Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 14.7s

Alternatives: 9

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% 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: 97.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 7.8 \cdot 10^{-22}:\\ \;\;\;\;\log z + \left(\log \left(x + y\right) + \log t \cdot \left(a + -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot a - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 7.8e-22)
   (+ (log z) (+ (log (+ x y)) (* (log t) (+ a -0.5))))
   (- (* (log t) a) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 7.8e-22) {
		tmp = log(z) + (log((x + y)) + (log(t) * (a + -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 (t <= 7.8d-22) then
        tmp = log(z) + (log((x + y)) + (log(t) * (a + (-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 (t <= 7.8e-22) {
		tmp = Math.log(z) + (Math.log((x + y)) + (Math.log(t) * (a + -0.5)));
	} else {
		tmp = (Math.log(t) * a) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 7.8e-22:
		tmp = math.log(z) + (math.log((x + y)) + (math.log(t) * (a + -0.5)))
	else:
		tmp = (math.log(t) * a) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 7.8e-22)
		tmp = Float64(log(z) + Float64(log(Float64(x + y)) + Float64(log(t) * Float64(a + -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 (t <= 7.8e-22)
		tmp = log(z) + (log((x + y)) + (log(t) * (a + -0.5)));
	else
		tmp = (log(t) * a) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 7.8e-22], N[(N[Log[z], $MachinePrecision] + N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 7.79999999999999996e-22

   1. Initial program 99.3%

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

   3. Taylor expanded in t around 0

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

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

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

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

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

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

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

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

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

      5. +-commutative N/A

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

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

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

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

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

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

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. +-lowering-+.f64 99.4%

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

   5. Simplified 99.4%

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

   if 7.79999999999999996e-22 < 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 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\log \left(x + y\right) + \left(\log z - \left(t + \log t \cdot \left(0.5 - a\right)\right)\right)} \]
   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 73.2%

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

   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 98.5%

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

   9. Simplified 98.5%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.8 \cdot 10^{-22}:\\ \;\;\;\;\log z + \left(\log \left(x + y\right) + \log t \cdot \left(a + -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot a - t\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 7.8 \cdot 10^{-22}:\\ \;\;\;\;\log z + \left(\log t \cdot \left(a + -0.5\right) + \log y\right)\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot a - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 7.8e-22)
   (+ (log z) (+ (* (log t) (+ a -0.5)) (log y)))
   (- (* (log t) a) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 7.8e-22) {
		tmp = log(z) + ((log(t) * (a + -0.5)) + log(y));
	} 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 (t <= 7.8d-22) then
        tmp = log(z) + ((log(t) * (a + (-0.5d0))) + log(y))
    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 (t <= 7.8e-22) {
		tmp = Math.log(z) + ((Math.log(t) * (a + -0.5)) + Math.log(y));
	} else {
		tmp = (Math.log(t) * a) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 7.8e-22:
		tmp = math.log(z) + ((math.log(t) * (a + -0.5)) + math.log(y))
	else:
		tmp = (math.log(t) * a) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 7.8e-22)
		tmp = Float64(log(z) + Float64(Float64(log(t) * Float64(a + -0.5)) + log(y)));
	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 (t <= 7.8e-22)
		tmp = log(z) + ((log(t) * (a + -0.5)) + log(y));
	else
		tmp = (log(t) * a) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 7.8e-22], N[(N[Log[z], $MachinePrecision] + N[(N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision]), $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 7.79999999999999996e-22

   1. Initial program 99.3%

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

   3. Taylor expanded in t around inf

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

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

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

      2. +-commutative N/A

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

      3. associate--l+ N/A

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

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

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

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

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

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

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

      7. +-commutative N/A

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

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

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

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

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

      12. /-lowering-/.f64 N/A

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

      13. log-lowering-log.f64 98.4%

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

   5. Simplified 98.4%

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

      1. *-commutative N/A

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

      2. fma-define N/A

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

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

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

   7. Applied egg-rr 73.2%

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

   8. Taylor expanded in t around 0

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

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

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

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

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

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

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

      4. +-lowering-+.f64 73.2%

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

   10. Simplified 73.2%

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

   11. Taylor expanded in y around inf

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

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

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

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

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

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

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

       4. mul-1-neg N/A

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

       5. log-rec N/A

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

       6. remove-double-neg N/A

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

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

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

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

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

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

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

       10. sub-neg N/A

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

       11. metadata-eval N/A

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

       12. +-lowering-+.f64 60.4%

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

   13. Simplified 60.4%

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

   if 7.79999999999999996e-22 < 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 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\log \left(x + y\right) + \left(\log z - \left(t + \log t \cdot \left(0.5 - a\right)\right)\right)} \]
   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 73.2%

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

   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 98.5%

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

   9. Simplified 98.5%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.8 \cdot 10^{-22}:\\ \;\;\;\;\log z + \left(\log t \cdot \left(a + -0.5\right) + \log y\right)\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot a - t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 4.8 \cdot 10^{-58}:\\ \;\;\;\;\log t \cdot \left(a + -0.5\right) + \log \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot \left(a - 0.5\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 4.8e-58)
   (+ (* (log t) (+ a -0.5)) (log (* y z)))
   (- (* (log t) (- a 0.5)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 4.8e-58) {
		tmp = (log(t) * (a + -0.5)) + log((y * z));
	} else {
		tmp = (log(t) * (a - 0.5)) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 4.8d-58) then
        tmp = (log(t) * (a + (-0.5d0))) + log((y * z))
    else
        tmp = (log(t) * (a - 0.5d0)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 4.8e-58) {
		tmp = (Math.log(t) * (a + -0.5)) + Math.log((y * z));
	} else {
		tmp = (Math.log(t) * (a - 0.5)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 4.8e-58:
		tmp = (math.log(t) * (a + -0.5)) + math.log((y * z))
	else:
		tmp = (math.log(t) * (a - 0.5)) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 4.8e-58)
		tmp = Float64(Float64(log(t) * Float64(a + -0.5)) + log(Float64(y * z)));
	else
		tmp = Float64(Float64(log(t) * Float64(a - 0.5)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 4.8e-58)
		tmp = (log(t) * (a + -0.5)) + log((y * z));
	else
		tmp = (log(t) * (a - 0.5)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 4.8e-58], N[(N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision]), $MachinePrecision] + N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 4.8000000000000001e-58

   1. Initial program 99.3%

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

   3. Taylor expanded in t around inf

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

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

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

      2. +-commutative N/A

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

      3. associate--l+ N/A

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

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

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

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

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

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

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

      7. +-commutative N/A

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

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

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

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

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

      12. /-lowering-/.f64 N/A

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

      13. log-lowering-log.f64 98.3%

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

   5. Simplified 98.3%

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

      1. *-commutative N/A

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

      2. fma-define N/A

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

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

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

   7. Applied egg-rr 73.9%

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

   8. Taylor expanded in t around 0

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

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

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

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

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

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

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

      4. +-lowering-+.f64 73.8%

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

   10. Simplified 73.8%

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

   11. Taylor expanded in x around 0

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

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

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

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

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

       3. *-commutative N/A

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

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

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

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

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

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

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

       7. sub-neg N/A

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

       8. metadata-eval N/A

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

       9. +-lowering-+.f64 39.7%

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

   13. Simplified 39.7%

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

   if 4.8000000000000001e-58 < t

   1. Initial program 99.9%

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

   3. Taylor expanded in t around inf

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

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

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

      2. +-commutative N/A

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

      3. associate--l+ N/A

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

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

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

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

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

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

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

      7. +-commutative N/A

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

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

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

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

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

      12. /-lowering-/.f64 N/A

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

      13. log-lowering-log.f64 99.8%

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

   5. Simplified 99.8%

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

   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 96.8%

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

   8. Simplified 96.8%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.8 \cdot 10^{-58}:\\ \;\;\;\;\log t \cdot \left(a + -0.5\right) + \log \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot \left(a - 0.5\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log t \cdot a - t\\ \mathbf{if}\;a \leq -1.3:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 0.65:\\ \;\;\;\;\log \left(x + y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* (log t) a) t)))
   (if (<= a -1.3) t_1 (if (<= a 0.65) (- (log (+ x 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 <= -1.3) {
		tmp = t_1;
	} else if (a <= 0.65) {
		tmp = log((x + y)) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (log(t) * a) - t
    if (a <= (-1.3d0)) then
        tmp = t_1
    else if (a <= 0.65d0) then
        tmp = log((x + y)) - t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (Math.log(t) * a) - t;
	double tmp;
	if (a <= -1.3) {
		tmp = t_1;
	} else if (a <= 0.65) {
		tmp = Math.log((x + 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 <= -1.3:
		tmp = t_1
	elif a <= 0.65:
		tmp = math.log((x + 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 <= -1.3)
		tmp = t_1;
	elseif (a <= 0.65)
		tmp = Float64(log(Float64(x + y)) - t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (log(t) * a) - t;
	tmp = 0.0;
	if (a <= -1.3)
		tmp = t_1;
	elseif (a <= 0.65)
		tmp = log((x + y)) - t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[a, -1.3], t$95$1, If[LessEqual[a, 0.65], N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.30000000000000004 or 0.650000000000000022 < a

   1. Initial program 99.7%

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

      1. associate--l+ N/A

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

      2. associate-+l+ N/A

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

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

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

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

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

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

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

      6. associate-+l- N/A

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

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

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

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

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

      9. sub-neg N/A

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

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

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

      11. *-commutative N/A

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

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

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

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

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

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

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

      15. neg-sub0 N/A

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

      16. associate--r- N/A

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

      17. neg-sub0 N/A

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

      18. +-commutative N/A

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

      19. unsub-neg N/A

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

      20. --lowering--.f64 99.7%

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

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\log \left(x + y\right) + \left(\log z - \left(t + \log t \cdot \left(0.5 - a\right)\right)\right)} \]
   4. Add Preprocessing
   5. 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 72.8%

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

   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%

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

   if -1.30000000000000004 < a < 0.650000000000000022

   1. Initial program 99.6%

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

      1. associate--l+ N/A

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

      2. associate-+l+ N/A

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

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

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

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

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

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

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

      6. associate-+l- N/A

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

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

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

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

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

      9. sub-neg N/A

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

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

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

      11. *-commutative N/A

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

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

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

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

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

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

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

      15. neg-sub0 N/A

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

      16. associate--r- N/A

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

      17. neg-sub0 N/A

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

      18. +-commutative N/A

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

      19. unsub-neg N/A

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

      20. --lowering--.f64 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in t around inf

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

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

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

      2. associate--r+ N/A

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

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

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

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

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

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

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

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

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

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

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

      12. log-rec N/A

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

      13. mul-1-neg N/A

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

      14. associate-*r* N/A

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

      15. metadata-eval N/A

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

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

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

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

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

      18. /-lowering-/.f64 N/A

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

      19. --lowering--.f64 98.8%

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

   7. Simplified 98.8%

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

   8. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 58.7%

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

   10. Simplified 58.7%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.3:\\ \;\;\;\;\log t \cdot a - t\\ \mathbf{elif}\;a \leq 0.65:\\ \;\;\;\;\log \left(x + y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot a - t\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log t \cdot a\\ \mathbf{if}\;a \leq -1.15 \cdot 10^{+39}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 6 \cdot 10^{+43}:\\ \;\;\;\;0 - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (log t) a)))
   (if (<= a -1.15e+39) t_1 (if (<= a 6e+43) (- 0.0 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 <= -1.15e+39) {
		tmp = t_1;
	} else if (a <= 6e+43) {
		tmp = 0.0 - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = log(t) * a
    if (a <= (-1.15d+39)) then
        tmp = t_1
    else if (a <= 6d+43) then
        tmp = 0.0d0 - t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = Math.log(t) * a;
	double tmp;
	if (a <= -1.15e+39) {
		tmp = t_1;
	} else if (a <= 6e+43) {
		tmp = 0.0 - 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 <= -1.15e+39:
		tmp = t_1
	elif a <= 6e+43:
		tmp = 0.0 - 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 <= -1.15e+39)
		tmp = t_1;
	elseif (a <= 6e+43)
		tmp = Float64(0.0 - t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = log(t) * a;
	tmp = 0.0;
	if (a <= -1.15e+39)
		tmp = t_1;
	elseif (a <= 6e+43)
		tmp = 0.0 - t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[a, -1.15e+39], t$95$1, If[LessEqual[a, 6e+43], N[(0.0 - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.15000000000000006e39 or 6.00000000000000033e43 < 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 83.8%

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

   7. Simplified 83.8%

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

   if -1.15000000000000006e39 < a < 6.00000000000000033e43

   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 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

      3. --lowering--.f64 53.8%

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

   7. Simplified 53.8%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 53.8%

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

   9. Applied egg-rr 53.8%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.15 \cdot 10^{+39}:\\ \;\;\;\;\log t \cdot a\\ \mathbf{elif}\;a \leq 6 \cdot 10^{+43}:\\ \;\;\;\;0 - t\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.4% 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

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

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

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

   2. +-commutative N/A

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

   3. associate--l+ N/A

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

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

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

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

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

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

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

   7. +-commutative N/A

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

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

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

   9. sub-neg N/A

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

   10. metadata-eval N/A

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

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

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

   12. /-lowering-/.f64 N/A

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

   13. log-lowering-log.f64 99.2%

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

5. Simplified 99.2%

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

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 76.2%

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

8. Simplified 76.2%

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

9. Final simplification 76.2%

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

Alternative 8: 74.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \log t \cdot a - t \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t, double a) {
	return (log(t) * a) - 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) - t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[Log[t], $MachinePrecision] * a), $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. 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 73.5%

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

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 73.5%

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

9. Simplified 73.5%

   \[\leadsto \color{blue}{\log t \cdot a} - t \]
10. Add Preprocessing

Alternative 9: 37.8% 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 38.9%

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

7. Simplified 38.9%

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

   1. sub0-neg N/A

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

   2. neg-lowering-neg.f64 38.9%

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

9. Applied egg-rr 38.9%

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

10. Final simplification 38.9%

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

Developer Target 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024160 
(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))))