Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 20.8s

Alternatives: 14

Speedup: 0.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. associate-+l- 99.7%

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

   2. associate--l+ 99.7%

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

   3. sub-neg 99.7%

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

   4. +-commutative 99.7%

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

   5. *-commutative 99.7%

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

   6. distribute-rgt-neg-in 99.7%

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

   7. fma-udef 99.7%

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

   8. sub-neg 99.7%

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

   9. +-commutative 99.7%

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

   10. distribute-neg-in 99.7%

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

   11. metadata-eval 99.7%

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

   12. metadata-eval 99.7%

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

   13. unsub-neg 99.7%

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

3. Simplified 99.7%

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

5. Final simplification 99.7%

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

Alternative 2: 80.8% accurate, 1.0× speedup

Math

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

FPCore

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

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 240.0d0) then
        tmp = log(y) + (log(z) + (log(t) * (a - 0.5d0)))
    else
        tmp = (log(t) * (a + (-0.5d0))) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 240.0) {
		tmp = Math.log(y) + (Math.log(z) + (Math.log(t) * (a - 0.5)));
	} else {
		tmp = (Math.log(t) * (a + -0.5)) - 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(t) * (a + -0.5)) - 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(t) * Float64(a + -0.5)) - t);
	end
	return tmp
end

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if t < 240

   1. Initial program 99.5%

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

      1. associate--l+ 99.5%

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

      2. sub-neg 99.5%

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

      3. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around 0 62.4%

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

      1. associate--l+ 62.4%

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

   7. Simplified 62.4%

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

   8. Taylor expanded in t around 0 62.1%

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

   if 240 < t

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. sub-neg 99.9%

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

      3. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 71.6%

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

      1. associate--l+ 71.6%

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

   7. Simplified 71.6%

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

   8. Taylor expanded in t around inf 97.8%

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

      1. mul-1-neg 97.8%

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

   10. Simplified 97.8%

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

4. Final simplification 79.1%

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

Alternative 3: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. associate--l+ 99.7%

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

   2. sub-neg 99.7%

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

   3. metadata-eval 99.7%

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

3. Simplified 99.7%

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

5. Final simplification 99.7%

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

Alternative 4: 68.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. associate--l+ 99.7%

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

   2. sub-neg 99.7%

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

   3. metadata-eval 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in x around 0 66.8%

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

   1. associate--l+ 66.8%

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

7. Simplified 66.8%

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

8. Final simplification 66.8%

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

Alternative 5: 75.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(\left(y \cdot z\right) \cdot {t}^{-0.5}\right) - t\\ t_2 := \log t \cdot \left(a + -0.5\right) - t\\ \mathbf{if}\;a \leq -0.017:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;a \leq -7.6 \cdot 10^{-238}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 6.8 \cdot 10^{-85}:\\ \;\;\;\;\log \left(x + y\right) + \left(\log z - t\right)\\ \mathbf{elif}\;a \leq 1.3 \cdot 10^{-47}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (log (* (* y z) (pow t -0.5))) t))
        (t_2 (- (* (log t) (+ a -0.5)) t)))
   (if (<= a -0.017)
     t_2
     (if (<= a -7.6e-238)
       t_1
       (if (<= a 6.8e-85)
         (+ (log (+ x y)) (- (log z) t))
         (if (<= a 1.3e-47) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log(((y * z) * pow(t, -0.5))) - t;
	double t_2 = (log(t) * (a + -0.5)) - t;
	double tmp;
	if (a <= -0.017) {
		tmp = t_2;
	} else if (a <= -7.6e-238) {
		tmp = t_1;
	} else if (a <= 6.8e-85) {
		tmp = log((x + y)) + (log(z) - t);
	} else if (a <= 1.3e-47) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = log(((y * z) * (t ** (-0.5d0)))) - t
    t_2 = (log(t) * (a + (-0.5d0))) - t
    if (a <= (-0.017d0)) then
        tmp = t_2
    else if (a <= (-7.6d-238)) then
        tmp = t_1
    else if (a <= 6.8d-85) then
        tmp = log((x + y)) + (log(z) - t)
    else if (a <= 1.3d-47) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = Math.log(((y * z) * Math.pow(t, -0.5))) - t;
	double t_2 = (Math.log(t) * (a + -0.5)) - t;
	double tmp;
	if (a <= -0.017) {
		tmp = t_2;
	} else if (a <= -7.6e-238) {
		tmp = t_1;
	} else if (a <= 6.8e-85) {
		tmp = Math.log((x + y)) + (Math.log(z) - t);
	} else if (a <= 1.3e-47) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = math.log(((y * z) * math.pow(t, -0.5))) - t
	t_2 = (math.log(t) * (a + -0.5)) - t
	tmp = 0
	if a <= -0.017:
		tmp = t_2
	elif a <= -7.6e-238:
		tmp = t_1
	elif a <= 6.8e-85:
		tmp = math.log((x + y)) + (math.log(z) - t)
	elif a <= 1.3e-47:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(Float64(Float64(y * z) * (t ^ -0.5))) - t)
	t_2 = Float64(Float64(log(t) * Float64(a + -0.5)) - t)
	tmp = 0.0
	if (a <= -0.017)
		tmp = t_2;
	elseif (a <= -7.6e-238)
		tmp = t_1;
	elseif (a <= 6.8e-85)
		tmp = Float64(log(Float64(x + y)) + Float64(log(z) - t));
	elseif (a <= 1.3e-47)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = log(((y * z) * (t ^ -0.5))) - t;
	t_2 = (log(t) * (a + -0.5)) - t;
	tmp = 0.0;
	if (a <= -0.017)
		tmp = t_2;
	elseif (a <= -7.6e-238)
		tmp = t_1;
	elseif (a <= 6.8e-85)
		tmp = log((x + y)) + (log(z) - t);
	elseif (a <= 1.3e-47)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[N[(N[(y * z), $MachinePrecision] * N[Power[t, -0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[a, -0.017], t$95$2, If[LessEqual[a, -7.6e-238], t$95$1, If[LessEqual[a, 6.8e-85], N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.3e-47], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if a < -0.017000000000000001 or 1.3e-47 < a

   1. Initial program 99.8%

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

      1. associate--l+ 99.8%

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

      2. sub-neg 99.8%

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

      3. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 74.0%

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

      1. associate--l+ 74.0%

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

   7. Simplified 74.0%

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

   8. Taylor expanded in t around inf 96.6%

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

      1. mul-1-neg 96.6%

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

   10. Simplified 96.6%

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

   if -0.017000000000000001 < a < -7.5999999999999994e-238 or 6.8e-85 < a < 1.3e-47

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. fma-def 99.5%

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

      3. sub-neg 99.5%

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

      4. metadata-eval 99.5%

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

      5. associate--l+ 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around 0 57.8%

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

      1. +-commutative 57.8%

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

   7. Simplified 57.8%

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

   8. Taylor expanded in a around 0 57.9%

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

      1. +-commutative 57.9%

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

      2. +-commutative 57.9%

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

      3. associate-+r+ 57.8%

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

      4. +-commutative 57.8%

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

      5. log-prod 49.4%

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

   10. Simplified 49.4%

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

       1. +-commutative 49.4%

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

       2. add-log-exp 49.4%

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

       3. sum-log 47.6%

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

       4. *-commutative 47.6%

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

       5. exp-to-pow 47.7%

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

   12. Applied egg-rr 47.7%

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

   if -7.5999999999999994e-238 < a < 6.8e-85

   1. Initial program 99.6%

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

      1. associate-+l- 99.6%

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

      2. associate--l+ 99.8%

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

      3. sub-neg 99.8%

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

      4. +-commutative 99.8%

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

      5. *-commutative 99.8%

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

      6. distribute-rgt-neg-in 99.8%

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

      7. fma-udef 99.8%

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

      8. sub-neg 99.8%

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

      9. +-commutative 99.8%

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

      10. distribute-neg-in 99.8%

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

      11. metadata-eval 99.8%

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

      12. metadata-eval 99.8%

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

      13. unsub-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t around inf 62.7%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.017:\\ \;\;\;\;\log t \cdot \left(a + -0.5\right) - t\\ \mathbf{elif}\;a \leq -7.6 \cdot 10^{-238}:\\ \;\;\;\;\log \left(\left(y \cdot z\right) \cdot {t}^{-0.5}\right) - t\\ \mathbf{elif}\;a \leq 6.8 \cdot 10^{-85}:\\ \;\;\;\;\log \left(x + y\right) + \left(\log z - t\right)\\ \mathbf{elif}\;a \leq 1.3 \cdot 10^{-47}:\\ \;\;\;\;\log \left(\left(y \cdot z\right) \cdot {t}^{-0.5}\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot \left(a + -0.5\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.2 \cdot 10^{-38} \lor \neg \left(a \leq 4.8 \cdot 10^{-27}\right):\\ \;\;\;\;\log t \cdot \left(a + -0.5\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{y}{\frac{{t}^{\left(0.5 - a\right)}}{z}}\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -3.2e-38) (not (<= a 4.8e-27)))
   (- (* (log t) (+ a -0.5)) t)
   (- (log (/ y (/ (pow t (- 0.5 a)) z))) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -3.2e-38) || !(a <= 4.8e-27)) {
		tmp = (log(t) * (a + -0.5)) - t;
	} else {
		tmp = log((y / (pow(t, (0.5 - a)) / 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 <= (-3.2d-38)) .or. (.not. (a <= 4.8d-27))) then
        tmp = (log(t) * (a + (-0.5d0))) - t
    else
        tmp = log((y / ((t ** (0.5d0 - a)) / 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 <= -3.2e-38) || !(a <= 4.8e-27)) {
		tmp = (Math.log(t) * (a + -0.5)) - t;
	} else {
		tmp = Math.log((y / (Math.pow(t, (0.5 - a)) / z))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -3.2e-38) or not (a <= 4.8e-27):
		tmp = (math.log(t) * (a + -0.5)) - t
	else:
		tmp = math.log((y / (math.pow(t, (0.5 - a)) / z))) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -3.2e-38) || !(a <= 4.8e-27))
		tmp = Float64(Float64(log(t) * Float64(a + -0.5)) - t);
	else
		tmp = Float64(log(Float64(y / Float64((t ^ Float64(0.5 - a)) / z))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -3.2e-38) || ~((a <= 4.8e-27)))
		tmp = (log(t) * (a + -0.5)) - t;
	else
		tmp = log((y / ((t ^ (0.5 - a)) / z))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -3.2e-38], N[Not[LessEqual[a, 4.8e-27]], $MachinePrecision]], N[(N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[Log[N[(y / N[(N[Power[t, N[(0.5 - a), $MachinePrecision]], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.19999999999999977e-38 or 4.80000000000000004e-27 < a

   1. Initial program 99.8%

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

      1. associate--l+ 99.8%

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

      2. sub-neg 99.8%

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

      3. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 73.5%

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

      1. associate--l+ 73.5%

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

   7. Simplified 73.5%

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

   8. Taylor expanded in t around inf 96.8%

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

      1. mul-1-neg 96.8%

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

   10. Simplified 96.8%

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

   if -3.19999999999999977e-38 < a < 4.80000000000000004e-27

   1. Initial program 99.6%

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

      1. associate-+l- 99.6%

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

      2. associate--l+ 99.6%

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

      3. sub-neg 99.6%

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

      4. +-commutative 99.6%

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

      5. *-commutative 99.6%

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

      6. distribute-rgt-neg-in 99.6%

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

      7. fma-udef 99.6%

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

      8. sub-neg 99.6%

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

      9. +-commutative 99.6%

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

      10. distribute-neg-in 99.6%

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

      11. metadata-eval 99.6%

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

      12. metadata-eval 99.6%

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

      13. unsub-neg 99.6%

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

   3. Simplified 99.6%

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

      1. associate-+r- 99.6%

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

      2. fma-udef 99.6%

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

      3. associate--r+ 99.6%

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

      4. +-commutative 99.6%

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

      5. sum-log 66.8%

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

   6. Applied egg-rr 66.8%

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

      1. add-log-exp 66.8%

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

      2. diff-log 64.7%

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

      3. +-commutative 64.7%

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

      4. exp-to-pow 64.8%

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

   8. Applied egg-rr 64.8%

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

   9. Taylor expanded in y around inf 35.8%

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

       1. exp-to-pow 35.8%

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

       2. associate-/l* 37.0%

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

   11. Simplified 37.0%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.2 \cdot 10^{-38} \lor \neg \left(a \leq 4.8 \cdot 10^{-27}\right):\\ \;\;\;\;\log t \cdot \left(a + -0.5\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{y}{\frac{{t}^{\left(0.5 - a\right)}}{z}}\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.00037 \lor \neg \left(a \leq 1.35 \cdot 10^{-47}\right):\\ \;\;\;\;\log t \cdot \left(a + -0.5\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot -0.5 + \left(\log \left(y \cdot z\right) - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -0.00037) (not (<= a 1.35e-47)))
   (- (* (log t) (+ a -0.5)) t)
   (+ (* (log t) -0.5) (- (log (* y z)) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -0.00037) || !(a <= 1.35e-47)) {
		tmp = (log(t) * (a + -0.5)) - t;
	} else {
		tmp = (log(t) * -0.5) + (log((y * 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 <= (-0.00037d0)) .or. (.not. (a <= 1.35d-47))) then
        tmp = (log(t) * (a + (-0.5d0))) - t
    else
        tmp = (log(t) * (-0.5d0)) + (log((y * 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 <= -0.00037) || !(a <= 1.35e-47)) {
		tmp = (Math.log(t) * (a + -0.5)) - t;
	} else {
		tmp = (Math.log(t) * -0.5) + (Math.log((y * z)) - t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -0.00037) or not (a <= 1.35e-47):
		tmp = (math.log(t) * (a + -0.5)) - t
	else:
		tmp = (math.log(t) * -0.5) + (math.log((y * z)) - t)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -0.00037) || !(a <= 1.35e-47))
		tmp = Float64(Float64(log(t) * Float64(a + -0.5)) - t);
	else
		tmp = Float64(Float64(log(t) * -0.5) + Float64(log(Float64(y * z)) - t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -0.00037) || ~((a <= 1.35e-47)))
		tmp = (log(t) * (a + -0.5)) - t;
	else
		tmp = (log(t) * -0.5) + (log((y * z)) - t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -0.00037], N[Not[LessEqual[a, 1.35e-47]], $MachinePrecision]], N[(N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[t], $MachinePrecision] * -0.5), $MachinePrecision] + N[(N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.6999999999999999e-4 or 1.3499999999999999e-47 < a

   1. Initial program 99.8%

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

      1. associate--l+ 99.8%

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

      2. sub-neg 99.8%

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

      3. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 74.0%

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

      1. associate--l+ 74.0%

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

   7. Simplified 74.0%

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

   8. Taylor expanded in t around inf 96.6%

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

      1. mul-1-neg 96.6%

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

   10. Simplified 96.6%

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

   if -3.6999999999999999e-4 < a < 1.3499999999999999e-47

   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-def 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. Taylor expanded in x around 0 57.6%

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

      1. +-commutative 57.6%

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

   7. Simplified 57.6%

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

   8. Taylor expanded in a around 0 57.6%

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

      1. +-commutative 57.6%

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

      2. +-commutative 57.6%

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

      3. associate-+r+ 57.6%

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

      4. +-commutative 57.6%

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

      5. log-prod 40.3%

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

   10. Simplified 40.3%

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

       1. associate--l+ 40.3%

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

   12. Applied egg-rr 40.3%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.00037 \lor \neg \left(a \leq 1.35 \cdot 10^{-47}\right):\\ \;\;\;\;\log t \cdot \left(a + -0.5\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot -0.5 + \left(\log \left(y \cdot z\right) - t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 87.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 425000000:\\ \;\;\;\;\left(\log \left(\left(x + y\right) \cdot z\right) + \log t \cdot \left(a - 0.5\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot \left(a + -0.5\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 425000000.0)
   (- (+ (log (* (+ x y) z)) (* (log t) (- a 0.5))) t)
   (- (* (log t) (+ a -0.5)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 425000000.0) {
		tmp = (log(((x + y) * z)) + (log(t) * (a - 0.5))) - t;
	} else {
		tmp = (log(t) * (a + -0.5)) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 425000000.0d0) then
        tmp = (log(((x + y) * z)) + (log(t) * (a - 0.5d0))) - t
    else
        tmp = (log(t) * (a + (-0.5d0))) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 425000000.0) {
		tmp = (Math.log(((x + y) * z)) + (Math.log(t) * (a - 0.5))) - t;
	} else {
		tmp = (Math.log(t) * (a + -0.5)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 425000000.0:
		tmp = (math.log(((x + y) * z)) + (math.log(t) * (a - 0.5))) - t
	else:
		tmp = (math.log(t) * (a + -0.5)) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 425000000.0)
		tmp = Float64(Float64(log(Float64(Float64(x + y) * z)) + Float64(log(t) * Float64(a - 0.5))) - t);
	else
		tmp = Float64(Float64(log(t) * Float64(a + -0.5)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 425000000.0)
		tmp = (log(((x + y) * z)) + (log(t) * (a - 0.5))) - t;
	else
		tmp = (log(t) * (a + -0.5)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 425000000.0], N[(N[(N[Log[N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 4.25e8

   1. Initial program 99.5%

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

      1. associate-+l- 99.5%

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

      2. associate--l+ 99.5%

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

      3. sub-neg 99.5%

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

      4. +-commutative 99.5%

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

      5. *-commutative 99.5%

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

      6. distribute-rgt-neg-in 99.5%

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

      7. fma-udef 99.5%

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

      8. sub-neg 99.5%

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

      9. +-commutative 99.5%

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

      10. distribute-neg-in 99.5%

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

      11. metadata-eval 99.5%

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

      12. metadata-eval 99.5%

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

      13. unsub-neg 99.5%

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

   3. Simplified 99.5%

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

      1. associate-+r- 99.5%

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

      2. fma-udef 99.5%

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

      3. associate--r+ 99.5%

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

      4. +-commutative 99.5%

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

      5. sum-log 70.4%

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

   6. Applied egg-rr 70.4%

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

   if 4.25e8 < t

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. sub-neg 99.9%

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

      3. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 71.3%

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

      1. associate--l+ 71.3%

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

   7. Simplified 71.3%

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

   8. Taylor expanded in t around inf 99.9%

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

      1. mul-1-neg 99.9%

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

   10. Simplified 99.9%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 425000000:\\ \;\;\;\;\left(\log \left(\left(x + y\right) \cdot z\right) + \log t \cdot \left(a - 0.5\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot \left(a + -0.5\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 86.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 7.5 \cdot 10^{-19}:\\ \;\;\;\;\log \left(\left(x + y\right) \cdot z\right) + \log t \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot \left(a + -0.5\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 7.5e-19)
   (+ (log (* (+ x y) z)) (* (log t) (- a 0.5)))
   (- (* (log t) (+ a -0.5)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 7.5e-19) {
		tmp = log(((x + y) * z)) + (log(t) * (a - 0.5));
	} else {
		tmp = (log(t) * (a + -0.5)) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 7.5d-19) then
        tmp = log(((x + y) * z)) + (log(t) * (a - 0.5d0))
    else
        tmp = (log(t) * (a + (-0.5d0))) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 7.5e-19) {
		tmp = Math.log(((x + y) * z)) + (Math.log(t) * (a - 0.5));
	} else {
		tmp = (Math.log(t) * (a + -0.5)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 7.5e-19:
		tmp = math.log(((x + y) * z)) + (math.log(t) * (a - 0.5))
	else:
		tmp = (math.log(t) * (a + -0.5)) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 7.5e-19)
		tmp = Float64(log(Float64(Float64(x + y) * z)) + Float64(log(t) * Float64(a - 0.5)));
	else
		tmp = Float64(Float64(log(t) * Float64(a + -0.5)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 7.5e-19)
		tmp = log(((x + y) * z)) + (log(t) * (a - 0.5));
	else
		tmp = (log(t) * (a + -0.5)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 7.5e-19], N[(N[Log[N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 7.49999999999999957e-19

   1. Initial program 99.5%

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

      1. associate-+l- 99.5%

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

      2. associate--l+ 99.5%

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

      3. sub-neg 99.5%

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

      4. +-commutative 99.5%

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

      5. *-commutative 99.5%

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

      6. distribute-rgt-neg-in 99.5%

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

      7. fma-udef 99.5%

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

      8. sub-neg 99.5%

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

      9. +-commutative 99.5%

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

      10. distribute-neg-in 99.5%

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

      11. metadata-eval 99.5%

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

      12. metadata-eval 99.5%

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

      13. unsub-neg 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in t around 0 99.5%

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

      1. log-prod 70.8%

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

      2. +-commutative 70.8%

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

   7. Simplified 70.8%

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

   if 7.49999999999999957e-19 < t

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. sub-neg 99.9%

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

      3. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 71.5%

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

      1. associate--l+ 71.5%

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

   7. Simplified 71.5%

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

   8. Taylor expanded in t around inf 97.1%

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

      1. mul-1-neg 97.1%

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

   10. Simplified 97.1%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.5 \cdot 10^{-19}:\\ \;\;\;\;\log \left(\left(x + y\right) \cdot z\right) + \log t \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot \left(a + -0.5\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 73.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log t \cdot \left(a + -0.5\right)\\ \mathbf{if}\;t \leq 7.5 \cdot 10^{-19}:\\ \;\;\;\;t\_1 + \log \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (log t) (+ a -0.5))))
   (if (<= t 7.5e-19) (+ t_1 (log (* y z))) (- t_1 t))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log(t) * (a + -0.5);
	double tmp;
	if (t <= 7.5e-19) {
		tmp = t_1 + log((y * z));
	} else {
		tmp = t_1 - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = log(t) * (a + (-0.5d0))
    if (t <= 7.5d-19) then
        tmp = t_1 + log((y * z))
    else
        tmp = t_1 - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = Math.log(t) * (a + -0.5);
	double tmp;
	if (t <= 7.5e-19) {
		tmp = t_1 + Math.log((y * z));
	} else {
		tmp = t_1 - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = math.log(t) * (a + -0.5)
	tmp = 0
	if t <= 7.5e-19:
		tmp = t_1 + math.log((y * z))
	else:
		tmp = t_1 - t
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(t) * Float64(a + -0.5))
	tmp = 0.0
	if (t <= 7.5e-19)
		tmp = Float64(t_1 + log(Float64(y * z)));
	else
		tmp = Float64(t_1 - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = log(t) * (a + -0.5);
	tmp = 0.0;
	if (t <= 7.5e-19)
		tmp = t_1 + log((y * z));
	else
		tmp = t_1 - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, 7.5e-19], N[(t$95$1 + N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$1 - t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 7.49999999999999957e-19

   1. Initial program 99.5%

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

      1. associate--l+ 99.5%

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

      2. sub-neg 99.5%

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

      3. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around 0 62.3%

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

      1. associate--l+ 62.3%

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

   7. Simplified 62.3%

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

   8. Taylor expanded in t around 0 62.4%

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

      1. associate-+r+ 62.3%

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

      2. log-prod 47.3%

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

      3. +-commutative 47.3%

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

      4. sub-neg 47.3%

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

      5. metadata-eval 47.3%

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

   10. Simplified 47.3%

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

   if 7.49999999999999957e-19 < t

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. sub-neg 99.9%

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

      3. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 71.5%

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

      1. associate--l+ 71.5%

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

   7. Simplified 71.5%

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

   8. Taylor expanded in t around inf 97.1%

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

      1. mul-1-neg 97.1%

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

   10. Simplified 97.1%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.5 \cdot 10^{-19}:\\ \;\;\;\;\log t \cdot \left(a + -0.5\right) + \log \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot \left(a + -0.5\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 62.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.2 \cdot 10^{+17} \lor \neg \left(t \leq 1.4 \cdot 10^{+55}\right) \land t \leq 7.5 \cdot 10^{+83}:\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t 1.2e+17) (and (not (<= t 1.4e+55)) (<= t 7.5e+83)))
   (* (log t) a)
   (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= 1.2e+17) || (!(t <= 1.4e+55) && (t <= 7.5e+83))) {
		tmp = log(t) * a;
	} else {
		tmp = -t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((t <= 1.2d+17) .or. (.not. (t <= 1.4d+55)) .and. (t <= 7.5d+83)) then
        tmp = log(t) * a
    else
        tmp = -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= 1.2e+17) || (!(t <= 1.4e+55) && (t <= 7.5e+83))) {
		tmp = Math.log(t) * a;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (t <= 1.2e+17) or (not (t <= 1.4e+55) and (t <= 7.5e+83)):
		tmp = math.log(t) * a
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((t <= 1.2e+17) || (!(t <= 1.4e+55) && (t <= 7.5e+83)))
		tmp = Float64(log(t) * a);
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((t <= 1.2e+17) || (~((t <= 1.4e+55)) && (t <= 7.5e+83)))
		tmp = log(t) * a;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[t, 1.2e+17], And[N[Not[LessEqual[t, 1.4e+55]], $MachinePrecision], LessEqual[t, 7.5e+83]]], N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if t < 1.2e17 or 1.4e55 < t < 7.49999999999999989e83

   1. Initial program 99.5%

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

      1. associate--l+ 99.5%

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

      2. sub-neg 99.5%

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

      3. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around 0 65.2%

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

      1. associate--l+ 65.2%

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

   7. Simplified 65.2%

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

   8. Taylor expanded in a around inf 56.3%

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

   if 1.2e17 < t < 1.4e55 or 7.49999999999999989e83 < t

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-def 99.9%

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

      3. sub-neg 99.9%

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

      4. metadata-eval 99.9%

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

      5. associate--l+ 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 69.3%

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

      1. +-commutative 69.3%

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

   7. Simplified 69.3%

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

   8. Taylor expanded in t around inf 81.2%

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

      1. mul-1-neg 81.2%

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

   10. Simplified 81.2%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.2 \cdot 10^{+17} \lor \neg \left(t \leq 1.4 \cdot 10^{+55}\right) \land t \leq 7.5 \cdot 10^{+83}:\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 65.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2700000000 \lor \neg \left(a \leq 2.5 \cdot 10^{+71}\right):\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -2700000000.0) (not (<= a 2.5e+71)))
   (* (log t) a)
   (- (log (+ x y)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -2700000000.0) || !(a <= 2.5e+71)) {
		tmp = log(t) * a;
	} else {
		tmp = log((x + y)) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((a <= (-2700000000.0d0)) .or. (.not. (a <= 2.5d+71))) then
        tmp = log(t) * a
    else
        tmp = log((x + y)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -2700000000.0) || !(a <= 2.5e+71)) {
		tmp = Math.log(t) * a;
	} else {
		tmp = Math.log((x + y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -2700000000.0) or not (a <= 2.5e+71):
		tmp = math.log(t) * a
	else:
		tmp = math.log((x + y)) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -2700000000.0) || !(a <= 2.5e+71))
		tmp = Float64(log(t) * a);
	else
		tmp = Float64(log(Float64(x + y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -2700000000.0) || ~((a <= 2.5e+71)))
		tmp = log(t) * a;
	else
		tmp = log((x + y)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -2700000000.0], N[Not[LessEqual[a, 2.5e+71]], $MachinePrecision]], N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision], N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.7e9 or 2.49999999999999986e71 < a

   1. Initial program 99.8%

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

      1. associate--l+ 99.8%

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

      2. sub-neg 99.8%

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

      3. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 75.9%

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

      1. associate--l+ 75.9%

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

   7. Simplified 75.9%

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

   8. Taylor expanded in a around inf 81.0%

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

   if -2.7e9 < a < 2.49999999999999986e71

   1. Initial program 99.6%

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

      1. associate-+l- 99.6%

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

      2. associate--l+ 99.6%

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

      3. sub-neg 99.6%

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

      4. +-commutative 99.6%

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

      5. *-commutative 99.6%

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

      6. distribute-rgt-neg-in 99.6%

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

      7. fma-udef 99.6%

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

      8. sub-neg 99.6%

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

      9. +-commutative 99.6%

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

      10. distribute-neg-in 99.6%

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

      11. metadata-eval 99.6%

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

      12. metadata-eval 99.6%

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

      13. unsub-neg 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in t around inf 54.9%

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

      1. neg-mul-1 54.9%

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

   7. Simplified 54.9%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2700000000 \lor \neg \left(a \leq 2.5 \cdot 10^{+71}\right):\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 77.6% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. associate--l+ 99.7%

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

   2. sub-neg 99.7%

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

   3. metadata-eval 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in x around 0 66.8%

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

   1. associate--l+ 66.8%

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

7. Simplified 66.8%

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

8. Taylor expanded in t around inf 77.9%

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

   1. mul-1-neg 77.9%

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

10. Simplified 77.9%

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

11. Final simplification 77.9%

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

Alternative 14: 38.3% accurate, 156.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. +-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. fma-def 99.7%

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

   3. sub-neg 99.7%

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

   4. metadata-eval 99.7%

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

   5. associate--l+ 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in x around 0 66.8%

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

   1. +-commutative 66.8%

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

7. Simplified 66.8%

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

8. Taylor expanded in t around inf 35.2%

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

   1. mul-1-neg 35.2%

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

10. Simplified 35.2%

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

11. Final simplification 35.2%

    \[\leadsto -t \]
12. 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 2024027 
(FPCore (x y z t a)
  :name "Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2"
  :precision binary64

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

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