

   if 700 < (+.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 99.5%

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

   3. Taylor expanded in a around 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 72.2

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

   5. Applied rewrites 72.2%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \leq -200000:\\ \;\;\;\;\mathsf{fma}\left(\left(-\log t\right) \cdot \frac{a - 0.5}{t}, -1, -1\right) \cdot t\\ \mathbf{elif}\;\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \leq 700:\\ \;\;\;\;\log \left({t}^{-0.5} \cdot \left(y \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \log \left(x + y\right) + \log z\\ \mathbf{if}\;t\_1 \leq -750 \lor \neg \left(t\_1 \leq 640\right):\\ \;\;\;\;\mathsf{fma}\left(\left(-\log t\right) \cdot \frac{a - 0.5}{t}, -1, -1\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log \left(z \cdot y\right) - t\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log (+ x y)) (log z))))
   (if (or (<= t_1 -750.0) (not (<= t_1 640.0)))
     (* (fma (* (- (log t)) (/ (- a 0.5) t)) -1.0 -1.0) t)
     (fma (- a 0.5) (log t) (- (log (* z y)) t)))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = log((x + y)) + log(z);
	double tmp;
	if ((t_1 <= -750.0) || !(t_1 <= 640.0)) {
		tmp = fma((-log(t) * ((a - 0.5) / t)), -1.0, -1.0) * t;
	} else {
		tmp = fma((a - 0.5), log(t), (log((z * y)) - t));
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(log(Float64(x + y)) + log(z))
	tmp = 0.0
	if ((t_1 <= -750.0) || !(t_1 <= 640.0))
		tmp = Float64(fma(Float64(Float64(-log(t)) * Float64(Float64(a - 0.5) / t)), -1.0, -1.0) * t);
	else
		tmp = fma(Float64(a - 0.5), log(t), Float64(log(Float64(z * y)) - t));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -750.0], N[Not[LessEqual[t$95$1, 640.0]], $MachinePrecision]], N[(N[(N[((-N[Log[t], $MachinePrecision]) * N[(N[(a - 0.5), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * -1.0 + -1.0), $MachinePrecision] * t), $MachinePrecision], N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[(N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -750 or 640 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z))

   1. Initial program 99.7%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. flip-+ N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. pow2 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower--.f64 N/A

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

      11. pow2 N/A

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

      12. lower-pow.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower--.f64 78.0

          \[\leadsto \frac{{\left(\log z - t\right)}^{2} - {\log \left(y + x\right)}^{2}}{\color{blue}{\left(\log z - t\right)} - \log \left(x + y\right)} + \left(a - 0.5\right) \cdot \log t \]

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 78.0

          \[\leadsto \frac{{\left(\log z - t\right)}^{2} - {\log \left(y + x\right)}^{2}}{\left(\log z - t\right) - \log \color{blue}{\left(y + x\right)}} + \left(a - 0.5\right) \cdot \log t \]

   4. Applied rewrites 78.0%

      \[\leadsto \color{blue}{\frac{{\left(\log z - t\right)}^{2} - {\log \left(y + x\right)}^{2}}{\left(\log z - t\right) - \log \left(y + x\right)}} + \left(a - 0.5\right) \cdot \log t \]

   5. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 96.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(-\log t\right) \cdot \frac{a - 0.5}{t}, -1, \frac{\mathsf{fma}\left(2, \log z, \log \left(y + x\right)\right)}{t} - \left(\frac{\log z}{t} + 1\right)\right) \cdot t} \]

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 74.5%

       \[\leadsto \mathsf{fma}\left(\left(-\log t\right) \cdot \frac{a - 0.5}{t}, -1, -1\right) \cdot t \]

   if -750 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 640

   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

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

      2. lift--.f64 N/A

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

      3. flip3-- N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. metadata-eval 62.7

          \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \frac{\left({a}^{3} - 0.125\right) \cdot \log t}{\mathsf{fma}\left(a, a, \mathsf{fma}\left(0.5, a, \color{blue}{0.25}\right)\right)} \]

   4. Applied rewrites 62.7%

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \color{blue}{\frac{\left({a}^{3} - 0.125\right) \cdot \log t}{\mathsf{fma}\left(a, a, \mathsf{fma}\left(0.5, a, 0.25\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift--.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. metadata-eval N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. flip3-- N/A

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

      15. lift--.f64 N/A

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 64.0

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

   9. Applied rewrites 64.0%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(x + y\right) + \log z \leq -750 \lor \neg \left(\log \left(x + y\right) + \log z \leq 640\right):\\ \;\;\;\;\mathsf{fma}\left(\left(-\log t\right) \cdot \frac{a - 0.5}{t}, -1, -1\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log \left(z \cdot y\right) - t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 155:\\ \;\;\;\;\mathsf{fma}\left(\log t, a - 0.5, \log z\right) + \log y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-\log t\right) \cdot \frac{a - 0.5}{t}, -1, -1\right) \cdot t\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= t 155.0)
   (+ (fma (log t) (- a 0.5) (log z)) (log y))
   (* (fma (* (- (log t)) (/ (- a 0.5) t)) -1.0 -1.0) t)))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 155.0) {
		tmp = fma(log(t), (a - 0.5), log(z)) + log(y);
	} else {
		tmp = fma((-log(t) * ((a - 0.5) / t)), -1.0, -1.0) * t;
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 155.0)
		tmp = Float64(fma(log(t), Float64(a - 0.5), log(z)) + log(y));
	else
		tmp = Float64(fma(Float64(Float64(-log(t)) * Float64(Float64(a - 0.5) / t)), -1.0, -1.0) * t);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[t, 155.0], N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision], N[(N[(N[((-N[Log[t], $MachinePrecision]) * N[(N[(a - 0.5), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * -1.0 + -1.0), $MachinePrecision] * t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 155

   1. Initial program 99.4%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-out-- N/A

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

      5. metadata-eval N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. +-commutative N/A

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

      8. distribute-rgt-out N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. lower-log.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-log.f64 98.4

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in x 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 58.9%

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

   if 155 < t

   1. Initial program 99.8%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. flip-+ N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. pow2 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower--.f64 N/A

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

      11. pow2 N/A

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

      12. lower-pow.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower--.f64 57.1

          \[\leadsto \frac{{\left(\log z - t\right)}^{2} - {\log \left(y + x\right)}^{2}}{\color{blue}{\left(\log z - t\right)} - \log \left(x + y\right)} + \left(a - 0.5\right) \cdot \log t \]

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 57.1

          \[\leadsto \frac{{\left(\log z - t\right)}^{2} - {\log \left(y + x\right)}^{2}}{\left(\log z - t\right) - \log \color{blue}{\left(y + x\right)}} + \left(a - 0.5\right) \cdot \log t \]

   4. Applied rewrites 57.1%

      \[\leadsto \color{blue}{\frac{{\left(\log z - t\right)}^{2} - {\log \left(y + x\right)}^{2}}{\left(\log z - t\right) - \log \left(y + x\right)}} + \left(a - 0.5\right) \cdot \log t \]

   5. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(-\log t\right) \cdot \frac{a - 0.5}{t}, -1, \frac{\mathsf{fma}\left(2, \log z, \log \left(y + x\right)\right)}{t} - \left(\frac{\log z}{t} + 1\right)\right) \cdot t} \]

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 98.4%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 155:\\ \;\;\;\;\mathsf{fma}\left(\log t, a - 0.5, \log z\right) + \log y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-\log t\right) \cdot \frac{a - 0.5}{t}, -1, -1\right) \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. associate--l+ N/A

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

   3. lower-+.f64 N/A

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

   4. +-commutative N/A

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

   5. distribute-rgt-out-- N/A

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

   6. metadata-eval N/A

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

   7. fp-cancel-sign-sub-inv N/A

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

   8. distribute-rgt-out N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. lower-+.f64 N/A

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

   13. lower-log.f64 N/A

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

   14. lower-log.f64 N/A

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

   15. lower--.f64 N/A

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

   16. lower-log.f64 66.5

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

5. Applied rewrites 66.5%

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

Alternative 7: 85.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 2.8 \cdot 10^{-33}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 + a, \log t, \log \left(\left(y + x\right) \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-\log t\right) \cdot \frac{a - 0.5}{t}, -1, -1\right) \cdot t\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= t 2.8e-33)
   (fma (+ -0.5 a) (log t) (log (* (+ y x) z)))
   (* (fma (* (- (log t)) (/ (- a 0.5) t)) -1.0 -1.0) t)))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 2.8e-33) {
		tmp = fma((-0.5 + a), log(t), log(((y + x) * z)));
	} else {
		tmp = fma((-log(t) * ((a - 0.5) / t)), -1.0, -1.0) * t;
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 2.8e-33)
		tmp = fma(Float64(-0.5 + a), log(t), log(Float64(Float64(y + x) * z)));
	else
		tmp = Float64(fma(Float64(Float64(-log(t)) * Float64(Float64(a - 0.5) / t)), -1.0, -1.0) * t);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[t, 2.8e-33], N[(N[(-0.5 + a), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[Log[N[(N[(y + x), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[((-N[Log[t], $MachinePrecision]) * N[(N[(a - 0.5), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * -1.0 + -1.0), $MachinePrecision] * t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.8e-33

   1. Initial program 99.3%

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

   3. Taylor expanded in t around 0

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

      1. +-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. distribute-rgt-out-- N/A

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

      5. metadata-eval N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. +-commutative N/A

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

      8. distribute-rgt-out N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. lower-log.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-log.f64 99.2

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

   5. Applied rewrites 99.2%

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

   7. Applied rewrites 78.6%

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

   if 2.8e-33 < t

   1. Initial program 99.8%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. flip-+ N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. pow2 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower--.f64 N/A

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

      11. pow2 N/A

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

      12. lower-pow.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower--.f64 61.9

          \[\leadsto \frac{{\left(\log z - t\right)}^{2} - {\log \left(y + x\right)}^{2}}{\color{blue}{\left(\log z - t\right)} - \log \left(x + y\right)} + \left(a - 0.5\right) \cdot \log t \]

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 61.9

          \[\leadsto \frac{{\left(\log z - t\right)}^{2} - {\log \left(y + x\right)}^{2}}{\left(\log z - t\right) - \log \color{blue}{\left(y + x\right)}} + \left(a - 0.5\right) \cdot \log t \]

   4. Applied rewrites 61.9%

      \[\leadsto \color{blue}{\frac{{\left(\log z - t\right)}^{2} - {\log \left(y + x\right)}^{2}}{\left(\log z - t\right) - \log \left(y + x\right)}} + \left(a - 0.5\right) \cdot \log t \]

   5. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(-\log t\right) \cdot \frac{a - 0.5}{t}, -1, \frac{\mathsf{fma}\left(2, \log z, \log \left(y + x\right)\right)}{t} - \left(\frac{\log z}{t} + 1\right)\right) \cdot t} \]

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 95.7%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.8 \cdot 10^{-33}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 + a, \log t, \log \left(\left(y + x\right) \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-\log t\right) \cdot \frac{a - 0.5}{t}, -1, -1\right) \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 2.15 \cdot 10^{-49}:\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-\log t\right) \cdot \frac{a - 0.5}{t}, -1, -1\right) \cdot t\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= t 2.15e-49)
   (* (log t) a)
   (* (fma (* (- (log t)) (/ (- a 0.5) t)) -1.0 -1.0) t)))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 2.15e-49) {
		tmp = log(t) * a;
	} else {
		tmp = fma((-log(t) * ((a - 0.5) / t)), -1.0, -1.0) * t;
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 2.15e-49)
		tmp = Float64(log(t) * a);
	else
		tmp = Float64(fma(Float64(Float64(-log(t)) * Float64(Float64(a - 0.5) / t)), -1.0, -1.0) * t);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[t, 2.15e-49], N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision], N[(N[(N[((-N[Log[t], $MachinePrecision]) * N[(N[(a - 0.5), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * -1.0 + -1.0), $MachinePrecision] * t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.15000000000000008e-49

   1. Initial program 99.4%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 53.6

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

   5. Applied rewrites 53.6%

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

   if 2.15000000000000008e-49 < t

   1. Initial program 99.8%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. flip-+ N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. pow2 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower--.f64 N/A

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

      11. pow2 N/A

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

      12. lower-pow.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower--.f64 63.7

          \[\leadsto \frac{{\left(\log z - t\right)}^{2} - {\log \left(y + x\right)}^{2}}{\color{blue}{\left(\log z - t\right)} - \log \left(x + y\right)} + \left(a - 0.5\right) \cdot \log t \]

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 63.7

          \[\leadsto \frac{{\left(\log z - t\right)}^{2} - {\log \left(y + x\right)}^{2}}{\left(\log z - t\right) - \log \color{blue}{\left(y + x\right)}} + \left(a - 0.5\right) \cdot \log t \]

   4. Applied rewrites 63.7%

      \[\leadsto \color{blue}{\frac{{\left(\log z - t\right)}^{2} - {\log \left(y + x\right)}^{2}}{\left(\log z - t\right) - \log \left(y + x\right)}} + \left(a - 0.5\right) \cdot \log t \]

   5. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(-\log t\right) \cdot \frac{a - 0.5}{t}, -1, \frac{\mathsf{fma}\left(2, \log z, \log \left(y + x\right)\right)}{t} - \left(\frac{\log z}{t} + 1\right)\right) \cdot t} \]

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 92.9%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.15 \cdot 10^{-49}:\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-\log t\right) \cdot \frac{a - 0.5}{t}, -1, -1\right) \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -6.5 \cdot 10^{+27} \lor \neg \left(a \leq 6 \cdot 10^{+47}\right):\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -6.5e+27) (not (<= a 6e+47))) (* (log t) a) (- t)))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -6.5e+27) || !(a <= 6e+47)) {
		tmp = log(t) * a;
	} else {
		tmp = -t;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((a <= (-6.5d+27)) .or. (.not. (a <= 6d+47))) then
        tmp = log(t) * a
    else
        tmp = -t
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -6.5e+27) || !(a <= 6e+47)) {
		tmp = Math.log(t) * a;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	tmp = 0
	if (a <= -6.5e+27) or not (a <= 6e+47):
		tmp = math.log(t) * a
	else:
		tmp = -t
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -6.5e+27) || !(a <= 6e+47))
		tmp = Float64(log(t) * a);
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -6.5e+27) || ~((a <= 6e+47)))
		tmp = log(t) * a;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -6.5e+27], N[Not[LessEqual[a, 6e+47]], $MachinePrecision]], N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if a < -6.5000000000000005e27 or 6.0000000000000003e47 < a

   1. Initial program 99.6%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 83.9

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

   5. Applied rewrites 83.9%

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

   if -6.5000000000000005e27 < a < 6.0000000000000003e47

   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 52.1

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

   5. Applied rewrites 52.1%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6.5 \cdot 10^{+27} \lor \neg \left(a \leq 6 \cdot 10^{+47}\right):\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 37.6% accurate, 107.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 36.1

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

5. Applied rewrites 36.1%

   \[\leadsto \color{blue}{-t} \]
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 2024326 
(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))))