Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 21.2s

Alternatives: 14

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, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[Log[z], $MachinePrecision] + N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] - N[(N[Log[t], $MachinePrecision] * N[(0.5 - a), $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- 99.6%

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

   2. +-commutative 99.6%

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

   3. associate--l+ 99.6%

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

   4. sub-neg 99.6%

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

   5. +-commutative 99.6%

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

   6. *-commutative 99.6%

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

   7. distribute-rgt-neg-in 99.6%

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

   8. fma-def 99.6%

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

   9. sub-neg 99.6%

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

   10. +-commutative 99.6%

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

   11. distribute-neg-in 99.6%

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

   12. metadata-eval 99.6%

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

   13. metadata-eval 99.6%

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

   14. unsub-neg 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate--l+ 99.6%

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

   2. sub-neg 99.6%

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

   3. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 3: 67.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 108:\\ \;\;\;\;\log y + \left(\log z + \log t \cdot \left(a - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log y + \left(\log t \cdot a - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 108.0)
   (+ (log y) (+ (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 <= 108.0) {
		tmp = log(y) + (log(z) + (log(t) * (a - 0.5)));
	} else {
		tmp = log(y) + ((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 <= 108.0d0) then
        tmp = log(y) + (log(z) + (log(t) * (a - 0.5d0)))
    else
        tmp = log(y) + ((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 <= 108.0) {
		tmp = Math.log(y) + (Math.log(z) + (Math.log(t) * (a - 0.5)));
	} else {
		tmp = Math.log(y) + ((Math.log(t) * a) - t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 108.0:
		tmp = math.log(y) + (math.log(z) + (math.log(t) * (a - 0.5)))
	else:
		tmp = math.log(y) + ((math.log(t) * a) - t)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 108.0)
		tmp = Float64(log(y) + Float64(log(z) + Float64(log(t) * Float64(a - 0.5))));
	else
		tmp = Float64(log(y) + 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 <= 108.0)
		tmp = log(y) + (log(z) + (log(t) * (a - 0.5)));
	else
		tmp = log(y) + ((log(t) * a) - t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 108.0], N[(N[Log[y], $MachinePrecision] + N[(N[Log[z], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Log[y], $MachinePrecision] + N[(N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 108

   1. Initial program 99.2%

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

      1. associate--l+ 99.3%

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

      2. sub-neg 99.3%

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

      3. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in x around 0 59.8%

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

      1. associate--l+ 59.9%

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

      2. sub-neg 59.9%

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

      3. metadata-eval 59.9%

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

   6. Simplified 59.9%

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

   7. Taylor expanded in t around 0 59.7%

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

   if 108 < 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+ 99.9%

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

      2. sub-neg 99.9%

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

      3. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 73.6%

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

      1. associate--l+ 73.6%

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

      2. sub-neg 73.6%

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

      3. metadata-eval 73.6%

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

   6. Simplified 73.6%

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

   7. Taylor expanded in a around inf 72.8%

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

      1. *-commutative 72.8%

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

   9. Simplified 72.8%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 108:\\ \;\;\;\;\log y + \left(\log z + \log t \cdot \left(a - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log y + \left(\log t \cdot a - t\right)\\ \end{array} \]

Alternative 4: 68.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate--l+ 99.6%

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

   2. sub-neg 99.6%

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

   3. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in x around 0 66.8%

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

   1. associate--l+ 66.8%

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

   2. sub-neg 66.8%

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

   3. metadata-eval 66.8%

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

6. Simplified 66.8%

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

7. Final simplification 66.8%

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

Alternative 5: 56.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y + \left(\log t \cdot a - t\right)\\ \mathbf{if}\;a \leq -7 \cdot 10^{-284}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 5.2 \cdot 10^{-293}:\\ \;\;\;\;\log \left(z \cdot \left(\left(x + y\right) \cdot {t}^{-0.5}\right)\right)\\ \mathbf{elif}\;a \leq 2.4 \cdot 10^{-246}:\\ \;\;\;\;\log z + \left(\log \left(x + y\right) - t\right)\\ \mathbf{elif}\;a \leq 3.4 \cdot 10^{-179}:\\ \;\;\;\;\log \left(z \cdot y\right) - \log t \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log y) (- (* (log t) a) t))))
   (if (<= a -7e-284)
     t_1
     (if (<= a 5.2e-293)
       (log (* z (* (+ x y) (pow t -0.5))))
       (if (<= a 2.4e-246)
         (+ (log z) (- (log (+ x y)) t))
         (if (<= a 3.4e-179) (- (log (* z y)) (* (log t) 0.5)) t_1))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log(y) + ((log(t) * a) - t);
	double tmp;
	if (a <= -7e-284) {
		tmp = t_1;
	} else if (a <= 5.2e-293) {
		tmp = log((z * ((x + y) * pow(t, -0.5))));
	} else if (a <= 2.4e-246) {
		tmp = log(z) + (log((x + y)) - t);
	} else if (a <= 3.4e-179) {
		tmp = log((z * y)) - (log(t) * 0.5);
	} 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(y) + ((log(t) * a) - t)
    if (a <= (-7d-284)) then
        tmp = t_1
    else if (a <= 5.2d-293) then
        tmp = log((z * ((x + y) * (t ** (-0.5d0)))))
    else if (a <= 2.4d-246) then
        tmp = log(z) + (log((x + y)) - t)
    else if (a <= 3.4d-179) then
        tmp = log((z * y)) - (log(t) * 0.5d0)
    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(y) + ((Math.log(t) * a) - t);
	double tmp;
	if (a <= -7e-284) {
		tmp = t_1;
	} else if (a <= 5.2e-293) {
		tmp = Math.log((z * ((x + y) * Math.pow(t, -0.5))));
	} else if (a <= 2.4e-246) {
		tmp = Math.log(z) + (Math.log((x + y)) - t);
	} else if (a <= 3.4e-179) {
		tmp = Math.log((z * y)) - (Math.log(t) * 0.5);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = math.log(y) + ((math.log(t) * a) - t)
	tmp = 0
	if a <= -7e-284:
		tmp = t_1
	elif a <= 5.2e-293:
		tmp = math.log((z * ((x + y) * math.pow(t, -0.5))))
	elif a <= 2.4e-246:
		tmp = math.log(z) + (math.log((x + y)) - t)
	elif a <= 3.4e-179:
		tmp = math.log((z * y)) - (math.log(t) * 0.5)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(y) + Float64(Float64(log(t) * a) - t))
	tmp = 0.0
	if (a <= -7e-284)
		tmp = t_1;
	elseif (a <= 5.2e-293)
		tmp = log(Float64(z * Float64(Float64(x + y) * (t ^ -0.5))));
	elseif (a <= 2.4e-246)
		tmp = Float64(log(z) + Float64(log(Float64(x + y)) - t));
	elseif (a <= 3.4e-179)
		tmp = Float64(log(Float64(z * y)) - Float64(log(t) * 0.5));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = log(y) + ((log(t) * a) - t);
	tmp = 0.0;
	if (a <= -7e-284)
		tmp = t_1;
	elseif (a <= 5.2e-293)
		tmp = log((z * ((x + y) * (t ^ -0.5))));
	elseif (a <= 2.4e-246)
		tmp = log(z) + (log((x + y)) - t);
	elseif (a <= 3.4e-179)
		tmp = log((z * y)) - (log(t) * 0.5);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] + N[(N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -7e-284], t$95$1, If[LessEqual[a, 5.2e-293], N[Log[N[(z * N[(N[(x + y), $MachinePrecision] * N[Power[t, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[a, 2.4e-246], N[(N[Log[z], $MachinePrecision] + N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 3.4e-179], N[(N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision] - N[(N[Log[t], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if a < -6.99999999999999951e-284 or 3.3999999999999997e-179 < 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+ 99.7%

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

      2. sub-neg 99.7%

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

      3. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 69.4%

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

      1. associate--l+ 69.4%

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

      2. sub-neg 69.4%

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

      3. metadata-eval 69.4%

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

   6. Simplified 69.4%

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

   7. Taylor expanded in a around inf 59.8%

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

      1. *-commutative 59.8%

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

   9. Simplified 59.8%

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

   if -6.99999999999999951e-284 < a < 5.1999999999999996e-293

   1. Initial program 98.8%

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

      1. associate--l+ 98.9%

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

      2. +-commutative 98.9%

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

      3. associate-+l+ 99.4%

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

      4. +-commutative 99.4%

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

      5. fma-def 99.4%

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

      6. sub-neg 99.4%

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

      7. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in a around 0 99.4%

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

      1. *-commutative 99.4%

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

   6. Simplified 99.4%

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

   7. Taylor expanded in t around 0 84.8%

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

      1. add-log-exp 84.8%

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

      2. sum-log 77.5%

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

      3. exp-sum 77.5%

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

      4. add-exp-log 77.6%

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

      5. +-commutative 77.6%

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

      6. exp-to-pow 78.0%

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

   9. Applied egg-rr 78.0%

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

   if 5.1999999999999996e-293 < a < 2.3999999999999998e-246

   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- 99.7%

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

      2. +-commutative 99.7%

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

      3. associate--l+ 99.7%

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

      4. sub-neg 99.7%

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

      5. +-commutative 99.7%

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

      6. *-commutative 99.7%

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

      7. distribute-rgt-neg-in 99.7%

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

      8. fma-def 99.7%

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

      9. sub-neg 99.7%

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

      10. +-commutative 99.7%

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

      11. distribute-neg-in 99.7%

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

      12. metadata-eval 99.7%

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

      13. metadata-eval 99.7%

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

      14. unsub-neg 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in t around inf 70.9%

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

   if 2.3999999999999998e-246 < a < 3.3999999999999997e-179

   1. Initial program 98.8%

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

      1. associate--l+ 98.9%

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

      2. sub-neg 98.9%

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

      3. metadata-eval 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in x around 0 50.5%

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

      1. associate--l+ 50.5%

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

      2. sub-neg 50.5%

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

      3. metadata-eval 50.5%

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

   6. Simplified 50.5%

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

   7. Taylor expanded in t around 0 38.2%

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

   8. Taylor expanded in t around inf 38.2%

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

      1. associate-+r+ 38.3%

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

      2. log-prod 38.6%

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

      3. mul-1-neg 38.6%

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

      4. unsub-neg 38.6%

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

      5. log-rec 38.6%

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

      6. sub-neg 38.6%

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

      7. metadata-eval 38.6%

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

      8. +-commutative 38.6%

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

   10. Simplified 38.6%

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

   11. Taylor expanded in a around 0 38.6%

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

       1. *-commutative 38.6%

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

   13. Simplified 38.6%

       \[\leadsto \log \left(y \cdot z\right) - \color{blue}{\log t \cdot 0.5} \]
3. Recombined 4 regimes into one program.

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -7 \cdot 10^{-284}:\\ \;\;\;\;\log y + \left(\log t \cdot a - t\right)\\ \mathbf{elif}\;a \leq 5.2 \cdot 10^{-293}:\\ \;\;\;\;\log \left(z \cdot \left(\left(x + y\right) \cdot {t}^{-0.5}\right)\right)\\ \mathbf{elif}\;a \leq 2.4 \cdot 10^{-246}:\\ \;\;\;\;\log z + \left(\log \left(x + y\right) - t\right)\\ \mathbf{elif}\;a \leq 3.4 \cdot 10^{-179}:\\ \;\;\;\;\log \left(z \cdot y\right) - \log t \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\log y + \left(\log t \cdot a - t\right)\\ \end{array} \]

Alternative 6: 74.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 6000000000000:\\ \;\;\;\;\left(\log \left(z \cdot \left(x + y\right)\right) + \log t \cdot \left(a - 0.5\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log y + \left(\log t \cdot a - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 6000000000000.0)
   (- (+ (log (* z (+ x y))) (* (log t) (- a 0.5))) t)
   (+ (log y) (- (* (log t) a) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 6000000000000.0) {
		tmp = (log((z * (x + y))) + (log(t) * (a - 0.5))) - t;
	} else {
		tmp = log(y) + ((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 <= 6000000000000.0d0) then
        tmp = (log((z * (x + y))) + (log(t) * (a - 0.5d0))) - t
    else
        tmp = log(y) + ((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 <= 6000000000000.0) {
		tmp = (Math.log((z * (x + y))) + (Math.log(t) * (a - 0.5))) - t;
	} else {
		tmp = Math.log(y) + ((Math.log(t) * a) - t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 6000000000000.0:
		tmp = (math.log((z * (x + y))) + (math.log(t) * (a - 0.5))) - t
	else:
		tmp = math.log(y) + ((math.log(t) * a) - t)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 6000000000000.0)
		tmp = Float64(Float64(log(Float64(z * Float64(x + y))) + Float64(log(t) * Float64(a - 0.5))) - t);
	else
		tmp = Float64(log(y) + 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 <= 6000000000000.0)
		tmp = (log((z * (x + y))) + (log(t) * (a - 0.5))) - t;
	else
		tmp = log(y) + ((log(t) * a) - t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 6000000000000.0], N[(N[(N[Log[N[(z * N[(x + y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[Log[y], $MachinePrecision] + N[(N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 6e12

   1. Initial program 99.2%

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

      1. associate-+l- 99.2%

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

      2. +-commutative 99.2%

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

      3. associate--l+ 99.3%

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

      4. sub-neg 99.3%

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

      5. +-commutative 99.3%

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

      6. *-commutative 99.3%

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

      7. distribute-rgt-neg-in 99.3%

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

      8. fma-def 99.3%

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

      9. sub-neg 99.3%

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

      10. +-commutative 99.3%

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

      11. distribute-neg-in 99.3%

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

      12. metadata-eval 99.3%

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

      13. metadata-eval 99.3%

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

      14. unsub-neg 99.3%

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

   3. Simplified 99.3%

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

      1. associate-+r- 99.2%

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

      2. fma-udef 99.2%

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

      3. associate--r+ 99.2%

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

      4. +-commutative 99.2%

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

      5. sum-log 78.2%

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

   5. Applied egg-rr 78.2%

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

   if 6e12 < 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+ 99.9%

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

      2. sub-neg 99.9%

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

      3. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 72.9%

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

      1. associate--l+ 72.9%

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

      2. sub-neg 72.9%

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

      3. metadata-eval 72.9%

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

   6. Simplified 72.9%

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

   7. Taylor expanded in a around inf 72.7%

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

      1. *-commutative 72.7%

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

   9. Simplified 72.7%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6000000000000:\\ \;\;\;\;\left(\log \left(z \cdot \left(x + y\right)\right) + \log t \cdot \left(a - 0.5\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log y + \left(\log t \cdot a - t\right)\\ \end{array} \]

Alternative 7: 72.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.35 \cdot 10^{-26}:\\ \;\;\;\;\log \left(z \cdot \left(x + y\right)\right) + \log t \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\log y + \left(\log t \cdot a - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 1.35e-26)
   (+ (log (* z (+ x y))) (* (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 <= 1.35e-26) {
		tmp = log((z * (x + y))) + (log(t) * (a - 0.5));
	} else {
		tmp = log(y) + ((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 <= 1.35d-26) then
        tmp = log((z * (x + y))) + (log(t) * (a - 0.5d0))
    else
        tmp = log(y) + ((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 <= 1.35e-26) {
		tmp = Math.log((z * (x + y))) + (Math.log(t) * (a - 0.5));
	} else {
		tmp = Math.log(y) + ((Math.log(t) * a) - t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 1.35e-26:
		tmp = math.log((z * (x + y))) + (math.log(t) * (a - 0.5))
	else:
		tmp = math.log(y) + ((math.log(t) * a) - t)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 1.35e-26)
		tmp = Float64(log(Float64(z * Float64(x + y))) + Float64(log(t) * Float64(a - 0.5)));
	else
		tmp = Float64(log(y) + 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 <= 1.35e-26)
		tmp = log((z * (x + y))) + (log(t) * (a - 0.5));
	else
		tmp = log(y) + ((log(t) * a) - t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 1.35e-26], N[(N[Log[N[(z * N[(x + y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Log[y], $MachinePrecision] + N[(N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.34999999999999991e-26

   1. Initial program 99.3%

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

      1. associate-+l- 99.3%

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

      2. +-commutative 99.3%

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

      3. associate--l+ 99.3%

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

      4. sub-neg 99.3%

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

      5. +-commutative 99.3%

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

      6. *-commutative 99.3%

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

      7. distribute-rgt-neg-in 99.3%

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

      8. fma-def 99.3%

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

      9. sub-neg 99.3%

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

      10. +-commutative 99.3%

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

      11. distribute-neg-in 99.3%

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

      12. metadata-eval 99.3%

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

      13. metadata-eval 99.3%

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

      14. unsub-neg 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in t around 0 99.3%

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

      1. log-prod 80.7%

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

   6. Simplified 80.7%

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

   if 1.34999999999999991e-26 < t

   1. Initial program 99.8%

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

      1. associate--l+ 99.9%

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

      2. sub-neg 99.9%

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

      3. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 72.2%

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

      1. associate--l+ 72.2%

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

      2. sub-neg 72.2%

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

      3. metadata-eval 72.2%

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

   6. Simplified 72.2%

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

   7. Taylor expanded in a around inf 69.3%

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

      1. *-commutative 69.3%

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

   9. Simplified 69.3%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.35 \cdot 10^{-26}:\\ \;\;\;\;\log \left(z \cdot \left(x + y\right)\right) + \log t \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\log y + \left(\log t \cdot a - t\right)\\ \end{array} \]

Alternative 8: 57.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3 \cdot 10^{-58} \lor \neg \left(a \leq 3.1 \cdot 10^{-66}\right):\\ \;\;\;\;\log y + \left(\log t \cdot a - t\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(z \cdot y\right) \cdot {t}^{-0.5}\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -3e-58) (not (<= a 3.1e-66)))
   (+ (log y) (- (* (log t) a) t))
   (- (log (* (* z y) (pow t -0.5))) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -3e-58) || !(a <= 3.1e-66)) {
		tmp = log(y) + ((log(t) * a) - t);
	} else {
		tmp = log(((z * y) * pow(t, -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 ((a <= (-3d-58)) .or. (.not. (a <= 3.1d-66))) then
        tmp = log(y) + ((log(t) * a) - t)
    else
        tmp = log(((z * y) * (t ** (-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 ((a <= -3e-58) || !(a <= 3.1e-66)) {
		tmp = Math.log(y) + ((Math.log(t) * a) - t);
	} else {
		tmp = Math.log(((z * y) * Math.pow(t, -0.5))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -3e-58) or not (a <= 3.1e-66):
		tmp = math.log(y) + ((math.log(t) * a) - t)
	else:
		tmp = math.log(((z * y) * math.pow(t, -0.5))) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -3e-58) || !(a <= 3.1e-66))
		tmp = Float64(log(y) + Float64(Float64(log(t) * a) - t));
	else
		tmp = Float64(log(Float64(Float64(z * y) * (t ^ -0.5))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -3e-58) || ~((a <= 3.1e-66)))
		tmp = log(y) + ((log(t) * a) - t);
	else
		tmp = log(((z * y) * (t ^ -0.5))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -3e-58], N[Not[LessEqual[a, 3.1e-66]], $MachinePrecision]], N[(N[Log[y], $MachinePrecision] + N[(N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], N[(N[Log[N[(N[(z * y), $MachinePrecision] * N[Power[t, -0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.00000000000000008e-58 or 3.0999999999999997e-66 < a

   1. Initial program 99.8%

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

      1. associate--l+ 99.8%

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

      2. sub-neg 99.8%

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

      3. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 70.7%

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

      1. associate--l+ 70.7%

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

      2. sub-neg 70.7%

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

      3. metadata-eval 70.7%

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

   6. Simplified 70.7%

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

   7. Taylor expanded in a around inf 67.7%

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

      1. *-commutative 67.7%

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

   9. Simplified 67.7%

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

   if -3.00000000000000008e-58 < a < 3.0999999999999997e-66

   1. Initial program 99.3%

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

      1. associate--l+ 99.3%

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

      2. sub-neg 99.3%

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

      3. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in a around 0 99.3%

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

      1. associate-+r+ 99.3%

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

      2. +-commutative 99.3%

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

      3. log-prod 77.2%

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

      4. *-commutative 77.2%

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

      5. *-commutative 77.2%

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

   6. Simplified 77.2%

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

   7. Taylor expanded in x around 0 47.9%

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

      1. +-commutative 47.9%

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

      2. *-commutative 47.9%

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

      3. *-commutative 47.9%

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

   9. Simplified 47.9%

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

   10. Taylor expanded in t around 0 47.9%

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

       1. log-pow 47.9%

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

       2. log-prod 44.6%

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

   12. Simplified 44.6%

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

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3 \cdot 10^{-58} \lor \neg \left(a \leq 3.1 \cdot 10^{-66}\right):\\ \;\;\;\;\log y + \left(\log t \cdot a - t\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(z \cdot y\right) \cdot {t}^{-0.5}\right) - t\\ \end{array} \]

Alternative 9: 60.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.6 \cdot 10^{-26}:\\ \;\;\;\;\log \left(z \cdot y\right) + \log t \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\log y + \left(\log t \cdot a - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 1.6e-26)
   (+ (log (* z y)) (* (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 <= 1.6e-26) {
		tmp = log((z * y)) + (log(t) * (a - 0.5));
	} else {
		tmp = log(y) + ((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 <= 1.6d-26) then
        tmp = log((z * y)) + (log(t) * (a - 0.5d0))
    else
        tmp = log(y) + ((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 <= 1.6e-26) {
		tmp = Math.log((z * y)) + (Math.log(t) * (a - 0.5));
	} else {
		tmp = Math.log(y) + ((Math.log(t) * a) - t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 1.6e-26:
		tmp = math.log((z * y)) + (math.log(t) * (a - 0.5))
	else:
		tmp = math.log(y) + ((math.log(t) * a) - t)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 1.6e-26)
		tmp = Float64(log(Float64(z * y)) + Float64(log(t) * Float64(a - 0.5)));
	else
		tmp = Float64(log(y) + 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 <= 1.6e-26)
		tmp = log((z * y)) + (log(t) * (a - 0.5));
	else
		tmp = log(y) + ((log(t) * a) - t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 1.6e-26], N[(N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Log[y], $MachinePrecision] + N[(N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.6000000000000001e-26

   1. Initial program 99.3%

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

      1. associate--l+ 99.3%

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

      2. sub-neg 99.3%

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

      3. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in x around 0 60.2%

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

      1. associate--l+ 60.2%

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

      2. sub-neg 60.2%

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

      3. metadata-eval 60.2%

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

   6. Simplified 60.2%

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

   7. Taylor expanded in t around 0 60.2%

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

   8. Taylor expanded in t around inf 60.2%

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

      1. associate-+r+ 60.2%

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

      2. log-prod 47.9%

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

      3. mul-1-neg 47.9%

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

      4. unsub-neg 47.9%

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

      5. log-rec 47.9%

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

      6. sub-neg 47.9%

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

      7. metadata-eval 47.9%

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

      8. +-commutative 47.9%

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

   10. Simplified 47.9%

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

   11. Taylor expanded in t around 0 47.9%

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

       1. mul-1-neg 47.9%

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

       2. sub-neg 47.9%

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

       3. metadata-eval 47.9%

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

       4. distribute-rgt-neg-in 47.9%

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

       5. neg-sub0 47.9%

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

       6. +-commutative 47.9%

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

       7. associate--r+ 47.9%

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

       8. metadata-eval 47.9%

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

   13. Simplified 47.9%

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

   if 1.6000000000000001e-26 < t

   1. Initial program 99.8%

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

      1. associate--l+ 99.9%

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

      2. sub-neg 99.9%

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

      3. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 72.2%

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

      1. associate--l+ 72.2%

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

      2. sub-neg 72.2%

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

      3. metadata-eval 72.2%

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

   6. Simplified 72.2%

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

   7. Taylor expanded in a around inf 69.3%

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

      1. *-commutative 69.3%

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

   9. Simplified 69.3%

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

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.6 \cdot 10^{-26}:\\ \;\;\;\;\log \left(z \cdot y\right) + \log t \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\log y + \left(\log t \cdot a - t\right)\\ \end{array} \]

Alternative 10: 48.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 8.6 \cdot 10^{+14}:\\ \;\;\;\;\log y + \log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;\log y - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 8.6e+14) (+ (log y) (* (log t) a)) (- (log y) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 8.6e+14) {
		tmp = log(y) + (log(t) * a);
	} else {
		tmp = log(y) - 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 <= 8.6d+14) then
        tmp = log(y) + (log(t) * a)
    else
        tmp = log(y) - 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 <= 8.6e+14) {
		tmp = Math.log(y) + (Math.log(t) * a);
	} else {
		tmp = Math.log(y) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 8.6e+14:
		tmp = math.log(y) + (math.log(t) * a)
	else:
		tmp = math.log(y) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 8.6e+14)
		tmp = Float64(log(y) + Float64(log(t) * a));
	else
		tmp = Float64(log(y) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 8.6e+14)
		tmp = log(y) + (log(t) * a);
	else
		tmp = log(y) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 8.6e+14], N[(N[Log[y], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], N[(N[Log[y], $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 8.6e14

   1. Initial program 99.2%

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

      1. associate--l+ 99.3%

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

      2. sub-neg 99.3%

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

      3. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in x around 0 61.7%

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

      1. associate--l+ 61.7%

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

      2. sub-neg 61.7%

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

      3. metadata-eval 61.7%

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

   6. Simplified 61.7%

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

   7. Taylor expanded in a around inf 38.3%

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

      1. *-commutative 38.3%

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

   9. Simplified 38.3%

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

   10. Taylor expanded in t around 0 37.4%

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

   if 8.6e14 < t

   1. Initial program 100.0%

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

      1. associate--l+ 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 72.5%

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

      1. associate--l+ 72.5%

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

      2. sub-neg 72.5%

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

      3. metadata-eval 72.5%

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

   6. Simplified 72.5%

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

   7. Taylor expanded in a around inf 72.3%

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

      1. *-commutative 72.3%

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

   9. Simplified 72.3%

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

   10. Taylor expanded in a around 0 61.8%

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

4. Final simplification 48.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8.6 \cdot 10^{+14}:\\ \;\;\;\;\log y + \log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;\log y - t\\ \end{array} \]

Alternative 11: 57.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate--l+ 99.6%

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

   2. sub-neg 99.6%

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

   3. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in x around 0 66.8%

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

   1. associate--l+ 66.8%

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

   2. sub-neg 66.8%

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

   3. metadata-eval 66.8%

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

6. Simplified 66.8%

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

7. Taylor expanded in a around inf 54.2%

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

   1. *-commutative 54.2%

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

9. Simplified 54.2%

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

10. Final simplification 54.2%

    \[\leadsto \log y + \left(\log t \cdot a - t\right) \]

Alternative 12: 62.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.3 \cdot 10^{+15}:\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 1.3e+15) (* (log t) a) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 1.3e+15) {
		tmp = log(t) * a;
	} else {
		tmp = -t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 1.3d+15) then
        tmp = log(t) * a
    else
        tmp = -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 1.3e+15) {
		tmp = Math.log(t) * a;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 1.3e+15:
		tmp = math.log(t) * a
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 1.3e+15)
		tmp = Float64(log(t) * a);
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 1.3e+15)
		tmp = log(t) * a;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 1.3e+15], N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if t < 1.3e15

   1. Initial program 99.2%

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

      1. associate-+l- 99.2%

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

      2. +-commutative 99.2%

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

      3. associate--l+ 99.3%

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

      4. sub-neg 99.3%

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

      5. +-commutative 99.3%

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

      6. *-commutative 99.3%

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

      7. distribute-rgt-neg-in 99.3%

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

      8. fma-def 99.3%

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

      9. sub-neg 99.3%

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

      10. +-commutative 99.3%

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

      11. distribute-neg-in 99.3%

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

      12. metadata-eval 99.3%

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

      13. metadata-eval 99.3%

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

      14. unsub-neg 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in a around inf 47.4%

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

      1. *-commutative 47.4%

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

   6. Simplified 47.4%

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

   if 1.3e15 < t

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

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

      2. +-commutative 100.0%

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

      3. associate--l+ 100.0%

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

      4. sub-neg 100.0%

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

      5. +-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. distribute-rgt-neg-in 100.0%

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

      8. fma-def 100.0%

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

      9. sub-neg 100.0%

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

      10. +-commutative 100.0%

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

      11. distribute-neg-in 100.0%

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

      12. metadata-eval 100.0%

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

      13. metadata-eval 100.0%

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

      14. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around inf 79.8%

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

      1. neg-mul-1 79.8%

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

   6. Simplified 79.8%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.3 \cdot 10^{+15}:\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 13: 52.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 5.5 \cdot 10^{+14}:\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;\log y - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 5.5e+14) (* (log t) a) (- (log y) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 5.5e+14) {
		tmp = log(t) * a;
	} else {
		tmp = log(y) - 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 <= 5.5d+14) then
        tmp = log(t) * a
    else
        tmp = log(y) - 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 <= 5.5e+14) {
		tmp = Math.log(t) * a;
	} else {
		tmp = Math.log(y) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 5.5e+14:
		tmp = math.log(t) * a
	else:
		tmp = math.log(y) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 5.5e+14)
		tmp = Float64(log(t) * a);
	else
		tmp = Float64(log(y) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 5.5e+14)
		tmp = log(t) * a;
	else
		tmp = log(y) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 5.5e+14], N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision], N[(N[Log[y], $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 5.5e14

   1. Initial program 99.2%

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

      1. associate-+l- 99.2%

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

      2. +-commutative 99.2%

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

      3. associate--l+ 99.3%

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

      4. sub-neg 99.3%

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

      5. +-commutative 99.3%

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

      6. *-commutative 99.3%

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

      7. distribute-rgt-neg-in 99.3%

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

      8. fma-def 99.3%

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

      9. sub-neg 99.3%

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

      10. +-commutative 99.3%

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

      11. distribute-neg-in 99.3%

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

      12. metadata-eval 99.3%

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

      13. metadata-eval 99.3%

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

      14. unsub-neg 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in a around inf 47.4%

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

      1. *-commutative 47.4%

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

   6. Simplified 47.4%

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

   if 5.5e14 < t

   1. Initial program 100.0%

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

      1. associate--l+ 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 72.5%

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

      1. associate--l+ 72.5%

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

      2. sub-neg 72.5%

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

      3. metadata-eval 72.5%

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

   6. Simplified 72.5%

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

   7. Taylor expanded in a around inf 72.3%

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

      1. *-commutative 72.3%

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

   9. Simplified 72.3%

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

   10. Taylor expanded in a around 0 61.8%

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

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.5 \cdot 10^{+14}:\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;\log y - t\\ \end{array} \]

Alternative 14: 37.6% accurate, 156.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = -t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate-+l- 99.6%

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

   2. +-commutative 99.6%

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

   3. associate--l+ 99.6%

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

   4. sub-neg 99.6%

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

   5. +-commutative 99.6%

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

   6. *-commutative 99.6%

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

   7. distribute-rgt-neg-in 99.6%

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

   8. fma-def 99.6%

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

   9. sub-neg 99.6%

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

   10. +-commutative 99.6%

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

   11. distribute-neg-in 99.6%

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

   12. metadata-eval 99.6%

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

   13. metadata-eval 99.6%

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

   14. unsub-neg 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in t around inf 39.5%

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

   1. neg-mul-1 39.5%

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

6. Simplified 39.5%

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

7. Final simplification 39.5%

   \[\leadsto -t \]

Developer target: 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 2023318 
(FPCore (x y z t a)
  :name "Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2"
  :precision binary64

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

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