Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 16.2s

Alternatives: 13

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

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

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 99.6%

   \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
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. Final simplification 99.6%

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

Alternative 2: 80.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -100000:\\ \;\;\;\;\log t \cdot \left(a - 0.5\right) - t\\ \mathbf{elif}\;a - 0.5 \leq -0.4:\\ \;\;\;\;\log y + \left(\log z - \left(t + \log t \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot a - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (- a 0.5) -100000.0)
   (- (* (log t) (- a 0.5)) t)
   (if (<= (- a 0.5) -0.4)
     (+ (log y) (- (log z) (+ t (* (log t) 0.5))))
     (- (* (log t) a) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a - 0.5) <= -100000.0) {
		tmp = (log(t) * (a - 0.5)) - t;
	} else if ((a - 0.5) <= -0.4) {
		tmp = log(y) + (log(z) - (t + (log(t) * 0.5)));
	} else {
		tmp = (log(t) * a) - 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 ((a - 0.5d0) <= (-100000.0d0)) then
        tmp = (log(t) * (a - 0.5d0)) - t
    else if ((a - 0.5d0) <= (-0.4d0)) then
        tmp = log(y) + (log(z) - (t + (log(t) * 0.5d0)))
    else
        tmp = (log(t) * a) - 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 ((a - 0.5) <= -100000.0) {
		tmp = (Math.log(t) * (a - 0.5)) - t;
	} else if ((a - 0.5) <= -0.4) {
		tmp = Math.log(y) + (Math.log(z) - (t + (Math.log(t) * 0.5)));
	} else {
		tmp = (Math.log(t) * a) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a - 0.5) <= -100000.0:
		tmp = (math.log(t) * (a - 0.5)) - t
	elif (a - 0.5) <= -0.4:
		tmp = math.log(y) + (math.log(z) - (t + (math.log(t) * 0.5)))
	else:
		tmp = (math.log(t) * a) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(a - 0.5) <= -100000.0)
		tmp = Float64(Float64(log(t) * Float64(a - 0.5)) - t);
	elseif (Float64(a - 0.5) <= -0.4)
		tmp = Float64(log(y) + Float64(log(z) - Float64(t + Float64(log(t) * 0.5))));
	else
		tmp = Float64(Float64(log(t) * a) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a - 0.5) <= -100000.0)
		tmp = (log(t) * (a - 0.5)) - t;
	elseif ((a - 0.5) <= -0.4)
		tmp = log(y) + (log(z) - (t + (log(t) * 0.5)));
	else
		tmp = (log(t) * a) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(a - 0.5), $MachinePrecision], -100000.0], N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[N[(a - 0.5), $MachinePrecision], -0.4], N[(N[Log[y], $MachinePrecision] + N[(N[Log[z], $MachinePrecision] - N[(t + N[(N[Log[t], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 a #s(literal 1/2 binary64)) < -1e5

   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

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

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

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

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

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

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

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

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

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

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

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

      12. +-commutative N/A

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

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

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

      14. neg-sub0 N/A

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

      15. --lowering--.f64 99.7%

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

   5. Simplified 99.7%

      \[\leadsto \color{blue}{\left(1 - \frac{\log z + \log \left(y + x\right)}{t}\right) \cdot \left(0 - t\right)} + \left(a - 0.5\right) \cdot \log t \]

   6. Taylor expanded in t around inf

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(-1 \cdot t\right)}, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]
   7. 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-lowering-neg.f64 98.5%

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

   8. Simplified 98.5%

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

   if -1e5 < (-.f64 a #s(literal 1/2 binary64)) < -0.40000000000000002

   1. Initial program 99.5%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   2. 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 x around 0

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

      1. log-lowering-log.f64 66.2%

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

   7. Simplified 66.2%

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

   8. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

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

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

      3. log-lowering-log.f64 65.7%

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

   10. Simplified 65.7%

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

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

   1. Initial program 99.7%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   2. 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. 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 82.6%

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

   7. Taylor expanded in a around inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(a \cdot \log t\right)}, t\right) \]
   8. 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 98.7%

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

   9. Simplified 98.7%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -100000:\\ \;\;\;\;\log t \cdot \left(a - 0.5\right) - t\\ \mathbf{elif}\;a - 0.5 \leq -0.4:\\ \;\;\;\;\log y + \left(\log z - \left(t + \log t \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot a - t\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 69.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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(t) * (a - 0.5d0)) - t)) + log(y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

   \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
2. 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 x around 0

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

   1. log-lowering-log.f64 70.6%

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

7. Simplified 70.6%

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

8. Final simplification 70.6%

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

Alternative 4: 86.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log t \cdot \left(a - 0.5\right)\\ \mathbf{if}\;t \leq 1.02 \cdot 10^{-8}:\\ \;\;\;\;\left(\log \left(\left(x + y\right) \cdot z\right) + t\_1\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1 - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (log t) (- a 0.5))))
   (if (<= t 1.02e-8) (- (+ (log (* (+ x y) z)) t_1) t) (- t_1 t))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log(t) * (a - 0.5);
	double tmp;
	if (t <= 1.02e-8) {
		tmp = (log(((x + y) * z)) + t_1) - t;
	} else {
		tmp = t_1 - 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(t) * (a - 0.5d0)
    if (t <= 1.02d-8) then
        tmp = (log(((x + y) * z)) + t_1) - t
    else
        tmp = t_1 - 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(t) * (a - 0.5);
	double tmp;
	if (t <= 1.02e-8) {
		tmp = (Math.log(((x + y) * z)) + t_1) - t;
	} else {
		tmp = t_1 - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = math.log(t) * (a - 0.5)
	tmp = 0
	if t <= 1.02e-8:
		tmp = (math.log(((x + y) * z)) + t_1) - t
	else:
		tmp = t_1 - t
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(t) * Float64(a - 0.5))
	tmp = 0.0
	if (t <= 1.02e-8)
		tmp = Float64(Float64(log(Float64(Float64(x + y) * z)) + t_1) - t);
	else
		tmp = Float64(t_1 - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = log(t) * (a - 0.5);
	tmp = 0.0;
	if (t <= 1.02e-8)
		tmp = (log(((x + y) * z)) + t_1) - t;
	else
		tmp = t_1 - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, 1.02e-8], N[(N[(N[Log[N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision] + t$95$1), $MachinePrecision] - t), $MachinePrecision], N[(t$95$1 - t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 1.02000000000000003e-8

   1. Initial program 99.3%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   2. 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 99.3%

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

   3. Simplified 99.3%

      \[\leadsto \color{blue}{\log \left(x + y\right) + \left(\log z - \left(t + \log t \cdot \left(0.5 - a\right)\right)\right)} \]
   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 82.7%

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

   if 1.02000000000000003e-8 < t

   1. Initial program 99.8%

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

   3. Taylor expanded in t around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

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

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

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

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

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

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

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

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

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

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

      12. +-commutative N/A

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

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

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

      14. neg-sub0 N/A

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

      15. --lowering--.f64 99.9%

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

   5. Simplified 99.9%

      \[\leadsto \color{blue}{\left(1 - \frac{\log z + \log \left(y + x\right)}{t}\right) \cdot \left(0 - t\right)} + \left(a - 0.5\right) \cdot \log t \]

   6. Taylor expanded in t around inf

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(-1 \cdot t\right)}, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]
   7. 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-lowering-neg.f64 98.1%

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

   8. Simplified 98.1%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.02 \cdot 10^{-8}:\\ \;\;\;\;\left(\log \left(\left(x + y\right) \cdot z\right) + \log t \cdot \left(a - 0.5\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot \left(a - 0.5\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log t \cdot \left(a - 0.5\right)\\ \mathbf{if}\;t \leq 6.2 \cdot 10^{-9}:\\ \;\;\;\;\log \left(\left(x + y\right) \cdot z\right) + t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_1 - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (log t) (- a 0.5))))
   (if (<= t 6.2e-9) (+ (log (* (+ x y) z)) t_1) (- t_1 t))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log(t) * (a - 0.5);
	double tmp;
	if (t <= 6.2e-9) {
		tmp = log(((x + y) * z)) + t_1;
	} else {
		tmp = t_1 - 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(t) * (a - 0.5d0)
    if (t <= 6.2d-9) then
        tmp = log(((x + y) * z)) + t_1
    else
        tmp = t_1 - 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(t) * (a - 0.5);
	double tmp;
	if (t <= 6.2e-9) {
		tmp = Math.log(((x + y) * z)) + t_1;
	} else {
		tmp = t_1 - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = math.log(t) * (a - 0.5)
	tmp = 0
	if t <= 6.2e-9:
		tmp = math.log(((x + y) * z)) + t_1
	else:
		tmp = t_1 - t
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(t) * Float64(a - 0.5))
	tmp = 0.0
	if (t <= 6.2e-9)
		tmp = Float64(log(Float64(Float64(x + y) * z)) + t_1);
	else
		tmp = Float64(t_1 - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = log(t) * (a - 0.5);
	tmp = 0.0;
	if (t <= 6.2e-9)
		tmp = log(((x + y) * z)) + t_1;
	else
		tmp = t_1 - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, 6.2e-9], N[(N[Log[N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision] + t$95$1), $MachinePrecision], N[(t$95$1 - t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 6.2000000000000001e-9

   1. Initial program 99.3%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   2. 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 99.3%

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

   3. Simplified 99.3%

      \[\leadsto \color{blue}{\log \left(x + y\right) + \left(\log z - \left(t + \log t \cdot \left(0.5 - a\right)\right)\right)} \]
   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 82.7%

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

   7. Taylor expanded in t around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. --lowering--.f64 82.6%

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

   9. Simplified 82.6%

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

   if 6.2000000000000001e-9 < t

   1. Initial program 99.8%

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

   3. Taylor expanded in t around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

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

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

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

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

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

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

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

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

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

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

      12. +-commutative N/A

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

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

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

      14. neg-sub0 N/A

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

      15. --lowering--.f64 99.9%

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

   5. Simplified 99.9%

      \[\leadsto \color{blue}{\left(1 - \frac{\log z + \log \left(y + x\right)}{t}\right) \cdot \left(0 - t\right)} + \left(a - 0.5\right) \cdot \log t \]

   6. Taylor expanded in t around inf

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(-1 \cdot t\right)}, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]
   7. 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-lowering-neg.f64 98.1%

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

   8. Simplified 98.1%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.2 \cdot 10^{-9}:\\ \;\;\;\;\log \left(\left(x + y\right) \cdot z\right) + \log t \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot \left(a - 0.5\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log t \cdot \left(a - 0.5\right)\\ \mathbf{if}\;t \leq 1.65 \cdot 10^{-10}:\\ \;\;\;\;\log \left(y \cdot z\right) + t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_1 - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (log t) (- a 0.5))))
   (if (<= t 1.65e-10) (+ (log (* y z)) t_1) (- t_1 t))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log(t) * (a - 0.5);
	double tmp;
	if (t <= 1.65e-10) {
		tmp = log((y * z)) + t_1;
	} else {
		tmp = t_1 - 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(t) * (a - 0.5d0)
    if (t <= 1.65d-10) then
        tmp = log((y * z)) + t_1
    else
        tmp = t_1 - 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(t) * (a - 0.5);
	double tmp;
	if (t <= 1.65e-10) {
		tmp = Math.log((y * z)) + t_1;
	} else {
		tmp = t_1 - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = math.log(t) * (a - 0.5)
	tmp = 0
	if t <= 1.65e-10:
		tmp = math.log((y * z)) + t_1
	else:
		tmp = t_1 - t
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(t) * Float64(a - 0.5))
	tmp = 0.0
	if (t <= 1.65e-10)
		tmp = Float64(log(Float64(y * z)) + t_1);
	else
		tmp = Float64(t_1 - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = log(t) * (a - 0.5);
	tmp = 0.0;
	if (t <= 1.65e-10)
		tmp = log((y * z)) + t_1;
	else
		tmp = t_1 - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, 1.65e-10], N[(N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision] + t$95$1), $MachinePrecision], N[(t$95$1 - t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 1.65e-10

   1. Initial program 99.3%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   2. 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 99.3%

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

   3. Simplified 99.3%

      \[\leadsto \color{blue}{\log \left(x + y\right) + \left(\log z - \left(t + \log t \cdot \left(0.5 - a\right)\right)\right)} \]
   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 82.7%

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

   7. Taylor expanded in x around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

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

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

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

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

      7. --lowering--.f64 48.7%

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

   9. Simplified 48.7%

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

   10. Taylor expanded in t around 0

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

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

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

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

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

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

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

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

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

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

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

       6. --lowering--.f64 48.7%

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

   12. Simplified 48.7%

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

   if 1.65e-10 < t

   1. Initial program 99.8%

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

   3. Taylor expanded in t around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

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

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

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

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

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

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

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

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

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

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

      12. +-commutative N/A

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

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

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

      14. neg-sub0 N/A

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

      15. --lowering--.f64 99.9%

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

   5. Simplified 99.9%

      \[\leadsto \color{blue}{\left(1 - \frac{\log z + \log \left(y + x\right)}{t}\right) \cdot \left(0 - t\right)} + \left(a - 0.5\right) \cdot \log t \]

   6. Taylor expanded in t around inf

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(-1 \cdot t\right)}, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]
   7. 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-lowering-neg.f64 98.1%

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

   8. Simplified 98.1%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.65 \cdot 10^{-10}:\\ \;\;\;\;\log \left(y \cdot z\right) + \log t \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot \left(a - 0.5\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log t \cdot a - t\\ \mathbf{if}\;a \leq -0.00017:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 0.36:\\ \;\;\;\;\log y - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* (log t) a) t)))
   (if (<= a -0.00017) t_1 (if (<= a 0.36) (- (log y) t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (log(t) * a) - t;
	double tmp;
	if (a <= -0.00017) {
		tmp = t_1;
	} else if (a <= 0.36) {
		tmp = log(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 = (log(t) * a) - t
    if (a <= (-0.00017d0)) then
        tmp = t_1
    else if (a <= 0.36d0) then
        tmp = log(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 = (Math.log(t) * a) - t;
	double tmp;
	if (a <= -0.00017) {
		tmp = t_1;
	} else if (a <= 0.36) {
		tmp = Math.log(y) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (math.log(t) * a) - t
	tmp = 0
	if a <= -0.00017:
		tmp = t_1
	elif a <= 0.36:
		tmp = math.log(y) - t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(log(t) * a) - t)
	tmp = 0.0
	if (a <= -0.00017)
		tmp = t_1;
	elseif (a <= 0.36)
		tmp = Float64(log(y) - t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (log(t) * a) - t;
	tmp = 0.0;
	if (a <= -0.00017)
		tmp = t_1;
	elseif (a <= 0.36)
		tmp = log(y) - t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[a, -0.00017], t$95$1, If[LessEqual[a, 0.36], N[(N[Log[y], $MachinePrecision] - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.7e-4 or 0.35999999999999999 < a

   1. Initial program 99.6%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   2. 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 79.1%

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

   7. Taylor expanded in a around inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(a \cdot \log t\right)}, t\right) \]
   8. 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 97.8%

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

   9. Simplified 97.8%

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

   if -1.7e-4 < a < 0.35999999999999999

   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 99.6%

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

   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 x around 0

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

      1. log-lowering-log.f64 65.9%

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

   7. Simplified 65.9%

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

   8. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

      3. --lowering--.f64 46.2%

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

   10. Simplified 46.2%

       \[\leadsto \log y + \color{blue}{\left(0 - t\right)} \]
   11. Step-by-step derivation

       1. sub0-neg N/A

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

       2. unsub-neg N/A

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

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

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

       4. log-lowering-log.f64 46.2%

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

   12. Applied egg-rr 46.2%

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

Alternative 8: 55.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log t \cdot a\\ \mathbf{if}\;a \leq -2.8 \cdot 10^{+78}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 3 \cdot 10^{+28}:\\ \;\;\;\;\log y - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (log t) a)))
   (if (<= a -2.8e+78) t_1 (if (<= a 3e+28) (- (log y) t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log(t) * a;
	double tmp;
	if (a <= -2.8e+78) {
		tmp = t_1;
	} else if (a <= 3e+28) {
		tmp = log(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 = log(t) * a
    if (a <= (-2.8d+78)) then
        tmp = t_1
    else if (a <= 3d+28) then
        tmp = log(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 = Math.log(t) * a;
	double tmp;
	if (a <= -2.8e+78) {
		tmp = t_1;
	} else if (a <= 3e+28) {
		tmp = Math.log(y) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = math.log(t) * a
	tmp = 0
	if a <= -2.8e+78:
		tmp = t_1
	elif a <= 3e+28:
		tmp = math.log(y) - t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(t) * a)
	tmp = 0.0
	if (a <= -2.8e+78)
		tmp = t_1;
	elseif (a <= 3e+28)
		tmp = Float64(log(y) - t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = log(t) * a;
	tmp = 0.0;
	if (a <= -2.8e+78)
		tmp = t_1;
	elseif (a <= 3e+28)
		tmp = log(y) - t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[a, -2.8e+78], t$95$1, If[LessEqual[a, 3e+28], N[(N[Log[y], $MachinePrecision] - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -2.8000000000000001e78 or 3.0000000000000001e28 < a

   1. Initial program 99.6%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   2. 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 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 78.6%

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

   7. Simplified 78.6%

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

   if -2.8000000000000001e78 < a < 3.0000000000000001e28

   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 99.6%

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

   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 x around 0

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

      1. log-lowering-log.f64 68.0%

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

   7. Simplified 68.0%

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

   8. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

      3. --lowering--.f64 46.4%

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

   10. Simplified 46.4%

       \[\leadsto \log y + \color{blue}{\left(0 - t\right)} \]
   11. Step-by-step derivation

       1. sub0-neg N/A

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

       2. unsub-neg N/A

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

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

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

       4. log-lowering-log.f64 46.4%

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

   12. Applied egg-rr 46.4%

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

Alternative 9: 62.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 6.8 \cdot 10^{+18}:\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;0 - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 6.8e+18) (* (log t) a) (- 0.0 t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 6.8e+18) {
		tmp = log(t) * a;
	} 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 <= 6.8d+18) then
        tmp = log(t) * a
    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 <= 6.8e+18) {
		tmp = Math.log(t) * a;
	} else {
		tmp = 0.0 - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 6.8e+18:
		tmp = math.log(t) * a
	else:
		tmp = 0.0 - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 6.8e+18)
		tmp = Float64(log(t) * a);
	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 <= 6.8e+18)
		tmp = log(t) * a;
	else
		tmp = 0.0 - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 6.8e+18], N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision], N[(0.0 - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 6.8e18

   1. Initial program 99.3%

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

      \[\leadsto \color{blue}{a \cdot \log t} \]
   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 51.4%

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

   7. Simplified 51.4%

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

   if 6.8e18 < 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 75.6%

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

   7. Simplified 75.6%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 75.6%

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

   9. Applied egg-rr 75.6%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.8 \cdot 10^{+18}:\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;0 - t\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 76.9% 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 99.6%

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

3. Taylor expanded in t around -inf

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

   1. mul-1-neg N/A

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

   2. *-commutative N/A

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

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

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

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

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

   5. mul-1-neg N/A

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

   6. unsub-neg N/A

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

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

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

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

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

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

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

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

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

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

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

   12. +-commutative N/A

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

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

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

   14. neg-sub0 N/A

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

   15. --lowering--.f64 99.6%

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

5. Simplified 99.6%

   \[\leadsto \color{blue}{\left(1 - \frac{\log z + \log \left(y + x\right)}{t}\right) \cdot \left(0 - t\right)} + \left(a - 0.5\right) \cdot \log t \]

6. Taylor expanded in t around inf

   \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(-1 \cdot t\right)}, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, \frac{1}{2}\right), \mathsf{log.f64}\left(t\right)\right)\right) \]
7. 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-lowering-neg.f64 77.5%

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

8. Simplified 77.5%

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

9. Final simplification 77.5%

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

Alternative 11: 39.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 0.00115:\\ \;\;\;\;\log y\\ \mathbf{else}:\\ \;\;\;\;0 - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (if (<= t 0.00115) (log y) (- 0.0 t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 0.00115) {
		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 <= 0.00115d0) 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 <= 0.00115) {
		tmp = Math.log(y);
	} else {
		tmp = 0.0 - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 0.00115:
		tmp = math.log(y)
	else:
		tmp = 0.0 - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 0.00115)
		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 <= 0.00115)
		tmp = log(y);
	else
		tmp = 0.0 - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 0.00115], N[Log[y], $MachinePrecision], N[(0.0 - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 0.00115

   1. Initial program 99.3%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   2. 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 x around 0

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

      1. log-lowering-log.f64 61.9%

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

   7. Simplified 61.9%

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

   8. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

      3. --lowering--.f64 6.4%

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

   10. Simplified 6.4%

       \[\leadsto \log y + \color{blue}{\left(0 - t\right)} \]

   11. Taylor expanded in t around 0

       \[\leadsto \color{blue}{\log y} \]
   12. Step-by-step derivation

       1. log-lowering-log.f64 6.4%

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

   13. Simplified 6.4%

       \[\leadsto \color{blue}{\log y} \]

   if 0.00115 < 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 71.5%

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

   7. Simplified 71.5%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 71.5%

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

   9. Applied egg-rr 71.5%

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

4. Final simplification 39.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 0.00115:\\ \;\;\;\;\log y\\ \mathbf{else}:\\ \;\;\;\;0 - t\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 38.0% 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 99.6%

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

   \[\leadsto \color{blue}{-1 \cdot t} \]
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 37.7%

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

7. Simplified 37.7%

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

   1. sub0-neg N/A

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

   2. neg-lowering-neg.f64 37.7%

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

9. Applied egg-rr 37.7%

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

10. Final simplification 37.7%

    \[\leadsto 0 - t \]
11. Add Preprocessing

Alternative 13: 2.5% accurate, 313.0× speedup

Math

\[\begin{array}{l} \\ t \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 t)

C

double code(double x, double y, double z, double t, double a) {
	return 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 = t
end function

Java

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

Python

def code(x, y, z, t, a):
	return t

Julia

function code(x, y, z, t, a)
	return t
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = t;
end

Wolfram

code[x_, y_, z_, t_, a_] := t

Derivation

1. Initial program 99.6%

   \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
2. 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 37.7%

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

7. Simplified 37.7%

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

   1. flip-- N/A

      \[\leadsto \frac{0 \cdot 0 - t \cdot t}{\color{blue}{0 + t}} \]

   2. metadata-eval N/A

      \[\leadsto \frac{0 - t \cdot t}{0 + t} \]

   3. neg-sub0 N/A

      \[\leadsto \frac{\mathsf{neg}\left(t \cdot t\right)}{\color{blue}{0} + t} \]

   4. +-lft-identity N/A

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

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

      \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(t \cdot t\right)\right), \color{blue}{t}\right) \]

   6. neg-sub0 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(0 - t \cdot t\right), t\right) \]

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

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

   8. *-lowering-*.f64 18.7%

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

9. Applied egg-rr 18.7%

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

    1. div-sub N/A

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

    2. flip3-- N/A

       \[\leadsto \frac{{\left(\frac{0}{t}\right)}^{3} - {\left(\frac{t \cdot t}{t}\right)}^{3}}{\color{blue}{\frac{0}{t} \cdot \frac{0}{t} + \left(\frac{t \cdot t}{t} \cdot \frac{t \cdot t}{t} + \frac{0}{t} \cdot \frac{t \cdot t}{t}\right)}} \]

11. Applied egg-rr 2.4%

    \[\leadsto \color{blue}{t} \]
12. 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 2024152 
(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))))