Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 19.5s

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, 1.0× speedup

Math

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

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(log(Float64(x + y)) + 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[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(N[(a + -0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

   1. associate--l+ 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. Final simplification 99.7%

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

Alternative 2: 81.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 0.00275:\\ \;\;\;\;\log z + \left(\log y + \log t \cdot \left(a - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 0.00275)
   (+ (log z) (+ (log y) (* (log t) (- a 0.5))))
   (- (* (+ a -0.5) (log t)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 0.00275) {
		tmp = log(z) + (log(y) + (log(t) * (a - 0.5)));
	} else {
		tmp = ((a + -0.5) * log(t)) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 0.00275d0) then
        tmp = log(z) + (log(y) + (log(t) * (a - 0.5d0)))
    else
        tmp = ((a + (-0.5d0)) * log(t)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 0.00275) {
		tmp = Math.log(z) + (Math.log(y) + (Math.log(t) * (a - 0.5)));
	} else {
		tmp = ((a + -0.5) * Math.log(t)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 0.00275:
		tmp = math.log(z) + (math.log(y) + (math.log(t) * (a - 0.5)))
	else:
		tmp = ((a + -0.5) * math.log(t)) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 0.00275)
		tmp = Float64(log(z) + Float64(log(y) + Float64(log(t) * Float64(a - 0.5))));
	else
		tmp = Float64(Float64(Float64(a + -0.5) * log(t)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 0.00275)
		tmp = log(z) + (log(y) + (log(t) * (a - 0.5)));
	else
		tmp = ((a + -0.5) * log(t)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 0.00275], N[(N[Log[z], $MachinePrecision] + N[(N[Log[y], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a + -0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 0.0027499999999999998

   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. +-commutative 99.6%

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

      3. associate-+l+ 99.5%

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

      4. +-commutative 99.5%

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

      5. fma-def 99.6%

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

      6. sub-neg 99.6%

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

      7. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 70.9%

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

   5. Taylor expanded in t around 0 69.7%

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

   if 0.0027499999999999998 < 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+ 99.8%

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

      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 70.3%

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

      1. associate--l+ 70.3%

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

   6. Simplified 70.3%

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

   7. Taylor expanded in t around inf 98.3%

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

      1. neg-mul-1 98.3%

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

   9. Simplified 98.3%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 0.00275:\\ \;\;\;\;\log z + \left(\log y + \log t \cdot \left(a - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t - t\\ \end{array} \]

Alternative 3: 81.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a + -0.5\right) \cdot \log t\\ \mathbf{if}\;t \leq 0.00275:\\ \;\;\;\;t_1 + \left(\log z + \log y\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (+ a -0.5) (log t))))
   (if (<= t 0.00275) (+ t_1 (+ (log z) (log y))) (- t_1 t))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (a + -0.5) * log(t);
	double tmp;
	if (t <= 0.00275) {
		tmp = t_1 + (log(z) + log(y));
	} else {
		tmp = t_1 - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (a + (-0.5d0)) * log(t)
    if (t <= 0.00275d0) then
        tmp = t_1 + (log(z) + log(y))
    else
        tmp = t_1 - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (a + -0.5) * Math.log(t);
	double tmp;
	if (t <= 0.00275) {
		tmp = t_1 + (Math.log(z) + Math.log(y));
	} else {
		tmp = t_1 - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (a + -0.5) * math.log(t)
	tmp = 0
	if t <= 0.00275:
		tmp = t_1 + (math.log(z) + math.log(y))
	else:
		tmp = t_1 - t
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(a + -0.5) * log(t))
	tmp = 0.0
	if (t <= 0.00275)
		tmp = Float64(t_1 + Float64(log(z) + log(y)));
	else
		tmp = Float64(t_1 - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (a + -0.5) * log(t);
	tmp = 0.0;
	if (t <= 0.00275)
		tmp = t_1 + (log(z) + log(y));
	else
		tmp = t_1 - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(a + -0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, 0.00275], N[(t$95$1 + N[(N[Log[z], $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 - t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 0.0027499999999999998

   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 70.9%

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

      1. associate--l+ 70.9%

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

   6. Simplified 70.9%

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

   7. Taylor expanded in t around 0 69.7%

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

   if 0.0027499999999999998 < 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+ 99.8%

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

      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 70.3%

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

      1. associate--l+ 70.3%

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

   6. Simplified 70.3%

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

   7. Taylor expanded in t around inf 98.3%

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

      1. neg-mul-1 98.3%

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

   9. Simplified 98.3%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 0.00275:\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t + \left(\log z + \log y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t - t\\ \end{array} \]

Alternative 4: 68.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. associate--l+ 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 70.5%

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

   1. associate--l+ 70.6%

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

6. Simplified 70.6%

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

7. Final simplification 70.6%

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

Alternative 5: 80.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a + -0.5\right) \cdot \log t\\ \mathbf{if}\;a \leq -2.35 \cdot 10^{-39}:\\ \;\;\;\;t_1 - t\\ \mathbf{elif}\;a \leq 1050:\\ \;\;\;\;\left(t_1 + \log \left(\left(x + y\right) \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log y - t\right) + a \cdot \log t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (+ a -0.5) (log t))))
   (if (<= a -2.35e-39)
     (- t_1 t)
     (if (<= a 1050.0)
       (- (+ t_1 (log (* (+ x y) z))) t)
       (+ (- (log y) t) (* a (log t)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (a + -0.5) * log(t);
	double tmp;
	if (a <= -2.35e-39) {
		tmp = t_1 - t;
	} else if (a <= 1050.0) {
		tmp = (t_1 + log(((x + y) * z))) - t;
	} else {
		tmp = (log(y) - t) + (a * log(t));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (a + (-0.5d0)) * log(t)
    if (a <= (-2.35d-39)) then
        tmp = t_1 - t
    else if (a <= 1050.0d0) then
        tmp = (t_1 + log(((x + y) * z))) - t
    else
        tmp = (log(y) - t) + (a * log(t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (a + -0.5) * Math.log(t);
	double tmp;
	if (a <= -2.35e-39) {
		tmp = t_1 - t;
	} else if (a <= 1050.0) {
		tmp = (t_1 + Math.log(((x + y) * z))) - t;
	} else {
		tmp = (Math.log(y) - t) + (a * Math.log(t));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (a + -0.5) * math.log(t)
	tmp = 0
	if a <= -2.35e-39:
		tmp = t_1 - t
	elif a <= 1050.0:
		tmp = (t_1 + math.log(((x + y) * z))) - t
	else:
		tmp = (math.log(y) - t) + (a * math.log(t))
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(a + -0.5) * log(t))
	tmp = 0.0
	if (a <= -2.35e-39)
		tmp = Float64(t_1 - t);
	elseif (a <= 1050.0)
		tmp = Float64(Float64(t_1 + log(Float64(Float64(x + y) * z))) - t);
	else
		tmp = Float64(Float64(log(y) - t) + Float64(a * log(t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (a + -0.5) * log(t);
	tmp = 0.0;
	if (a <= -2.35e-39)
		tmp = t_1 - t;
	elseif (a <= 1050.0)
		tmp = (t_1 + log(((x + y) * z))) - t;
	else
		tmp = (log(y) - t) + (a * log(t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(a + -0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -2.35e-39], N[(t$95$1 - t), $MachinePrecision], If[LessEqual[a, 1050.0], N[(N[(t$95$1 + N[Log[N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] - t), $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.3500000000000001e-39

   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 79.3%

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

      1. associate--l+ 79.3%

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

   6. Simplified 79.3%

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

   7. Taylor expanded in t around inf 93.9%

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

      1. neg-mul-1 93.9%

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

   9. Simplified 93.9%

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

   if -2.3500000000000001e-39 < a < 1050

   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. associate-+l+ 99.7%

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

      3. +-commutative 99.7%

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

      4. fma-def 99.7%

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

      5. remove-double-neg 99.7%

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

      6. remove-double-neg 99.7%

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

      7. sub-neg 99.7%

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

      8. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. +-commutative 99.7%

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

      2. fma-udef 99.7%

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

      3. associate-+r+ 99.8%

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

      4. +-commutative 99.8%

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

      5. associate-+r- 99.8%

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

      6. associate-+r- 99.8%

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

      7. sum-log 79.1%

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

   5. Applied egg-rr 79.1%

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

   if 1050 < 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. associate-+l+ 99.7%

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

      3. +-commutative 99.7%

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

      4. fma-def 99.7%

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

      5. remove-double-neg 99.7%

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

      6. remove-double-neg 99.7%

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

      7. sub-neg 99.7%

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

      8. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 73.7%

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

      1. associate-+r+ 73.7%

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

      2. associate--l+ 73.7%

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

      3. fma-def 73.7%

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

      4. sub-neg 73.7%

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

      5. metadata-eval 73.7%

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

      6. +-commutative 73.7%

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

   6. Simplified 73.7%

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

   7. Taylor expanded in a around inf 73.3%

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

      1. *-commutative 73.3%

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

   9. Simplified 73.3%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.35 \cdot 10^{-39}:\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t - t\\ \mathbf{elif}\;a \leq 1050:\\ \;\;\;\;\left(\left(a + -0.5\right) \cdot \log t + \log \left(\left(x + y\right) \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log y - t\right) + a \cdot \log t\\ \end{array} \]

Alternative 6: 58.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(z \cdot \left(y \cdot {t}^{-0.5}\right)\right) - t\\ t_2 := \left(\log y - t\right) + a \cdot \log t\\ \mathbf{if}\;a \leq -1.65 \cdot 10^{-56}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 1.08 \cdot 10^{-189}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 2.45 \cdot 10^{-144}:\\ \;\;\;\;\log \left(x + y\right) - t\\ \mathbf{elif}\;a \leq 1.7 \cdot 10^{-17}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (log (* z (* y (pow t -0.5)))) t))
        (t_2 (+ (- (log y) t) (* a (log t)))))
   (if (<= a -1.65e-56)
     t_2
     (if (<= a 1.08e-189)
       t_1
       (if (<= a 2.45e-144)
         (- (log (+ x y)) t)
         (if (<= a 1.7e-17) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((z * (y * pow(t, -0.5)))) - t;
	double t_2 = (log(y) - t) + (a * log(t));
	double tmp;
	if (a <= -1.65e-56) {
		tmp = t_2;
	} else if (a <= 1.08e-189) {
		tmp = t_1;
	} else if (a <= 2.45e-144) {
		tmp = log((x + y)) - t;
	} else if (a <= 1.7e-17) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = log((z * (y * (t ** (-0.5d0))))) - t
    t_2 = (log(y) - t) + (a * log(t))
    if (a <= (-1.65d-56)) then
        tmp = t_2
    else if (a <= 1.08d-189) then
        tmp = t_1
    else if (a <= 2.45d-144) then
        tmp = log((x + y)) - t
    else if (a <= 1.7d-17) then
        tmp = t_1
    else
        tmp = t_2
    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((z * (y * Math.pow(t, -0.5)))) - t;
	double t_2 = (Math.log(y) - t) + (a * Math.log(t));
	double tmp;
	if (a <= -1.65e-56) {
		tmp = t_2;
	} else if (a <= 1.08e-189) {
		tmp = t_1;
	} else if (a <= 2.45e-144) {
		tmp = Math.log((x + y)) - t;
	} else if (a <= 1.7e-17) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = math.log((z * (y * math.pow(t, -0.5)))) - t
	t_2 = (math.log(y) - t) + (a * math.log(t))
	tmp = 0
	if a <= -1.65e-56:
		tmp = t_2
	elif a <= 1.08e-189:
		tmp = t_1
	elif a <= 2.45e-144:
		tmp = math.log((x + y)) - t
	elif a <= 1.7e-17:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(Float64(z * Float64(y * (t ^ -0.5)))) - t)
	t_2 = Float64(Float64(log(y) - t) + Float64(a * log(t)))
	tmp = 0.0
	if (a <= -1.65e-56)
		tmp = t_2;
	elseif (a <= 1.08e-189)
		tmp = t_1;
	elseif (a <= 2.45e-144)
		tmp = Float64(log(Float64(x + y)) - t);
	elseif (a <= 1.7e-17)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = log((z * (y * (t ^ -0.5)))) - t;
	t_2 = (log(y) - t) + (a * log(t));
	tmp = 0.0;
	if (a <= -1.65e-56)
		tmp = t_2;
	elseif (a <= 1.08e-189)
		tmp = t_1;
	elseif (a <= 2.45e-144)
		tmp = log((x + y)) - t;
	elseif (a <= 1.7e-17)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[N[(z * N[(y * N[Power[t, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Log[y], $MachinePrecision] - t), $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -1.65e-56], t$95$2, If[LessEqual[a, 1.08e-189], t$95$1, If[LessEqual[a, 2.45e-144], N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision], If[LessEqual[a, 1.7e-17], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.64999999999999992e-56 or 1.6999999999999999e-17 < 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. associate-+l+ 99.6%

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

      3. +-commutative 99.6%

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

      4. fma-def 99.7%

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

      5. remove-double-neg 99.7%

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

      6. remove-double-neg 99.7%

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

      7. sub-neg 99.7%

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

      8. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 75.2%

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

      1. associate-+r+ 75.2%

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

      2. associate--l+ 75.2%

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

      3. fma-def 75.3%

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

      4. sub-neg 75.3%

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

      5. metadata-eval 75.3%

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

      6. +-commutative 75.3%

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

   6. Simplified 75.3%

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

   7. Taylor expanded in a around inf 72.7%

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

      1. *-commutative 72.7%

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

   9. Simplified 72.7%

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

   if -1.64999999999999992e-56 < a < 1.08e-189 or 2.45000000000000005e-144 < a < 1.6999999999999999e-17

   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. associate-+l+ 99.7%

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

      3. +-commutative 99.7%

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

      4. fma-def 99.7%

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

      5. remove-double-neg 99.7%

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

      6. remove-double-neg 99.7%

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

      7. sub-neg 99.7%

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

      8. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 66.0%

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

      1. associate-+r+ 66.0%

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

      2. associate--l+ 65.9%

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

      3. fma-def 65.9%

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

      4. sub-neg 65.9%

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

      5. metadata-eval 65.9%

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

      6. +-commutative 65.9%

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

   6. Simplified 65.9%

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

   7. Taylor expanded in a around 0 66.0%

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

      1. rem-log-exp 52.3%

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

      2. exp-sum 52.3%

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

      3. rem-exp-log 52.3%

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

      4. exp-sum 52.4%

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

      5. rem-exp-log 52.7%

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

      6. *-commutative 52.7%

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

      7. exp-to-pow 52.7%

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

   9. Simplified 52.7%

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

   if 1.08e-189 < a < 2.45000000000000005e-144

   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. associate-+l+ 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-def 100.0%

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

      5. remove-double-neg 100.0%

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

      6. remove-double-neg 100.0%

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

      7. sub-neg 100.0%

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

      8. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around inf 78.4%

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

      1. neg-mul-1 78.4%

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

   6. Simplified 78.4%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.65 \cdot 10^{-56}:\\ \;\;\;\;\left(\log y - t\right) + a \cdot \log t\\ \mathbf{elif}\;a \leq 1.08 \cdot 10^{-189}:\\ \;\;\;\;\log \left(z \cdot \left(y \cdot {t}^{-0.5}\right)\right) - t\\ \mathbf{elif}\;a \leq 2.45 \cdot 10^{-144}:\\ \;\;\;\;\log \left(x + y\right) - t\\ \mathbf{elif}\;a \leq 1.7 \cdot 10^{-17}:\\ \;\;\;\;\log \left(z \cdot \left(y \cdot {t}^{-0.5}\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log y - t\right) + a \cdot \log t\\ \end{array} \]

Alternative 7: 80.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.8 \cdot 10^{-39}:\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t - t\\ \mathbf{elif}\;a \leq 4.4 \cdot 10^{-15}:\\ \;\;\;\;\log \left(\left(x + y\right) \cdot z\right) + \left(-0.5 \cdot \log t - t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log y - t\right) + a \cdot \log t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -2.8e-39)
   (- (* (+ a -0.5) (log t)) t)
   (if (<= a 4.4e-15)
     (+ (log (* (+ x y) z)) (- (* -0.5 (log t)) t))
     (+ (- (log y) t) (* a (log t))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -2.8e-39) {
		tmp = ((a + -0.5) * log(t)) - t;
	} else if (a <= 4.4e-15) {
		tmp = log(((x + y) * z)) + ((-0.5 * log(t)) - t);
	} else {
		tmp = (log(y) - t) + (a * log(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 <= (-2.8d-39)) then
        tmp = ((a + (-0.5d0)) * log(t)) - t
    else if (a <= 4.4d-15) then
        tmp = log(((x + y) * z)) + (((-0.5d0) * log(t)) - t)
    else
        tmp = (log(y) - t) + (a * log(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 <= -2.8e-39) {
		tmp = ((a + -0.5) * Math.log(t)) - t;
	} else if (a <= 4.4e-15) {
		tmp = Math.log(((x + y) * z)) + ((-0.5 * Math.log(t)) - t);
	} else {
		tmp = (Math.log(y) - t) + (a * Math.log(t));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if a <= -2.8e-39:
		tmp = ((a + -0.5) * math.log(t)) - t
	elif a <= 4.4e-15:
		tmp = math.log(((x + y) * z)) + ((-0.5 * math.log(t)) - t)
	else:
		tmp = (math.log(y) - t) + (a * math.log(t))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -2.8e-39)
		tmp = Float64(Float64(Float64(a + -0.5) * log(t)) - t);
	elseif (a <= 4.4e-15)
		tmp = Float64(log(Float64(Float64(x + y) * z)) + Float64(Float64(-0.5 * log(t)) - t));
	else
		tmp = Float64(Float64(log(y) - t) + Float64(a * log(t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (a <= -2.8e-39)
		tmp = ((a + -0.5) * log(t)) - t;
	elseif (a <= 4.4e-15)
		tmp = log(((x + y) * z)) + ((-0.5 * log(t)) - t);
	else
		tmp = (log(y) - t) + (a * log(t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -2.8e-39], N[(N[(N[(a + -0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[a, 4.4e-15], N[(N[Log[N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision] + N[(N[(-0.5 * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] - t), $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.8000000000000001e-39

   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 79.3%

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

      1. associate--l+ 79.3%

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

   6. Simplified 79.3%

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

   7. Taylor expanded in t around inf 93.9%

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

      1. neg-mul-1 93.9%

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

   9. Simplified 93.9%

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

   if -2.8000000000000001e-39 < a < 4.39999999999999971e-15

   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. associate-+l+ 99.8%

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

      3. +-commutative 99.8%

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

      4. fma-def 99.8%

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

      5. remove-double-neg 99.8%

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

      6. remove-double-neg 99.8%

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

      7. sub-neg 99.8%

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

      8. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in a around 0 99.8%

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

      1. associate-+r+ 99.8%

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

      2. +-commutative 99.8%

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

      3. log-prod 79.3%

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

      4. associate--l+ 79.2%

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

      5. *-commutative 79.2%

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

      6. +-commutative 79.2%

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

   6. Simplified 79.2%

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

   if 4.39999999999999971e-15 < 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. associate-+l+ 99.6%

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

      3. +-commutative 99.6%

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

      4. fma-def 99.7%

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

      5. remove-double-neg 99.7%

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

      6. remove-double-neg 99.7%

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

      7. sub-neg 99.7%

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

      8. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 71.3%

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

      1. associate-+r+ 71.3%

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

      2. associate--l+ 71.3%

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

      3. fma-def 71.3%

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

      4. sub-neg 71.3%

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

      5. metadata-eval 71.3%

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

      6. +-commutative 71.3%

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

   6. Simplified 71.3%

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

   7. Taylor expanded in a around inf 69.8%

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

      1. *-commutative 69.8%

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

   9. Simplified 69.8%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.8 \cdot 10^{-39}:\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t - t\\ \mathbf{elif}\;a \leq 4.4 \cdot 10^{-15}:\\ \;\;\;\;\log \left(\left(x + y\right) \cdot z\right) + \left(-0.5 \cdot \log t - t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log y - t\right) + a \cdot \log t\\ \end{array} \]

Alternative 8: 80.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -7.5 \cdot 10^{-40}:\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t - t\\ \mathbf{elif}\;a \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\left(\log \left(\left(x + y\right) \cdot z\right) + -0.5 \cdot \log t\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log y - t\right) + a \cdot \log t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -7.5e-40)
   (- (* (+ a -0.5) (log t)) t)
   (if (<= a 5e-15)
     (- (+ (log (* (+ x y) z)) (* -0.5 (log t))) t)
     (+ (- (log y) t) (* a (log t))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -7.5e-40) {
		tmp = ((a + -0.5) * log(t)) - t;
	} else if (a <= 5e-15) {
		tmp = (log(((x + y) * z)) + (-0.5 * log(t))) - t;
	} else {
		tmp = (log(y) - t) + (a * log(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 <= (-7.5d-40)) then
        tmp = ((a + (-0.5d0)) * log(t)) - t
    else if (a <= 5d-15) then
        tmp = (log(((x + y) * z)) + ((-0.5d0) * log(t))) - t
    else
        tmp = (log(y) - t) + (a * log(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 <= -7.5e-40) {
		tmp = ((a + -0.5) * Math.log(t)) - t;
	} else if (a <= 5e-15) {
		tmp = (Math.log(((x + y) * z)) + (-0.5 * Math.log(t))) - t;
	} else {
		tmp = (Math.log(y) - t) + (a * Math.log(t));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if a <= -7.5e-40:
		tmp = ((a + -0.5) * math.log(t)) - t
	elif a <= 5e-15:
		tmp = (math.log(((x + y) * z)) + (-0.5 * math.log(t))) - t
	else:
		tmp = (math.log(y) - t) + (a * math.log(t))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -7.5e-40)
		tmp = Float64(Float64(Float64(a + -0.5) * log(t)) - t);
	elseif (a <= 5e-15)
		tmp = Float64(Float64(log(Float64(Float64(x + y) * z)) + Float64(-0.5 * log(t))) - t);
	else
		tmp = Float64(Float64(log(y) - t) + Float64(a * log(t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (a <= -7.5e-40)
		tmp = ((a + -0.5) * log(t)) - t;
	elseif (a <= 5e-15)
		tmp = (log(((x + y) * z)) + (-0.5 * log(t))) - t;
	else
		tmp = (log(y) - t) + (a * log(t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -7.5e-40], N[(N[(N[(a + -0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[a, 5e-15], N[(N[(N[Log[N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision] + N[(-0.5 * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] - t), $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -7.50000000000000069e-40

   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 79.3%

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

      1. associate--l+ 79.3%

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

   6. Simplified 79.3%

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

   7. Taylor expanded in t around inf 93.9%

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

      1. neg-mul-1 93.9%

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

   9. Simplified 93.9%

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

   if -7.50000000000000069e-40 < a < 4.99999999999999999e-15

   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. associate-+l+ 99.8%

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

      3. +-commutative 99.8%

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

      4. fma-def 99.8%

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

      5. remove-double-neg 99.8%

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

      6. remove-double-neg 99.8%

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

      7. sub-neg 99.8%

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

      8. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in a around 0 99.8%

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

      1. associate-+r+ 99.8%

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

      2. +-commutative 99.8%

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

      3. log-prod 79.3%

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

      4. *-commutative 79.3%

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

      5. +-commutative 79.3%

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

   6. Simplified 79.3%

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

   if 4.99999999999999999e-15 < 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. associate-+l+ 99.6%

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

      3. +-commutative 99.6%

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

      4. fma-def 99.7%

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

      5. remove-double-neg 99.7%

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

      6. remove-double-neg 99.7%

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

      7. sub-neg 99.7%

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

      8. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 71.3%

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

      1. associate-+r+ 71.3%

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

      2. associate--l+ 71.3%

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

      3. fma-def 71.3%

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

      4. sub-neg 71.3%

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

      5. metadata-eval 71.3%

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

      6. +-commutative 71.3%

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

   6. Simplified 71.3%

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

   7. Taylor expanded in a around inf 69.8%

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

      1. *-commutative 69.8%

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

   9. Simplified 69.8%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -7.5 \cdot 10^{-40}:\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t - t\\ \mathbf{elif}\;a \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\left(\log \left(\left(x + y\right) \cdot z\right) + -0.5 \cdot \log t\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log y - t\right) + a \cdot \log t\\ \end{array} \]

Alternative 9: 67.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.1 \cdot 10^{-40}:\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t - t\\ \mathbf{elif}\;a \leq 1.75 \cdot 10^{-15}:\\ \;\;\;\;\left(-0.5 \cdot \log t + \log \left(y \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log y - t\right) + a \cdot \log t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -1.1e-40)
   (- (* (+ a -0.5) (log t)) t)
   (if (<= a 1.75e-15)
     (- (+ (* -0.5 (log t)) (log (* y z))) t)
     (+ (- (log y) t) (* a (log t))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -1.1e-40) {
		tmp = ((a + -0.5) * log(t)) - t;
	} else if (a <= 1.75e-15) {
		tmp = ((-0.5 * log(t)) + log((y * z))) - t;
	} else {
		tmp = (log(y) - t) + (a * log(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 <= (-1.1d-40)) then
        tmp = ((a + (-0.5d0)) * log(t)) - t
    else if (a <= 1.75d-15) then
        tmp = (((-0.5d0) * log(t)) + log((y * z))) - t
    else
        tmp = (log(y) - t) + (a * log(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 <= -1.1e-40) {
		tmp = ((a + -0.5) * Math.log(t)) - t;
	} else if (a <= 1.75e-15) {
		tmp = ((-0.5 * Math.log(t)) + Math.log((y * z))) - t;
	} else {
		tmp = (Math.log(y) - t) + (a * Math.log(t));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if a <= -1.1e-40:
		tmp = ((a + -0.5) * math.log(t)) - t
	elif a <= 1.75e-15:
		tmp = ((-0.5 * math.log(t)) + math.log((y * z))) - t
	else:
		tmp = (math.log(y) - t) + (a * math.log(t))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -1.1e-40)
		tmp = Float64(Float64(Float64(a + -0.5) * log(t)) - t);
	elseif (a <= 1.75e-15)
		tmp = Float64(Float64(Float64(-0.5 * log(t)) + log(Float64(y * z))) - t);
	else
		tmp = Float64(Float64(log(y) - t) + Float64(a * log(t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (a <= -1.1e-40)
		tmp = ((a + -0.5) * log(t)) - t;
	elseif (a <= 1.75e-15)
		tmp = ((-0.5 * log(t)) + log((y * z))) - t;
	else
		tmp = (log(y) - t) + (a * log(t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -1.1e-40], N[(N[(N[(a + -0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[a, 1.75e-15], N[(N[(N[(-0.5 * N[Log[t], $MachinePrecision]), $MachinePrecision] + N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] - t), $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.10000000000000004e-40

   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 79.3%

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

      1. associate--l+ 79.3%

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

   6. Simplified 79.3%

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

   7. Taylor expanded in t around inf 93.9%

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

      1. neg-mul-1 93.9%

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

   9. Simplified 93.9%

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

   if -1.10000000000000004e-40 < a < 1.75e-15

   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 62.7%

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

      1. associate--l+ 62.7%

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

   6. Simplified 62.7%

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

   7. Taylor expanded in a around 0 62.7%

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

      1. associate-+r+ 62.7%

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

      2. log-prod 54.5%

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

      3. *-commutative 54.5%

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

   9. Simplified 54.5%

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

   if 1.75e-15 < 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. associate-+l+ 99.6%

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

      3. +-commutative 99.6%

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

      4. fma-def 99.7%

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

      5. remove-double-neg 99.7%

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

      6. remove-double-neg 99.7%

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

      7. sub-neg 99.7%

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

      8. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 71.3%

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

      1. associate-+r+ 71.3%

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

      2. associate--l+ 71.3%

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

      3. fma-def 71.3%

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

      4. sub-neg 71.3%

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

      5. metadata-eval 71.3%

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

      6. +-commutative 71.3%

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

   6. Simplified 71.3%

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

   7. Taylor expanded in a around inf 69.8%

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

      1. *-commutative 69.8%

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

   9. Simplified 69.8%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.1 \cdot 10^{-40}:\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t - t\\ \mathbf{elif}\;a \leq 1.75 \cdot 10^{-15}:\\ \;\;\;\;\left(-0.5 \cdot \log t + \log \left(y \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log y - t\right) + a \cdot \log t\\ \end{array} \]

Alternative 10: 60.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 2.7 \cdot 10^{-72}:\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t + \log \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log y - t\right) + a \cdot \log t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 2.7e-72)
   (+ (* (+ a -0.5) (log t)) (log (* y z)))
   (+ (- (log y) t) (* a (log t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 2.7e-72) {
		tmp = ((a + -0.5) * log(t)) + log((y * z));
	} else {
		tmp = (log(y) - t) + (a * log(t));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 2.7d-72) then
        tmp = ((a + (-0.5d0)) * log(t)) + log((y * z))
    else
        tmp = (log(y) - t) + (a * log(t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 2.7e-72) {
		tmp = ((a + -0.5) * Math.log(t)) + Math.log((y * z));
	} else {
		tmp = (Math.log(y) - t) + (a * Math.log(t));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 2.7e-72:
		tmp = ((a + -0.5) * math.log(t)) + math.log((y * z))
	else:
		tmp = (math.log(y) - t) + (a * math.log(t))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 2.7e-72)
		tmp = Float64(Float64(Float64(a + -0.5) * log(t)) + log(Float64(y * z)));
	else
		tmp = Float64(Float64(log(y) - t) + Float64(a * log(t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 2.7e-72)
		tmp = ((a + -0.5) * log(t)) + log((y * z));
	else
		tmp = (log(y) - t) + (a * log(t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 2.7e-72], N[(N[(N[(a + -0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision] + N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] - t), $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.7e-72

   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 71.9%

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

      1. associate--l+ 71.9%

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

   6. Simplified 71.9%

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

   7. Taylor expanded in t around 0 71.9%

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

      1. log-prod 57.7%

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

   9. Simplified 57.7%

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

   if 2.7e-72 < t

   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+ 99.8%

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

      2. associate-+l+ 99.8%

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

      3. +-commutative 99.8%

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

      4. fma-def 99.8%

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

      5. remove-double-neg 99.8%

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

      6. remove-double-neg 99.8%

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

      7. sub-neg 99.8%

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

      8. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 69.9%

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

      1. associate-+r+ 69.9%

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

      2. associate--l+ 69.9%

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

      3. fma-def 69.9%

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

      4. sub-neg 69.9%

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

      5. metadata-eval 69.9%

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

      6. +-commutative 69.9%

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

   6. Simplified 69.9%

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

   7. Taylor expanded in a around inf 64.9%

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

      1. *-commutative 64.9%

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

   9. Simplified 64.9%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.7 \cdot 10^{-72}:\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t + \log \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log y - t\right) + a \cdot \log t\\ \end{array} \]

Alternative 11: 57.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. associate--l+ 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. associate-+l+ 99.7%

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

   3. +-commutative 99.7%

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

   4. fma-def 99.7%

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

   5. remove-double-neg 99.7%

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

   6. remove-double-neg 99.7%

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

   7. sub-neg 99.7%

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

   8. metadata-eval 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in x around 0 70.6%

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

   1. associate-+r+ 70.6%

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

   2. associate--l+ 70.6%

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

   3. fma-def 70.6%

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

   4. sub-neg 70.6%

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

   5. metadata-eval 70.6%

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

   6. +-commutative 70.6%

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

6. Simplified 70.6%

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

7. Taylor expanded in a around inf 60.2%

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

   1. *-commutative 60.2%

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

9. Simplified 60.2%

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

10. Final simplification 60.2%

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

Alternative 12: 77.3% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

   \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
2. 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 70.5%

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

   1. associate--l+ 70.6%

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

6. Simplified 70.6%

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

7. Taylor expanded in t around inf 80.7%

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

   1. neg-mul-1 80.7%

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

9. Simplified 80.7%

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

10. Final simplification 80.7%

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

Alternative 13: 63.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 7.6 \cdot 10^{+27}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 7.6e+27) (* a (log t)) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 7.6e+27) {
		tmp = a * log(t);
	} 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 <= 7.6d+27) then
        tmp = a * log(t)
    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 <= 7.6e+27) {
		tmp = a * Math.log(t);
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 7.6e+27:
		tmp = a * math.log(t)
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 7.6e+27)
		tmp = Float64(a * log(t));
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 7.6e+27)
		tmp = a * log(t);
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 7.6e+27], N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if t < 7.60000000000000043e27

   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 70.5%

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

      1. associate--l+ 70.5%

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

   6. Simplified 70.5%

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

   7. Taylor expanded in a around inf 54.9%

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

   if 7.60000000000000043e27 < 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. associate-+l+ 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-def 99.9%

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

      5. remove-double-neg 99.9%

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

      6. remove-double-neg 99.9%

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

      7. sub-neg 99.9%

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

      8. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 70.6%

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

      1. associate-+r+ 70.6%

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

      2. associate--l+ 70.6%

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

      3. fma-def 70.6%

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

      4. sub-neg 70.6%

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

      5. metadata-eval 70.6%

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

      6. +-commutative 70.6%

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

   6. Simplified 70.6%

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

   7. Taylor expanded in t around inf 75.2%

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

      1. neg-mul-1 75.2%

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

   9. Simplified 75.2%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.6 \cdot 10^{+27}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 14: 38.1% 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 99.7%

   \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
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. associate-+l+ 99.7%

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

   3. +-commutative 99.7%

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

   4. fma-def 99.7%

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

   5. remove-double-neg 99.7%

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

   6. remove-double-neg 99.7%

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

   7. sub-neg 99.7%

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

   8. metadata-eval 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in x around 0 70.6%

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

   1. associate-+r+ 70.6%

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

   2. associate--l+ 70.6%

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

   3. fma-def 70.6%

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

   4. sub-neg 70.6%

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

   5. metadata-eval 70.6%

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

   6. +-commutative 70.6%

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

6. Simplified 70.6%

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

7. Taylor expanded in t around inf 39.5%

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

   1. neg-mul-1 39.5%

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

9. Simplified 39.5%

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

10. 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 2023240 
(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))))