Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 15.3s

Alternatives: 15

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, 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. Final simplification 99.6%

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

Alternative 2: 85.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log z \leq 230:\\ \;\;\;\;\left(\log \left(\left(x + y\right) \cdot z\right) + \log t \cdot \left(a - 0.5\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (log z) 230.0)
   (- (+ (log (* (+ x y) z)) (* (log t) (- a 0.5))) t)
   (- (* a (log t)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (log(z) <= 230.0) {
		tmp = (log(((x + y) * z)) + (log(t) * (a - 0.5))) - t;
	} else {
		tmp = (a * log(t)) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (log(z) <= 230.0d0) then
        tmp = (log(((x + y) * z)) + (log(t) * (a - 0.5d0))) - t
    else
        tmp = (a * log(t)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (Math.log(z) <= 230.0) {
		tmp = (Math.log(((x + y) * z)) + (Math.log(t) * (a - 0.5))) - t;
	} else {
		tmp = (a * Math.log(t)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if math.log(z) <= 230.0:
		tmp = (math.log(((x + y) * z)) + (math.log(t) * (a - 0.5))) - t
	else:
		tmp = (a * math.log(t)) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (log(z) <= 230.0)
		tmp = Float64(Float64(log(Float64(Float64(x + y) * z)) + Float64(log(t) * Float64(a - 0.5))) - t);
	else
		tmp = Float64(Float64(a * log(t)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (log(z) <= 230.0)
		tmp = (log(((x + y) * z)) + (log(t) * (a - 0.5))) - t;
	else
		tmp = (a * log(t)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[Log[z], $MachinePrecision], 230.0], 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[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (log.f64 z) < 230

   1. Initial program 99.7%

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

      1. associate-+l- 99.7%

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

      2. +-commutative 99.7%

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

      3. associate--l+ 99.6%

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

      4. sub-neg 99.6%

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

      5. +-commutative 99.6%

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

      6. *-commutative 99.6%

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

      7. distribute-rgt-neg-in 99.6%

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

      8. fma-def 99.6%

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

      9. sub-neg 99.6%

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

      10. +-commutative 99.6%

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

      11. distribute-neg-in 99.6%

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

      12. metadata-eval 99.6%

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

      13. metadata-eval 99.6%

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

      14. unsub-neg 99.6%

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

   3. Simplified 99.6%

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

      1. associate-+r- 99.7%

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

      2. +-commutative 99.7%

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

      3. fma-udef 99.7%

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

      4. associate--r+ 99.7%

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

      5. +-commutative 99.7%

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

      6. sum-log 96.2%

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

   5. Applied egg-rr 96.2%

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

   if 230 < (log.f64 z)

   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. Taylor expanded in x around 0 63.0%

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

   3. Taylor expanded in a around inf 80.5%

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

      1. *-commutative 80.5%

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

   5. Simplified 80.5%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log z \leq 230:\\ \;\;\;\;\left(\log \left(\left(x + y\right) \cdot z\right) + \log t \cdot \left(a - 0.5\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \]

Alternative 3: 76.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1450000000 \lor \neg \left(a \leq 0.64\right):\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + \log \left(y \cdot \sqrt{\frac{1}{t}}\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -1450000000.0) (not (<= a 0.64)))
   (- (* a (log t)) t)
   (- (+ (log z) (log (* y (sqrt (/ 1.0 t))))) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -1450000000.0) || !(a <= 0.64)) {
		tmp = (a * log(t)) - t;
	} else {
		tmp = (log(z) + log((y * sqrt((1.0 / t))))) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((a <= (-1450000000.0d0)) .or. (.not. (a <= 0.64d0))) then
        tmp = (a * log(t)) - t
    else
        tmp = (log(z) + log((y * sqrt((1.0d0 / t))))) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -1450000000.0) || !(a <= 0.64)) {
		tmp = (a * Math.log(t)) - t;
	} else {
		tmp = (Math.log(z) + Math.log((y * Math.sqrt((1.0 / t))))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -1450000000.0) or not (a <= 0.64):
		tmp = (a * math.log(t)) - t
	else:
		tmp = (math.log(z) + math.log((y * math.sqrt((1.0 / t))))) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -1450000000.0) || !(a <= 0.64))
		tmp = Float64(Float64(a * log(t)) - t);
	else
		tmp = Float64(Float64(log(z) + log(Float64(y * sqrt(Float64(1.0 / t))))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -1450000000.0) || ~((a <= 0.64)))
		tmp = (a * log(t)) - t;
	else
		tmp = (log(z) + log((y * sqrt((1.0 / t))))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -1450000000.0], N[Not[LessEqual[a, 0.64]], $MachinePrecision]], N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[z], $MachinePrecision] + N[Log[N[(y * N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.45e9 or 0.640000000000000013 < a

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 63.9%

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

   3. Taylor expanded in a around inf 98.7%

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

      1. *-commutative 98.7%

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

   5. Simplified 98.7%

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

   if -1.45e9 < a < 0.640000000000000013

   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. Taylor expanded in a around 0 97.6%

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

      1. associate-+r+ 97.7%

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

      2. log-prod 77.3%

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

      3. add-log-exp 77.3%

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

      4. sum-log 69.4%

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

      5. +-commutative 69.4%

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

      6. *-commutative 69.4%

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

      7. exp-to-pow 69.5%

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

   4. Applied egg-rr 69.5%

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

   5. Taylor expanded in y around inf 46.0%

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

   6. Taylor expanded in z around 0 59.0%

      \[\leadsto \color{blue}{\left(\log z + \log \left(\sqrt{\frac{1}{t}} \cdot y\right)\right)} - t \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1450000000 \lor \neg \left(a \leq 0.64\right):\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + \log \left(y \cdot \sqrt{\frac{1}{t}}\right)\right) - t\\ \end{array} \]

Alternative 4: 80.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1450000000 \lor \neg \left(a \leq 2.2\right):\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + \left(\log y + \log t \cdot -0.5\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -1450000000.0) (not (<= a 2.2)))
   (- (* a (log t)) t)
   (- (+ (log z) (+ (log y) (* (log t) -0.5))) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -1450000000.0) || !(a <= 2.2)) {
		tmp = (a * log(t)) - t;
	} else {
		tmp = (log(z) + (log(y) + (log(t) * -0.5))) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((a <= (-1450000000.0d0)) .or. (.not. (a <= 2.2d0))) then
        tmp = (a * log(t)) - t
    else
        tmp = (log(z) + (log(y) + (log(t) * (-0.5d0)))) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -1450000000.0) || !(a <= 2.2)) {
		tmp = (a * Math.log(t)) - t;
	} else {
		tmp = (Math.log(z) + (Math.log(y) + (Math.log(t) * -0.5))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -1450000000.0) or not (a <= 2.2):
		tmp = (a * math.log(t)) - t
	else:
		tmp = (math.log(z) + (math.log(y) + (math.log(t) * -0.5))) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -1450000000.0) || !(a <= 2.2))
		tmp = Float64(Float64(a * log(t)) - t);
	else
		tmp = Float64(Float64(log(z) + Float64(log(y) + Float64(log(t) * -0.5))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -1450000000.0) || ~((a <= 2.2)))
		tmp = (a * log(t)) - t;
	else
		tmp = (log(z) + (log(y) + (log(t) * -0.5))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -1450000000.0], N[Not[LessEqual[a, 2.2]], $MachinePrecision]], N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[z], $MachinePrecision] + N[(N[Log[y], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.45e9 or 2.2000000000000002 < a

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 63.9%

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

   3. Taylor expanded in a around inf 98.7%

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

      1. *-commutative 98.7%

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

   5. Simplified 98.7%

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

   if -1.45e9 < a < 2.2000000000000002

   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. Taylor expanded in a around 0 97.6%

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

   3. Taylor expanded in x around 0 66.0%

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

      1. *-commutative 66.0%

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

   5. Simplified 66.0%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1450000000 \lor \neg \left(a \leq 2.2\right):\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + \left(\log y + \log t \cdot -0.5\right)\right) - t\\ \end{array} \]

Alternative 5: 64.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log z \leq 230:\\ \;\;\;\;\left(\log \left(y \cdot z\right) + \log t \cdot \left(a - 0.5\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (log z) 230.0)
   (- (+ (log (* y z)) (* (log t) (- a 0.5))) t)
   (- (* a (log t)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (log(z) <= 230.0) {
		tmp = (log((y * z)) + (log(t) * (a - 0.5))) - t;
	} else {
		tmp = (a * log(t)) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (log(z) <= 230.0d0) then
        tmp = (log((y * z)) + (log(t) * (a - 0.5d0))) - t
    else
        tmp = (a * log(t)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (Math.log(z) <= 230.0) {
		tmp = (Math.log((y * z)) + (Math.log(t) * (a - 0.5))) - t;
	} else {
		tmp = (a * Math.log(t)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if math.log(z) <= 230.0:
		tmp = (math.log((y * z)) + (math.log(t) * (a - 0.5))) - t
	else:
		tmp = (a * math.log(t)) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (log(z) <= 230.0)
		tmp = Float64(Float64(log(Float64(y * z)) + Float64(log(t) * Float64(a - 0.5))) - t);
	else
		tmp = Float64(Float64(a * log(t)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (log(z) <= 230.0)
		tmp = (log((y * z)) + (log(t) * (a - 0.5))) - t;
	else
		tmp = (a * log(t)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[Log[z], $MachinePrecision], 230.0], N[(N[(N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (log.f64 z) < 230

   1. Initial program 99.7%

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

      1. associate-+l- 99.7%

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

      2. +-commutative 99.7%

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

      3. associate--l+ 99.6%

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

      4. sub-neg 99.6%

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

      5. +-commutative 99.6%

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

      6. *-commutative 99.6%

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

      7. distribute-rgt-neg-in 99.6%

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

      8. fma-def 99.6%

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

      9. sub-neg 99.6%

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

      10. +-commutative 99.6%

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

      11. distribute-neg-in 99.6%

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

      12. metadata-eval 99.6%

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

      13. metadata-eval 99.6%

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

      14. unsub-neg 99.6%

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

   3. Simplified 99.6%

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

      1. associate-+r- 99.7%

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

      2. +-commutative 99.7%

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

      3. fma-udef 99.7%

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

      4. associate--r+ 99.7%

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

      5. +-commutative 99.7%

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

      6. sum-log 96.2%

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

   5. Applied egg-rr 96.2%

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

   6. Taylor expanded in x around 0 62.7%

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

   if 230 < (log.f64 z)

   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. Taylor expanded in x around 0 63.0%

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

   3. Taylor expanded in a around inf 80.5%

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

      1. *-commutative 80.5%

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

   5. Simplified 80.5%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log z \leq 230:\\ \;\;\;\;\left(\log \left(y \cdot z\right) + \log t \cdot \left(a - 0.5\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \]

Alternative 6: 69.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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) * (0.5d0 - a))))
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) * (0.5 - a))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate-+l- 99.6%

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

   2. +-commutative 99.6%

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

   3. associate--l+ 99.5%

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

   4. sub-neg 99.5%

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

   5. +-commutative 99.5%

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

   6. *-commutative 99.5%

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

   7. distribute-rgt-neg-in 99.5%

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

   8. fma-def 99.6%

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

   9. sub-neg 99.6%

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

   10. +-commutative 99.6%

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

   11. distribute-neg-in 99.6%

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

   12. metadata-eval 99.6%

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

   13. metadata-eval 99.6%

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

   14. unsub-neg 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in x around 0 70.4%

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

5. Final simplification 70.4%

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

Alternative 7: 69.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[Log[y], $MachinePrecision] + N[(N[Log[z], $MachinePrecision] - N[(N[Log[t], $MachinePrecision] * N[(0.5 - a), $MachinePrecision]), $MachinePrecision]), $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. Taylor expanded in x around 0 61.4%

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

3. Taylor expanded in x around 0 70.5%

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

4. Final simplification 70.5%

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

Alternative 8: 73.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.6 \lor \neg \left(a \leq 0.49\right):\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log t \cdot -0.5 + \log \left(y \cdot z\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -1.6) (not (<= a 0.49)))
   (- (* a (log t)) 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 <= -1.6) || !(a <= 0.49)) {
		tmp = (a * log(t)) - 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 <= (-1.6d0)) .or. (.not. (a <= 0.49d0))) then
        tmp = (a * log(t)) - 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 <= -1.6) || !(a <= 0.49)) {
		tmp = (a * Math.log(t)) - 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 <= -1.6) or not (a <= 0.49):
		tmp = (a * math.log(t)) - 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 <= -1.6) || !(a <= 0.49))
		tmp = Float64(Float64(a * log(t)) - t);
	else
		tmp = Float64(Float64(Float64(log(t) * -0.5) + log(Float64(y * z))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -1.6) || ~((a <= 0.49)))
		tmp = (a * log(t)) - 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, -1.6], N[Not[LessEqual[a, 0.49]], $MachinePrecision]], N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[(N[Log[t], $MachinePrecision] * -0.5), $MachinePrecision] + N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.6000000000000001 or 0.48999999999999999 < a

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 64.2%

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

   3. Taylor expanded in a around inf 98.7%

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

      1. *-commutative 98.7%

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

   5. Simplified 98.7%

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

   if -1.6000000000000001 < a < 0.48999999999999999

   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. Taylor expanded in a around 0 97.5%

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

   3. Taylor expanded in x around 0 65.8%

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

      1. associate-+r+ 65.8%

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

      2. log-prod 50.1%

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

      3. *-commutative 50.1%

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

   5. Simplified 50.1%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.6 \lor \neg \left(a \leq 0.49\right):\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log t \cdot -0.5 + \log \left(y \cdot z\right)\right) - t\\ \end{array} \]

Alternative 9: 87.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 0.135:\\ \;\;\;\;\log \left(\left(x + y\right) \cdot z\right) + \log t \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 0.135)
   (+ (log (* (+ x y) z)) (* (log t) (- a 0.5)))
   (- (* a (log t)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 0.135) {
		tmp = log(((x + y) * z)) + (log(t) * (a - 0.5));
	} else {
		tmp = (a * log(t)) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 0.135d0) then
        tmp = log(((x + y) * z)) + (log(t) * (a - 0.5d0))
    else
        tmp = (a * log(t)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 0.135) {
		tmp = Math.log(((x + y) * z)) + (Math.log(t) * (a - 0.5));
	} else {
		tmp = (a * Math.log(t)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 0.135:
		tmp = math.log(((x + y) * z)) + (math.log(t) * (a - 0.5))
	else:
		tmp = (a * math.log(t)) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 0.135)
		tmp = Float64(log(Float64(Float64(x + y) * z)) + Float64(log(t) * Float64(a - 0.5)));
	else
		tmp = Float64(Float64(a * log(t)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 0.135)
		tmp = log(((x + y) * z)) + (log(t) * (a - 0.5));
	else
		tmp = (a * log(t)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 0.135], 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[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 0.13500000000000001

   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. associate-+l- 99.4%

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

      2. +-commutative 99.4%

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

      3. associate--l+ 99.2%

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

      4. sub-neg 99.2%

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

      5. +-commutative 99.2%

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

      6. *-commutative 99.2%

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

      7. distribute-rgt-neg-in 99.2%

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

      8. fma-def 99.2%

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

      9. sub-neg 99.2%

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

      10. +-commutative 99.2%

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

      11. distribute-neg-in 99.2%

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

      12. metadata-eval 99.2%

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

      13. metadata-eval 99.2%

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

      14. unsub-neg 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in t around 0 98.3%

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

      1. log-prod 80.6%

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

   6. Simplified 80.6%

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

   if 0.13500000000000001 < 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. Taylor expanded in x around 0 59.2%

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

   3. Taylor expanded in a around inf 99.0%

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

      1. *-commutative 99.0%

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

   5. Simplified 99.0%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 0.135:\\ \;\;\;\;\log \left(\left(x + y\right) \cdot z\right) + \log t \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \]

Alternative 10: 72.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.0075 \lor \neg \left(a \leq 0.41\right):\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\log \left(y \cdot \left(z \cdot {t}^{-0.5}\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -0.0075) (not (<= a 0.41)))
   (- (* a (log t)) t)
   (- (log (* y (* z (pow t -0.5)))) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -0.0075) || !(a <= 0.41)) {
		tmp = (a * log(t)) - t;
	} else {
		tmp = log((y * (z * pow(t, -0.5)))) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((a <= (-0.0075d0)) .or. (.not. (a <= 0.41d0))) then
        tmp = (a * log(t)) - t
    else
        tmp = log((y * (z * (t ** (-0.5d0))))) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -0.0075) || !(a <= 0.41)) {
		tmp = (a * Math.log(t)) - t;
	} else {
		tmp = Math.log((y * (z * Math.pow(t, -0.5)))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -0.0075) or not (a <= 0.41):
		tmp = (a * math.log(t)) - t
	else:
		tmp = math.log((y * (z * math.pow(t, -0.5)))) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -0.0075) || !(a <= 0.41))
		tmp = Float64(Float64(a * log(t)) - t);
	else
		tmp = Float64(log(Float64(y * Float64(z * (t ^ -0.5)))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -0.0075) || ~((a <= 0.41)))
		tmp = (a * log(t)) - t;
	else
		tmp = log((y * (z * (t ^ -0.5)))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -0.0075], N[Not[LessEqual[a, 0.41]], $MachinePrecision]], N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[Log[N[(y * N[(z * N[Power[t, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -0.0074999999999999997 or 0.409999999999999976 < a

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 63.7%

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

   3. Taylor expanded in a around inf 98.8%

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

      1. *-commutative 98.8%

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

   5. Simplified 98.8%

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

   if -0.0074999999999999997 < a < 0.409999999999999976

   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. Taylor expanded in a around 0 97.5%

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

      1. associate-+r+ 97.7%

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

      2. log-prod 77.7%

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

      3. add-log-exp 77.7%

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

      4. sum-log 70.5%

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

      5. +-commutative 70.5%

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

      6. *-commutative 70.5%

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

      7. exp-to-pow 70.6%

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

   4. Applied egg-rr 70.6%

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

   5. Taylor expanded in y around inf 46.7%

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

      1. expm1-log1p-u 46.7%

         \[\leadsto \log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left(y \cdot z\right) \cdot {t}^{-0.5}\right)\right)\right)} - t \]

      2. expm1-udef 28.6%

         \[\leadsto \log \color{blue}{\left(e^{\mathsf{log1p}\left(\left(y \cdot z\right) \cdot {t}^{-0.5}\right)} - 1\right)} - t \]

      3. associate-*l* 31.8%

         \[\leadsto \log \left(e^{\mathsf{log1p}\left(\color{blue}{y \cdot \left(z \cdot {t}^{-0.5}\right)}\right)} - 1\right) - t \]

   7. Applied egg-rr 31.8%

      \[\leadsto \log \color{blue}{\left(e^{\mathsf{log1p}\left(y \cdot \left(z \cdot {t}^{-0.5}\right)\right)} - 1\right)} - t \]
   8. Step-by-step derivation

      1. expm1-def 49.1%

         \[\leadsto \log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(y \cdot \left(z \cdot {t}^{-0.5}\right)\right)\right)\right)} - t \]

      2. expm1-log1p 49.1%

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

   9. Simplified 49.1%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.0075 \lor \neg \left(a \leq 0.41\right):\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\log \left(y \cdot \left(z \cdot {t}^{-0.5}\right)\right) - t\\ \end{array} \]

Alternative 11: 78.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1450000000 \lor \neg \left(a \leq 2.2\right):\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\log z + \left(\log \left(x + y\right) - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -1450000000.0) (not (<= a 2.2)))
   (- (* a (log t)) t)
   (+ (log z) (- (log (+ x y)) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -1450000000.0) || !(a <= 2.2)) {
		tmp = (a * log(t)) - t;
	} else {
		tmp = log(z) + (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 <= (-1450000000.0d0)) .or. (.not. (a <= 2.2d0))) then
        tmp = (a * log(t)) - t
    else
        tmp = log(z) + (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 <= -1450000000.0) || !(a <= 2.2)) {
		tmp = (a * Math.log(t)) - t;
	} else {
		tmp = Math.log(z) + (Math.log((x + y)) - t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -1450000000.0) or not (a <= 2.2):
		tmp = (a * math.log(t)) - t
	else:
		tmp = math.log(z) + (math.log((x + y)) - t)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -1450000000.0) || !(a <= 2.2))
		tmp = Float64(Float64(a * log(t)) - t);
	else
		tmp = Float64(log(z) + 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 <= -1450000000.0) || ~((a <= 2.2)))
		tmp = (a * log(t)) - t;
	else
		tmp = log(z) + (log((x + y)) - t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -1450000000.0], N[Not[LessEqual[a, 2.2]], $MachinePrecision]], N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[Log[z], $MachinePrecision] + N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.45e9 or 2.2000000000000002 < a

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 63.9%

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

   3. Taylor expanded in a around inf 98.7%

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

      1. *-commutative 98.7%

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

   5. Simplified 98.7%

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

   if -1.45e9 < a < 2.2000000000000002

   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. +-commutative 99.5%

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

      3. associate--l+ 99.4%

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

      4. sub-neg 99.4%

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

      5. +-commutative 99.4%

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

      6. *-commutative 99.4%

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

      7. distribute-rgt-neg-in 99.4%

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

      8. fma-def 99.4%

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

      9. sub-neg 99.4%

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

      10. +-commutative 99.4%

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

      11. distribute-neg-in 99.4%

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

      12. metadata-eval 99.4%

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

      13. metadata-eval 99.4%

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

      14. unsub-neg 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in t around inf 59.6%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1450000000 \lor \neg \left(a \leq 2.2\right):\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\log z + \left(\log \left(x + y\right) - t\right)\\ \end{array} \]

Alternative 12: 62.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.52 \cdot 10^{+18}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 1.52e+18) (* a (log t)) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 1.52e+18) {
		tmp = a * log(t);
	} else {
		tmp = -t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 1.52d+18) then
        tmp = a * log(t)
    else
        tmp = -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 1.52e+18) {
		tmp = a * Math.log(t);
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 1.52e+18:
		tmp = a * math.log(t)
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 1.52e+18)
		tmp = Float64(a * log(t));
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 1.52e+18)
		tmp = a * log(t);
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 1.52e+18], N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if t < 1.52e18

   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. associate-+l- 99.4%

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

      2. +-commutative 99.4%

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

      3. associate--l+ 99.3%

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

      4. sub-neg 99.3%

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

      5. +-commutative 99.3%

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

      6. *-commutative 99.3%

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

      7. distribute-rgt-neg-in 99.3%

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

      8. fma-def 99.3%

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

      9. sub-neg 99.3%

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

      10. +-commutative 99.3%

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

      11. distribute-neg-in 99.3%

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

      12. metadata-eval 99.3%

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

      13. metadata-eval 99.3%

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

      14. unsub-neg 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in a around inf 57.8%

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

      1. *-commutative 57.8%

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

   6. Simplified 57.8%

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

   if 1.52e18 < 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%

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

      2. +-commutative 99.9%

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

      3. associate--l+ 99.9%

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

      4. sub-neg 99.9%

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

      5. +-commutative 99.9%

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

      6. *-commutative 99.9%

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

      7. distribute-rgt-neg-in 99.9%

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

      8. fma-def 99.9%

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

      9. sub-neg 99.9%

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

      10. +-commutative 99.9%

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

      11. distribute-neg-in 99.9%

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

      12. metadata-eval 99.9%

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

      13. metadata-eval 99.9%

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

      14. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t around inf 82.3%

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

      1. neg-mul-1 82.3%

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

   6. Simplified 82.3%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.52 \cdot 10^{+18}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 13: 74.9% 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. Taylor expanded in x around 0 61.4%

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

3. Taylor expanded in a around inf 77.0%

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

   1. *-commutative 77.0%

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

5. Simplified 77.0%

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

6. Final simplification 77.0%

   \[\leadsto a \cdot \log t - t \]

Alternative 14: 29.1% accurate, 62.6× speedup

Math

\[\begin{array}{l} \\ \frac{x}{y} - t \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(x / y), $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. Taylor expanded in x around 0 61.4%

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

3. Taylor expanded in y around 0 27.6%

   \[\leadsto \color{blue}{\frac{x}{y}} - t \]

4. Final simplification 27.6%

   \[\leadsto \frac{x}{y} - t \]

Alternative 15: 37.9% accurate, 156.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate-+l- 99.6%

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

   2. +-commutative 99.6%

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

   3. associate--l+ 99.5%

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

   4. sub-neg 99.5%

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

   5. +-commutative 99.5%

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

   6. *-commutative 99.5%

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

   7. distribute-rgt-neg-in 99.5%

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

   8. fma-def 99.6%

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

   9. sub-neg 99.6%

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

   10. +-commutative 99.6%

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

   11. distribute-neg-in 99.6%

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

   12. metadata-eval 99.6%

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

   13. metadata-eval 99.6%

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

   14. unsub-neg 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in t around inf 38.5%

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

   1. neg-mul-1 38.5%

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

6. Simplified 38.5%

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

7. Final simplification 38.5%

   \[\leadsto -t \]

Developer target: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023336 
(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))))