Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.2%

Time: 13.4s

Alternatives: 10

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 1.0× speedup

Math

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

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
 (+ (log y) (fma (log t) (+ a -0.5) (- (log z) 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) + fma(log(t), (a + -0.5), (log(z) - t));
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	return Float64(log(y) + fma(log(t), Float64(a + -0.5), Float64(log(z) - 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[Log[y], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision] + N[(N[Log[z], $MachinePrecision] - 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

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

   1. associate--l+ N/A

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

   2. lower-+.f64 N/A

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

   3. lower-log.f64 N/A

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

   4. +-commutative N/A

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

   5. associate--l+ N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-log.f64 N/A

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

   8. sub-neg N/A

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

   9. metadata-eval N/A

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

   10. lower-+.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lower-log.f64 66.7

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

5. Applied rewrites 66.7%

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

Alternative 2: 91.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \left(\left(\log z + \log \left(x + y\right)\right) - t\right) + \log t \cdot \left(a - 0.5\right)\\ t_2 := a \cdot \log t\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+27}:\\ \;\;\;\;t\_2 - t\\ \mathbf{elif}\;t\_1 \leq 860:\\ \;\;\;\;\log \left(y \cdot z\right) - \mathsf{fma}\left(\log t, 0.5, t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + t\_2\right) - t\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- (+ (log z) (log (+ x y))) t) (* (log t) (- a 0.5))))
        (t_2 (* a (log t))))
   (if (<= t_1 -5e+27)
     (- t_2 t)
     (if (<= t_1 860.0)
       (- (log (* y z)) (fma (log t) 0.5 t))
       (- (+ (log z) t_2) t)))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = ((log(z) + log((x + y))) - t) + (log(t) * (a - 0.5));
	double t_2 = a * log(t);
	double tmp;
	if (t_1 <= -5e+27) {
		tmp = t_2 - t;
	} else if (t_1 <= 860.0) {
		tmp = log((y * z)) - fma(log(t), 0.5, t);
	} else {
		tmp = (log(z) + t_2) - t;
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(log(z) + log(Float64(x + y))) - t) + Float64(log(t) * Float64(a - 0.5)))
	t_2 = Float64(a * log(t))
	tmp = 0.0
	if (t_1 <= -5e+27)
		tmp = Float64(t_2 - t);
	elseif (t_1 <= 860.0)
		tmp = Float64(log(Float64(y * z)) - fma(log(t), 0.5, t));
	else
		tmp = Float64(Float64(log(z) + t_2) - 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[(N[(N[Log[z], $MachinePrecision] + N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+27], N[(t$95$2 - t), $MachinePrecision], If[LessEqual[t$95$1, 860.0], N[(N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision] - N[(N[Log[t], $MachinePrecision] * 0.5 + t), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[z], $MachinePrecision] + t$95$2), $MachinePrecision] - t), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < -4.99999999999999979e27

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-log.f64 72.2

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

   5. Applied rewrites 72.2%

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

   7. Applied rewrites 72.2%

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

   8. Taylor expanded in a around inf

      \[\leadsto a \cdot \log t - t \]
   9. Step-by-step derivation

   10. Applied rewrites 99.9%

       \[\leadsto \log t \cdot a - t \]

   if -4.99999999999999979e27 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 860

   1. Initial program 99.0%

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

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. sum-log N/A

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

      9. lower-log.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 92.5

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. lower-+.f64 N/A

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

      15. metadata-eval 92.5

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

   4. Applied rewrites 92.5%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 49.6

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

   7. Applied rewrites 49.6%

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

   8. Taylor expanded in a around 0

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-log.f64 46.5

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

   10. Applied rewrites 46.5%

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

   if 860 < (+.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 x around 0

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-log.f64 62.7

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

   5. Applied rewrites 62.7%

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

   7. Applied rewrites 62.6%

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

   8. Taylor expanded in a around inf

      \[\leadsto \left(a \cdot \log t + \log z\right) - t \]
   9. Step-by-step derivation

   10. Applied rewrites 82.9%

       \[\leadsto \left(\log t \cdot a + \log z\right) - t \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\log z + \log \left(x + y\right)\right) - t\right) + \log t \cdot \left(a - 0.5\right) \leq -5 \cdot 10^{+27}:\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{elif}\;\left(\left(\log z + \log \left(x + y\right)\right) - t\right) + \log t \cdot \left(a - 0.5\right) \leq 860:\\ \;\;\;\;\log \left(y \cdot z\right) - \mathsf{fma}\left(\log t, 0.5, t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + a \cdot \log t\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \log z + \log \left(x + y\right)\\ t_2 := \left(\log z + a \cdot \log t\right) - t\\ \mathbf{if}\;t\_1 \leq -800:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 608:\\ \;\;\;\;\mathsf{fma}\left(\log t, a + -0.5, \log \left(y \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log z) (log (+ x y)))) (t_2 (- (+ (log z) (* a (log t))) t)))
   (if (<= t_1 -800.0)
     t_2
     (if (<= t_1 608.0) (- (fma (log t) (+ a -0.5) (log (* y z))) t) t_2))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = log(z) + log((x + y));
	double t_2 = (log(z) + (a * log(t))) - t;
	double tmp;
	if (t_1 <= -800.0) {
		tmp = t_2;
	} else if (t_1 <= 608.0) {
		tmp = fma(log(t), (a + -0.5), log((y * z))) - t;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(log(z) + log(Float64(x + y)))
	t_2 = Float64(Float64(log(z) + Float64(a * log(t))) - t)
	tmp = 0.0
	if (t_1 <= -800.0)
		tmp = t_2;
	elseif (t_1 <= 608.0)
		tmp = Float64(fma(log(t), Float64(a + -0.5), log(Float64(y * z))) - t);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[z], $MachinePrecision] + N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Log[z], $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[t$95$1, -800.0], t$95$2, If[LessEqual[t$95$1, 608.0], N[(N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision] + N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -800 or 608 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z))

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-log.f64 58.4

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

   5. Applied rewrites 58.4%

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

   7. Applied rewrites 58.4%

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

   8. Taylor expanded in a around inf

      \[\leadsto \left(a \cdot \log t + \log z\right) - t \]
   9. Step-by-step derivation

   10. Applied rewrites 78.3%

       \[\leadsto \left(\log t \cdot a + \log z\right) - t \]

   if -800 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 608

   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

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. sum-log N/A

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

      9. lower-log.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 99.6

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. lower-+.f64 N/A

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

      15. metadata-eval 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-*.f64 67.1

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

   7. Applied rewrites 67.1%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log z + \log \left(x + y\right) \leq -800:\\ \;\;\;\;\left(\log z + a \cdot \log t\right) - t\\ \mathbf{elif}\;\log z + \log \left(x + y\right) \leq 608:\\ \;\;\;\;\mathsf{fma}\left(\log t, a + -0.5, \log \left(y \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + a \cdot \log t\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 10^{-17}:\\ \;\;\;\;\log y + \mathsf{fma}\left(\log t, a + -0.5, \log z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + a \cdot \log t\right) - t\\ \end{array} \end{array} \]

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 1e-17)
   (+ (log y) (fma (log t) (+ a -0.5) (log z)))
   (- (+ (log z) (* a (log t))) t)))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 1e-17) {
		tmp = log(y) + fma(log(t), (a + -0.5), log(z));
	} else {
		tmp = (log(z) + (a * log(t))) - t;
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 1e-17)
		tmp = Float64(log(y) + fma(log(t), Float64(a + -0.5), log(z)));
	else
		tmp = Float64(Float64(log(z) + Float64(a * log(t))) - t);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[t, 1e-17], N[(N[Log[y], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[z], $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.00000000000000007e-17

   1. Initial program 99.3%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-log.f64 58.3

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

   5. Applied rewrites 58.3%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 58.3%

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

   if 1.00000000000000007e-17 < t

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-log.f64 72.9

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

   5. Applied rewrites 72.9%

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

   7. Applied rewrites 72.9%

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

   8. Taylor expanded in a around inf

      \[\leadsto \left(a \cdot \log t + \log z\right) - t \]
   9. Step-by-step derivation

   10. Applied rewrites 98.5%

       \[\leadsto \left(\log t \cdot a + \log z\right) - t \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 10^{-17}:\\ \;\;\;\;\log y + \mathsf{fma}\left(\log t, a + -0.5, \log z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + a \cdot \log t\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \left(-t\right) + \left(a \cdot \log t\right) \cdot \left(1 + \frac{-0.5}{a}\right) \end{array} \]

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
 (+ (- t) (* (* a (log t)) (+ 1.0 (/ -0.5 a)))))

C

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

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 + ((a * log(t)) * (1.0d0 + ((-0.5d0) / a)))
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 + ((a * Math.log(t)) * (1.0 + (-0.5 / a)));
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	return -t + ((a * math.log(t)) * (1.0 + (-0.5 / a)))

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	return Float64(Float64(-t) + Float64(Float64(a * log(t)) * Float64(1.0 + Float64(-0.5 / a))))
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp = code(x, y, z, t, a)
	tmp = -t + ((a * log(t)) * (1.0 + (-0.5 / a)));
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[((-t) + N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-0.5 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in t around -inf

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

   1. mul-1-neg N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. *-lft-identity N/A

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

   5. distribute-neg-in N/A

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

   6. *-commutative N/A

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

   7. mul-1-neg N/A

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

   8. distribute-rgt-neg-out N/A

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

   9. remove-double-neg N/A

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

   10. lower-fma.f64 N/A

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

5. Applied rewrites 99.6%

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

6. Taylor expanded in a around inf

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

   1. *-lft-identity N/A

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

   2. *-commutative N/A

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

   3. associate-*r/ N/A

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

   4. *-commutative N/A

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

   5. associate-/l* N/A

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

   6. distribute-lft-in N/A

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

   7. metadata-eval N/A

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

   8. distribute-neg-frac N/A

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

   9. metadata-eval N/A

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

   10. associate-*r/ N/A

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

   11. sub-neg N/A

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

   12. associate-*r* N/A

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

   13. lower-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lower-log.f64 N/A

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

   17. sub-neg N/A

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

   18. associate-*r/ N/A

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

   19. metadata-eval N/A

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

   20. distribute-neg-frac N/A

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

   21. metadata-eval N/A

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

8. Applied rewrites 99.6%

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

9. Taylor expanded in t around inf

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

    1. mul-1-neg N/A

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

    2. lower-neg.f64 80.8

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

11. Applied rewrites 80.8%

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

12. Final simplification 80.8%

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

Alternative 6: 62.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := a \cdot \log t\\ \mathbf{if}\;a \leq -2.2 \cdot 10^{+113}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 2.85 \cdot 10^{+85}:\\ \;\;\;\;\log y + \left(-t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* a (log t))))
   (if (<= a -2.2e+113) t_1 (if (<= a 2.85e+85) (+ (log y) (- t)) t_1))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = a * log(t);
	double tmp;
	if (a <= -2.2e+113) {
		tmp = t_1;
	} else if (a <= 2.85e+85) {
		tmp = log(y) + -t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = a * log(t)
    if (a <= (-2.2d+113)) then
        tmp = t_1
    else if (a <= 2.85d+85) then
        tmp = log(y) + -t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = a * Math.log(t);
	double tmp;
	if (a <= -2.2e+113) {
		tmp = t_1;
	} else if (a <= 2.85e+85) {
		tmp = Math.log(y) + -t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	t_1 = a * math.log(t)
	tmp = 0
	if a <= -2.2e+113:
		tmp = t_1
	elif a <= 2.85e+85:
		tmp = math.log(y) + -t
	else:
		tmp = t_1
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(a * log(t))
	tmp = 0.0
	if (a <= -2.2e+113)
		tmp = t_1;
	elseif (a <= 2.85e+85)
		tmp = Float64(log(y) + Float64(-t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = a * log(t);
	tmp = 0.0;
	if (a <= -2.2e+113)
		tmp = t_1;
	elseif (a <= 2.85e+85)
		tmp = log(y) + -t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -2.2e+113], t$95$1, If[LessEqual[a, 2.85e+85], N[(N[Log[y], $MachinePrecision] + (-t)), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -2.2000000000000001e113 or 2.8500000000000001e85 < a

   1. Initial program 99.7%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 87.7

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

   5. Applied rewrites 87.7%

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

   if -2.2000000000000001e113 < a < 2.8500000000000001e85

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-log.f64 63.7

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

   5. Applied rewrites 63.7%

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

   6. Taylor expanded in t around inf

      \[\leadsto \log y + -1 \cdot \color{blue}{t} \]
   7. Step-by-step derivation

   8. Applied rewrites 43.9%

      \[\leadsto \log y + \left(-t\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.2 \cdot 10^{+113}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{elif}\;a \leq 2.85 \cdot 10^{+85}:\\ \;\;\;\;\log y + \left(-t\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.6% accurate, 2.8× speedup

Math

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

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

Python

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

Julia

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 80.8

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

5. Applied rewrites 80.8%

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

6. Final simplification 80.8%

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

Alternative 8: 61.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 1.4 \cdot 10^{+49}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

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 (<= t 1.4e+49) (* a (log t)) (- t)))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 1.4e+49) {
		tmp = a * log(t);
	} 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 (t <= 1.4d+49) then
        tmp = a * log(t)
    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 (t <= 1.4e+49) {
		tmp = a * Math.log(t);
	} 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 t <= 1.4e+49:
		tmp = a * math.log(t)
	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 (t <= 1.4e+49)
		tmp = Float64(a * log(t));
	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 (t <= 1.4e+49)
		tmp = a * log(t);
	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[LessEqual[t, 1.4e+49], N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if t < 1.3999999999999999e49

   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 53.5

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

   5. Applied rewrites 53.5%

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

   if 1.3999999999999999e49 < t

   1. Initial program 100.0%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   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 81.7

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

   5. Applied rewrites 81.7%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.4 \cdot 10^{+49}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.0% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in x around 0

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

   1. associate--l+ N/A

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

   2. lower-+.f64 N/A

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

   3. lower-log.f64 N/A

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

   4. +-commutative N/A

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

   5. associate--l+ N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-log.f64 N/A

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

   8. sub-neg N/A

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

   9. metadata-eval N/A

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

   10. lower-+.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lower-log.f64 66.7

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

5. Applied rewrites 66.7%

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

7. Applied rewrites 66.7%

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

8. Taylor expanded in a around inf

   \[\leadsto a \cdot \log t - t \]
9. Step-by-step derivation

10. Applied rewrites 78.0%

    \[\leadsto \log t \cdot a - t \]

11. Final simplification 78.0%

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

Alternative 10: 38.3% 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 41.2

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

5. Applied rewrites 41.2%

   \[\leadsto \color{blue}{-t} \]
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 2024233 
(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))))