Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 26.6s

Alternatives: 16

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. +-commutative 99.6%

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

   2. fma-define 99.6%

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

   3. sub-neg 99.6%

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

   4. metadata-eval 99.6%

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

   5. associate--l+ 99.6%

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

3. Simplified 99.6%

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

5. Final simplification 99.6%

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

Alternative 2: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 98:\\ \;\;\;\;\log z + \left(\log \left(x + y\right) + \log t \cdot \left(a - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 98.0)
   (+ (log z) (+ (log (+ x y)) (* (log t) (- a 0.5))))
   (+ (- (log z) t) (* a (log t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 98.0) {
		tmp = log(z) + (log((x + y)) + (log(t) * (a - 0.5)));
	} else {
		tmp = (log(z) - 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 <= 98.0d0) then
        tmp = log(z) + (log((x + y)) + (log(t) * (a - 0.5d0)))
    else
        tmp = (log(z) - 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 <= 98.0) {
		tmp = Math.log(z) + (Math.log((x + y)) + (Math.log(t) * (a - 0.5)));
	} else {
		tmp = (Math.log(z) - t) + (a * Math.log(t));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 98.0:
		tmp = math.log(z) + (math.log((x + y)) + (math.log(t) * (a - 0.5)))
	else:
		tmp = (math.log(z) - t) + (a * math.log(t))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 98.0)
		tmp = Float64(log(z) + Float64(log(Float64(x + y)) + Float64(log(t) * Float64(a - 0.5))));
	else
		tmp = Float64(Float64(log(z) - t) + Float64(a * log(t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 98.0)
		tmp = log(z) + (log((x + y)) + (log(t) * (a - 0.5)));
	else
		tmp = (log(z) - t) + (a * log(t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 98.0], N[(N[Log[z], $MachinePrecision] + N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 98

   1. Initial program 99.4%

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

      1. +-commutative 99.4%

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

      2. associate--l+ 99.4%

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

      3. associate-+r+ 99.3%

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

      4. +-commutative 99.3%

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

      5. fma-define 99.3%

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

      6. sub-neg 99.3%

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

      7. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in t around 0 98.2%

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

      1. associate-+r+ 98.3%

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

      2. +-commutative 98.3%

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

      3. log-prod 69.5%

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

      4. sub-neg 69.5%

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

      5. metadata-eval 69.5%

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

      6. *-commutative 69.5%

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

      7. +-commutative 69.5%

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

      8. fma-define 69.5%

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

      9. +-commutative 69.5%

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

   7. Simplified 69.5%

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

   8. Taylor expanded in z around 0 98.2%

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

   if 98 < 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. +-commutative 99.8%

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

      2. associate--l+ 99.8%

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

      3. associate-+r+ 99.8%

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

      4. +-commutative 99.8%

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

      5. fma-define 99.8%

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

      6. sub-neg 99.8%

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

      7. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in a around inf 98.4%

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

      1. *-commutative 98.4%

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

   7. Simplified 98.4%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 98:\\ \;\;\;\;\log z + \left(\log \left(x + y\right) + \log t \cdot \left(a - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 240:\\ \;\;\;\;\log y + \left(\log z + \log t \cdot \left(a - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 240.0)
   (+ (log y) (+ (log z) (* (log t) (- a 0.5))))
   (+ (- (log z) t) (* a (log t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 240.0) {
		tmp = log(y) + (log(z) + (log(t) * (a - 0.5)));
	} else {
		tmp = (log(z) - 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 <= 240.0d0) then
        tmp = log(y) + (log(z) + (log(t) * (a - 0.5d0)))
    else
        tmp = (log(z) - 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 <= 240.0) {
		tmp = Math.log(y) + (Math.log(z) + (Math.log(t) * (a - 0.5)));
	} else {
		tmp = (Math.log(z) - t) + (a * Math.log(t));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 240.0:
		tmp = math.log(y) + (math.log(z) + (math.log(t) * (a - 0.5)))
	else:
		tmp = (math.log(z) - t) + (a * math.log(t))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 240.0)
		tmp = Float64(log(y) + Float64(log(z) + Float64(log(t) * Float64(a - 0.5))));
	else
		tmp = Float64(Float64(log(z) - t) + Float64(a * log(t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 240.0)
		tmp = log(y) + (log(z) + (log(t) * (a - 0.5)));
	else
		tmp = (log(z) - t) + (a * log(t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 240.0], N[(N[Log[y], $MachinePrecision] + N[(N[Log[z], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 240

   1. Initial program 99.4%

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

   3. Taylor expanded in x around 0 56.0%

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

   4. Taylor expanded in t around 0 55.5%

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

   if 240 < 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. +-commutative 99.8%

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

      2. associate--l+ 99.8%

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

      3. associate-+r+ 99.8%

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

      4. +-commutative 99.8%

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

      5. fma-define 99.8%

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

      6. sub-neg 99.8%

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

      7. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in a around inf 98.4%

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

      1. *-commutative 98.4%

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

   7. Simplified 98.4%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 240:\\ \;\;\;\;\log y + \left(\log z + \log t \cdot \left(a - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Final simplification 99.6%

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

Alternative 5: 69.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in x around 0 63.5%

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

4. Final simplification 63.5%

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

Alternative 6: 67.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \log t\\ t_2 := \log \left(y \cdot z\right) + -0.5 \cdot \log t\\ \mathbf{if}\;t \leq 6.4 \cdot 10^{-231}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t \leq 1.12 \cdot 10^{-169}:\\ \;\;\;\;\log z + t\_1\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{-89}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1 - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* a (log t))) (t_2 (+ (log (* y z)) (* -0.5 (log t)))))
   (if (<= t 6.4e-231)
     t_2
     (if (<= t 1.12e-169) (+ (log z) t_1) (if (<= t 4.5e-89) t_2 (- t_1 t))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = a * log(t);
	double t_2 = log((y * z)) + (-0.5 * log(t));
	double tmp;
	if (t <= 6.4e-231) {
		tmp = t_2;
	} else if (t <= 1.12e-169) {
		tmp = log(z) + t_1;
	} else if (t <= 4.5e-89) {
		tmp = t_2;
	} 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) :: t_2
    real(8) :: tmp
    t_1 = a * log(t)
    t_2 = log((y * z)) + ((-0.5d0) * log(t))
    if (t <= 6.4d-231) then
        tmp = t_2
    else if (t <= 1.12d-169) then
        tmp = log(z) + t_1
    else if (t <= 4.5d-89) then
        tmp = t_2
    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 * Math.log(t);
	double t_2 = Math.log((y * z)) + (-0.5 * Math.log(t));
	double tmp;
	if (t <= 6.4e-231) {
		tmp = t_2;
	} else if (t <= 1.12e-169) {
		tmp = Math.log(z) + t_1;
	} else if (t <= 4.5e-89) {
		tmp = t_2;
	} else {
		tmp = t_1 - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = a * math.log(t)
	t_2 = math.log((y * z)) + (-0.5 * math.log(t))
	tmp = 0
	if t <= 6.4e-231:
		tmp = t_2
	elif t <= 1.12e-169:
		tmp = math.log(z) + t_1
	elif t <= 4.5e-89:
		tmp = t_2
	else:
		tmp = t_1 - t
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(a * log(t))
	t_2 = Float64(log(Float64(y * z)) + Float64(-0.5 * log(t)))
	tmp = 0.0
	if (t <= 6.4e-231)
		tmp = t_2;
	elseif (t <= 1.12e-169)
		tmp = Float64(log(z) + t_1);
	elseif (t <= 4.5e-89)
		tmp = t_2;
	else
		tmp = Float64(t_1 - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = a * log(t);
	t_2 = log((y * z)) + (-0.5 * log(t));
	tmp = 0.0;
	if (t <= 6.4e-231)
		tmp = t_2;
	elseif (t <= 1.12e-169)
		tmp = log(z) + t_1;
	elseif (t <= 4.5e-89)
		tmp = t_2;
	else
		tmp = t_1 - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision] + N[(-0.5 * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, 6.4e-231], t$95$2, If[LessEqual[t, 1.12e-169], N[(N[Log[z], $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[t, 4.5e-89], t$95$2, N[(t$95$1 - t), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < 6.40000000000000016e-231 or 1.11999999999999998e-169 < t < 4.4999999999999999e-89

   1. Initial program 99.2%

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

      1. +-commutative 99.2%

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

      2. associate--l+ 99.2%

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

      3. associate-+r+ 99.2%

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

      4. +-commutative 99.2%

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

      5. fma-define 99.2%

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

      6. sub-neg 99.2%

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

      7. metadata-eval 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in t around 0 99.2%

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

      1. associate-+r+ 99.2%

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

      2. +-commutative 99.2%

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

      3. log-prod 76.7%

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

      4. sub-neg 76.7%

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

      5. metadata-eval 76.7%

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

      6. *-commutative 76.7%

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

      7. +-commutative 76.7%

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

      8. fma-define 76.7%

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

      9. +-commutative 76.7%

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

   7. Simplified 76.7%

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

   8. Taylor expanded in a around 0 58.2%

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

   9. Taylor expanded in x around 0 24.4%

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

       1. +-commutative 24.4%

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

       2. *-commutative 24.4%

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

       3. *-commutative 24.4%

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

   11. Simplified 24.4%

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

   if 6.40000000000000016e-231 < t < 1.11999999999999998e-169

   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. +-commutative 99.7%

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

      2. associate--l+ 99.7%

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

      3. associate-+r+ 99.8%

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

      4. +-commutative 99.8%

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

      5. fma-define 99.8%

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

      6. sub-neg 99.8%

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

      7. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in a around inf 50.1%

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

      1. *-commutative 50.1%

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

   7. Simplified 50.1%

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

   8. Taylor expanded in t around 0 50.1%

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

   if 4.4999999999999999e-89 < 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. +-commutative 99.8%

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

      2. associate--l+ 99.8%

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

      3. associate-+r+ 99.7%

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

      4. +-commutative 99.7%

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

      5. fma-define 99.7%

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

      6. sub-neg 99.7%

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

      7. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in a around 0 99.7%

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

   6. Taylor expanded in a around inf 89.8%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.4 \cdot 10^{-231}:\\ \;\;\;\;\log \left(y \cdot z\right) + -0.5 \cdot \log t\\ \mathbf{elif}\;t \leq 1.12 \cdot 10^{-169}:\\ \;\;\;\;\log z + a \cdot \log t\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{-89}:\\ \;\;\;\;\log \left(y \cdot z\right) + -0.5 \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \log t\\ t_2 := \log \left(y \cdot z\right) + -0.5 \cdot \log t\\ \mathbf{if}\;t \leq 5.3 \cdot 10^{-230}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t \leq 1.85 \cdot 10^{-169}:\\ \;\;\;\;\log z + t\_1\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{-89}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* a (log t))) (t_2 (+ (log (* y z)) (* -0.5 (log t)))))
   (if (<= t 5.3e-230)
     t_2
     (if (<= t 1.85e-169)
       (+ (log z) t_1)
       (if (<= t 3.5e-89) t_2 (+ (- (log z) t) t_1))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = a * log(t);
	double t_2 = log((y * z)) + (-0.5 * log(t));
	double tmp;
	if (t <= 5.3e-230) {
		tmp = t_2;
	} else if (t <= 1.85e-169) {
		tmp = log(z) + t_1;
	} else if (t <= 3.5e-89) {
		tmp = t_2;
	} else {
		tmp = (log(z) - t) + t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = a * log(t)
    t_2 = log((y * z)) + ((-0.5d0) * log(t))
    if (t <= 5.3d-230) then
        tmp = t_2
    else if (t <= 1.85d-169) then
        tmp = log(z) + t_1
    else if (t <= 3.5d-89) then
        tmp = t_2
    else
        tmp = (log(z) - t) + t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = a * Math.log(t);
	double t_2 = Math.log((y * z)) + (-0.5 * Math.log(t));
	double tmp;
	if (t <= 5.3e-230) {
		tmp = t_2;
	} else if (t <= 1.85e-169) {
		tmp = Math.log(z) + t_1;
	} else if (t <= 3.5e-89) {
		tmp = t_2;
	} else {
		tmp = (Math.log(z) - t) + t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = a * math.log(t)
	t_2 = math.log((y * z)) + (-0.5 * math.log(t))
	tmp = 0
	if t <= 5.3e-230:
		tmp = t_2
	elif t <= 1.85e-169:
		tmp = math.log(z) + t_1
	elif t <= 3.5e-89:
		tmp = t_2
	else:
		tmp = (math.log(z) - t) + t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(a * log(t))
	t_2 = Float64(log(Float64(y * z)) + Float64(-0.5 * log(t)))
	tmp = 0.0
	if (t <= 5.3e-230)
		tmp = t_2;
	elseif (t <= 1.85e-169)
		tmp = Float64(log(z) + t_1);
	elseif (t <= 3.5e-89)
		tmp = t_2;
	else
		tmp = Float64(Float64(log(z) - t) + t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = a * log(t);
	t_2 = log((y * z)) + (-0.5 * log(t));
	tmp = 0.0;
	if (t <= 5.3e-230)
		tmp = t_2;
	elseif (t <= 1.85e-169)
		tmp = log(z) + t_1;
	elseif (t <= 3.5e-89)
		tmp = t_2;
	else
		tmp = (log(z) - t) + t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision] + N[(-0.5 * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, 5.3e-230], t$95$2, If[LessEqual[t, 1.85e-169], N[(N[Log[z], $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[t, 3.5e-89], t$95$2, N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] + t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < 5.2999999999999998e-230 or 1.8499999999999999e-169 < t < 3.4999999999999997e-89

   1. Initial program 99.2%

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

      1. +-commutative 99.2%

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

      2. associate--l+ 99.2%

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

      3. associate-+r+ 99.2%

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

      4. +-commutative 99.2%

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

      5. fma-define 99.2%

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

      6. sub-neg 99.2%

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

      7. metadata-eval 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in t around 0 99.2%

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

      1. associate-+r+ 99.2%

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

      2. +-commutative 99.2%

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

      3. log-prod 76.7%

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

      4. sub-neg 76.7%

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

      5. metadata-eval 76.7%

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

      6. *-commutative 76.7%

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

      7. +-commutative 76.7%

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

      8. fma-define 76.7%

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

      9. +-commutative 76.7%

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

   7. Simplified 76.7%

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

   8. Taylor expanded in a around 0 58.2%

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

   9. Taylor expanded in x around 0 24.4%

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

       1. +-commutative 24.4%

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

       2. *-commutative 24.4%

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

       3. *-commutative 24.4%

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

   11. Simplified 24.4%

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

   if 5.2999999999999998e-230 < t < 1.8499999999999999e-169

   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. +-commutative 99.7%

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

      2. associate--l+ 99.7%

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

      3. associate-+r+ 99.8%

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

      4. +-commutative 99.8%

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

      5. fma-define 99.8%

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

      6. sub-neg 99.8%

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

      7. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in a around inf 50.1%

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

      1. *-commutative 50.1%

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

   7. Simplified 50.1%

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

   8. Taylor expanded in t around 0 50.1%

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

   if 3.4999999999999997e-89 < 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. +-commutative 99.8%

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

      2. associate--l+ 99.8%

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

      3. associate-+r+ 99.7%

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

      4. +-commutative 99.7%

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

      5. fma-define 99.7%

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

      6. sub-neg 99.7%

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

      7. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in a around inf 90.6%

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

      1. *-commutative 90.6%

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

   7. Simplified 90.6%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.3 \cdot 10^{-230}:\\ \;\;\;\;\log \left(y \cdot z\right) + -0.5 \cdot \log t\\ \mathbf{elif}\;t \leq 1.85 \cdot 10^{-169}:\\ \;\;\;\;\log z + a \cdot \log t\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{-89}:\\ \;\;\;\;\log \left(y \cdot z\right) + -0.5 \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 87.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 11000000000:\\ \;\;\;\;\left(\left(a + -0.5\right) \cdot \log t + \log \left(\left(x + y\right) \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 11000000000.0)
   (- (+ (* (+ a -0.5) (log t)) (log (* (+ x y) z))) t)
   (+ (- (log z) t) (* a (log t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 11000000000.0) {
		tmp = (((a + -0.5) * log(t)) + log(((x + y) * z))) - t;
	} else {
		tmp = (log(z) - 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 <= 11000000000.0d0) then
        tmp = (((a + (-0.5d0)) * log(t)) + log(((x + y) * z))) - t
    else
        tmp = (log(z) - 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 <= 11000000000.0) {
		tmp = (((a + -0.5) * Math.log(t)) + Math.log(((x + y) * z))) - t;
	} else {
		tmp = (Math.log(z) - t) + (a * Math.log(t));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 11000000000.0:
		tmp = (((a + -0.5) * math.log(t)) + math.log(((x + y) * z))) - t
	else:
		tmp = (math.log(z) - t) + (a * math.log(t))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 11000000000.0)
		tmp = Float64(Float64(Float64(Float64(a + -0.5) * log(t)) + log(Float64(Float64(x + y) * z))) - t);
	else
		tmp = Float64(Float64(log(z) - t) + Float64(a * log(t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 11000000000.0)
		tmp = (((a + -0.5) * log(t)) + log(((x + y) * z))) - t;
	else
		tmp = (log(z) - t) + (a * log(t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 11000000000.0], N[(N[(N[(N[(a + -0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision] + N[Log[N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.1e10

   1. Initial program 99.4%

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

      1. +-commutative 99.4%

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

      2. associate--l+ 99.4%

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

      3. associate-+r+ 99.3%

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

      4. +-commutative 99.3%

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

      5. fma-define 99.4%

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

      6. sub-neg 99.4%

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

      7. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

      1. +-commutative 99.4%

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

      2. fma-undefine 99.3%

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

      3. metadata-eval 99.3%

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

      4. sub-neg 99.3%

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

      5. associate-+r+ 99.4%

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

      6. associate-+r- 99.4%

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

      7. associate-+r- 99.4%

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

      8. sub-neg 99.4%

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

      9. metadata-eval 99.4%

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

      10. sum-log 71.3%

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

   6. Applied egg-rr 71.3%

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

   if 1.1e10 < 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. +-commutative 99.8%

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

      2. associate--l+ 99.8%

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

      3. associate-+r+ 99.8%

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

      4. +-commutative 99.8%

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

      5. fma-define 99.8%

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

      6. sub-neg 99.8%

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

      7. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in a around inf 99.3%

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

      1. *-commutative 99.3%

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

   7. Simplified 99.3%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 11000000000:\\ \;\;\;\;\left(\left(a + -0.5\right) \cdot \log t + \log \left(\left(x + y\right) \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 71.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \log t\\ \mathbf{if}\;t \leq 2.1 \cdot 10^{-169}:\\ \;\;\;\;\log z + t\_1\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-89}:\\ \;\;\;\;\log \left(\left(y \cdot z\right) \cdot {t}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* a (log t))))
   (if (<= t 2.1e-169)
     (+ (log z) t_1)
     (if (<= t 5e-89) (log (* (* y z) (pow t -0.5))) (- t_1 t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = a * log(t);
	double tmp;
	if (t <= 2.1e-169) {
		tmp = log(z) + t_1;
	} else if (t <= 5e-89) {
		tmp = log(((y * z) * pow(t, -0.5)));
	} 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 * log(t)
    if (t <= 2.1d-169) then
        tmp = log(z) + t_1
    else if (t <= 5d-89) then
        tmp = log(((y * z) * (t ** (-0.5d0))))
    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 * Math.log(t);
	double tmp;
	if (t <= 2.1e-169) {
		tmp = Math.log(z) + t_1;
	} else if (t <= 5e-89) {
		tmp = Math.log(((y * z) * Math.pow(t, -0.5)));
	} else {
		tmp = t_1 - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = a * math.log(t)
	tmp = 0
	if t <= 2.1e-169:
		tmp = math.log(z) + t_1
	elif t <= 5e-89:
		tmp = math.log(((y * z) * math.pow(t, -0.5)))
	else:
		tmp = t_1 - t
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(a * log(t))
	tmp = 0.0
	if (t <= 2.1e-169)
		tmp = Float64(log(z) + t_1);
	elseif (t <= 5e-89)
		tmp = log(Float64(Float64(y * z) * (t ^ -0.5)));
	else
		tmp = Float64(t_1 - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = a * log(t);
	tmp = 0.0;
	if (t <= 2.1e-169)
		tmp = log(z) + t_1;
	elseif (t <= 5e-89)
		tmp = log(((y * z) * (t ^ -0.5)));
	else
		tmp = t_1 - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, 2.1e-169], N[(N[Log[z], $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[t, 5e-89], N[Log[N[(N[(y * z), $MachinePrecision] * N[Power[t, -0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(t$95$1 - t), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 2.1000000000000001e-169

   1. Initial program 99.4%

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

      1. +-commutative 99.4%

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

      2. associate--l+ 99.4%

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

      3. associate-+r+ 99.5%

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

      4. +-commutative 99.5%

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

      5. fma-define 99.5%

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

      6. sub-neg 99.5%

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

      7. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in a around inf 42.8%

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

      1. *-commutative 42.8%

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

   7. Simplified 42.8%

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

   8. Taylor expanded in t around 0 42.8%

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

   if 2.1000000000000001e-169 < t < 4.99999999999999967e-89

   1. Initial program 99.2%

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

      1. +-commutative 99.2%

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

      2. associate--l+ 99.2%

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

      3. associate-+r+ 99.1%

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

      4. +-commutative 99.1%

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

      5. fma-define 99.1%

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

      6. sub-neg 99.1%

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

      7. metadata-eval 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in t around 0 99.1%

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

      1. associate-+r+ 99.2%

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

      2. +-commutative 99.2%

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

      3. log-prod 79.9%

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

      4. sub-neg 79.9%

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

      5. metadata-eval 79.9%

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

      6. *-commutative 79.9%

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

      7. +-commutative 79.9%

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

      8. fma-define 79.8%

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

      9. +-commutative 79.8%

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

   7. Simplified 79.8%

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

   8. Taylor expanded in a around 0 57.5%

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

      1. add-log-exp 57.5%

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

      2. sum-log 52.2%

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

      3. +-commutative 52.2%

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

      4. *-commutative 52.2%

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

      5. pow-to-exp 52.4%

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

   10. Applied egg-rr 52.4%

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

   11. Taylor expanded in y around inf 23.4%

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

       1. *-commutative 23.4%

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

   13. Simplified 23.4%

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

   if 4.99999999999999967e-89 < 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. +-commutative 99.8%

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

      2. associate--l+ 99.8%

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

      3. associate-+r+ 99.7%

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

      4. +-commutative 99.7%

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

      5. fma-define 99.7%

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

      6. sub-neg 99.7%

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

      7. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in a around 0 99.7%

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

   6. Taylor expanded in a around inf 89.8%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.1 \cdot 10^{-169}:\\ \;\;\;\;\log z + a \cdot \log t\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-89}:\\ \;\;\;\;\log \left(\left(y \cdot z\right) \cdot {t}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 73.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 24000000000:\\ \;\;\;\;\left(\left(a + -0.5\right) \cdot \log t + \log \left(y \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 24000000000.0)
   (- (+ (* (+ a -0.5) (log t)) (log (* y z))) t)
   (+ (- (log z) t) (* a (log t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 24000000000.0) {
		tmp = (((a + -0.5) * log(t)) + log((y * z))) - t;
	} else {
		tmp = (log(z) - 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 <= 24000000000.0d0) then
        tmp = (((a + (-0.5d0)) * log(t)) + log((y * z))) - t
    else
        tmp = (log(z) - 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 <= 24000000000.0) {
		tmp = (((a + -0.5) * Math.log(t)) + Math.log((y * z))) - t;
	} else {
		tmp = (Math.log(z) - t) + (a * Math.log(t));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 24000000000.0:
		tmp = (((a + -0.5) * math.log(t)) + math.log((y * z))) - t
	else:
		tmp = (math.log(z) - t) + (a * math.log(t))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 24000000000.0)
		tmp = Float64(Float64(Float64(Float64(a + -0.5) * log(t)) + log(Float64(y * z))) - t);
	else
		tmp = Float64(Float64(log(z) - t) + Float64(a * log(t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 24000000000.0)
		tmp = (((a + -0.5) * log(t)) + log((y * z))) - t;
	else
		tmp = (log(z) - t) + (a * log(t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 24000000000.0], N[(N[(N[(N[(a + -0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision] + N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.4e10

   1. Initial program 99.4%

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

      1. +-commutative 99.4%

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

      2. associate--l+ 99.4%

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

      3. associate-+r+ 99.3%

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

      4. +-commutative 99.3%

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

      5. fma-define 99.4%

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

      6. sub-neg 99.4%

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

      7. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in a around 0 99.4%

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

   6. Taylor expanded in x around 0 56.2%

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

      1. distribute-rgt-in 56.2%

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

      2. associate-+r+ 56.3%

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

      3. log-prod 39.9%

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

   8. Simplified 39.9%

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

   if 2.4e10 < 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. +-commutative 99.8%

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

      2. associate--l+ 99.8%

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

      3. associate-+r+ 99.8%

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

      4. +-commutative 99.8%

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

      5. fma-define 99.8%

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

      6. sub-neg 99.8%

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

      7. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in a around inf 99.3%

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

      1. *-commutative 99.3%

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

   7. Simplified 99.3%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 24000000000:\\ \;\;\;\;\left(\left(a + -0.5\right) \cdot \log t + \log \left(y \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 73.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 2.15 \cdot 10^{-5}:\\ \;\;\;\;\log t \cdot \left(a - 0.5\right) + \log \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 2.15e-5)
   (+ (* (log t) (- a 0.5)) (log (* y z)))
   (+ (- (log z) t) (* a (log t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 2.15e-5) {
		tmp = (log(t) * (a - 0.5)) + log((y * z));
	} else {
		tmp = (log(z) - 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 <= 2.15d-5) then
        tmp = (log(t) * (a - 0.5d0)) + log((y * z))
    else
        tmp = (log(z) - 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 <= 2.15e-5) {
		tmp = (Math.log(t) * (a - 0.5)) + Math.log((y * z));
	} else {
		tmp = (Math.log(z) - t) + (a * Math.log(t));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 2.15e-5:
		tmp = (math.log(t) * (a - 0.5)) + math.log((y * z))
	else:
		tmp = (math.log(z) - t) + (a * math.log(t))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 2.15e-5)
		tmp = Float64(Float64(log(t) * Float64(a - 0.5)) + log(Float64(y * z)));
	else
		tmp = Float64(Float64(log(z) - t) + Float64(a * log(t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 2.15e-5)
		tmp = (log(t) * (a - 0.5)) + log((y * z));
	else
		tmp = (log(z) - t) + (a * log(t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 2.15e-5], N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision] + N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.1500000000000001e-5

   1. Initial program 99.4%

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

      1. +-commutative 99.4%

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

      2. associate--l+ 99.4%

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

      3. associate-+r+ 99.3%

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

      4. +-commutative 99.3%

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

      5. fma-define 99.3%

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

      6. sub-neg 99.3%

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

      7. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in t around 0 98.6%

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

      1. associate-+r+ 98.7%

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

      2. +-commutative 98.7%

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

      3. log-prod 70.3%

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

      4. sub-neg 70.3%

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

      5. metadata-eval 70.3%

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

      6. *-commutative 70.3%

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

      7. +-commutative 70.3%

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

      8. fma-define 70.3%

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

      9. +-commutative 70.3%

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

   7. Simplified 70.3%

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

   8. Taylor expanded in x around 0 38.9%

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

   if 2.1500000000000001e-5 < 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. +-commutative 99.8%

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

      2. associate--l+ 99.8%

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

      3. associate-+r+ 99.8%

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

      4. +-commutative 99.8%

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

      5. fma-define 99.8%

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

      6. sub-neg 99.8%

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

      7. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in a around inf 97.9%

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

      1. *-commutative 97.9%

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

   7. Simplified 97.9%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.15 \cdot 10^{-5}:\\ \;\;\;\;\log t \cdot \left(a - 0.5\right) + \log \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 77.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \log t\\ \mathbf{if}\;t \leq 600:\\ \;\;\;\;\log z + t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_1 - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* a (log t)))) (if (<= t 600.0) (+ (log z) t_1) (- t_1 t))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = a * log(t);
	double tmp;
	if (t <= 600.0) {
		tmp = log(z) + t_1;
	} else {
		tmp = t_1 - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = a * log(t)
    if (t <= 600.0d0) then
        tmp = log(z) + t_1
    else
        tmp = t_1 - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = a * Math.log(t);
	double tmp;
	if (t <= 600.0) {
		tmp = Math.log(z) + t_1;
	} else {
		tmp = t_1 - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = a * math.log(t)
	tmp = 0
	if t <= 600.0:
		tmp = math.log(z) + t_1
	else:
		tmp = t_1 - t
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(a * log(t))
	tmp = 0.0
	if (t <= 600.0)
		tmp = Float64(log(z) + t_1);
	else
		tmp = Float64(t_1 - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = a * log(t);
	tmp = 0.0;
	if (t <= 600.0)
		tmp = log(z) + t_1;
	else
		tmp = t_1 - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, 600.0], N[(N[Log[z], $MachinePrecision] + t$95$1), $MachinePrecision], N[(t$95$1 - t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 600

   1. Initial program 99.4%

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

      1. +-commutative 99.4%

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

      2. associate--l+ 99.4%

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

      3. associate-+r+ 99.3%

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

      4. +-commutative 99.3%

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

      5. fma-define 99.3%

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

      6. sub-neg 99.3%

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

      7. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in a around inf 46.4%

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

      1. *-commutative 46.4%

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

   7. Simplified 46.4%

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

   8. Taylor expanded in t around 0 46.4%

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

   if 600 < 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. +-commutative 99.8%

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

      2. associate--l+ 99.8%

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

      3. associate-+r+ 99.8%

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

      4. +-commutative 99.8%

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

      5. fma-define 99.8%

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

      6. sub-neg 99.8%

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

      7. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in a around 0 99.8%

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

   6. Taylor expanded in a around inf 98.4%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 600:\\ \;\;\;\;\log z + a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 59.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.85 \cdot 10^{+64} \lor \neg \left(a \leq 4.2 \cdot 10^{+154}\right):\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -3.85e+64) (not (<= a 4.2e+154))) (* a (log t)) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -3.85e+64) || !(a <= 4.2e+154)) {
		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 ((a <= (-3.85d+64)) .or. (.not. (a <= 4.2d+154))) 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 ((a <= -3.85e+64) || !(a <= 4.2e+154)) {
		tmp = a * Math.log(t);
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -3.85e+64) or not (a <= 4.2e+154):
		tmp = a * math.log(t)
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -3.85e+64) || !(a <= 4.2e+154))
		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 ((a <= -3.85e+64) || ~((a <= 4.2e+154)))
		tmp = a * log(t);
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -3.85e+64], N[Not[LessEqual[a, 4.2e+154]], $MachinePrecision]], N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if a < -3.85e64 or 4.19999999999999989e154 < a

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 73.5%

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

   4. Taylor expanded in a around inf 86.1%

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

      1. *-commutative 86.1%

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

   6. Simplified 86.1%

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

   if -3.85e64 < a < 4.19999999999999989e154

   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. +-commutative 99.6%

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

      2. associate--l+ 99.6%

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

      3. associate-+r+ 99.6%

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

      4. +-commutative 99.6%

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

      5. fma-define 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. Add Preprocessing

   5. Taylor expanded in t around inf 50.0%

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

      1. neg-mul-1 50.0%

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

   7. Simplified 50.0%

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

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.85 \cdot 10^{+64} \lor \neg \left(a \leq 4.2 \cdot 10^{+154}\right):\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 61.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -9.4 \cdot 10^{+76} \lor \neg \left(a \leq 4.2 \cdot 10^{+154}\right):\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\log z - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -9.4e+76) (not (<= a 4.2e+154))) (* a (log t)) (- (log z) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -9.4e+76) || !(a <= 4.2e+154)) {
		tmp = a * log(t);
	} else {
		tmp = log(z) - 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 <= (-9.4d+76)) .or. (.not. (a <= 4.2d+154))) then
        tmp = a * log(t)
    else
        tmp = log(z) - 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 <= -9.4e+76) || !(a <= 4.2e+154)) {
		tmp = a * Math.log(t);
	} else {
		tmp = Math.log(z) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -9.4e+76) or not (a <= 4.2e+154):
		tmp = a * math.log(t)
	else:
		tmp = math.log(z) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -9.4e+76) || !(a <= 4.2e+154))
		tmp = Float64(a * log(t));
	else
		tmp = Float64(log(z) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -9.4e+76) || ~((a <= 4.2e+154)))
		tmp = a * log(t);
	else
		tmp = log(z) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -9.4e+76], N[Not[LessEqual[a, 4.2e+154]], $MachinePrecision]], N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision], N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -9.4000000000000006e76 or 4.19999999999999989e154 < a

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 73.4%

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

   4. Taylor expanded in a around inf 87.9%

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

      1. *-commutative 87.9%

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

   6. Simplified 87.9%

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

   if -9.4000000000000006e76 < a < 4.19999999999999989e154

   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. +-commutative 99.6%

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

      2. associate--l+ 99.6%

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

      3. associate-+r+ 99.6%

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

      4. +-commutative 99.6%

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

      5. fma-define 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. Add Preprocessing

   5. Taylor expanded in a around inf 62.9%

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

      1. *-commutative 62.9%

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

   7. Simplified 62.9%

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

   8. Taylor expanded in a around 0 54.6%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -9.4 \cdot 10^{+76} \lor \neg \left(a \leq 4.2 \cdot 10^{+154}\right):\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\log z - t\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 74.6% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. +-commutative 99.6%

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

   2. associate--l+ 99.6%

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

   3. associate-+r+ 99.6%

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

   4. +-commutative 99.6%

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

   5. fma-define 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. Add Preprocessing

5. Taylor expanded in a around 0 99.6%

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

6. Taylor expanded in a around inf 70.9%

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

7. Final simplification 70.9%

   \[\leadsto a \cdot \log t - t \]
8. Add Preprocessing

Alternative 16: 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 99.6%

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

   1. +-commutative 99.6%

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

   2. associate--l+ 99.6%

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

   3. associate-+r+ 99.6%

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

   4. +-commutative 99.6%

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

   5. fma-define 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. Add Preprocessing

5. Taylor expanded in t around inf 39.4%

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

   1. neg-mul-1 39.4%

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

7. Simplified 39.4%

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

8. Final simplification 39.4%

   \[\leadsto -t \]
9. Add Preprocessing

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 2024046 
(FPCore (x y z t a)
  :name "Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2"
  :precision binary64

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

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