Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 15.2s

Alternatives: 11

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

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

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]

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. Final simplification 99.7%

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

Alternative 2: 65.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log z \leq 230:\\ \;\;\;\;\left(\frac{\log t}{\frac{1}{a + -0.5}} + \log \left(y \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (log z) 230.0)
   (- (+ (/ (log t) (/ 1.0 (+ a -0.5))) (log (* y z))) t)
   (+ (- (log z) t) (* a (log t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (log(z) <= 230.0) {
		tmp = ((log(t) / (1.0 / (a + -0.5))) + log((y * z))) - t;
	} else {
		tmp = (log(z) - t) + (a * log(t));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (log(z) <= 230.0d0) then
        tmp = ((log(t) / (1.0d0 / (a + (-0.5d0)))) + log((y * z))) - t
    else
        tmp = (log(z) - t) + (a * log(t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (Math.log(z) <= 230.0) {
		tmp = ((Math.log(t) / (1.0 / (a + -0.5))) + Math.log((y * z))) - t;
	} else {
		tmp = (Math.log(z) - t) + (a * Math.log(t));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if math.log(z) <= 230.0:
		tmp = ((math.log(t) / (1.0 / (a + -0.5))) + math.log((y * z))) - t
	else:
		tmp = (math.log(z) - t) + (a * math.log(t))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (log(z) <= 230.0)
		tmp = Float64(Float64(Float64(log(t) / Float64(1.0 / Float64(a + -0.5))) + log(Float64(y * z))) - t);
	else
		tmp = Float64(Float64(log(z) - t) + Float64(a * log(t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (log(z) <= 230.0)
		tmp = ((log(t) / (1.0 / (a + -0.5))) + log((y * z))) - t;
	else
		tmp = (log(z) - t) + (a * log(t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[Log[z], $MachinePrecision], 230.0], N[(N[(N[(N[Log[t], $MachinePrecision] / N[(1.0 / N[(a + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (log.f64 z) < 230

   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. sub-neg 99.6%

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

      2. metadata-eval 99.6%

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

      3. flip-+ 73.6%

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

      4. associate-*l/ 73.6%

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

      5. fma-neg 73.6%

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

      6. metadata-eval 73.6%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \frac{\mathsf{fma}\left(a, a, -\color{blue}{0.25}\right) \cdot \log t}{a - -0.5} \]

      7. metadata-eval 73.6%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \frac{\mathsf{fma}\left(a, a, \color{blue}{-0.25}\right) \cdot \log t}{a - -0.5} \]

      8. sub-neg 73.6%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \frac{\mathsf{fma}\left(a, a, -0.25\right) \cdot \log t}{\color{blue}{a + \left(--0.5\right)}} \]

      9. metadata-eval 73.6%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \frac{\mathsf{fma}\left(a, a, -0.25\right) \cdot \log t}{a + \color{blue}{0.5}} \]

   3. Applied egg-rr 73.6%

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \color{blue}{\frac{\mathsf{fma}\left(a, a, -0.25\right) \cdot \log t}{a + 0.5}} \]

   4. Taylor expanded in x around 0 47.8%

      \[\leadsto \color{blue}{\left(\log y + \left(\log z + \frac{\log t \cdot \left({a}^{2} - 0.25\right)}{0.5 + a}\right)\right) - t} \]
   5. Step-by-step derivation

      1. associate-+r+ 47.8%

         \[\leadsto \color{blue}{\left(\left(\log y + \log z\right) + \frac{\log t \cdot \left({a}^{2} - 0.25\right)}{0.5 + a}\right)} - t \]

      2. log-prod 43.7%

         \[\leadsto \left(\color{blue}{\log \left(y \cdot z\right)} + \frac{\log t \cdot \left({a}^{2} - 0.25\right)}{0.5 + a}\right) - t \]

      3. *-commutative 43.7%

         \[\leadsto \left(\log \color{blue}{\left(z \cdot y\right)} + \frac{\log t \cdot \left({a}^{2} - 0.25\right)}{0.5 + a}\right) - t \]

      4. *-commutative 43.7%

         \[\leadsto \left(\log \left(z \cdot y\right) + \frac{\color{blue}{\left({a}^{2} - 0.25\right) \cdot \log t}}{0.5 + a}\right) - t \]

      5. unpow2 43.7%

         \[\leadsto \left(\log \left(z \cdot y\right) + \frac{\left(\color{blue}{a \cdot a} - 0.25\right) \cdot \log t}{0.5 + a}\right) - t \]

      6. fma-neg 43.7%

         \[\leadsto \left(\log \left(z \cdot y\right) + \frac{\color{blue}{\mathsf{fma}\left(a, a, -0.25\right)} \cdot \log t}{0.5 + a}\right) - t \]

      7. metadata-eval 43.7%

         \[\leadsto \left(\log \left(z \cdot y\right) + \frac{\mathsf{fma}\left(a, a, \color{blue}{-0.25}\right) \cdot \log t}{0.5 + a}\right) - t \]

      8. +-commutative 43.7%

         \[\leadsto \left(\log \left(z \cdot y\right) + \frac{\mathsf{fma}\left(a, a, -0.25\right) \cdot \log t}{\color{blue}{a + 0.5}}\right) - t \]

      9. associate-*r/ 43.7%

         \[\leadsto \left(\log \left(z \cdot y\right) + \color{blue}{\mathsf{fma}\left(a, a, -0.25\right) \cdot \frac{\log t}{a + 0.5}}\right) - t \]

   6. Simplified 43.7%

      \[\leadsto \color{blue}{\left(\log \left(z \cdot y\right) + \mathsf{fma}\left(a, a, -0.25\right) \cdot \frac{\log t}{a + 0.5}\right) - t} \]
   7. Step-by-step derivation

      1. *-commutative 43.7%

         \[\leadsto \left(\log \left(z \cdot y\right) + \color{blue}{\frac{\log t}{a + 0.5} \cdot \mathsf{fma}\left(a, a, -0.25\right)}\right) - t \]

      2. associate-/r/ 43.7%

         \[\leadsto \left(\log \left(z \cdot y\right) + \color{blue}{\frac{\log t}{\frac{a + 0.5}{\mathsf{fma}\left(a, a, -0.25\right)}}}\right) - t \]

      3. clear-num 43.7%

         \[\leadsto \left(\log \left(z \cdot y\right) + \frac{\log t}{\color{blue}{\frac{1}{\frac{\mathsf{fma}\left(a, a, -0.25\right)}{a + 0.5}}}}\right) - t \]

      4. metadata-eval 43.7%

         \[\leadsto \left(\log \left(z \cdot y\right) + \frac{\log t}{\frac{1}{\frac{\mathsf{fma}\left(a, a, \color{blue}{-0.25}\right)}{a + 0.5}}}\right) - t \]

      5. metadata-eval 43.7%

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

      6. fma-neg 43.7%

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

      7. metadata-eval 43.7%

         \[\leadsto \left(\log \left(z \cdot y\right) + \frac{\log t}{\frac{1}{\frac{a \cdot a - \color{blue}{0.25}}{a + 0.5}}}\right) - t \]

      8. metadata-eval 43.7%

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

      9. *-un-lft-identity 43.7%

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

      10. fma-def 43.7%

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

      11. metadata-eval 43.7%

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

      12. fma-neg 43.7%

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

      13. *-un-lft-identity 43.7%

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

      14. flip-+ 61.4%

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

   8. Applied egg-rr 61.4%

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

   if 230 < (log.f64 z)

   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. +-commutative 99.8%

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

      3. associate-+l+ 99.8%

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

      4. +-commutative 99.8%

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

      5. fma-def 99.8%

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

      6. sub-neg 99.8%

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

      7. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in a around inf 84.4%

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

      1. *-commutative 84.4%

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

   6. Simplified 84.4%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log z \leq 230:\\ \;\;\;\;\left(\frac{\log t}{\frac{1}{a + -0.5}} + \log \left(y \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \end{array} \]

Alternative 3: 86.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (log.f64 z) < 230

   1. Initial program 99.6%

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

      1. associate--l+ 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-+l+ 99.5%

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

      4. +-commutative 99.5%

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

      5. fma-def 99.5%

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

      6. sub-neg 99.5%

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

      7. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

      1. +-commutative 99.5%

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

      2. fma-udef 99.5%

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

      3. metadata-eval 99.5%

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

      4. sub-neg 99.5%

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

      5. associate-+r+ 99.6%

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

      6. associate--l+ 99.6%

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

      7. associate-+r- 99.6%

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

      8. sub-neg 99.6%

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

      9. metadata-eval 99.6%

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

      10. sum-log 93.9%

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

   5. Applied egg-rr 93.9%

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

   if 230 < (log.f64 z)

   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. +-commutative 99.8%

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

      3. associate-+l+ 99.8%

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

      4. +-commutative 99.8%

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

      5. fma-def 99.8%

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

      6. sub-neg 99.8%

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

      7. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in a around inf 84.4%

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

      1. *-commutative 84.4%

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

   6. Simplified 84.4%

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

4. Final simplification 90.7%

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

Alternative 4: 80.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 9:\\ \;\;\;\;\log y + \left(\log z + \left(a - 0.5\right) \cdot \log t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 9.0)
   (+ (log y) (+ (log z) (* (- a 0.5) (log t))))
   (+ (- (log z) t) (* a (log t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 9.0) {
		tmp = log(y) + (log(z) + ((a - 0.5) * log(t)));
	} else {
		tmp = (log(z) - t) + (a * log(t));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 9.0d0) then
        tmp = log(y) + (log(z) + ((a - 0.5d0) * log(t)))
    else
        tmp = (log(z) - t) + (a * log(t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 9.0) {
		tmp = Math.log(y) + (Math.log(z) + ((a - 0.5) * Math.log(t)));
	} else {
		tmp = (Math.log(z) - t) + (a * Math.log(t));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 9.0:
		tmp = math.log(y) + (math.log(z) + ((a - 0.5) * math.log(t)))
	else:
		tmp = (math.log(z) - t) + (a * math.log(t))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 9.0)
		tmp = Float64(log(y) + Float64(log(z) + Float64(Float64(a - 0.5) * log(t))));
	else
		tmp = Float64(Float64(log(z) - t) + Float64(a * log(t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 9.0)
		tmp = log(y) + (log(z) + ((a - 0.5) * log(t)));
	else
		tmp = (log(z) - t) + (a * log(t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 9.0], N[(N[Log[y], $MachinePrecision] + N[(N[Log[z], $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 9

   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%

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

      2. +-commutative 99.4%

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

      3. associate-+l+ 99.3%

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

      4. +-commutative 99.3%

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

      5. fma-def 99.3%

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

      6. sub-neg 99.3%

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

      7. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in t around 0 98.7%

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

      1. associate-+r+ 98.8%

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

      2. sub-neg 98.8%

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

      3. metadata-eval 98.8%

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

      4. *-commutative 98.8%

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

      5. +-commutative 98.8%

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

      6. log-prod 80.3%

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

      7. +-commutative 80.3%

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

      8. fma-def 80.3%

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

      9. +-commutative 80.3%

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

   6. Simplified 80.3%

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

   7. Taylor expanded in x around 0 50.2%

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

   8. Taylor expanded in y around 0 63.8%

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

   if 9 < 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. +-commutative 99.9%

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

      3. associate-+l+ 99.9%

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

      4. +-commutative 99.9%

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

      5. fma-def 99.9%

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

      6. sub-neg 99.9%

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

      7. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in a around inf 98.3%

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

      1. *-commutative 98.3%

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

   6. Simplified 98.3%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9:\\ \;\;\;\;\log y + \left(\log z + \left(a - 0.5\right) \cdot \log t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \end{array} \]

Alternative 5: 68.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[Log[y], $MachinePrecision] + N[(N[Log[z], $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $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. +-commutative 99.7%

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

   3. associate-+l+ 99.6%

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

   4. +-commutative 99.6%

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

   5. fma-def 99.6%

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

   6. sub-neg 99.6%

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

   7. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in x around 0 71.3%

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

5. Final simplification 71.3%

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

Alternative 6: 68.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(y \cdot z\right) + \log t \cdot -0.5\\ t_2 := a \cdot \log t\\ t_3 := t_2 - t\\ \mathbf{if}\;a \leq -2.2 \cdot 10^{-23}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq -5.3 \cdot 10^{-216}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.1 \cdot 10^{-64}:\\ \;\;\;\;\left|t_2\right| - t\\ \mathbf{elif}\;a \leq 0.23:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log (* y z)) (* (log t) -0.5)))
        (t_2 (* a (log t)))
        (t_3 (- t_2 t)))
   (if (<= a -2.2e-23)
     t_3
     (if (<= a -5.3e-216)
       t_1
       (if (<= a 1.1e-64) (- (fabs t_2) t) (if (<= a 0.23) t_1 t_3))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((y * z)) + (log(t) * -0.5);
	double t_2 = a * log(t);
	double t_3 = t_2 - t;
	double tmp;
	if (a <= -2.2e-23) {
		tmp = t_3;
	} else if (a <= -5.3e-216) {
		tmp = t_1;
	} else if (a <= 1.1e-64) {
		tmp = fabs(t_2) - t;
	} else if (a <= 0.23) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	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) :: t_3
    real(8) :: tmp
    t_1 = log((y * z)) + (log(t) * (-0.5d0))
    t_2 = a * log(t)
    t_3 = t_2 - t
    if (a <= (-2.2d-23)) then
        tmp = t_3
    else if (a <= (-5.3d-216)) then
        tmp = t_1
    else if (a <= 1.1d-64) then
        tmp = abs(t_2) - t
    else if (a <= 0.23d0) then
        tmp = t_1
    else
        tmp = t_3
    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.log(t) * -0.5);
	double t_2 = a * Math.log(t);
	double t_3 = t_2 - t;
	double tmp;
	if (a <= -2.2e-23) {
		tmp = t_3;
	} else if (a <= -5.3e-216) {
		tmp = t_1;
	} else if (a <= 1.1e-64) {
		tmp = Math.abs(t_2) - t;
	} else if (a <= 0.23) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = math.log((y * z)) + (math.log(t) * -0.5)
	t_2 = a * math.log(t)
	t_3 = t_2 - t
	tmp = 0
	if a <= -2.2e-23:
		tmp = t_3
	elif a <= -5.3e-216:
		tmp = t_1
	elif a <= 1.1e-64:
		tmp = math.fabs(t_2) - t
	elif a <= 0.23:
		tmp = t_1
	else:
		tmp = t_3
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(Float64(y * z)) + Float64(log(t) * -0.5))
	t_2 = Float64(a * log(t))
	t_3 = Float64(t_2 - t)
	tmp = 0.0
	if (a <= -2.2e-23)
		tmp = t_3;
	elseif (a <= -5.3e-216)
		tmp = t_1;
	elseif (a <= 1.1e-64)
		tmp = Float64(abs(t_2) - t);
	elseif (a <= 0.23)
		tmp = t_1;
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = log((y * z)) + (log(t) * -0.5);
	t_2 = a * log(t);
	t_3 = t_2 - t;
	tmp = 0.0;
	if (a <= -2.2e-23)
		tmp = t_3;
	elseif (a <= -5.3e-216)
		tmp = t_1;
	elseif (a <= 1.1e-64)
		tmp = abs(t_2) - t;
	elseif (a <= 0.23)
		tmp = t_1;
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 - t), $MachinePrecision]}, If[LessEqual[a, -2.2e-23], t$95$3, If[LessEqual[a, -5.3e-216], t$95$1, If[LessEqual[a, 1.1e-64], N[(N[Abs[t$95$2], $MachinePrecision] - t), $MachinePrecision], If[LessEqual[a, 0.23], t$95$1, t$95$3]]]]]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.1999999999999999e-23 or 0.23000000000000001 < 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. +-commutative 99.8%

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

      3. associate-+l+ 99.8%

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

      4. +-commutative 99.8%

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

      5. fma-def 99.8%

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

      6. sub-neg 99.8%

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

      7. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 80.7%

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

   5. Taylor expanded in a around inf 98.3%

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

      1. *-commutative 98.3%

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

   7. Simplified 98.3%

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

   if -2.1999999999999999e-23 < a < -5.29999999999999977e-216 or 1.1e-64 < a < 0.23000000000000001

   1. Initial program 99.3%

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

      1. associate--l+ 99.3%

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

      2. +-commutative 99.3%

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

      3. associate-+l+ 98.8%

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

      4. +-commutative 98.8%

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

      5. fma-def 98.8%

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

      6. sub-neg 98.8%

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

      7. metadata-eval 98.8%

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

   3. Simplified 98.8%

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

   4. Taylor expanded in t around 0 77.5%

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

      1. associate-+r+ 78.0%

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

      2. sub-neg 78.0%

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

      3. metadata-eval 78.0%

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

      4. *-commutative 78.0%

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

      5. +-commutative 78.0%

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

      6. log-prod 68.8%

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

      7. +-commutative 68.8%

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

      8. fma-def 68.8%

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

      9. +-commutative 68.8%

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

   6. Simplified 68.8%

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

   7. Taylor expanded in x around 0 26.4%

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

   8. Taylor expanded in a around 0 26.4%

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

      1. *-commutative 26.4%

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

   10. Simplified 26.4%

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

   if -5.29999999999999977e-216 < a < 1.1e-64

   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. +-commutative 99.5%

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

      3. associate-+l+ 99.6%

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

      4. +-commutative 99.6%

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

      5. fma-def 99.6%

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

      6. sub-neg 99.6%

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

      7. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 63.7%

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

   5. Taylor expanded in a around inf 59.3%

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

      1. *-commutative 59.3%

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

   7. Simplified 59.3%

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

      1. add-sqr-sqrt 38.3%

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

      2. sqrt-unprod 59.4%

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

      3. pow2 59.4%

         \[\leadsto \sqrt{\color{blue}{{\left(\log t \cdot a\right)}^{2}}} - t \]

      4. *-commutative 59.4%

         \[\leadsto \sqrt{{\color{blue}{\left(a \cdot \log t\right)}}^{2}} - t \]

   9. Applied egg-rr 59.4%

      \[\leadsto \color{blue}{\sqrt{{\left(a \cdot \log t\right)}^{2}}} - t \]
   10. Step-by-step derivation

       1. unpow2 59.4%

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

       2. rem-sqrt-square 59.5%

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

       3. *-commutative 59.5%

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

   11. Simplified 59.5%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.2 \cdot 10^{-23}:\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{elif}\;a \leq -5.3 \cdot 10^{-216}:\\ \;\;\;\;\log \left(y \cdot z\right) + \log t \cdot -0.5\\ \mathbf{elif}\;a \leq 1.1 \cdot 10^{-64}:\\ \;\;\;\;\left|a \cdot \log t\right| - t\\ \mathbf{elif}\;a \leq 0.23:\\ \;\;\;\;\log \left(y \cdot z\right) + \log t \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \]

Alternative 7: 70.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.4 \cdot 10^{-18} \lor \neg \left(a \leq 2.6 \cdot 10^{-8}\right):\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(y \cdot z\right) \cdot {t}^{-0.5}\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -5.4e-18) (not (<= a 2.6e-8)))
   (+ (- (log z) t) (* a (log 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 <= -5.4e-18) || !(a <= 2.6e-8)) {
		tmp = (log(z) - t) + (a * log(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 <= (-5.4d-18)) .or. (.not. (a <= 2.6d-8))) then
        tmp = (log(z) - t) + (a * log(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 <= -5.4e-18) || !(a <= 2.6e-8)) {
		tmp = (Math.log(z) - t) + (a * Math.log(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 <= -5.4e-18) or not (a <= 2.6e-8):
		tmp = (math.log(z) - t) + (a * math.log(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 <= -5.4e-18) || !(a <= 2.6e-8))
		tmp = Float64(Float64(log(z) - t) + Float64(a * log(t)));
	else
		tmp = Float64(log(Float64(Float64(y * z) * (t ^ -0.5))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -5.4e-18) || ~((a <= 2.6e-8)))
		tmp = (log(z) - t) + (a * log(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, -5.4e-18], N[Not[LessEqual[a, 2.6e-8]], $MachinePrecision]], N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Log[N[(N[(y * z), $MachinePrecision] * N[Power[t, -0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5.39999999999999977e-18 or 2.6000000000000001e-8 < 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. +-commutative 99.8%

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

      3. associate-+l+ 99.8%

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

      4. +-commutative 99.8%

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

      5. fma-def 99.8%

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

      6. sub-neg 99.8%

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

      7. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in a around inf 97.3%

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

      1. *-commutative 97.3%

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

   6. Simplified 97.3%

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

   if -5.39999999999999977e-18 < a < 2.6000000000000001e-8

   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%

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

      2. +-commutative 99.4%

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

      3. associate-+l+ 99.2%

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

      4. +-commutative 99.2%

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

      5. fma-def 99.2%

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

      6. sub-neg 99.2%

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

      7. metadata-eval 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in x around 0 56.9%

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

   5. Taylor expanded in a around 0 56.9%

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

      1. associate-+r+ 56.9%

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

      2. log-prod 45.3%

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

      3. *-commutative 45.3%

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

      4. *-commutative 45.3%

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

   7. Simplified 45.3%

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

      1. add-log-exp 45.3%

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

      2. sum-log 41.3%

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

      3. exp-to-pow 41.3%

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

   9. Applied egg-rr 41.3%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.4 \cdot 10^{-18} \lor \neg \left(a \leq 2.6 \cdot 10^{-8}\right):\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(y \cdot z\right) \cdot {t}^{-0.5}\right) - t\\ \end{array} \]

Alternative 8: 73.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 6.8 \cdot 10^{-11}:\\ \;\;\;\;\left(a - 0.5\right) \cdot \log t + \log \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 6.8e-11)
   (+ (* (- a 0.5) (log t)) (log (* y z)))
   (- (* a (log t)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 6.8e-11) {
		tmp = ((a - 0.5) * log(t)) + log((y * z));
	} 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 <= 6.8d-11) then
        tmp = ((a - 0.5d0) * log(t)) + log((y * z))
    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 <= 6.8e-11) {
		tmp = ((a - 0.5) * Math.log(t)) + Math.log((y * z));
	} else {
		tmp = (a * Math.log(t)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 6.8e-11:
		tmp = ((a - 0.5) * math.log(t)) + math.log((y * z))
	else:
		tmp = (a * math.log(t)) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 6.8e-11)
		tmp = Float64(Float64(Float64(a - 0.5) * log(t)) + log(Float64(y * z)));
	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 <= 6.8e-11)
		tmp = ((a - 0.5) * log(t)) + log((y * z));
	else
		tmp = (a * log(t)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 6.8e-11], N[(N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision] + N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 6.7999999999999998e-11

   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%

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

      2. +-commutative 99.4%

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

      3. associate-+l+ 99.3%

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

      4. +-commutative 99.3%

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

      5. fma-def 99.3%

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

      6. sub-neg 99.3%

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

      7. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in t around 0 99.3%

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

      1. associate-+r+ 99.4%

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

      2. sub-neg 99.4%

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

      3. metadata-eval 99.4%

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

      4. *-commutative 99.4%

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

      5. +-commutative 99.4%

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

      6. log-prod 81.5%

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

      7. +-commutative 81.5%

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

      8. fma-def 81.5%

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

      9. +-commutative 81.5%

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

   6. Simplified 81.5%

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

   7. Taylor expanded in x around 0 50.6%

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

   if 6.7999999999999998e-11 < 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. +-commutative 99.9%

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

      3. associate-+l+ 99.9%

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

      4. +-commutative 99.9%

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

      5. fma-def 99.9%

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

      6. sub-neg 99.9%

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

      7. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 78.6%

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

   5. Taylor expanded in a around inf 96.9%

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

      1. *-commutative 96.9%

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

   7. Simplified 96.9%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.8 \cdot 10^{-11}:\\ \;\;\;\;\left(a - 0.5\right) \cdot \log t + \log \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \]

Alternative 9: 77.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. associate--l+ 99.7%

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

   2. +-commutative 99.7%

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

   3. associate-+l+ 99.6%

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

   4. +-commutative 99.6%

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

   5. fma-def 99.6%

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

   6. sub-neg 99.6%

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

   7. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in a around inf 80.0%

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

   1. *-commutative 80.0%

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

6. Simplified 80.0%

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

7. Final simplification 80.0%

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

Alternative 10: 75.3% 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 99.7%

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

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

   3. associate-+l+ 99.6%

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

   4. +-commutative 99.6%

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

   5. fma-def 99.6%

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

   6. sub-neg 99.6%

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

   7. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in x around 0 71.3%

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

5. Taylor expanded in a around inf 78.3%

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

   1. *-commutative 78.3%

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

7. Simplified 78.3%

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

8. Final simplification 78.3%

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

Alternative 11: 37.6% accurate, 156.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

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

   3. associate-+l+ 99.6%

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

   4. +-commutative 99.6%

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

   5. fma-def 99.6%

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

   6. sub-neg 99.6%

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

   7. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in t around inf 36.1%

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

   1. neg-mul-1 36.1%

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

6. Simplified 36.1%

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

7. Final simplification 36.1%

   \[\leadsto -t \]

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 2023333 
(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))))