Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.5%

Time: 12.7s

Alternatives: 14

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, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \left(\mathsf{fma}\left(-0.5 + a, \log t, \mathsf{fma}\left({\left(\frac{x}{y}\right)}^{3}, 0.3333333333333333, \frac{\mathsf{fma}\left(x \cdot \frac{x}{y}, -0.5, x\right)}{y}\right)\right) + \left(\log z + \log y\right)\right) - t \end{array} \]

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
 (-
  (+
   (fma
    (+ -0.5 a)
    (log t)
    (fma
     (pow (/ x y) 3.0)
     0.3333333333333333
     (/ (fma (* x (/ x y)) -0.5 x) y)))
   (+ (log z) (log y)))
  t))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	return (fma((-0.5 + a), log(t), fma(pow((x / y), 3.0), 0.3333333333333333, (fma((x * (x / y)), -0.5, x) / y))) + (log(z) + log(y))) - t;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	return Float64(Float64(fma(Float64(-0.5 + a), log(t), fma((Float64(x / y) ^ 3.0), 0.3333333333333333, Float64(fma(Float64(x * Float64(x / y)), -0.5, x) / y))) + Float64(log(z) + log(y))) - t)
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := N[(N[(N[(N[(-0.5 + a), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[(N[Power[N[(x / y), $MachinePrecision], 3.0], $MachinePrecision] * 0.3333333333333333 + N[(N[(N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision] * -0.5 + x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Log[z], $MachinePrecision] + N[Log[y], $MachinePrecision]), $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 y around inf

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

5. Applied rewrites 48.9%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-0.5 + a, \log t, \mathsf{fma}\left({\left(\frac{x}{y}\right)}^{3}, 0.3333333333333333, \frac{\mathsf{fma}\left(x \cdot \frac{x}{y}, -0.5, x\right)}{y}\right)\right) + \left(\log z + \log y\right)\right) - t} \]
6. Add Preprocessing

Alternative 2: 70.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ \mathbf{if}\;t\_1 \leq -500:\\ \;\;\;\;-t\\ \mathbf{elif}\;t\_1 \leq 700:\\ \;\;\;\;\log \left(\frac{\left(y + x\right) \cdot z}{\sqrt{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot a\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t)))))
   (if (<= t_1 -500.0)
     (- t)
     (if (<= t_1 700.0) (log (/ (* (+ y x) z) (sqrt t))) (* (log t) a)))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
	double tmp;
	if (t_1 <= -500.0) {
		tmp = -t;
	} else if (t_1 <= 700.0) {
		tmp = log((((y + x) * z) / sqrt(t)));
	} else {
		tmp = log(t) * a;
	}
	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 = ((log((x + y)) + log(z)) - t) + ((a - 0.5d0) * log(t))
    if (t_1 <= (-500.0d0)) then
        tmp = -t
    else if (t_1 <= 700.0d0) then
        tmp = log((((y + x) * z) / sqrt(t)))
    else
        tmp = log(t) * a
    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 = ((Math.log((x + y)) + Math.log(z)) - t) + ((a - 0.5) * Math.log(t));
	double tmp;
	if (t_1 <= -500.0) {
		tmp = -t;
	} else if (t_1 <= 700.0) {
		tmp = Math.log((((y + x) * z) / Math.sqrt(t)));
	} else {
		tmp = Math.log(t) * a;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	t_1 = ((math.log((x + y)) + math.log(z)) - t) + ((a - 0.5) * math.log(t))
	tmp = 0
	if t_1 <= -500.0:
		tmp = -t
	elif t_1 <= 700.0:
		tmp = math.log((((y + x) * z) / math.sqrt(t)))
	else:
		tmp = math.log(t) * a
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + Float64(Float64(a - 0.5) * log(t)))
	tmp = 0.0
	if (t_1 <= -500.0)
		tmp = Float64(-t);
	elseif (t_1 <= 700.0)
		tmp = log(Float64(Float64(Float64(y + x) * z) / sqrt(t)));
	else
		tmp = Float64(log(t) * a);
	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 = ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
	tmp = 0.0;
	if (t_1 <= -500.0)
		tmp = -t;
	elseif (t_1 <= 700.0)
		tmp = log((((y + x) * z) / sqrt(t)));
	else
		tmp = log(t) * a;
	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[(N[(N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -500.0], (-t), If[LessEqual[t$95$1, 700.0], N[Log[N[(N[(N[(y + x), $MachinePrecision] * z), $MachinePrecision] / N[Sqrt[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < -500

   1. Initial program 99.9%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 62.5

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

   5. Applied rewrites 62.5%

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

   if -500 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 700

   1. Initial program 98.8%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

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

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

      3. associate--l- N/A

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

      4. lower--.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-log.f64 98.3

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 97.1%

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

   10. Applied rewrites 98.3%

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

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

   1. Initial program 99.5%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 68.2

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

   5. Applied rewrites 68.2%

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

Alternative 3: 88.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;\log \left(x + y\right) + \log z \leq 680:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log \left(z \cdot \left(y + x\right)\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log z + \log y\right)\\ \end{array} \end{array} \]

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 (<= (+ (log (+ x y)) (log z)) 680.0)
   (fma (- a 0.5) (log t) (- (log (* z (+ y x))) t))
   (fma (- a 0.5) (log t) (+ (log z) (log y)))))

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 ((log((x + y)) + log(z)) <= 680.0) {
		tmp = fma((a - 0.5), log(t), (log((z * (y + x))) - t));
	} else {
		tmp = fma((a - 0.5), log(t), (log(z) + log(y)));
	}
	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(log(Float64(x + y)) + log(z)) <= 680.0)
		tmp = fma(Float64(a - 0.5), log(t), Float64(log(Float64(z * Float64(y + x))) - t));
	else
		tmp = fma(Float64(a - 0.5), log(t), Float64(log(z) + log(y)));
	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[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision], 680.0], N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[(N[Log[N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[(N[Log[z], $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 680

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.6

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

      5. lift-+.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. sum-log N/A

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

      9. lower-log.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 96.7

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 96.7

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

   4. Applied rewrites 96.7%

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

   if 680 < (+.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

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

      1. lower-log.f64 64.1

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

   5. Applied rewrites 64.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 64.1

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

      5. lift--.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate--l+ N/A

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

      8. lift--.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 64.1

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

   7. Applied rewrites 64.1%

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

   8. Taylor expanded in t around 0

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

      1. lower-log.f64 50.1

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

   10. Applied rewrites 50.1%

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

Alternative 4: 83.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.6

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

      5. lift-+.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. sum-log N/A

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

      9. lower-log.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 96.7

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 96.7

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

   4. Applied rewrites 96.7%

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

   if 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 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 45.6

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

   5. Applied rewrites 45.6%

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

Alternative 5: 87.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \log t \cdot a\\ \mathbf{if}\;a \leq -1 \cdot 10^{+20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\left(\log y + \log z\right) - \mathsf{fma}\left(0.5, \log t, t\right)\\ \mathbf{elif}\;a \leq 2.9 \cdot 10^{+91}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log \left(z \cdot \left(y + x\right)\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (* (log t) a)))
   (if (<= a -1e+20)
     t_1
     (if (<= a 5e-6)
       (- (+ (log y) (log z)) (fma 0.5 (log t) t))
       (if (<= a 2.9e+91)
         (fma (- a 0.5) (log t) (- (log (* z (+ y x))) 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 = log(t) * a;
	double tmp;
	if (a <= -1e+20) {
		tmp = t_1;
	} else if (a <= 5e-6) {
		tmp = (log(y) + log(z)) - fma(0.5, log(t), t);
	} else if (a <= 2.9e+91) {
		tmp = fma((a - 0.5), log(t), (log((z * (y + x))) - 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(log(t) * a)
	tmp = 0.0
	if (a <= -1e+20)
		tmp = t_1;
	elseif (a <= 5e-6)
		tmp = Float64(Float64(log(y) + log(z)) - fma(0.5, log(t), t));
	elseif (a <= 2.9e+91)
		tmp = fma(Float64(a - 0.5), log(t), Float64(log(Float64(z * Float64(y + x))) - t));
	else
		tmp = t_1;
	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[t], $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[a, -1e+20], t$95$1, If[LessEqual[a, 5e-6], N[(N[(N[Log[y], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - N[(0.5 * N[Log[t], $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 2.9e+91], N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[(N[Log[N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1e20 or 2.90000000000000014e91 < a

   1. Initial program 99.6%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 90.5

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

   5. Applied rewrites 90.5%

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

   if -1e20 < a < 5.00000000000000041e-6

   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 0

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

      1. associate-+r+ N/A

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

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

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

      3. associate--l- N/A

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

      4. lower--.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-log.f64 98.3

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 60.0%

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

   if 5.00000000000000041e-6 < a < 2.90000000000000014e91

   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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.8

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

      5. lift-+.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. sum-log N/A

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

      9. lower-log.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 86.2

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 86.2

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

   4. Applied rewrites 86.2%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{+20}:\\ \;\;\;\;\log t \cdot a\\ \mathbf{elif}\;a \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\left(\log y + \log z\right) - \mathsf{fma}\left(0.5, \log t, t\right)\\ \mathbf{elif}\;a \leq 2.9 \cdot 10^{+91}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log \left(z \cdot \left(y + x\right)\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

Alternative 7: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in x around 0

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

   1. lower-log.f64 65.3

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

5. Applied rewrites 65.3%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 65.3

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

   5. lift--.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. associate--l+ N/A

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

   8. lift--.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 65.3

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

7. Applied rewrites 65.3%

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

Alternative 8: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. lower-+.f64 N/A

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

   4. +-commutative N/A

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

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

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

   6. metadata-eval N/A

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

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

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

   8. distribute-rgt-out N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. lower-+.f64 N/A

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

   13. lower-log.f64 N/A

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

   14. lower-log.f64 N/A

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

   15. lower--.f64 N/A

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

   16. lower-log.f64 65.3

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

5. Applied rewrites 65.3%

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

Alternative 9: 83.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \log t \cdot a\\ \mathbf{if}\;a \leq -1 \cdot 10^{+20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 2.8 \cdot 10^{-6}:\\ \;\;\;\;\left(\log \left(\frac{z}{\sqrt{t}}\right) - t\right) + \log \left(y + x\right)\\ \mathbf{elif}\;a \leq 2.9 \cdot 10^{+91}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log \left(z \cdot \left(y + x\right)\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (* (log t) a)))
   (if (<= a -1e+20)
     t_1
     (if (<= a 2.8e-6)
       (+ (- (log (/ z (sqrt t))) t) (log (+ y x)))
       (if (<= a 2.9e+91)
         (fma (- a 0.5) (log t) (- (log (* z (+ y x))) 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 = log(t) * a;
	double tmp;
	if (a <= -1e+20) {
		tmp = t_1;
	} else if (a <= 2.8e-6) {
		tmp = (log((z / sqrt(t))) - t) + log((y + x));
	} else if (a <= 2.9e+91) {
		tmp = fma((a - 0.5), log(t), (log((z * (y + x))) - 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(log(t) * a)
	tmp = 0.0
	if (a <= -1e+20)
		tmp = t_1;
	elseif (a <= 2.8e-6)
		tmp = Float64(Float64(log(Float64(z / sqrt(t))) - t) + log(Float64(y + x)));
	elseif (a <= 2.9e+91)
		tmp = fma(Float64(a - 0.5), log(t), Float64(log(Float64(z * Float64(y + x))) - t));
	else
		tmp = t_1;
	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[t], $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[a, -1e+20], t$95$1, If[LessEqual[a, 2.8e-6], N[(N[(N[Log[N[(z / N[Sqrt[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision] + N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 2.9e+91], N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[(N[Log[N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1e20 or 2.90000000000000014e91 < a

   1. Initial program 99.6%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 90.5

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

   5. Applied rewrites 90.5%

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

   if -1e20 < a < 2.79999999999999987e-6

   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 0

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

      1. associate-+r+ N/A

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

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

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

      3. associate--l- N/A

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

      4. lower--.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-log.f64 98.3

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

   5. Applied rewrites 98.3%

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

   7. Applied rewrites 89.3%

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

   if 2.79999999999999987e-6 < a < 2.90000000000000014e91

   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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.8

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

      5. lift-+.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. sum-log N/A

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

      9. lower-log.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 86.2

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 86.2

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

   4. Applied rewrites 86.2%

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

Alternative 10: 74.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -4 \cdot 10^{+14} \lor \neg \left(a - 0.5 \leq -0.4\right):\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(y + x\right) \cdot z\right) - \mathsf{fma}\left(\log t, 0.5, t\right)\\ \end{array} \end{array} \]

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 (or (<= (- a 0.5) -4e+14) (not (<= (- a 0.5) -0.4)))
   (* (log t) a)
   (- (log (* (+ y x) z)) (fma (log t) 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) <= -4e+14) || !((a - 0.5) <= -0.4)) {
		tmp = log(t) * a;
	} else {
		tmp = log(((y + x) * z)) - fma(log(t), 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) <= -4e+14) || !(Float64(a - 0.5) <= -0.4))
		tmp = Float64(log(t) * a);
	else
		tmp = Float64(log(Float64(Float64(y + x) * z)) - fma(log(t), 0.5, 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[Or[LessEqual[N[(a - 0.5), $MachinePrecision], -4e+14], N[Not[LessEqual[N[(a - 0.5), $MachinePrecision], -0.4]], $MachinePrecision]], N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision], N[(N[Log[N[(N[(y + x), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision] - N[(N[Log[t], $MachinePrecision] * 0.5 + t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 a #s(literal 1/2 binary64)) < -4e14 or -0.40000000000000002 < (-.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 84.8

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

   5. Applied rewrites 84.8%

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

   if -4e14 < (-.f64 a #s(literal 1/2 binary64)) < -0.40000000000000002

   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 0

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

      1. associate-+r+ N/A

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

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

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

      3. associate--l- N/A

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

      4. lower--.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-log.f64 98.3

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

   5. Applied rewrites 98.3%

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

   7. Applied rewrites 74.3%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -4 \cdot 10^{+14} \lor \neg \left(a - 0.5 \leq -0.4\right):\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(y + x\right) \cdot z\right) - \mathsf{fma}\left(\log t, 0.5, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 74.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -4 \cdot 10^{+14} \lor \neg \left(a - 0.5 \leq -0.4\right):\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;\log \left(y \cdot z\right) - \mathsf{fma}\left(\log t, 0.5, t\right)\\ \end{array} \end{array} \]

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 (or (<= (- a 0.5) -4e+14) (not (<= (- a 0.5) -0.4)))
   (* (log t) a)
   (- (log (* y z)) (fma (log t) 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) <= -4e+14) || !((a - 0.5) <= -0.4)) {
		tmp = log(t) * a;
	} else {
		tmp = log((y * z)) - fma(log(t), 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) <= -4e+14) || !(Float64(a - 0.5) <= -0.4))
		tmp = Float64(log(t) * a);
	else
		tmp = Float64(log(Float64(y * z)) - fma(log(t), 0.5, 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[Or[LessEqual[N[(a - 0.5), $MachinePrecision], -4e+14], N[Not[LessEqual[N[(a - 0.5), $MachinePrecision], -0.4]], $MachinePrecision]], N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision], N[(N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision] - N[(N[Log[t], $MachinePrecision] * 0.5 + t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 a #s(literal 1/2 binary64)) < -4e14 or -0.40000000000000002 < (-.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 84.8

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

   5. Applied rewrites 84.8%

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

   if -4e14 < (-.f64 a #s(literal 1/2 binary64)) < -0.40000000000000002

   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 0

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

      1. associate-+r+ N/A

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

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

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

      3. associate--l- N/A

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

      4. lower--.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-log.f64 98.3

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

   5. Applied rewrites 98.3%

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

   7. Applied rewrites 74.3%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 46.3%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -4 \cdot 10^{+14} \lor \neg \left(a - 0.5 \leq -0.4\right):\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;\log \left(y \cdot z\right) - \mathsf{fma}\left(\log t, 0.5, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 71.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -4 \cdot 10^{+14} \lor \neg \left(a - 0.5 \leq -0.4\right):\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{\left(y + x\right) \cdot z}{\sqrt{t}}\right) - t\\ \end{array} \end{array} \]

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 (or (<= (- a 0.5) -4e+14) (not (<= (- a 0.5) -0.4)))
   (* (log t) a)
   (- (log (/ (* (+ y x) z) (sqrt 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 (((a - 0.5) <= -4e+14) || !((a - 0.5) <= -0.4)) {
		tmp = log(t) * a;
	} else {
		tmp = log((((y + x) * z) / sqrt(t))) - t;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (((a - 0.5d0) <= (-4d+14)) .or. (.not. ((a - 0.5d0) <= (-0.4d0)))) then
        tmp = log(t) * a
    else
        tmp = log((((y + x) * z) / sqrt(t))) - t
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((a - 0.5) <= -4e+14) || !((a - 0.5) <= -0.4)) {
		tmp = Math.log(t) * a;
	} else {
		tmp = Math.log((((y + x) * z) / Math.sqrt(t))) - t;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	tmp = 0
	if ((a - 0.5) <= -4e+14) or not ((a - 0.5) <= -0.4):
		tmp = math.log(t) * a
	else:
		tmp = math.log((((y + x) * z) / math.sqrt(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 ((Float64(a - 0.5) <= -4e+14) || !(Float64(a - 0.5) <= -0.4))
		tmp = Float64(log(t) * a);
	else
		tmp = Float64(log(Float64(Float64(Float64(y + x) * z) / sqrt(t))) - t);
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (((a - 0.5) <= -4e+14) || ~(((a - 0.5) <= -0.4)))
		tmp = log(t) * a;
	else
		tmp = log((((y + x) * z) / sqrt(t))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[Or[LessEqual[N[(a - 0.5), $MachinePrecision], -4e+14], N[Not[LessEqual[N[(a - 0.5), $MachinePrecision], -0.4]], $MachinePrecision]], N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision], N[(N[Log[N[(N[(N[(y + x), $MachinePrecision] * z), $MachinePrecision] / N[Sqrt[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 a #s(literal 1/2 binary64)) < -4e14 or -0.40000000000000002 < (-.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 84.8

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

   5. Applied rewrites 84.8%

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

   if -4e14 < (-.f64 a #s(literal 1/2 binary64)) < -0.40000000000000002

   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 0

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

      1. associate-+r+ N/A

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

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

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

      3. associate--l- N/A

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

      4. lower--.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-log.f64 98.3

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

   5. Applied rewrites 98.3%

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

   7. Applied rewrites 70.3%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -4 \cdot 10^{+14} \lor \neg \left(a - 0.5 \leq -0.4\right):\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{\left(y + x\right) \cdot z}{\sqrt{t}}\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 61.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{+14} \lor \neg \left(a \leq 710000000000\right):\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -5e+14) (not (<= a 710000000000.0))) (* (log t) a) (- t)))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -5e+14) || !(a <= 710000000000.0)) {
		tmp = log(t) * a;
	} else {
		tmp = -t;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((a <= (-5d+14)) .or. (.not. (a <= 710000000000.0d0))) then
        tmp = log(t) * a
    else
        tmp = -t
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -5e+14) || !(a <= 710000000000.0)) {
		tmp = Math.log(t) * a;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	tmp = 0
	if (a <= -5e+14) or not (a <= 710000000000.0):
		tmp = math.log(t) * a
	else:
		tmp = -t
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -5e+14) || !(a <= 710000000000.0))
		tmp = Float64(log(t) * a);
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -5e+14) || ~((a <= 710000000000.0)))
		tmp = log(t) * a;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -5e+14], N[Not[LessEqual[a, 710000000000.0]], $MachinePrecision]], N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if a < -5e14 or 7.1e11 < a

   1. Initial program 99.6%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 85.5

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

   5. Applied rewrites 85.5%

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

   if -5e14 < a < 7.1e11

   1. Initial program 99.5%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 53.9

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

   5. Applied rewrites 53.9%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{+14} \lor \neg \left(a \leq 710000000000\right):\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 37.4% 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 35.8

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

5. Applied rewrites 35.8%

   \[\leadsto \color{blue}{-t} \]
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 2024338 
(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))))