Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.5%

Time: 14.0s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

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

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

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	return Float64(log(y) + Float64(log(z) + fma(log(t), Float64(a + -0.5), Float64(Float64(x / 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[Log[y], $MachinePrecision] + N[(N[Log[z], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision] + N[(N[(x / y), $MachinePrecision] - t), $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 x around 0

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

   1. associate--l+ N/A

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

   2. lower-+.f64 N/A

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

   3. lower-log.f64 N/A

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

   4. associate--l+ N/A

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

   5. lower-+.f64 N/A

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

   6. lower-log.f64 N/A

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

   7. associate--l+ N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-log.f64 N/A

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

   12. lower-+.f64 N/A

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

   13. lower--.f64 N/A

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

   14. lower-/.f64 57.6

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

5. Applied rewrites 57.6%

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

Alternative 2: 94.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \log z + \log \left(x + y\right)\\ t_2 := \log y + \mathsf{fma}\left(\log t, a + -0.5, -t\right)\\ \mathbf{if}\;t\_1 \leq -750:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 700:\\ \;\;\;\;\log \left(z \cdot \left(x + y\right)\right) + \left(\log t \cdot \left(a + -0.5\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 y) (fma (log t) (+ a -0.5) (- t)))))
   (if (<= t_1 -750.0)
     t_2
     (if (<= t_1 700.0)
       (+ (log (* z (+ x y))) (- (* (log t) (+ a -0.5)) 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(y) + fma(log(t), (a + -0.5), -t);
	double tmp;
	if (t_1 <= -750.0) {
		tmp = t_2;
	} else if (t_1 <= 700.0) {
		tmp = log((z * (x + y))) + ((log(t) * (a + -0.5)) - 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(log(y) + fma(log(t), Float64(a + -0.5), Float64(-t)))
	tmp = 0.0
	if (t_1 <= -750.0)
		tmp = t_2;
	elseif (t_1 <= 700.0)
		tmp = Float64(log(Float64(z * Float64(x + y))) + Float64(Float64(log(t) * Float64(a + -0.5)) - 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[Log[y], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision] + (-t)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -750.0], t$95$2, If[LessEqual[t$95$1, 700.0], N[(N[Log[N[(z * N[(x + y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -750 or 700 < (+.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 58.7

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

   5. Applied rewrites 58.7%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 44.6%

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

   if -750 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 700

   1. Initial program 99.5%

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

      1. lift-+.f64 N/A

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

      2. 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 99.5

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

      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 99.5

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

   4. Applied rewrites 99.5%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log z + \log \left(x + y\right) \leq -750:\\ \;\;\;\;\log y + \mathsf{fma}\left(\log t, a + -0.5, -t\right)\\ \mathbf{elif}\;\log z + \log \left(x + y\right) \leq 700:\\ \;\;\;\;\log \left(z \cdot \left(x + y\right)\right) + \left(\log t \cdot \left(a + -0.5\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;\log y + \mathsf{fma}\left(\log t, a + -0.5, -t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \log z + \log \left(x + y\right)\\ t_2 := \log y + \mathsf{fma}\left(\log t, a + -0.5, -t\right)\\ \mathbf{if}\;t\_1 \leq -750:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 700:\\ \;\;\;\;\mathsf{fma}\left(\log t, a + -0.5, \log \left(y \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 y) (fma (log t) (+ a -0.5) (- t)))))
   (if (<= t_1 -750.0)
     t_2
     (if (<= t_1 700.0) (- (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(y) + fma(log(t), (a + -0.5), -t);
	double tmp;
	if (t_1 <= -750.0) {
		tmp = t_2;
	} else if (t_1 <= 700.0) {
		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(log(y) + fma(log(t), Float64(a + -0.5), Float64(-t)))
	tmp = 0.0
	if (t_1 <= -750.0)
		tmp = t_2;
	elseif (t_1 <= 700.0)
		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[Log[y], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision] + (-t)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -750.0], t$95$2, If[LessEqual[t$95$1, 700.0], 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)) < -750 or 700 < (+.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 58.7

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

   5. Applied rewrites 58.7%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 44.6%

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

   if -750 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 700

   1. Initial program 99.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. flip-- N/A

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

      6. associate-*l/ N/A

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

      7. div-inv N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. lower-+.f64 77.4

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

   4. Applied rewrites 77.4%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lift-+.f64 N/A

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

      7. associate-/r/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lift-fma.f64 75.8

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

   6. Applied rewrites 75.8%

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

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\log \left(y \cdot z\right) + \log t \cdot \left(a - \frac{1}{2}\right)\right) - t} \]
   8. 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 66.9

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

   9. Applied rewrites 66.9%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log z + \log \left(x + y\right) \leq -750:\\ \;\;\;\;\log y + \mathsf{fma}\left(\log t, a + -0.5, -t\right)\\ \mathbf{elif}\;\log z + \log \left(x + y\right) \leq 700:\\ \;\;\;\;\mathsf{fma}\left(\log t, a + -0.5, \log \left(y \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log y + \mathsf{fma}\left(\log t, a + -0.5, -t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -1:\\ \;\;\;\;\log y + \mathsf{fma}\left(\log t, a + -0.5, -t\right)\\ \mathbf{elif}\;a - 0.5 \leq -0.499:\\ \;\;\;\;\log y + \left(\mathsf{fma}\left(\log t, -0.5, \log z\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot \left(a - 0.5\right) + \left(-t\right)\\ \end{array} \end{array} \]

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
 (if (<= (- a 0.5) -1.0)
   (+ (log y) (fma (log t) (+ a -0.5) (- t)))
   (if (<= (- a 0.5) -0.499)
     (+ (log y) (- (fma (log t) -0.5 (log z)) t))
     (+ (* (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) {
	double tmp;
	if ((a - 0.5) <= -1.0) {
		tmp = log(y) + fma(log(t), (a + -0.5), -t);
	} else if ((a - 0.5) <= -0.499) {
		tmp = log(y) + (fma(log(t), -0.5, log(z)) - t);
	} else {
		tmp = (log(t) * (a - 0.5)) + -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 (Float64(a - 0.5) <= -1.0)
		tmp = Float64(log(y) + fma(log(t), Float64(a + -0.5), Float64(-t)));
	elseif (Float64(a - 0.5) <= -0.499)
		tmp = Float64(log(y) + Float64(fma(log(t), -0.5, log(z)) - t));
	else
		tmp = Float64(Float64(log(t) * Float64(a - 0.5)) + Float64(-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[N[(a - 0.5), $MachinePrecision], -1.0], N[(N[Log[y], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision] + (-t)), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(a - 0.5), $MachinePrecision], -0.499], N[(N[Log[y], $MachinePrecision] + N[(N[(N[Log[t], $MachinePrecision] * -0.5 + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision] + (-t)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 a #s(literal 1/2 binary64)) < -1

   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 71.6

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

   5. Applied rewrites 71.6%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 71.6%

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

   if -1 < (-.f64 a #s(literal 1/2 binary64)) < -0.499

   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 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 63.3

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

   5. Applied rewrites 63.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 61.3%

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

   if -0.499 < (-.f64 a #s(literal 1/2 binary64))

   1. Initial program 99.7%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 99.6

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

   5. Applied rewrites 99.6%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -1:\\ \;\;\;\;\log y + \mathsf{fma}\left(\log t, a + -0.5, -t\right)\\ \mathbf{elif}\;a - 0.5 \leq -0.499:\\ \;\;\;\;\log y + \left(\mathsf{fma}\left(\log t, -0.5, \log z\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot \left(a - 0.5\right) + \left(-t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 1.0× speedup

Math

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

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 z) (log (+ x y))) t) (* (log t) (- a 0.5))))

C

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

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Final simplification 99.6%

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

Alternative 6: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 29.5:\\ \;\;\;\;\log y + \mathsf{fma}\left(\log t, a + -0.5, \log z\right)\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot \left(a - 0.5\right) + \left(-t\right)\\ \end{array} \end{array} \]

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
 (if (<= t 29.5)
   (+ (log y) (fma (log t) (+ a -0.5) (log z)))
   (+ (* (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) {
	double tmp;
	if (t <= 29.5) {
		tmp = log(y) + fma(log(t), (a + -0.5), log(z));
	} else {
		tmp = (log(t) * (a - 0.5)) + -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 <= 29.5)
		tmp = Float64(log(y) + fma(log(t), Float64(a + -0.5), log(z)));
	else
		tmp = Float64(Float64(log(t) * Float64(a - 0.5)) + Float64(-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, 29.5], 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[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision] + (-t)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 29.5

   1. Initial program 99.2%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\log y + \left(\log z + \log t \cdot \left(a - \frac{1}{2}\right)\right)\right) - t} \]
   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 60.9

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

   5. Applied rewrites 60.9%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 60.9%

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

   if 29.5 < t

   1. Initial program 99.9%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{-1 \cdot t} + \left(a - \frac{1}{2}\right) \cdot \log t \]
   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 98.2

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

   5. Applied rewrites 98.2%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 29.5:\\ \;\;\;\;\log y + \mathsf{fma}\left(\log t, a + -0.5, \log z\right)\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot \left(a - 0.5\right) + \left(-t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.3% 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 67.6

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

5. Applied rewrites 67.6%

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

Alternative 8: 78.1% accurate, 1.5× speedup

Math

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

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) (- 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), -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(-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] + (-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. 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 67.6

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

5. Applied rewrites 67.6%

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

6. Taylor expanded in t around inf

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

8. Applied rewrites 54.5%

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

Alternative 9: 64.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := a \cdot \log t\\ \mathbf{if}\;a - 0.5 \leq -0.502:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a - 0.5 \leq 5 \cdot 10^{+91}:\\ \;\;\;\;\log y + \left(-t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 0.5) -0.502)
     t_1
     (if (<= (- a 0.5) 5e+91) (+ (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 - 0.5) <= -0.502) {
		tmp = t_1;
	} else if ((a - 0.5) <= 5e+91) {
		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 - 0.5d0) <= (-0.502d0)) then
        tmp = t_1
    else if ((a - 0.5d0) <= 5d+91) 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 - 0.5) <= -0.502) {
		tmp = t_1;
	} else if ((a - 0.5) <= 5e+91) {
		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 - 0.5) <= -0.502:
		tmp = t_1
	elif (a - 0.5) <= 5e+91:
		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 (Float64(a - 0.5) <= -0.502)
		tmp = t_1;
	elseif (Float64(a - 0.5) <= 5e+91)
		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 - 0.5) <= -0.502)
		tmp = t_1;
	elseif ((a - 0.5) <= 5e+91)
		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[N[(a - 0.5), $MachinePrecision], -0.502], t$95$1, If[LessEqual[N[(a - 0.5), $MachinePrecision], 5e+91], N[(N[Log[y], $MachinePrecision] + (-t)), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 a #s(literal 1/2 binary64)) < -0.502 or 5.0000000000000002e91 < (-.f64 a #s(literal 1/2 binary64))

   1. Initial program 99.6%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 83.5

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

   5. Applied rewrites 83.5%

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

   if -0.502 < (-.f64 a #s(literal 1/2 binary64)) < 5.0000000000000002e91

   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

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. associate--l+ N/A

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

      5. lower-+.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. associate--l+ N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 54.0

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

   5. Applied rewrites 54.0%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 40.5%

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

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -0.502:\\ \;\;\;\;a \cdot \log t\\ \mathbf{elif}\;a - 0.5 \leq 5 \cdot 10^{+91}:\\ \;\;\;\;\log y + \left(-t\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 61.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := a \cdot \log t\\ \mathbf{if}\;a - 0.5 \leq -0.5001:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a - 0.5 \leq 5 \cdot 10^{+91}:\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* a (log t))))
   (if (<= (- a 0.5) -0.5001) t_1 (if (<= (- a 0.5) 5e+91) (- t) t_1))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = a * log(t);
	double tmp;
	if ((a - 0.5) <= -0.5001) {
		tmp = t_1;
	} else if ((a - 0.5) <= 5e+91) {
		tmp = -t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = a * log(t)
    if ((a - 0.5d0) <= (-0.5001d0)) then
        tmp = t_1
    else if ((a - 0.5d0) <= 5d+91) then
        tmp = -t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = a * Math.log(t);
	double tmp;
	if ((a - 0.5) <= -0.5001) {
		tmp = t_1;
	} else if ((a - 0.5) <= 5e+91) {
		tmp = -t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	t_1 = a * math.log(t)
	tmp = 0
	if (a - 0.5) <= -0.5001:
		tmp = t_1
	elif (a - 0.5) <= 5e+91:
		tmp = -t
	else:
		tmp = t_1
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(a * log(t))
	tmp = 0.0
	if (Float64(a - 0.5) <= -0.5001)
		tmp = t_1;
	elseif (Float64(a - 0.5) <= 5e+91)
		tmp = 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 - 0.5) <= -0.5001)
		tmp = t_1;
	elseif ((a - 0.5) <= 5e+91)
		tmp = -t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a - 0.5), $MachinePrecision], -0.5001], t$95$1, If[LessEqual[N[(a - 0.5), $MachinePrecision], 5e+91], (-t), t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 a #s(literal 1/2 binary64)) < -0.50009999999999999 or 5.0000000000000002e91 < (-.f64 a #s(literal 1/2 binary64))

   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 82.8

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

   5. Applied rewrites 82.8%

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

   if -0.50009999999999999 < (-.f64 a #s(literal 1/2 binary64)) < 5.0000000000000002e91

   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 48.6

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

   5. Applied rewrites 48.6%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -0.5001:\\ \;\;\;\;a \cdot \log t\\ \mathbf{elif}\;a - 0.5 \leq 5 \cdot 10^{+91}:\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 77.6% 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 73.4

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

5. Applied rewrites 73.4%

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

6. Final simplification 73.4%

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

Alternative 12: 38.2% 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 34.9

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

5. Applied rewrites 34.9%

   \[\leadsto \color{blue}{-t} \]
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 2024222 
(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))))