Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 16.5s

Alternatives: 16

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: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

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. Final simplification 99.7%

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

Alternative 2: 94.2% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log (+ x y)) (log z))) (t_2 (* (log t) (- a 0.5))))
   (if (<= t_1 -750.0)
     (- (* a (log t)) t)
     (if (<= t_1 700.0) (- (+ (log (* (+ x y) z)) t_2) t) (- t_2 t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((x + y)) + log(z);
	double t_2 = log(t) * (a - 0.5);
	double tmp;
	if (t_1 <= -750.0) {
		tmp = (a * log(t)) - t;
	} else if (t_1 <= 700.0) {
		tmp = (log(((x + y) * z)) + t_2) - t;
	} else {
		tmp = t_2 - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = log((x + y)) + log(z)
    t_2 = log(t) * (a - 0.5d0)
    if (t_1 <= (-750.0d0)) then
        tmp = (a * log(t)) - t
    else if (t_1 <= 700.0d0) then
        tmp = (log(((x + y) * z)) + t_2) - t
    else
        tmp = t_2 - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = Math.log((x + y)) + Math.log(z);
	double t_2 = Math.log(t) * (a - 0.5);
	double tmp;
	if (t_1 <= -750.0) {
		tmp = (a * Math.log(t)) - t;
	} else if (t_1 <= 700.0) {
		tmp = (Math.log(((x + y) * z)) + t_2) - t;
	} else {
		tmp = t_2 - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = math.log((x + y)) + math.log(z)
	t_2 = math.log(t) * (a - 0.5)
	tmp = 0
	if t_1 <= -750.0:
		tmp = (a * math.log(t)) - t
	elif t_1 <= 700.0:
		tmp = (math.log(((x + y) * z)) + t_2) - t
	else:
		tmp = t_2 - t
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(Float64(x + y)) + log(z))
	t_2 = Float64(log(t) * Float64(a - 0.5))
	tmp = 0.0
	if (t_1 <= -750.0)
		tmp = Float64(Float64(a * log(t)) - t);
	elseif (t_1 <= 700.0)
		tmp = Float64(Float64(log(Float64(Float64(x + y) * z)) + t_2) - t);
	else
		tmp = Float64(t_2 - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = log((x + y)) + log(z);
	t_2 = log(t) * (a - 0.5);
	tmp = 0.0;
	if (t_1 <= -750.0)
		tmp = (a * log(t)) - t;
	elseif (t_1 <= 700.0)
		tmp = (log(((x + y) * z)) + t_2) - t;
	else
		tmp = t_2 - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -750.0], N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[t$95$1, 700.0], N[(N[(N[Log[N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision] + t$95$2), $MachinePrecision] - t), $MachinePrecision], N[(t$95$2 - t), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\color{blue}{\left(\log y + \log z\right)}, t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\left(\log z + \log y\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\color{blue}{a}, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      2. remove-double-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\left(\log z + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log y\right)\right)\right)\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      3. log-rec N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\left(\log z + \left(\mathsf{neg}\left(\log \left(\frac{1}{y}\right)\right)\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\left(\log z + -1 \cdot \log \left(\frac{1}{y}\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\log z, \left(-1 \cdot \log \left(\frac{1}{y}\right)\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\color{blue}{a}, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      6. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \left(-1 \cdot \log \left(\frac{1}{y}\right)\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \left(\mathsf{neg}\left(\log \left(\frac{1}{y}\right)\right)\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      8. log-rec N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log y\right)\right)\right)\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      9. remove-double-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \log y\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      10. log-lowering-log.f64 49.7%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{log.f64}\left(y\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

   5. Simplified 49.7%

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

      1. +-commutative N/A

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

      2. associate-+r- N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(a - \frac{1}{2}\right) \cdot \log t + \left(\log z + \log y\right)\right), \color{blue}{t}\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(\left(a - \frac{1}{2}\right) \cdot \log t\right), \left(\log z + \log y\right)\right), t\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{2}\right), \log t\right), \left(\log z + \log y\right)\right), t\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log t\right), \left(\log z + \log y\right)\right), t\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a + \frac{-1}{2}\right), \log t\right), \left(\log z + \log y\right)\right), t\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{2}\right), \log t\right), \left(\log z + \log y\right)\right), t\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{2}\right), \mathsf{log.f64}\left(t\right)\right), \left(\log z + \log y\right)\right), t\right) \]

      10. sum-log N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{2}\right), \mathsf{log.f64}\left(t\right)\right), \log \left(z \cdot y\right)\right), t\right) \]

      11. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{2}\right), \mathsf{log.f64}\left(t\right)\right), \mathsf{log.f64}\left(\left(z \cdot y\right)\right)\right), t\right) \]

      12. *-lowering-*.f64 4.0%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{2}\right), \mathsf{log.f64}\left(t\right)\right), \mathsf{log.f64}\left(\mathsf{*.f64}\left(z, y\right)\right)\right), t\right) \]

   7. Applied egg-rr 4.0%

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

   8. Taylor expanded in a around inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(a \cdot \log t\right)}, t\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\log t \cdot a\right), t\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\log t, a\right), t\right) \]

      3. log-lowering-log.f64 99.5%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), a\right), t\right) \]

   10. Simplified 99.5%

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

   if -750 < (+.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. Step-by-step derivation

      1. associate--l+ N/A

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

      2. associate-+l+ N/A

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

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\log \left(x + y\right), \color{blue}{\left(\left(\log z - t\right) + \left(a - \frac{1}{2}\right) \cdot \log t\right)}\right) \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\left(x + y\right)\right), \left(\color{blue}{\left(\log z - t\right)} + \left(a - \frac{1}{2}\right) \cdot \log t\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \left(\left(\color{blue}{\log z} - t\right) + \left(a - \frac{1}{2}\right) \cdot \log t\right)\right) \]

      6. associate-+l- N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \left(\log z - \color{blue}{\left(t - \left(a - \frac{1}{2}\right) \cdot \log t\right)}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\log z, \color{blue}{\left(t - \left(a - \frac{1}{2}\right) \cdot \log t\right)}\right)\right) \]

      8. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \left(\color{blue}{t} - \left(a - \frac{1}{2}\right) \cdot \log t\right)\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \left(t + \color{blue}{\left(\mathsf{neg}\left(\left(a - \frac{1}{2}\right) \cdot \log t\right)\right)}\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \color{blue}{\left(\mathsf{neg}\left(\left(a - \frac{1}{2}\right) \cdot \log t\right)\right)}\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \left(\mathsf{neg}\left(\log t \cdot \left(a - \frac{1}{2}\right)\right)\right)\right)\right)\right) \]

      12. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \left(\log t \cdot \color{blue}{\left(\mathsf{neg}\left(\left(a - \frac{1}{2}\right)\right)\right)}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\log t, \color{blue}{\left(\mathsf{neg}\left(\left(a - \frac{1}{2}\right)\right)\right)}\right)\right)\right)\right) \]

      14. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(\mathsf{neg}\left(\color{blue}{\left(a - \frac{1}{2}\right)}\right)\right)\right)\right)\right)\right) \]

      15. neg-sub0 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(0 - \color{blue}{\left(a - \frac{1}{2}\right)}\right)\right)\right)\right)\right) \]

      16. associate--r- N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(\left(0 - a\right) + \color{blue}{\frac{1}{2}}\right)\right)\right)\right)\right) \]

      17. neg-sub0 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(\left(\mathsf{neg}\left(a\right)\right) + \frac{1}{2}\right)\right)\right)\right)\right) \]

      18. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(a\right)\right)}\right)\right)\right)\right)\right) \]

      19. unsub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(\frac{1}{2} - \color{blue}{a}\right)\right)\right)\right)\right) \]

      20. --lowering--.f64 99.6%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \mathsf{\_.f64}\left(\frac{1}{2}, \color{blue}{a}\right)\right)\right)\right)\right) \]

   3. Simplified 99.6%

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

      1. associate-+r- N/A

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

      2. +-commutative N/A

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

      3. associate--r+ N/A

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

      4. flip3-+ N/A

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

      5. div-inv N/A

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

      6. fmm-def N/A

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

      7. *-commutative N/A

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

   6. Applied egg-rr 99.7%

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

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

   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

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(-1 \cdot t\right)}, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{neg}\left(t\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{\_.f64}\left(a, \frac{1}{2}\right)}, \mathsf{log.f64}\left(t\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(0 - t\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{\_.f64}\left(a, \frac{1}{2}\right)}, \mathsf{log.f64}\left(t\right)\right)\right) \]

      3. --lowering--.f64 75.5%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(0, t\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{\_.f64}\left(a, \frac{1}{2}\right)}, \mathsf{log.f64}\left(t\right)\right)\right) \]

   5. Simplified 75.5%

      \[\leadsto \color{blue}{\left(0 - t\right)} + \left(a - 0.5\right) \cdot \log t \]
3. Recombined 3 regimes into one program.

4. Final simplification 94.6%

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

Alternative 3: 68.6% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log (+ x y)) (log z))))
   (if (<= t_1 -750.0)
     (- (* a (log t)) t)
     (if (<= t_1 700.0)
       (- (+ (* (log t) (+ a -0.5)) (log (* y z))) t)
       (- (* (log t) (- a 0.5)) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((x + y)) + log(z);
	double tmp;
	if (t_1 <= -750.0) {
		tmp = (a * log(t)) - t;
	} else if (t_1 <= 700.0) {
		tmp = ((log(t) * (a + -0.5)) + log((y * z))) - t;
	} else {
		tmp = (log(t) * (a - 0.5)) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = log((x + y)) + log(z)
    if (t_1 <= (-750.0d0)) then
        tmp = (a * log(t)) - t
    else if (t_1 <= 700.0d0) then
        tmp = ((log(t) * (a + (-0.5d0))) + log((y * z))) - t
    else
        tmp = (log(t) * (a - 0.5d0)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = Math.log((x + y)) + Math.log(z);
	double tmp;
	if (t_1 <= -750.0) {
		tmp = (a * Math.log(t)) - t;
	} else if (t_1 <= 700.0) {
		tmp = ((Math.log(t) * (a + -0.5)) + Math.log((y * z))) - t;
	} else {
		tmp = (Math.log(t) * (a - 0.5)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = math.log((x + y)) + math.log(z)
	tmp = 0
	if t_1 <= -750.0:
		tmp = (a * math.log(t)) - t
	elif t_1 <= 700.0:
		tmp = ((math.log(t) * (a + -0.5)) + math.log((y * z))) - t
	else:
		tmp = (math.log(t) * (a - 0.5)) - t
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(Float64(x + y)) + log(z))
	tmp = 0.0
	if (t_1 <= -750.0)
		tmp = Float64(Float64(a * log(t)) - t);
	elseif (t_1 <= 700.0)
		tmp = Float64(Float64(Float64(log(t) * Float64(a + -0.5)) + log(Float64(y * z))) - t);
	else
		tmp = Float64(Float64(log(t) * Float64(a - 0.5)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = log((x + y)) + log(z);
	tmp = 0.0;
	if (t_1 <= -750.0)
		tmp = (a * log(t)) - t;
	elseif (t_1 <= 700.0)
		tmp = ((log(t) * (a + -0.5)) + log((y * z))) - t;
	else
		tmp = (log(t) * (a - 0.5)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -750.0], N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[t$95$1, 700.0], N[(N[(N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision]), $MachinePrecision] + N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $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 (log.f64 (+.f64 x y)) (log.f64 z)) < -750

   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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\color{blue}{\left(\log y + \log z\right)}, t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\left(\log z + \log y\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\color{blue}{a}, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      2. remove-double-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\left(\log z + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log y\right)\right)\right)\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      3. log-rec N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\left(\log z + \left(\mathsf{neg}\left(\log \left(\frac{1}{y}\right)\right)\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\left(\log z + -1 \cdot \log \left(\frac{1}{y}\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\log z, \left(-1 \cdot \log \left(\frac{1}{y}\right)\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\color{blue}{a}, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      6. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \left(-1 \cdot \log \left(\frac{1}{y}\right)\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \left(\mathsf{neg}\left(\log \left(\frac{1}{y}\right)\right)\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      8. log-rec N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log y\right)\right)\right)\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      9. remove-double-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \log y\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      10. log-lowering-log.f64 49.7%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{log.f64}\left(y\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

   5. Simplified 49.7%

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

      1. +-commutative N/A

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

      2. associate-+r- N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(a - \frac{1}{2}\right) \cdot \log t + \left(\log z + \log y\right)\right), \color{blue}{t}\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(\left(a - \frac{1}{2}\right) \cdot \log t\right), \left(\log z + \log y\right)\right), t\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{2}\right), \log t\right), \left(\log z + \log y\right)\right), t\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log t\right), \left(\log z + \log y\right)\right), t\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a + \frac{-1}{2}\right), \log t\right), \left(\log z + \log y\right)\right), t\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{2}\right), \log t\right), \left(\log z + \log y\right)\right), t\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{2}\right), \mathsf{log.f64}\left(t\right)\right), \left(\log z + \log y\right)\right), t\right) \]

      10. sum-log N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{2}\right), \mathsf{log.f64}\left(t\right)\right), \log \left(z \cdot y\right)\right), t\right) \]

      11. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{2}\right), \mathsf{log.f64}\left(t\right)\right), \mathsf{log.f64}\left(\left(z \cdot y\right)\right)\right), t\right) \]

      12. *-lowering-*.f64 4.0%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{2}\right), \mathsf{log.f64}\left(t\right)\right), \mathsf{log.f64}\left(\mathsf{*.f64}\left(z, y\right)\right)\right), t\right) \]

   7. Applied egg-rr 4.0%

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

   8. Taylor expanded in a around inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(a \cdot \log t\right)}, t\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\log t \cdot a\right), t\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\log t, a\right), t\right) \]

      3. log-lowering-log.f64 99.5%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), a\right), t\right) \]

   10. Simplified 99.5%

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

   if -750 < (+.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. Taylor expanded in x around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\color{blue}{\left(\log y + \log z\right)}, t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\left(\log z + \log y\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\color{blue}{a}, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      2. remove-double-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\left(\log z + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log y\right)\right)\right)\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      3. log-rec N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\left(\log z + \left(\mathsf{neg}\left(\log \left(\frac{1}{y}\right)\right)\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\left(\log z + -1 \cdot \log \left(\frac{1}{y}\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\log z, \left(-1 \cdot \log \left(\frac{1}{y}\right)\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\color{blue}{a}, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      6. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \left(-1 \cdot \log \left(\frac{1}{y}\right)\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \left(\mathsf{neg}\left(\log \left(\frac{1}{y}\right)\right)\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      8. log-rec N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log y\right)\right)\right)\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      9. remove-double-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \log y\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      10. log-lowering-log.f64 67.1%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{log.f64}\left(y\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

   5. Simplified 67.1%

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

      1. +-commutative N/A

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

      2. associate-+r- N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(a - \frac{1}{2}\right) \cdot \log t + \left(\log z + \log y\right)\right), \color{blue}{t}\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(\left(a - \frac{1}{2}\right) \cdot \log t\right), \left(\log z + \log y\right)\right), t\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{2}\right), \log t\right), \left(\log z + \log y\right)\right), t\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log t\right), \left(\log z + \log y\right)\right), t\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a + \frac{-1}{2}\right), \log t\right), \left(\log z + \log y\right)\right), t\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{2}\right), \log t\right), \left(\log z + \log y\right)\right), t\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{2}\right), \mathsf{log.f64}\left(t\right)\right), \left(\log z + \log y\right)\right), t\right) \]

      10. sum-log N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{2}\right), \mathsf{log.f64}\left(t\right)\right), \log \left(z \cdot y\right)\right), t\right) \]

      11. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{2}\right), \mathsf{log.f64}\left(t\right)\right), \mathsf{log.f64}\left(\left(z \cdot y\right)\right)\right), t\right) \]

      12. *-lowering-*.f64 65.9%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{2}\right), \mathsf{log.f64}\left(t\right)\right), \mathsf{log.f64}\left(\mathsf{*.f64}\left(z, y\right)\right)\right), t\right) \]

   7. Applied egg-rr 65.9%

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

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

   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

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(-1 \cdot t\right)}, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{neg}\left(t\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{\_.f64}\left(a, \frac{1}{2}\right)}, \mathsf{log.f64}\left(t\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(0 - t\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{\_.f64}\left(a, \frac{1}{2}\right)}, \mathsf{log.f64}\left(t\right)\right)\right) \]

      3. --lowering--.f64 75.5%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(0, t\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{\_.f64}\left(a, \frac{1}{2}\right)}, \mathsf{log.f64}\left(t\right)\right)\right) \]

   5. Simplified 75.5%

      \[\leadsto \color{blue}{\left(0 - t\right)} + \left(a - 0.5\right) \cdot \log t \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.7%

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

Alternative 4: 80.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 2.3 \cdot 10^{-16}:\\ \;\;\;\;\log y + \left(\log z + \log t \cdot \left(a + -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 2.3e-16)
   (+ (log y) (+ (log z) (* (log t) (+ a -0.5))))
   (- (* a (log t)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 2.3e-16) {
		tmp = log(y) + (log(z) + (log(t) * (a + -0.5)));
	} else {
		tmp = (a * log(t)) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 2.3d-16) then
        tmp = log(y) + (log(z) + (log(t) * (a + (-0.5d0))))
    else
        tmp = (a * log(t)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 2.3e-16) {
		tmp = Math.log(y) + (Math.log(z) + (Math.log(t) * (a + -0.5)));
	} else {
		tmp = (a * Math.log(t)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 2.3e-16:
		tmp = math.log(y) + (math.log(z) + (math.log(t) * (a + -0.5)))
	else:
		tmp = (a * math.log(t)) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 2.3e-16)
		tmp = Float64(log(y) + Float64(log(z) + Float64(log(t) * Float64(a + -0.5))));
	else
		tmp = Float64(Float64(a * log(t)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 2.3e-16)
		tmp = log(y) + (log(z) + (log(t) * (a + -0.5)));
	else
		tmp = (a * log(t)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 2.3e-16], N[(N[Log[y], $MachinePrecision] + N[(N[Log[z], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.2999999999999999e-16

   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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\color{blue}{\left(\log y + \log z\right)}, t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\left(\log z + \log y\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\color{blue}{a}, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      2. remove-double-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\left(\log z + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log y\right)\right)\right)\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      3. log-rec N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\left(\log z + \left(\mathsf{neg}\left(\log \left(\frac{1}{y}\right)\right)\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\left(\log z + -1 \cdot \log \left(\frac{1}{y}\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\log z, \left(-1 \cdot \log \left(\frac{1}{y}\right)\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\color{blue}{a}, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      6. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \left(-1 \cdot \log \left(\frac{1}{y}\right)\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \left(\mathsf{neg}\left(\log \left(\frac{1}{y}\right)\right)\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      8. log-rec N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log y\right)\right)\right)\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      9. remove-double-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \log y\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      10. log-lowering-log.f64 63.4%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{log.f64}\left(y\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

   5. Simplified 63.4%

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

   6. Taylor expanded in t around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\log y, \color{blue}{\left(\log z + \log t \cdot \left(a - \frac{1}{2}\right)\right)}\right) \]

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(y\right), \left(\color{blue}{\log z} + \log t \cdot \left(a - \frac{1}{2}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(\log z, \color{blue}{\left(\log t \cdot \left(a - \frac{1}{2}\right)\right)}\right)\right) \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \left(\color{blue}{\log t} \cdot \left(a - \frac{1}{2}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{*.f64}\left(\log t, \color{blue}{\left(a - \frac{1}{2}\right)}\right)\right)\right) \]

      6. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(\color{blue}{a} - \frac{1}{2}\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(a + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(a + \frac{-1}{2}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 63.4%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \mathsf{+.f64}\left(a, \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

   8. Simplified 63.4%

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

   if 2.2999999999999999e-16 < t

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\color{blue}{\left(\log y + \log z\right)}, t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\left(\log z + \log y\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\color{blue}{a}, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      2. remove-double-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\left(\log z + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log y\right)\right)\right)\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      3. log-rec N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\left(\log z + \left(\mathsf{neg}\left(\log \left(\frac{1}{y}\right)\right)\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\left(\log z + -1 \cdot \log \left(\frac{1}{y}\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\log z, \left(-1 \cdot \log \left(\frac{1}{y}\right)\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\color{blue}{a}, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      6. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \left(-1 \cdot \log \left(\frac{1}{y}\right)\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \left(\mathsf{neg}\left(\log \left(\frac{1}{y}\right)\right)\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      8. log-rec N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log y\right)\right)\right)\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      9. remove-double-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \log y\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      10. log-lowering-log.f64 68.8%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{log.f64}\left(y\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

   5. Simplified 68.8%

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

      1. +-commutative N/A

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

      2. associate-+r- N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(a - \frac{1}{2}\right) \cdot \log t + \left(\log z + \log y\right)\right), \color{blue}{t}\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(\left(a - \frac{1}{2}\right) \cdot \log t\right), \left(\log z + \log y\right)\right), t\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{2}\right), \log t\right), \left(\log z + \log y\right)\right), t\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log t\right), \left(\log z + \log y\right)\right), t\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a + \frac{-1}{2}\right), \log t\right), \left(\log z + \log y\right)\right), t\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{2}\right), \log t\right), \left(\log z + \log y\right)\right), t\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{2}\right), \mathsf{log.f64}\left(t\right)\right), \left(\log z + \log y\right)\right), t\right) \]

      10. sum-log N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{2}\right), \mathsf{log.f64}\left(t\right)\right), \log \left(z \cdot y\right)\right), t\right) \]

      11. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{2}\right), \mathsf{log.f64}\left(t\right)\right), \mathsf{log.f64}\left(\left(z \cdot y\right)\right)\right), t\right) \]

      12. *-lowering-*.f64 57.0%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{2}\right), \mathsf{log.f64}\left(t\right)\right), \mathsf{log.f64}\left(\mathsf{*.f64}\left(z, y\right)\right)\right), t\right) \]

   7. Applied egg-rr 57.0%

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

   8. Taylor expanded in a around inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(a \cdot \log t\right)}, t\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\log t \cdot a\right), t\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\log t, a\right), t\right) \]

      3. log-lowering-log.f64 97.9%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), a\right), t\right) \]

   10. Simplified 97.9%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.3 \cdot 10^{-16}:\\ \;\;\;\;\log y + \left(\log z + \log t \cdot \left(a + -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t, double a) {
	return log((x + y)) + (log(z) + ((log(t) * (a - 0.5)) - 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) + ((log(t) * (a - 0.5d0)) - 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) + ((Math.log(t) * (a - 0.5)) - t));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. associate--l+ N/A

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

   2. associate-+l+ N/A

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

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\log \left(x + y\right), \color{blue}{\left(\left(\log z - t\right) + \left(a - \frac{1}{2}\right) \cdot \log t\right)}\right) \]

   4. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\left(x + y\right)\right), \left(\color{blue}{\left(\log z - t\right)} + \left(a - \frac{1}{2}\right) \cdot \log t\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \left(\left(\color{blue}{\log z} - t\right) + \left(a - \frac{1}{2}\right) \cdot \log t\right)\right) \]

   6. associate-+l- N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \left(\log z - \color{blue}{\left(t - \left(a - \frac{1}{2}\right) \cdot \log t\right)}\right)\right) \]

   7. --lowering--.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\log z, \color{blue}{\left(t - \left(a - \frac{1}{2}\right) \cdot \log t\right)}\right)\right) \]

   8. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \left(\color{blue}{t} - \left(a - \frac{1}{2}\right) \cdot \log t\right)\right)\right) \]

   9. sub-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \left(t + \color{blue}{\left(\mathsf{neg}\left(\left(a - \frac{1}{2}\right) \cdot \log t\right)\right)}\right)\right)\right) \]

   10. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \color{blue}{\left(\mathsf{neg}\left(\left(a - \frac{1}{2}\right) \cdot \log t\right)\right)}\right)\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \left(\mathsf{neg}\left(\log t \cdot \left(a - \frac{1}{2}\right)\right)\right)\right)\right)\right) \]

   12. distribute-rgt-neg-in N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \left(\log t \cdot \color{blue}{\left(\mathsf{neg}\left(\left(a - \frac{1}{2}\right)\right)\right)}\right)\right)\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\log t, \color{blue}{\left(\mathsf{neg}\left(\left(a - \frac{1}{2}\right)\right)\right)}\right)\right)\right)\right) \]

   14. log-lowering-log.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(\mathsf{neg}\left(\color{blue}{\left(a - \frac{1}{2}\right)}\right)\right)\right)\right)\right)\right) \]

   15. neg-sub0 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(0 - \color{blue}{\left(a - \frac{1}{2}\right)}\right)\right)\right)\right)\right) \]

   16. associate--r- N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(\left(0 - a\right) + \color{blue}{\frac{1}{2}}\right)\right)\right)\right)\right) \]

   17. neg-sub0 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(\left(\mathsf{neg}\left(a\right)\right) + \frac{1}{2}\right)\right)\right)\right)\right) \]

   18. +-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(a\right)\right)}\right)\right)\right)\right)\right) \]

   19. unsub-neg N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(\frac{1}{2} - \color{blue}{a}\right)\right)\right)\right)\right) \]

   20. --lowering--.f64 99.7%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \mathsf{\_.f64}\left(\frac{1}{2}, \color{blue}{a}\right)\right)\right)\right)\right) \]

3. Simplified 99.7%

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

5. Final simplification 99.7%

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

Alternative 6: 68.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 x around 0

   \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\color{blue}{\left(\log y + \log z\right)}, t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\left(\log z + \log y\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\color{blue}{a}, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

   2. remove-double-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\left(\log z + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log y\right)\right)\right)\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

   3. log-rec N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\left(\log z + \left(\mathsf{neg}\left(\log \left(\frac{1}{y}\right)\right)\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

   4. mul-1-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\left(\log z + -1 \cdot \log \left(\frac{1}{y}\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\log z, \left(-1 \cdot \log \left(\frac{1}{y}\right)\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\color{blue}{a}, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

   6. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \left(-1 \cdot \log \left(\frac{1}{y}\right)\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

   7. mul-1-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \left(\mathsf{neg}\left(\log \left(\frac{1}{y}\right)\right)\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

   8. log-rec N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log y\right)\right)\right)\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

   9. remove-double-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \log y\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

   10. log-lowering-log.f64 66.1%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{log.f64}\left(y\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

5. Simplified 66.1%

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

6. Final simplification 66.1%

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

Alternative 7: 68.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. associate--l+ N/A

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

   2. associate-+l+ N/A

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

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\log \left(x + y\right), \color{blue}{\left(\left(\log z - t\right) + \left(a - \frac{1}{2}\right) \cdot \log t\right)}\right) \]

   4. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\left(x + y\right)\right), \left(\color{blue}{\left(\log z - t\right)} + \left(a - \frac{1}{2}\right) \cdot \log t\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \left(\left(\color{blue}{\log z} - t\right) + \left(a - \frac{1}{2}\right) \cdot \log t\right)\right) \]

   6. associate-+l- N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \left(\log z - \color{blue}{\left(t - \left(a - \frac{1}{2}\right) \cdot \log t\right)}\right)\right) \]

   7. --lowering--.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\log z, \color{blue}{\left(t - \left(a - \frac{1}{2}\right) \cdot \log t\right)}\right)\right) \]

   8. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \left(\color{blue}{t} - \left(a - \frac{1}{2}\right) \cdot \log t\right)\right)\right) \]

   9. sub-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \left(t + \color{blue}{\left(\mathsf{neg}\left(\left(a - \frac{1}{2}\right) \cdot \log t\right)\right)}\right)\right)\right) \]

   10. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \color{blue}{\left(\mathsf{neg}\left(\left(a - \frac{1}{2}\right) \cdot \log t\right)\right)}\right)\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \left(\mathsf{neg}\left(\log t \cdot \left(a - \frac{1}{2}\right)\right)\right)\right)\right)\right) \]

   12. distribute-rgt-neg-in N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \left(\log t \cdot \color{blue}{\left(\mathsf{neg}\left(\left(a - \frac{1}{2}\right)\right)\right)}\right)\right)\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\log t, \color{blue}{\left(\mathsf{neg}\left(\left(a - \frac{1}{2}\right)\right)\right)}\right)\right)\right)\right) \]

   14. log-lowering-log.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(\mathsf{neg}\left(\color{blue}{\left(a - \frac{1}{2}\right)}\right)\right)\right)\right)\right)\right) \]

   15. neg-sub0 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(0 - \color{blue}{\left(a - \frac{1}{2}\right)}\right)\right)\right)\right)\right) \]

   16. associate--r- N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(\left(0 - a\right) + \color{blue}{\frac{1}{2}}\right)\right)\right)\right)\right) \]

   17. neg-sub0 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(\left(\mathsf{neg}\left(a\right)\right) + \frac{1}{2}\right)\right)\right)\right)\right) \]

   18. +-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(a\right)\right)}\right)\right)\right)\right)\right) \]

   19. unsub-neg N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(\frac{1}{2} - \color{blue}{a}\right)\right)\right)\right)\right) \]

   20. --lowering--.f64 99.7%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \mathsf{\_.f64}\left(\frac{1}{2}, \color{blue}{a}\right)\right)\right)\right)\right) \]

3. Simplified 99.7%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. remove-double-neg N/A

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

   3. log-rec N/A

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

   4. mul-1-neg N/A

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

   5. associate--l+ N/A

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

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\log z, \color{blue}{\left(-1 \cdot \log \left(\frac{1}{y}\right) - \left(t + \log t \cdot \left(\frac{1}{2} - a\right)\right)\right)}\right) \]

   7. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \left(\color{blue}{-1 \cdot \log \left(\frac{1}{y}\right)} - \left(t + \log t \cdot \left(\frac{1}{2} - a\right)\right)\right)\right) \]

   8. associate--r+ N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \left(\left(-1 \cdot \log \left(\frac{1}{y}\right) - t\right) - \color{blue}{\log t \cdot \left(\frac{1}{2} - a\right)}\right)\right) \]

   9. --lowering--.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{\_.f64}\left(\left(-1 \cdot \log \left(\frac{1}{y}\right) - t\right), \color{blue}{\left(\log t \cdot \left(\frac{1}{2} - a\right)\right)}\right)\right) \]

   10. --lowering--.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\left(-1 \cdot \log \left(\frac{1}{y}\right)\right), t\right), \left(\color{blue}{\log t} \cdot \left(\frac{1}{2} - a\right)\right)\right)\right) \]

   11. mul-1-neg N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(\log \left(\frac{1}{y}\right)\right)\right), t\right), \left(\log \color{blue}{t} \cdot \left(\frac{1}{2} - a\right)\right)\right)\right) \]

   12. log-rec N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log y\right)\right)\right)\right), t\right), \left(\log t \cdot \left(\frac{1}{2} - a\right)\right)\right)\right) \]

   13. remove-double-neg N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\log y, t\right), \left(\log \color{blue}{t} \cdot \left(\frac{1}{2} - a\right)\right)\right)\right) \]

   14. log-lowering-log.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{log.f64}\left(y\right), t\right), \left(\log \color{blue}{t} \cdot \left(\frac{1}{2} - a\right)\right)\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{log.f64}\left(y\right), t\right), \mathsf{*.f64}\left(\log t, \color{blue}{\left(\frac{1}{2} - a\right)}\right)\right)\right) \]

   16. log-lowering-log.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{log.f64}\left(y\right), t\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(\color{blue}{\frac{1}{2}} - a\right)\right)\right)\right) \]

   17. --lowering--.f64 66.1%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{log.f64}\left(y\right), t\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \mathsf{\_.f64}\left(\frac{1}{2}, \color{blue}{a}\right)\right)\right)\right) \]

7. Simplified 66.1%

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

8. Final simplification 66.1%

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

Alternative 8: 74.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 4.7 \cdot 10^{-17}:\\ \;\;\;\;\log t \cdot \left(a + -0.5\right) + \log \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot \left(a - 0.5\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 4.7e-17)
   (+ (* (log t) (+ a -0.5)) (log (* y z)))
   (- (* (log t) (- a 0.5)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 4.7e-17) {
		tmp = (log(t) * (a + -0.5)) + log((y * z));
	} else {
		tmp = (log(t) * (a - 0.5)) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 4.7d-17) then
        tmp = (log(t) * (a + (-0.5d0))) + log((y * z))
    else
        tmp = (log(t) * (a - 0.5d0)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 4.7e-17) {
		tmp = (Math.log(t) * (a + -0.5)) + Math.log((y * z));
	} else {
		tmp = (Math.log(t) * (a - 0.5)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 4.7e-17:
		tmp = (math.log(t) * (a + -0.5)) + math.log((y * z))
	else:
		tmp = (math.log(t) * (a - 0.5)) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 4.7e-17)
		tmp = Float64(Float64(log(t) * Float64(a + -0.5)) + log(Float64(y * z)));
	else
		tmp = Float64(Float64(log(t) * Float64(a - 0.5)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 4.7e-17)
		tmp = (log(t) * (a + -0.5)) + log((y * z));
	else
		tmp = (log(t) * (a - 0.5)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 4.7e-17], N[(N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision]), $MachinePrecision] + N[Log[N[(y * 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 < 4.7e-17

   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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\color{blue}{\left(\log y + \log z\right)}, t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\left(\log z + \log y\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\color{blue}{a}, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      2. remove-double-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\left(\log z + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log y\right)\right)\right)\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      3. log-rec N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\left(\log z + \left(\mathsf{neg}\left(\log \left(\frac{1}{y}\right)\right)\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\left(\log z + -1 \cdot \log \left(\frac{1}{y}\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\log z, \left(-1 \cdot \log \left(\frac{1}{y}\right)\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\color{blue}{a}, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      6. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \left(-1 \cdot \log \left(\frac{1}{y}\right)\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \left(\mathsf{neg}\left(\log \left(\frac{1}{y}\right)\right)\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      8. log-rec N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log y\right)\right)\right)\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      9. remove-double-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \log y\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      10. log-lowering-log.f64 63.2%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{log.f64}\left(y\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

   5. Simplified 63.2%

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

      1. +-commutative N/A

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

      2. associate-+r- N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(a - \frac{1}{2}\right) \cdot \log t + \left(\log z + \log y\right)\right), \color{blue}{t}\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(\left(a - \frac{1}{2}\right) \cdot \log t\right), \left(\log z + \log y\right)\right), t\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{2}\right), \log t\right), \left(\log z + \log y\right)\right), t\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log t\right), \left(\log z + \log y\right)\right), t\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a + \frac{-1}{2}\right), \log t\right), \left(\log z + \log y\right)\right), t\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{2}\right), \log t\right), \left(\log z + \log y\right)\right), t\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{2}\right), \mathsf{log.f64}\left(t\right)\right), \left(\log z + \log y\right)\right), t\right) \]

      10. sum-log N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{2}\right), \mathsf{log.f64}\left(t\right)\right), \log \left(z \cdot y\right)\right), t\right) \]

      11. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{2}\right), \mathsf{log.f64}\left(t\right)\right), \mathsf{log.f64}\left(\left(z \cdot y\right)\right)\right), t\right) \]

      12. *-lowering-*.f64 53.7%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{2}\right), \mathsf{log.f64}\left(t\right)\right), \mathsf{log.f64}\left(\mathsf{*.f64}\left(z, y\right)\right)\right), t\right) \]

   7. Applied egg-rr 53.7%

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

   8. Taylor expanded in t around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\log \left(y \cdot z\right), \color{blue}{\left(\log t \cdot \left(a - \frac{1}{2}\right)\right)}\right) \]

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\left(y \cdot z\right)\right), \left(\color{blue}{\log t} \cdot \left(a - \frac{1}{2}\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\left(z \cdot y\right)\right), \left(\log \color{blue}{t} \cdot \left(a - \frac{1}{2}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(z, y\right)\right), \left(\log \color{blue}{t} \cdot \left(a - \frac{1}{2}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(z, y\right)\right), \mathsf{*.f64}\left(\log t, \color{blue}{\left(a - \frac{1}{2}\right)}\right)\right) \]

      6. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(z, y\right)\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(\color{blue}{a} - \frac{1}{2}\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(z, y\right)\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(a + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(z, y\right)\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(a + \frac{-1}{2}\right)\right)\right) \]

      9. +-lowering-+.f64 53.7%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(z, y\right)\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \mathsf{+.f64}\left(a, \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

   10. Simplified 53.7%

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

   if 4.7e-17 < t

   1. Initial program 99.9%

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

   3. Taylor expanded in t around inf

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(-1 \cdot t\right)}, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{neg}\left(t\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{\_.f64}\left(a, \frac{1}{2}\right)}, \mathsf{log.f64}\left(t\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(0 - t\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{\_.f64}\left(a, \frac{1}{2}\right)}, \mathsf{log.f64}\left(t\right)\right)\right) \]

      3. --lowering--.f64 97.3%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(0, t\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{\_.f64}\left(a, \frac{1}{2}\right)}, \mathsf{log.f64}\left(t\right)\right)\right) \]

   5. Simplified 97.3%

      \[\leadsto \color{blue}{\left(0 - t\right)} + \left(a - 0.5\right) \cdot \log t \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.7 \cdot 10^{-17}:\\ \;\;\;\;\log t \cdot \left(a + -0.5\right) + \log \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot \left(a - 0.5\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \log t - t\\ \mathbf{if}\;a \leq -0.45:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 1.9:\\ \;\;\;\;\log \left(x + y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* a (log t)) t)))
   (if (<= a -0.45) t_1 (if (<= a 1.9) (- (log (+ x y)) t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (a * log(t)) - t;
	double tmp;
	if (a <= -0.45) {
		tmp = t_1;
	} else if (a <= 1.9) {
		tmp = log((x + y)) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (a * log(t)) - t
    if (a <= (-0.45d0)) then
        tmp = t_1
    else if (a <= 1.9d0) then
        tmp = log((x + y)) - t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (a * Math.log(t)) - t;
	double tmp;
	if (a <= -0.45) {
		tmp = t_1;
	} else if (a <= 1.9) {
		tmp = Math.log((x + y)) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (a * math.log(t)) - t
	tmp = 0
	if a <= -0.45:
		tmp = t_1
	elif a <= 1.9:
		tmp = math.log((x + y)) - t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(a * log(t)) - t)
	tmp = 0.0
	if (a <= -0.45)
		tmp = t_1;
	elseif (a <= 1.9)
		tmp = Float64(log(Float64(x + y)) - t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (a * log(t)) - t;
	tmp = 0.0;
	if (a <= -0.45)
		tmp = t_1;
	elseif (a <= 1.9)
		tmp = log((x + y)) - t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[a, -0.45], t$95$1, If[LessEqual[a, 1.9], N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -0.450000000000000011 or 1.8999999999999999 < a

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\color{blue}{\left(\log y + \log z\right)}, t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\left(\log z + \log y\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\color{blue}{a}, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      2. remove-double-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\left(\log z + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log y\right)\right)\right)\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      3. log-rec N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\left(\log z + \left(\mathsf{neg}\left(\log \left(\frac{1}{y}\right)\right)\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\left(\log z + -1 \cdot \log \left(\frac{1}{y}\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\log z, \left(-1 \cdot \log \left(\frac{1}{y}\right)\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\color{blue}{a}, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      6. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \left(-1 \cdot \log \left(\frac{1}{y}\right)\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \left(\mathsf{neg}\left(\log \left(\frac{1}{y}\right)\right)\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      8. log-rec N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log y\right)\right)\right)\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      9. remove-double-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \log y\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      10. log-lowering-log.f64 69.1%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{log.f64}\left(y\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

   5. Simplified 69.1%

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

      1. +-commutative N/A

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

      2. associate-+r- N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(a - \frac{1}{2}\right) \cdot \log t + \left(\log z + \log y\right)\right), \color{blue}{t}\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(\left(a - \frac{1}{2}\right) \cdot \log t\right), \left(\log z + \log y\right)\right), t\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{2}\right), \log t\right), \left(\log z + \log y\right)\right), t\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log t\right), \left(\log z + \log y\right)\right), t\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a + \frac{-1}{2}\right), \log t\right), \left(\log z + \log y\right)\right), t\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{2}\right), \log t\right), \left(\log z + \log y\right)\right), t\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{2}\right), \mathsf{log.f64}\left(t\right)\right), \left(\log z + \log y\right)\right), t\right) \]

      10. sum-log N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{2}\right), \mathsf{log.f64}\left(t\right)\right), \log \left(z \cdot y\right)\right), t\right) \]

      11. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{2}\right), \mathsf{log.f64}\left(t\right)\right), \mathsf{log.f64}\left(\left(z \cdot y\right)\right)\right), t\right) \]

      12. *-lowering-*.f64 58.1%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{2}\right), \mathsf{log.f64}\left(t\right)\right), \mathsf{log.f64}\left(\mathsf{*.f64}\left(z, y\right)\right)\right), t\right) \]

   7. Applied egg-rr 58.1%

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

   8. Taylor expanded in a around inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(a \cdot \log t\right)}, t\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\log t \cdot a\right), t\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\log t, a\right), t\right) \]

      3. log-lowering-log.f64 98.6%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), a\right), t\right) \]

   10. Simplified 98.6%

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

   if -0.450000000000000011 < a < 1.8999999999999999

   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. Step-by-step derivation

      1. associate--l+ N/A

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

      2. associate-+l+ N/A

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

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\log \left(x + y\right), \color{blue}{\left(\left(\log z - t\right) + \left(a - \frac{1}{2}\right) \cdot \log t\right)}\right) \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\left(x + y\right)\right), \left(\color{blue}{\left(\log z - t\right)} + \left(a - \frac{1}{2}\right) \cdot \log t\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \left(\left(\color{blue}{\log z} - t\right) + \left(a - \frac{1}{2}\right) \cdot \log t\right)\right) \]

      6. associate-+l- N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \left(\log z - \color{blue}{\left(t - \left(a - \frac{1}{2}\right) \cdot \log t\right)}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\log z, \color{blue}{\left(t - \left(a - \frac{1}{2}\right) \cdot \log t\right)}\right)\right) \]

      8. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \left(\color{blue}{t} - \left(a - \frac{1}{2}\right) \cdot \log t\right)\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \left(t + \color{blue}{\left(\mathsf{neg}\left(\left(a - \frac{1}{2}\right) \cdot \log t\right)\right)}\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \color{blue}{\left(\mathsf{neg}\left(\left(a - \frac{1}{2}\right) \cdot \log t\right)\right)}\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \left(\mathsf{neg}\left(\log t \cdot \left(a - \frac{1}{2}\right)\right)\right)\right)\right)\right) \]

      12. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \left(\log t \cdot \color{blue}{\left(\mathsf{neg}\left(\left(a - \frac{1}{2}\right)\right)\right)}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\log t, \color{blue}{\left(\mathsf{neg}\left(\left(a - \frac{1}{2}\right)\right)\right)}\right)\right)\right)\right) \]

      14. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(\mathsf{neg}\left(\color{blue}{\left(a - \frac{1}{2}\right)}\right)\right)\right)\right)\right)\right) \]

      15. neg-sub0 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(0 - \color{blue}{\left(a - \frac{1}{2}\right)}\right)\right)\right)\right)\right) \]

      16. associate--r- N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(\left(0 - a\right) + \color{blue}{\frac{1}{2}}\right)\right)\right)\right)\right) \]

      17. neg-sub0 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(\left(\mathsf{neg}\left(a\right)\right) + \frac{1}{2}\right)\right)\right)\right)\right) \]

      18. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(a\right)\right)}\right)\right)\right)\right)\right) \]

      19. unsub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(\frac{1}{2} - \color{blue}{a}\right)\right)\right)\right)\right) \]

      20. --lowering--.f64 99.6%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \mathsf{\_.f64}\left(\frac{1}{2}, \color{blue}{a}\right)\right)\right)\right)\right) \]

   3. Simplified 99.6%

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

   5. Taylor expanded in t around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \color{blue}{\left(-1 \cdot t\right)}\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \left(\mathsf{neg}\left(t\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \left(0 - \color{blue}{t}\right)\right) \]

      3. --lowering--.f64 54.8%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(0, \color{blue}{t}\right)\right) \]

   7. Simplified 54.8%

      \[\leadsto \log \left(x + y\right) + \color{blue}{\left(0 - t\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.45:\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{elif}\;a \leq 1.9:\\ \;\;\;\;\log \left(x + y\right) - t\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \log t\\ \mathbf{if}\;a \leq -1.1 \cdot 10^{+60}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 1.5 \cdot 10^{+80}:\\ \;\;\;\;\log z - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* a (log t))))
   (if (<= a -1.1e+60) t_1 (if (<= a 1.5e+80) (- (log z) t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = a * log(t);
	double tmp;
	if (a <= -1.1e+60) {
		tmp = t_1;
	} else if (a <= 1.5e+80) {
		tmp = log(z) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = a * log(t)
    if (a <= (-1.1d+60)) then
        tmp = t_1
    else if (a <= 1.5d+80) then
        tmp = log(z) - t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = a * Math.log(t);
	double tmp;
	if (a <= -1.1e+60) {
		tmp = t_1;
	} else if (a <= 1.5e+80) {
		tmp = Math.log(z) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = a * math.log(t)
	tmp = 0
	if a <= -1.1e+60:
		tmp = t_1
	elif a <= 1.5e+80:
		tmp = math.log(z) - t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(a * log(t))
	tmp = 0.0
	if (a <= -1.1e+60)
		tmp = t_1;
	elseif (a <= 1.5e+80)
		tmp = Float64(log(z) - t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = a * log(t);
	tmp = 0.0;
	if (a <= -1.1e+60)
		tmp = t_1;
	elseif (a <= 1.5e+80)
		tmp = log(z) - t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -1.1e+60], t$95$1, If[LessEqual[a, 1.5e+80], N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.09999999999999998e60 or 1.49999999999999993e80 < a

   1. Initial program 99.7%

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

      1. associate--l+ N/A

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

      2. associate-+l+ N/A

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

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\log \left(x + y\right), \color{blue}{\left(\left(\log z - t\right) + \left(a - \frac{1}{2}\right) \cdot \log t\right)}\right) \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\left(x + y\right)\right), \left(\color{blue}{\left(\log z - t\right)} + \left(a - \frac{1}{2}\right) \cdot \log t\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \left(\left(\color{blue}{\log z} - t\right) + \left(a - \frac{1}{2}\right) \cdot \log t\right)\right) \]

      6. associate-+l- N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \left(\log z - \color{blue}{\left(t - \left(a - \frac{1}{2}\right) \cdot \log t\right)}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\log z, \color{blue}{\left(t - \left(a - \frac{1}{2}\right) \cdot \log t\right)}\right)\right) \]

      8. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \left(\color{blue}{t} - \left(a - \frac{1}{2}\right) \cdot \log t\right)\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \left(t + \color{blue}{\left(\mathsf{neg}\left(\left(a - \frac{1}{2}\right) \cdot \log t\right)\right)}\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \color{blue}{\left(\mathsf{neg}\left(\left(a - \frac{1}{2}\right) \cdot \log t\right)\right)}\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \left(\mathsf{neg}\left(\log t \cdot \left(a - \frac{1}{2}\right)\right)\right)\right)\right)\right) \]

      12. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \left(\log t \cdot \color{blue}{\left(\mathsf{neg}\left(\left(a - \frac{1}{2}\right)\right)\right)}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\log t, \color{blue}{\left(\mathsf{neg}\left(\left(a - \frac{1}{2}\right)\right)\right)}\right)\right)\right)\right) \]

      14. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(\mathsf{neg}\left(\color{blue}{\left(a - \frac{1}{2}\right)}\right)\right)\right)\right)\right)\right) \]

      15. neg-sub0 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(0 - \color{blue}{\left(a - \frac{1}{2}\right)}\right)\right)\right)\right)\right) \]

      16. associate--r- N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(\left(0 - a\right) + \color{blue}{\frac{1}{2}}\right)\right)\right)\right)\right) \]

      17. neg-sub0 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(\left(\mathsf{neg}\left(a\right)\right) + \frac{1}{2}\right)\right)\right)\right)\right) \]

      18. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(a\right)\right)}\right)\right)\right)\right)\right) \]

      19. unsub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(\frac{1}{2} - \color{blue}{a}\right)\right)\right)\right)\right) \]

      20. --lowering--.f64 99.7%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \mathsf{\_.f64}\left(\frac{1}{2}, \color{blue}{a}\right)\right)\right)\right)\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\log t, \color{blue}{a}\right) \]

      3. log-lowering-log.f64 86.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), a\right) \]

   7. Simplified 86.0%

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

   if -1.09999999999999998e60 < a < 1.49999999999999993e80

   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 x around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\color{blue}{\left(\log y + \log z\right)}, t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\left(\log z + \log y\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\color{blue}{a}, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      2. remove-double-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\left(\log z + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log y\right)\right)\right)\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      3. log-rec N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\left(\log z + \left(\mathsf{neg}\left(\log \left(\frac{1}{y}\right)\right)\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\left(\log z + -1 \cdot \log \left(\frac{1}{y}\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\log z, \left(-1 \cdot \log \left(\frac{1}{y}\right)\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\color{blue}{a}, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      6. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \left(-1 \cdot \log \left(\frac{1}{y}\right)\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \left(\mathsf{neg}\left(\log \left(\frac{1}{y}\right)\right)\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      8. log-rec N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log y\right)\right)\right)\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      9. remove-double-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \log y\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

      10. log-lowering-log.f64 62.9%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{log.f64}\left(y\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

   5. Simplified 62.9%

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

   6. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. remove-double-neg N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

      7. associate--l+ N/A

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

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\log z, \color{blue}{\left(\left(-1 \cdot \log \left(\frac{1}{y}\right) + \frac{-1}{2} \cdot \log t\right) - t\right)}\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \left(\color{blue}{\left(-1 \cdot \log \left(\frac{1}{y}\right) + \frac{-1}{2} \cdot \log t\right)} - t\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \left(\left(\frac{-1}{2} \cdot \log t + -1 \cdot \log \left(\frac{1}{y}\right)\right) - t\right)\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \left(\left(\frac{-1}{2} \cdot \log t + \left(\mathsf{neg}\left(\log \left(\frac{1}{y}\right)\right)\right)\right) - t\right)\right) \]

      12. log-rec N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \left(\left(\frac{-1}{2} \cdot \log t + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log y\right)\right)\right)\right)\right) - t\right)\right) \]

      13. remove-double-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \left(\left(\frac{-1}{2} \cdot \log t + \log y\right) - t\right)\right) \]

      14. associate--l+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \left(\frac{-1}{2} \cdot \log t + \color{blue}{\left(\log y - t\right)}\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \log t\right), \color{blue}{\left(\log y - t\right)}\right)\right) \]

   8. Simplified 58.5%

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

   9. Taylor expanded in t around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \color{blue}{\left(-1 \cdot t\right)}\right) \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \left(\mathsf{neg}\left(t\right)\right)\right) \]

       2. neg-sub0 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \left(0 - \color{blue}{t}\right)\right) \]

       3. --lowering--.f64 54.9%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{\_.f64}\left(0, \color{blue}{t}\right)\right) \]

   11. Simplified 54.9%

       \[\leadsto \log z + \color{blue}{\left(0 - t\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.1 \cdot 10^{+60}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{elif}\;a \leq 1.5 \cdot 10^{+80}:\\ \;\;\;\;\log z - t\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 61.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \log t\\ \mathbf{if}\;a \leq -5.2 \cdot 10^{+36}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 2.15 \cdot 10^{+82}:\\ \;\;\;\;0 - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* a (log t))))
   (if (<= a -5.2e+36) t_1 (if (<= a 2.15e+82) (- 0.0 t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = a * log(t);
	double tmp;
	if (a <= -5.2e+36) {
		tmp = t_1;
	} else if (a <= 2.15e+82) {
		tmp = 0.0 - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = a * log(t)
    if (a <= (-5.2d+36)) then
        tmp = t_1
    else if (a <= 2.15d+82) then
        tmp = 0.0d0 - t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = a * Math.log(t);
	double tmp;
	if (a <= -5.2e+36) {
		tmp = t_1;
	} else if (a <= 2.15e+82) {
		tmp = 0.0 - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = a * math.log(t)
	tmp = 0
	if a <= -5.2e+36:
		tmp = t_1
	elif a <= 2.15e+82:
		tmp = 0.0 - t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(a * log(t))
	tmp = 0.0
	if (a <= -5.2e+36)
		tmp = t_1;
	elseif (a <= 2.15e+82)
		tmp = Float64(0.0 - t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = a * log(t);
	tmp = 0.0;
	if (a <= -5.2e+36)
		tmp = t_1;
	elseif (a <= 2.15e+82)
		tmp = 0.0 - t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -5.2e+36], t$95$1, If[LessEqual[a, 2.15e+82], N[(0.0 - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -5.2000000000000003e36 or 2.15000000000000007e82 < a

   1. Initial program 99.7%

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

      1. associate--l+ N/A

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

      2. associate-+l+ N/A

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

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\log \left(x + y\right), \color{blue}{\left(\left(\log z - t\right) + \left(a - \frac{1}{2}\right) \cdot \log t\right)}\right) \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\left(x + y\right)\right), \left(\color{blue}{\left(\log z - t\right)} + \left(a - \frac{1}{2}\right) \cdot \log t\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \left(\left(\color{blue}{\log z} - t\right) + \left(a - \frac{1}{2}\right) \cdot \log t\right)\right) \]

      6. associate-+l- N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \left(\log z - \color{blue}{\left(t - \left(a - \frac{1}{2}\right) \cdot \log t\right)}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\log z, \color{blue}{\left(t - \left(a - \frac{1}{2}\right) \cdot \log t\right)}\right)\right) \]

      8. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \left(\color{blue}{t} - \left(a - \frac{1}{2}\right) \cdot \log t\right)\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \left(t + \color{blue}{\left(\mathsf{neg}\left(\left(a - \frac{1}{2}\right) \cdot \log t\right)\right)}\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \color{blue}{\left(\mathsf{neg}\left(\left(a - \frac{1}{2}\right) \cdot \log t\right)\right)}\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \left(\mathsf{neg}\left(\log t \cdot \left(a - \frac{1}{2}\right)\right)\right)\right)\right)\right) \]

      12. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \left(\log t \cdot \color{blue}{\left(\mathsf{neg}\left(\left(a - \frac{1}{2}\right)\right)\right)}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\log t, \color{blue}{\left(\mathsf{neg}\left(\left(a - \frac{1}{2}\right)\right)\right)}\right)\right)\right)\right) \]

      14. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(\mathsf{neg}\left(\color{blue}{\left(a - \frac{1}{2}\right)}\right)\right)\right)\right)\right)\right) \]

      15. neg-sub0 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(0 - \color{blue}{\left(a - \frac{1}{2}\right)}\right)\right)\right)\right)\right) \]

      16. associate--r- N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(\left(0 - a\right) + \color{blue}{\frac{1}{2}}\right)\right)\right)\right)\right) \]

      17. neg-sub0 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(\left(\mathsf{neg}\left(a\right)\right) + \frac{1}{2}\right)\right)\right)\right)\right) \]

      18. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(a\right)\right)}\right)\right)\right)\right)\right) \]

      19. unsub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(\frac{1}{2} - \color{blue}{a}\right)\right)\right)\right)\right) \]

      20. --lowering--.f64 99.7%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \mathsf{\_.f64}\left(\frac{1}{2}, \color{blue}{a}\right)\right)\right)\right)\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\log t, \color{blue}{a}\right) \]

      3. log-lowering-log.f64 85.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), a\right) \]

   7. Simplified 85.4%

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

   if -5.2000000000000003e36 < a < 2.15000000000000007e82

   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. Step-by-step derivation

      1. associate--l+ N/A

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

      2. associate-+l+ N/A

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

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\log \left(x + y\right), \color{blue}{\left(\left(\log z - t\right) + \left(a - \frac{1}{2}\right) \cdot \log t\right)}\right) \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\left(x + y\right)\right), \left(\color{blue}{\left(\log z - t\right)} + \left(a - \frac{1}{2}\right) \cdot \log t\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \left(\left(\color{blue}{\log z} - t\right) + \left(a - \frac{1}{2}\right) \cdot \log t\right)\right) \]

      6. associate-+l- N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \left(\log z - \color{blue}{\left(t - \left(a - \frac{1}{2}\right) \cdot \log t\right)}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\log z, \color{blue}{\left(t - \left(a - \frac{1}{2}\right) \cdot \log t\right)}\right)\right) \]

      8. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \left(\color{blue}{t} - \left(a - \frac{1}{2}\right) \cdot \log t\right)\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \left(t + \color{blue}{\left(\mathsf{neg}\left(\left(a - \frac{1}{2}\right) \cdot \log t\right)\right)}\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \color{blue}{\left(\mathsf{neg}\left(\left(a - \frac{1}{2}\right) \cdot \log t\right)\right)}\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \left(\mathsf{neg}\left(\log t \cdot \left(a - \frac{1}{2}\right)\right)\right)\right)\right)\right) \]

      12. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \left(\log t \cdot \color{blue}{\left(\mathsf{neg}\left(\left(a - \frac{1}{2}\right)\right)\right)}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\log t, \color{blue}{\left(\mathsf{neg}\left(\left(a - \frac{1}{2}\right)\right)\right)}\right)\right)\right)\right) \]

      14. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(\mathsf{neg}\left(\color{blue}{\left(a - \frac{1}{2}\right)}\right)\right)\right)\right)\right)\right) \]

      15. neg-sub0 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(0 - \color{blue}{\left(a - \frac{1}{2}\right)}\right)\right)\right)\right)\right) \]

      16. associate--r- N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(\left(0 - a\right) + \color{blue}{\frac{1}{2}}\right)\right)\right)\right)\right) \]

      17. neg-sub0 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(\left(\mathsf{neg}\left(a\right)\right) + \frac{1}{2}\right)\right)\right)\right)\right) \]

      18. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(a\right)\right)}\right)\right)\right)\right)\right) \]

      19. unsub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(\frac{1}{2} - \color{blue}{a}\right)\right)\right)\right)\right) \]

      20. --lowering--.f64 99.6%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \mathsf{\_.f64}\left(\frac{1}{2}, \color{blue}{a}\right)\right)\right)\right)\right) \]

   3. Simplified 99.6%

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

   5. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{t} \]

      3. --lowering--.f64 50.6%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{t}\right) \]

   7. Simplified 50.6%

      \[\leadsto \color{blue}{0 - t} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 50.6%

         \[\leadsto \mathsf{neg.f64}\left(t\right) \]

   9. Applied egg-rr 50.6%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.2 \cdot 10^{+36}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{elif}\;a \leq 2.15 \cdot 10^{+82}:\\ \;\;\;\;0 - t\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 41.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.8 \cdot 10^{-12}:\\ \;\;\;\;\log \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;0 - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 1.8e-12) (log (+ x y)) (- 0.0 t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 1.8e-12) {
		tmp = log((x + y));
	} else {
		tmp = 0.0 - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 1.8d-12) then
        tmp = log((x + y))
    else
        tmp = 0.0d0 - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 1.8e-12) {
		tmp = Math.log((x + y));
	} else {
		tmp = 0.0 - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 1.8e-12:
		tmp = math.log((x + y))
	else:
		tmp = 0.0 - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 1.8e-12)
		tmp = log(Float64(x + y));
	else
		tmp = Float64(0.0 - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 1.8e-12)
		tmp = log((x + y));
	else
		tmp = 0.0 - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 1.8e-12], N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision], N[(0.0 - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.8e-12

   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. Step-by-step derivation

      1. associate--l+ N/A

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

      2. associate-+l+ N/A

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

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\log \left(x + y\right), \color{blue}{\left(\left(\log z - t\right) + \left(a - \frac{1}{2}\right) \cdot \log t\right)}\right) \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\left(x + y\right)\right), \left(\color{blue}{\left(\log z - t\right)} + \left(a - \frac{1}{2}\right) \cdot \log t\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \left(\left(\color{blue}{\log z} - t\right) + \left(a - \frac{1}{2}\right) \cdot \log t\right)\right) \]

      6. associate-+l- N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \left(\log z - \color{blue}{\left(t - \left(a - \frac{1}{2}\right) \cdot \log t\right)}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\log z, \color{blue}{\left(t - \left(a - \frac{1}{2}\right) \cdot \log t\right)}\right)\right) \]

      8. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \left(\color{blue}{t} - \left(a - \frac{1}{2}\right) \cdot \log t\right)\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \left(t + \color{blue}{\left(\mathsf{neg}\left(\left(a - \frac{1}{2}\right) \cdot \log t\right)\right)}\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \color{blue}{\left(\mathsf{neg}\left(\left(a - \frac{1}{2}\right) \cdot \log t\right)\right)}\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \left(\mathsf{neg}\left(\log t \cdot \left(a - \frac{1}{2}\right)\right)\right)\right)\right)\right) \]

      12. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \left(\log t \cdot \color{blue}{\left(\mathsf{neg}\left(\left(a - \frac{1}{2}\right)\right)\right)}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\log t, \color{blue}{\left(\mathsf{neg}\left(\left(a - \frac{1}{2}\right)\right)\right)}\right)\right)\right)\right) \]

      14. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(\mathsf{neg}\left(\color{blue}{\left(a - \frac{1}{2}\right)}\right)\right)\right)\right)\right)\right) \]

      15. neg-sub0 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(0 - \color{blue}{\left(a - \frac{1}{2}\right)}\right)\right)\right)\right)\right) \]

      16. associate--r- N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(\left(0 - a\right) + \color{blue}{\frac{1}{2}}\right)\right)\right)\right)\right) \]

      17. neg-sub0 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(\left(\mathsf{neg}\left(a\right)\right) + \frac{1}{2}\right)\right)\right)\right)\right) \]

      18. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(a\right)\right)}\right)\right)\right)\right)\right) \]

      19. unsub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(\frac{1}{2} - \color{blue}{a}\right)\right)\right)\right)\right) \]

      20. --lowering--.f64 99.4%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \mathsf{\_.f64}\left(\frac{1}{2}, \color{blue}{a}\right)\right)\right)\right)\right) \]

   3. Simplified 99.4%

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

   5. Taylor expanded in a around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \color{blue}{\left(a \cdot \log t\right)}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \left(\log t \cdot \color{blue}{a}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{*.f64}\left(\log t, \color{blue}{a}\right)\right) \]

      3. log-lowering-log.f64 57.9%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), a\right)\right) \]

   7. Simplified 57.9%

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

   8. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\log \left(x + y\right)} \]
   9. Step-by-step derivation

      1. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\left(x + y\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{log.f64}\left(\left(y + x\right)\right) \]

      3. +-lowering-+.f64 9.9%

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(y, x\right)\right) \]

   10. Simplified 9.9%

       \[\leadsto \color{blue}{\log \left(y + x\right)} \]

   if 1.8e-12 < 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. Step-by-step derivation

      1. associate--l+ N/A

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

      2. associate-+l+ N/A

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

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\log \left(x + y\right), \color{blue}{\left(\left(\log z - t\right) + \left(a - \frac{1}{2}\right) \cdot \log t\right)}\right) \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\left(x + y\right)\right), \left(\color{blue}{\left(\log z - t\right)} + \left(a - \frac{1}{2}\right) \cdot \log t\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \left(\left(\color{blue}{\log z} - t\right) + \left(a - \frac{1}{2}\right) \cdot \log t\right)\right) \]

      6. associate-+l- N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \left(\log z - \color{blue}{\left(t - \left(a - \frac{1}{2}\right) \cdot \log t\right)}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\log z, \color{blue}{\left(t - \left(a - \frac{1}{2}\right) \cdot \log t\right)}\right)\right) \]

      8. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \left(\color{blue}{t} - \left(a - \frac{1}{2}\right) \cdot \log t\right)\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \left(t + \color{blue}{\left(\mathsf{neg}\left(\left(a - \frac{1}{2}\right) \cdot \log t\right)\right)}\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \color{blue}{\left(\mathsf{neg}\left(\left(a - \frac{1}{2}\right) \cdot \log t\right)\right)}\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \left(\mathsf{neg}\left(\log t \cdot \left(a - \frac{1}{2}\right)\right)\right)\right)\right)\right) \]

      12. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \left(\log t \cdot \color{blue}{\left(\mathsf{neg}\left(\left(a - \frac{1}{2}\right)\right)\right)}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\log t, \color{blue}{\left(\mathsf{neg}\left(\left(a - \frac{1}{2}\right)\right)\right)}\right)\right)\right)\right) \]

      14. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(\mathsf{neg}\left(\color{blue}{\left(a - \frac{1}{2}\right)}\right)\right)\right)\right)\right)\right) \]

      15. neg-sub0 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(0 - \color{blue}{\left(a - \frac{1}{2}\right)}\right)\right)\right)\right)\right) \]

      16. associate--r- N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(\left(0 - a\right) + \color{blue}{\frac{1}{2}}\right)\right)\right)\right)\right) \]

      17. neg-sub0 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(\left(\mathsf{neg}\left(a\right)\right) + \frac{1}{2}\right)\right)\right)\right)\right) \]

      18. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(a\right)\right)}\right)\right)\right)\right)\right) \]

      19. unsub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(\frac{1}{2} - \color{blue}{a}\right)\right)\right)\right)\right) \]

      20. --lowering--.f64 99.9%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \mathsf{\_.f64}\left(\frac{1}{2}, \color{blue}{a}\right)\right)\right)\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{t} \]

      3. --lowering--.f64 68.8%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{t}\right) \]

   7. Simplified 68.8%

      \[\leadsto \color{blue}{0 - t} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 68.8%

         \[\leadsto \mathsf{neg.f64}\left(t\right) \]

   9. Applied egg-rr 68.8%

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

4. Final simplification 38.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.8 \cdot 10^{-12}:\\ \;\;\;\;\log \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;0 - t\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 77.2% accurate, 2.9× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t, double a) {
	return (log(t) * (a - 0.5)) - 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(t) * (a - 0.5d0)) - t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in t around inf

   \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(-1 \cdot t\right)}, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{neg}\left(t\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{\_.f64}\left(a, \frac{1}{2}\right)}, \mathsf{log.f64}\left(t\right)\right)\right) \]

   2. neg-sub0 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(0 - t\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{\_.f64}\left(a, \frac{1}{2}\right)}, \mathsf{log.f64}\left(t\right)\right)\right) \]

   3. --lowering--.f64 76.9%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(0, t\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{\_.f64}\left(a, \frac{1}{2}\right)}, \mathsf{log.f64}\left(t\right)\right)\right) \]

5. Simplified 76.9%

   \[\leadsto \color{blue}{\left(0 - t\right)} + \left(a - 0.5\right) \cdot \log t \]

6. Final simplification 76.9%

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

Alternative 14: 39.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.05 \cdot 10^{-12}:\\ \;\;\;\;\log y\\ \mathbf{else}:\\ \;\;\;\;0 - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (if (<= t 1.05e-12) (log y) (- 0.0 t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 1.05e-12) {
		tmp = log(y);
	} else {
		tmp = 0.0 - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 1.05d-12) then
        tmp = log(y)
    else
        tmp = 0.0d0 - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 1.05e-12) {
		tmp = Math.log(y);
	} else {
		tmp = 0.0 - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 1.05e-12:
		tmp = math.log(y)
	else:
		tmp = 0.0 - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 1.05e-12)
		tmp = log(y);
	else
		tmp = Float64(0.0 - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 1.05e-12)
		tmp = log(y);
	else
		tmp = 0.0 - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 1.05e-12], N[Log[y], $MachinePrecision], N[(0.0 - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.04999999999999997e-12

   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. Step-by-step derivation

      1. associate--l+ N/A

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

      2. associate-+l+ N/A

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

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\log \left(x + y\right), \color{blue}{\left(\left(\log z - t\right) + \left(a - \frac{1}{2}\right) \cdot \log t\right)}\right) \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\left(x + y\right)\right), \left(\color{blue}{\left(\log z - t\right)} + \left(a - \frac{1}{2}\right) \cdot \log t\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \left(\left(\color{blue}{\log z} - t\right) + \left(a - \frac{1}{2}\right) \cdot \log t\right)\right) \]

      6. associate-+l- N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \left(\log z - \color{blue}{\left(t - \left(a - \frac{1}{2}\right) \cdot \log t\right)}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\log z, \color{blue}{\left(t - \left(a - \frac{1}{2}\right) \cdot \log t\right)}\right)\right) \]

      8. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \left(\color{blue}{t} - \left(a - \frac{1}{2}\right) \cdot \log t\right)\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \left(t + \color{blue}{\left(\mathsf{neg}\left(\left(a - \frac{1}{2}\right) \cdot \log t\right)\right)}\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \color{blue}{\left(\mathsf{neg}\left(\left(a - \frac{1}{2}\right) \cdot \log t\right)\right)}\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \left(\mathsf{neg}\left(\log t \cdot \left(a - \frac{1}{2}\right)\right)\right)\right)\right)\right) \]

      12. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \left(\log t \cdot \color{blue}{\left(\mathsf{neg}\left(\left(a - \frac{1}{2}\right)\right)\right)}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\log t, \color{blue}{\left(\mathsf{neg}\left(\left(a - \frac{1}{2}\right)\right)\right)}\right)\right)\right)\right) \]

      14. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(\mathsf{neg}\left(\color{blue}{\left(a - \frac{1}{2}\right)}\right)\right)\right)\right)\right)\right) \]

      15. neg-sub0 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(0 - \color{blue}{\left(a - \frac{1}{2}\right)}\right)\right)\right)\right)\right) \]

      16. associate--r- N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(\left(0 - a\right) + \color{blue}{\frac{1}{2}}\right)\right)\right)\right)\right) \]

      17. neg-sub0 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(\left(\mathsf{neg}\left(a\right)\right) + \frac{1}{2}\right)\right)\right)\right)\right) \]

      18. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(a\right)\right)}\right)\right)\right)\right)\right) \]

      19. unsub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(\frac{1}{2} - \color{blue}{a}\right)\right)\right)\right)\right) \]

      20. --lowering--.f64 99.4%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \mathsf{\_.f64}\left(\frac{1}{2}, \color{blue}{a}\right)\right)\right)\right)\right) \]

   3. Simplified 99.4%

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

   5. Taylor expanded in a around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \color{blue}{\left(a \cdot \log t\right)}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \left(\log t \cdot \color{blue}{a}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{*.f64}\left(\log t, \color{blue}{a}\right)\right) \]

      3. log-lowering-log.f64 57.9%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), a\right)\right) \]

   7. Simplified 57.9%

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

   8. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\log \left(x + y\right)} \]
   9. Step-by-step derivation

      1. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\left(x + y\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{log.f64}\left(\left(y + x\right)\right) \]

      3. +-lowering-+.f64 9.9%

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(y, x\right)\right) \]

   10. Simplified 9.9%

       \[\leadsto \color{blue}{\log \left(y + x\right)} \]

   11. Taylor expanded in y around inf

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

       1. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\log \left(\frac{1}{y}\right)\right) \]

       2. log-rec N/A

          \[\leadsto \mathsf{neg}\left(\left(\mathsf{neg}\left(\log y\right)\right)\right) \]

       3. remove-double-neg N/A

          \[\leadsto \log y \]

       4. log-lowering-log.f64 6.3%

          \[\leadsto \mathsf{log.f64}\left(y\right) \]

   13. Simplified 6.3%

       \[\leadsto \color{blue}{\log y} \]

   if 1.04999999999999997e-12 < 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. Step-by-step derivation

      1. associate--l+ N/A

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

      2. associate-+l+ N/A

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

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\log \left(x + y\right), \color{blue}{\left(\left(\log z - t\right) + \left(a - \frac{1}{2}\right) \cdot \log t\right)}\right) \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\left(x + y\right)\right), \left(\color{blue}{\left(\log z - t\right)} + \left(a - \frac{1}{2}\right) \cdot \log t\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \left(\left(\color{blue}{\log z} - t\right) + \left(a - \frac{1}{2}\right) \cdot \log t\right)\right) \]

      6. associate-+l- N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \left(\log z - \color{blue}{\left(t - \left(a - \frac{1}{2}\right) \cdot \log t\right)}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\log z, \color{blue}{\left(t - \left(a - \frac{1}{2}\right) \cdot \log t\right)}\right)\right) \]

      8. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \left(\color{blue}{t} - \left(a - \frac{1}{2}\right) \cdot \log t\right)\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \left(t + \color{blue}{\left(\mathsf{neg}\left(\left(a - \frac{1}{2}\right) \cdot \log t\right)\right)}\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \color{blue}{\left(\mathsf{neg}\left(\left(a - \frac{1}{2}\right) \cdot \log t\right)\right)}\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \left(\mathsf{neg}\left(\log t \cdot \left(a - \frac{1}{2}\right)\right)\right)\right)\right)\right) \]

      12. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \left(\log t \cdot \color{blue}{\left(\mathsf{neg}\left(\left(a - \frac{1}{2}\right)\right)\right)}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\log t, \color{blue}{\left(\mathsf{neg}\left(\left(a - \frac{1}{2}\right)\right)\right)}\right)\right)\right)\right) \]

      14. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(\mathsf{neg}\left(\color{blue}{\left(a - \frac{1}{2}\right)}\right)\right)\right)\right)\right)\right) \]

      15. neg-sub0 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(0 - \color{blue}{\left(a - \frac{1}{2}\right)}\right)\right)\right)\right)\right) \]

      16. associate--r- N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(\left(0 - a\right) + \color{blue}{\frac{1}{2}}\right)\right)\right)\right)\right) \]

      17. neg-sub0 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(\left(\mathsf{neg}\left(a\right)\right) + \frac{1}{2}\right)\right)\right)\right)\right) \]

      18. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(a\right)\right)}\right)\right)\right)\right)\right) \]

      19. unsub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(\frac{1}{2} - \color{blue}{a}\right)\right)\right)\right)\right) \]

      20. --lowering--.f64 99.9%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \mathsf{\_.f64}\left(\frac{1}{2}, \color{blue}{a}\right)\right)\right)\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{t} \]

      3. --lowering--.f64 68.8%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{t}\right) \]

   7. Simplified 68.8%

      \[\leadsto \color{blue}{0 - t} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 68.8%

         \[\leadsto \mathsf{neg.f64}\left(t\right) \]

   9. Applied egg-rr 68.8%

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

4. Final simplification 37.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.05 \cdot 10^{-12}:\\ \;\;\;\;\log y\\ \mathbf{else}:\\ \;\;\;\;0 - t\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 74.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ a \cdot \log t - t \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t, double a) {
	return (a * log(t)) - 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 = (a * log(t)) - t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 x around 0

   \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\color{blue}{\left(\log y + \log z\right)}, t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\left(\log z + \log y\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\color{blue}{a}, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

   2. remove-double-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\left(\log z + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log y\right)\right)\right)\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

   3. log-rec N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\left(\log z + \left(\mathsf{neg}\left(\log \left(\frac{1}{y}\right)\right)\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

   4. mul-1-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\left(\log z + -1 \cdot \log \left(\frac{1}{y}\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\log z, \left(-1 \cdot \log \left(\frac{1}{y}\right)\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\color{blue}{a}, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

   6. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \left(-1 \cdot \log \left(\frac{1}{y}\right)\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

   7. mul-1-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \left(\mathsf{neg}\left(\log \left(\frac{1}{y}\right)\right)\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

   8. log-rec N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log y\right)\right)\right)\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

   9. remove-double-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \log y\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

   10. log-lowering-log.f64 66.1%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{log.f64}\left(y\right)\right), t\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]

5. Simplified 66.1%

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

   1. +-commutative N/A

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

   2. associate-+r- N/A

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

   3. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\left(a - \frac{1}{2}\right) \cdot \log t + \left(\log z + \log y\right)\right), \color{blue}{t}\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(\left(a - \frac{1}{2}\right) \cdot \log t\right), \left(\log z + \log y\right)\right), t\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{2}\right), \log t\right), \left(\log z + \log y\right)\right), t\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log t\right), \left(\log z + \log y\right)\right), t\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a + \frac{-1}{2}\right), \log t\right), \left(\log z + \log y\right)\right), t\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{2}\right), \log t\right), \left(\log z + \log y\right)\right), t\right) \]

   9. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{2}\right), \mathsf{log.f64}\left(t\right)\right), \left(\log z + \log y\right)\right), t\right) \]

   10. sum-log N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{2}\right), \mathsf{log.f64}\left(t\right)\right), \log \left(z \cdot y\right)\right), t\right) \]

   11. log-lowering-log.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{2}\right), \mathsf{log.f64}\left(t\right)\right), \mathsf{log.f64}\left(\left(z \cdot y\right)\right)\right), t\right) \]

   12. *-lowering-*.f64 55.2%

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{2}\right), \mathsf{log.f64}\left(t\right)\right), \mathsf{log.f64}\left(\mathsf{*.f64}\left(z, y\right)\right)\right), t\right) \]

7. Applied egg-rr 55.2%

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

8. Taylor expanded in a around inf

   \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(a \cdot \log t\right)}, t\right) \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\log t \cdot a\right), t\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\log t, a\right), t\right) \]

   3. log-lowering-log.f64 74.2%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), a\right), t\right) \]

10. Simplified 74.2%

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

11. Final simplification 74.2%

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

Alternative 16: 38.1% accurate, 104.3× speedup

Math

\[\begin{array}{l} \\ 0 - t \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (- 0.0 t))

C

double code(double x, double y, double z, double t, double a) {
	return 0.0 - 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 = 0.0d0 - t
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return 0.0 - t;
}

Python

def code(x, y, z, t, a):
	return 0.0 - t

Julia

function code(x, y, z, t, a)
	return Float64(0.0 - t)
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = 0.0 - t;
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(0.0 - t), $MachinePrecision]

Derivation

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. Step-by-step derivation

   1. associate--l+ N/A

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

   2. associate-+l+ N/A

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

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\log \left(x + y\right), \color{blue}{\left(\left(\log z - t\right) + \left(a - \frac{1}{2}\right) \cdot \log t\right)}\right) \]

   4. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\left(x + y\right)\right), \left(\color{blue}{\left(\log z - t\right)} + \left(a - \frac{1}{2}\right) \cdot \log t\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \left(\left(\color{blue}{\log z} - t\right) + \left(a - \frac{1}{2}\right) \cdot \log t\right)\right) \]

   6. associate-+l- N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \left(\log z - \color{blue}{\left(t - \left(a - \frac{1}{2}\right) \cdot \log t\right)}\right)\right) \]

   7. --lowering--.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\log z, \color{blue}{\left(t - \left(a - \frac{1}{2}\right) \cdot \log t\right)}\right)\right) \]

   8. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \left(\color{blue}{t} - \left(a - \frac{1}{2}\right) \cdot \log t\right)\right)\right) \]

   9. sub-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \left(t + \color{blue}{\left(\mathsf{neg}\left(\left(a - \frac{1}{2}\right) \cdot \log t\right)\right)}\right)\right)\right) \]

   10. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \color{blue}{\left(\mathsf{neg}\left(\left(a - \frac{1}{2}\right) \cdot \log t\right)\right)}\right)\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \left(\mathsf{neg}\left(\log t \cdot \left(a - \frac{1}{2}\right)\right)\right)\right)\right)\right) \]

   12. distribute-rgt-neg-in N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \left(\log t \cdot \color{blue}{\left(\mathsf{neg}\left(\left(a - \frac{1}{2}\right)\right)\right)}\right)\right)\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\log t, \color{blue}{\left(\mathsf{neg}\left(\left(a - \frac{1}{2}\right)\right)\right)}\right)\right)\right)\right) \]

   14. log-lowering-log.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(\mathsf{neg}\left(\color{blue}{\left(a - \frac{1}{2}\right)}\right)\right)\right)\right)\right)\right) \]

   15. neg-sub0 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(0 - \color{blue}{\left(a - \frac{1}{2}\right)}\right)\right)\right)\right)\right) \]

   16. associate--r- N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(\left(0 - a\right) + \color{blue}{\frac{1}{2}}\right)\right)\right)\right)\right) \]

   17. neg-sub0 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(\left(\mathsf{neg}\left(a\right)\right) + \frac{1}{2}\right)\right)\right)\right)\right) \]

   18. +-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(a\right)\right)}\right)\right)\right)\right)\right) \]

   19. unsub-neg N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \left(\frac{1}{2} - \color{blue}{a}\right)\right)\right)\right)\right) \]

   20. --lowering--.f64 99.7%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(\mathsf{log.f64}\left(z\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{log.f64}\left(t\right), \mathsf{\_.f64}\left(\frac{1}{2}, \color{blue}{a}\right)\right)\right)\right)\right) \]

3. Simplified 99.7%

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

5. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

      \[\leadsto 0 - \color{blue}{t} \]

   3. --lowering--.f64 35.2%

      \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{t}\right) \]

7. Simplified 35.2%

   \[\leadsto \color{blue}{0 - t} \]
8. Step-by-step derivation

   1. sub0-neg N/A

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

   2. neg-lowering-neg.f64 35.2%

      \[\leadsto \mathsf{neg.f64}\left(t\right) \]

9. Applied egg-rr 35.2%

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

10. Final simplification 35.2%

    \[\leadsto 0 - t \]
11. 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 2024158 
(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))))