Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 19.1s

Alternatives: 18

Speedup: 0.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. +-commutative 99.7%

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

   2. fma-def 99.7%

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

   3. sub-neg 99.7%

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

   4. metadata-eval 99.7%

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

   5. associate--l+ 99.7%

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

3. Simplified 99.7%

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

5. Final simplification 99.7%

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

Alternative 2: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 210:\\ \;\;\;\;\log z + \left(\log \left(x + y\right) + \log t \cdot \left(a - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 210.0)
   (+ (log z) (+ (log (+ x y)) (* (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 <= 210.0) {
		tmp = log(z) + (log((x + y)) + (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 <= 210.0d0) then
        tmp = log(z) + (log((x + y)) + (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 <= 210.0) {
		tmp = Math.log(z) + (Math.log((x + y)) + (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 <= 210.0:
		tmp = math.log(z) + (math.log((x + y)) + (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 <= 210.0)
		tmp = Float64(log(z) + Float64(log(Float64(x + y)) + 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 <= 210.0)
		tmp = log(z) + (log((x + y)) + (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, 210.0], N[(N[Log[z], $MachinePrecision] + N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 210

   1. Initial program 99.5%

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

      1. associate--l+ 99.5%

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

      2. sub-neg 99.5%

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

      3. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in t around 0 98.3%

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

   if 210 < 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. associate--l+ 99.9%

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

      3. sub-neg 99.9%

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

      4. +-commutative 99.9%

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

      5. *-commutative 99.9%

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

      6. distribute-rgt-neg-in 99.9%

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

      7. fma-udef 99.9%

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

      8. sub-neg 99.9%

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

      9. +-commutative 99.9%

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

      10. distribute-neg-in 99.9%

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

      11. metadata-eval 99.9%

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

      12. metadata-eval 99.9%

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

      13. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

      1. associate-+r- 99.9%

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

      2. fma-udef 99.9%

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

      3. associate--r+ 99.9%

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

      4. +-commutative 99.9%

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

      5. sum-log 81.4%

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

   6. Applied egg-rr 81.4%

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

   7. Taylor expanded in x around inf 71.8%

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

      1. mul-1-neg 71.8%

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

      2. log-rec 71.8%

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

      3. remove-double-neg 71.8%

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

      4. log-prod 55.3%

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

      5. *-commutative 55.3%

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

   9. Simplified 55.3%

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

   10. Taylor expanded in a around inf 98.8%

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

       1. *-commutative 98.8%

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

   12. Simplified 98.8%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 210:\\ \;\;\;\;\log z + \left(\log \left(x + y\right) + \log t \cdot \left(a - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 210:\\ \;\;\;\;\left(\log z + \log y\right) + \log t \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 210.0)
   (+ (+ (log z) (log y)) (* (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 <= 210.0) {
		tmp = (log(z) + log(y)) + (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 <= 210.0d0) then
        tmp = (log(z) + log(y)) + (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 <= 210.0) {
		tmp = (Math.log(z) + Math.log(y)) + (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 <= 210.0:
		tmp = (math.log(z) + math.log(y)) + (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 <= 210.0)
		tmp = Float64(Float64(log(z) + log(y)) + 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 <= 210.0)
		tmp = (log(z) + log(y)) + (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, 210.0], N[(N[(N[Log[z], $MachinePrecision] + N[Log[y], $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 < 210

   1. Initial program 99.5%

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

      1. associate-+l- 99.5%

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

      2. associate--l+ 99.5%

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

      3. sub-neg 99.5%

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

      4. +-commutative 99.5%

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

      5. *-commutative 99.5%

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

      6. distribute-rgt-neg-in 99.5%

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

      7. fma-udef 99.5%

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

      8. sub-neg 99.5%

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

      9. +-commutative 99.5%

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

      10. distribute-neg-in 99.5%

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

      11. metadata-eval 99.5%

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

      12. metadata-eval 99.5%

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

      13. unsub-neg 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around 0 70.6%

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

   6. Taylor expanded in t around 0 70.1%

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

   if 210 < 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. associate--l+ 99.9%

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

      3. sub-neg 99.9%

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

      4. +-commutative 99.9%

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

      5. *-commutative 99.9%

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

      6. distribute-rgt-neg-in 99.9%

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

      7. fma-udef 99.9%

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

      8. sub-neg 99.9%

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

      9. +-commutative 99.9%

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

      10. distribute-neg-in 99.9%

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

      11. metadata-eval 99.9%

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

      12. metadata-eval 99.9%

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

      13. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

      1. associate-+r- 99.9%

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

      2. fma-udef 99.9%

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

      3. associate--r+ 99.9%

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

      4. +-commutative 99.9%

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

      5. sum-log 81.4%

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

   6. Applied egg-rr 81.4%

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

   7. Taylor expanded in x around inf 71.8%

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

      1. mul-1-neg 71.8%

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

      2. log-rec 71.8%

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

      3. remove-double-neg 71.8%

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

      4. log-prod 55.3%

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

      5. *-commutative 55.3%

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

   9. Simplified 55.3%

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

   10. Taylor expanded in a around inf 98.8%

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

       1. *-commutative 98.8%

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

   12. Simplified 98.8%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 210:\\ \;\;\;\;\left(\log z + \log y\right) + \log t \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. associate--l+ 99.7%

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

   2. sub-neg 99.7%

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

   3. metadata-eval 99.7%

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

3. Simplified 99.7%

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

5. Final simplification 99.7%

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

Alternative 5: 69.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. associate-+l- 99.7%

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

   2. associate--l+ 99.7%

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

   3. sub-neg 99.7%

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

   4. +-commutative 99.7%

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

   5. *-commutative 99.7%

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

   6. distribute-rgt-neg-in 99.7%

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

   7. fma-udef 99.7%

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

   8. sub-neg 99.7%

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

   9. +-commutative 99.7%

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

   10. distribute-neg-in 99.7%

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

   11. metadata-eval 99.7%

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

   12. metadata-eval 99.7%

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

   13. unsub-neg 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in x around 0 72.4%

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

6. Final simplification 72.4%

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

Alternative 6: 77.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \log t - t\\ \mathbf{if}\;a \leq -1.4:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 7.2 \cdot 10^{-72}:\\ \;\;\;\;\log \left(x + y\right) + \left(\log z - t\right)\\ \mathbf{elif}\;a \leq 6.2 \cdot 10^{-33}:\\ \;\;\;\;\log \left(\frac{z}{\frac{{t}^{\left(0.5 - a\right)}}{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* a (log t)) t)))
   (if (<= a -1.4)
     t_1
     (if (<= a 7.2e-72)
       (+ (log (+ x y)) (- (log z) t))
       (if (<= a 6.2e-33) (log (/ z (/ (pow t (- 0.5 a)) y))) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (a * log(t)) - t;
	double tmp;
	if (a <= -1.4) {
		tmp = t_1;
	} else if (a <= 7.2e-72) {
		tmp = log((x + y)) + (log(z) - t);
	} else if (a <= 6.2e-33) {
		tmp = log((z / (pow(t, (0.5 - a)) / y)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (a * log(t)) - t
    if (a <= (-1.4d0)) then
        tmp = t_1
    else if (a <= 7.2d-72) then
        tmp = log((x + y)) + (log(z) - t)
    else if (a <= 6.2d-33) then
        tmp = log((z / ((t ** (0.5d0 - a)) / y)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (a * Math.log(t)) - t;
	double tmp;
	if (a <= -1.4) {
		tmp = t_1;
	} else if (a <= 7.2e-72) {
		tmp = Math.log((x + y)) + (Math.log(z) - t);
	} else if (a <= 6.2e-33) {
		tmp = Math.log((z / (Math.pow(t, (0.5 - a)) / y)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (a * math.log(t)) - t
	tmp = 0
	if a <= -1.4:
		tmp = t_1
	elif a <= 7.2e-72:
		tmp = math.log((x + y)) + (math.log(z) - t)
	elif a <= 6.2e-33:
		tmp = math.log((z / (math.pow(t, (0.5 - a)) / y)))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(a * log(t)) - t)
	tmp = 0.0
	if (a <= -1.4)
		tmp = t_1;
	elseif (a <= 7.2e-72)
		tmp = Float64(log(Float64(x + y)) + Float64(log(z) - t));
	elseif (a <= 6.2e-33)
		tmp = log(Float64(z / Float64((t ^ Float64(0.5 - a)) / y)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (a * log(t)) - t;
	tmp = 0.0;
	if (a <= -1.4)
		tmp = t_1;
	elseif (a <= 7.2e-72)
		tmp = log((x + y)) + (log(z) - t);
	elseif (a <= 6.2e-33)
		tmp = log((z / ((t ^ (0.5 - a)) / y)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[a, -1.4], t$95$1, If[LessEqual[a, 7.2e-72], N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 6.2e-33], N[Log[N[(z / N[(N[Power[t, N[(0.5 - a), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.3999999999999999 or 6.19999999999999994e-33 < a

   1. Initial program 99.7%

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

      1. associate-+l- 99.7%

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

      2. associate--l+ 99.6%

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

      3. sub-neg 99.6%

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

      4. +-commutative 99.6%

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

      5. *-commutative 99.6%

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

      6. distribute-rgt-neg-in 99.6%

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

      7. fma-udef 99.6%

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

      8. sub-neg 99.6%

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

      9. +-commutative 99.6%

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

      10. distribute-neg-in 99.6%

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

      11. metadata-eval 99.6%

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

      12. metadata-eval 99.6%

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

      13. unsub-neg 99.6%

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

   3. Simplified 99.6%

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

      1. associate-+r- 99.7%

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

      2. fma-udef 99.7%

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

      3. associate--r+ 99.7%

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

      4. +-commutative 99.7%

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

      5. sum-log 80.9%

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

   6. Applied egg-rr 80.9%

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

   7. Taylor expanded in x around inf 72.0%

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

      1. mul-1-neg 72.0%

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

      2. log-rec 72.0%

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

      3. remove-double-neg 72.0%

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

      4. log-prod 56.2%

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

      5. *-commutative 56.2%

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

   9. Simplified 56.2%

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

   10. Taylor expanded in a around inf 98.2%

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

       1. *-commutative 98.2%

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

   12. Simplified 98.2%

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

   if -1.3999999999999999 < a < 7.2e-72

   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%

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

      2. associate--l+ 99.8%

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

      3. sub-neg 99.8%

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

      4. +-commutative 99.8%

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

      5. *-commutative 99.8%

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

      6. distribute-rgt-neg-in 99.8%

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

      7. fma-udef 99.8%

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

      8. sub-neg 99.8%

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

      9. +-commutative 99.8%

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

      10. distribute-neg-in 99.8%

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

      11. metadata-eval 99.8%

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

      12. metadata-eval 99.8%

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

      13. unsub-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t around inf 60.0%

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

   if 7.2e-72 < a < 6.19999999999999994e-33

   1. Initial program 98.9%

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

      1. associate-+l- 98.9%

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

      2. associate--l+ 99.2%

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

      3. sub-neg 99.2%

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

      4. +-commutative 99.2%

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

      5. *-commutative 99.2%

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

      6. distribute-rgt-neg-in 99.2%

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

      7. fma-udef 99.2%

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

      8. sub-neg 99.2%

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

      9. +-commutative 99.2%

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

      10. distribute-neg-in 99.2%

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

      11. metadata-eval 99.2%

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

      12. metadata-eval 99.2%

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

      13. unsub-neg 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in x around 0 90.1%

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

   6. Taylor expanded in t around 0 80.2%

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

   7. Taylor expanded in y around inf 80.2%

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

      1. *-commutative 80.2%

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

      2. mul-1-neg 80.2%

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

      3. log-rec 80.2%

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

      4. remove-double-neg 80.2%

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

      5. log-prod 61.1%

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

      6. log-pow 61.1%

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

      7. log-div 61.1%

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

      8. associate-/l* 61.1%

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

   9. Simplified 61.1%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.4:\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{elif}\;a \leq 7.2 \cdot 10^{-72}:\\ \;\;\;\;\log \left(x + y\right) + \left(\log z - t\right)\\ \mathbf{elif}\;a \leq 6.2 \cdot 10^{-33}:\\ \;\;\;\;\log \left(\frac{z}{\frac{{t}^{\left(0.5 - a\right)}}{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \log t - t\\ \mathbf{if}\;a \leq -1.85:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 6.5 \cdot 10^{-72}:\\ \;\;\;\;\log \left(x + y\right) + \left(\log z - t\right)\\ \mathbf{elif}\;a \leq 5.2 \cdot 10^{-33}:\\ \;\;\;\;\log \left(\frac{y \cdot z}{{t}^{\left(0.5 - a\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* a (log t)) t)))
   (if (<= a -1.85)
     t_1
     (if (<= a 6.5e-72)
       (+ (log (+ x y)) (- (log z) t))
       (if (<= a 5.2e-33) (log (/ (* y z) (pow t (- 0.5 a)))) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (a * log(t)) - t;
	double tmp;
	if (a <= -1.85) {
		tmp = t_1;
	} else if (a <= 6.5e-72) {
		tmp = log((x + y)) + (log(z) - t);
	} else if (a <= 5.2e-33) {
		tmp = log(((y * z) / pow(t, (0.5 - a))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (a * log(t)) - t
    if (a <= (-1.85d0)) then
        tmp = t_1
    else if (a <= 6.5d-72) then
        tmp = log((x + y)) + (log(z) - t)
    else if (a <= 5.2d-33) then
        tmp = log(((y * z) / (t ** (0.5d0 - a))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (a * Math.log(t)) - t;
	double tmp;
	if (a <= -1.85) {
		tmp = t_1;
	} else if (a <= 6.5e-72) {
		tmp = Math.log((x + y)) + (Math.log(z) - t);
	} else if (a <= 5.2e-33) {
		tmp = Math.log(((y * z) / Math.pow(t, (0.5 - a))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (a * math.log(t)) - t
	tmp = 0
	if a <= -1.85:
		tmp = t_1
	elif a <= 6.5e-72:
		tmp = math.log((x + y)) + (math.log(z) - t)
	elif a <= 5.2e-33:
		tmp = math.log(((y * z) / math.pow(t, (0.5 - a))))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(a * log(t)) - t)
	tmp = 0.0
	if (a <= -1.85)
		tmp = t_1;
	elseif (a <= 6.5e-72)
		tmp = Float64(log(Float64(x + y)) + Float64(log(z) - t));
	elseif (a <= 5.2e-33)
		tmp = log(Float64(Float64(y * z) / (t ^ Float64(0.5 - a))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (a * log(t)) - t;
	tmp = 0.0;
	if (a <= -1.85)
		tmp = t_1;
	elseif (a <= 6.5e-72)
		tmp = log((x + y)) + (log(z) - t);
	elseif (a <= 5.2e-33)
		tmp = log(((y * z) / (t ^ (0.5 - a))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[a, -1.85], t$95$1, If[LessEqual[a, 6.5e-72], N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 5.2e-33], N[Log[N[(N[(y * z), $MachinePrecision] / N[Power[t, N[(0.5 - a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.8500000000000001 or 5.19999999999999988e-33 < a

   1. Initial program 99.7%

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

      1. associate-+l- 99.7%

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

      2. associate--l+ 99.6%

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

      3. sub-neg 99.6%

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

      4. +-commutative 99.6%

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

      5. *-commutative 99.6%

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

      6. distribute-rgt-neg-in 99.6%

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

      7. fma-udef 99.6%

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

      8. sub-neg 99.6%

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

      9. +-commutative 99.6%

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

      10. distribute-neg-in 99.6%

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

      11. metadata-eval 99.6%

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

      12. metadata-eval 99.6%

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

      13. unsub-neg 99.6%

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

   3. Simplified 99.6%

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

      1. associate-+r- 99.7%

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

      2. fma-udef 99.7%

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

      3. associate--r+ 99.7%

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

      4. +-commutative 99.7%

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

      5. sum-log 80.9%

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

   6. Applied egg-rr 80.9%

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

   7. Taylor expanded in x around inf 72.0%

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

      1. mul-1-neg 72.0%

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

      2. log-rec 72.0%

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

      3. remove-double-neg 72.0%

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

      4. log-prod 56.2%

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

      5. *-commutative 56.2%

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

   9. Simplified 56.2%

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

   10. Taylor expanded in a around inf 98.2%

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

       1. *-commutative 98.2%

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

   12. Simplified 98.2%

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

   if -1.8500000000000001 < a < 6.4999999999999997e-72

   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%

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

      2. associate--l+ 99.8%

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

      3. sub-neg 99.8%

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

      4. +-commutative 99.8%

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

      5. *-commutative 99.8%

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

      6. distribute-rgt-neg-in 99.8%

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

      7. fma-udef 99.8%

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

      8. sub-neg 99.8%

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

      9. +-commutative 99.8%

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

      10. distribute-neg-in 99.8%

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

      11. metadata-eval 99.8%

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

      12. metadata-eval 99.8%

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

      13. unsub-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t around inf 60.0%

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

   if 6.4999999999999997e-72 < a < 5.19999999999999988e-33

   1. Initial program 98.9%

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

      1. associate-+l- 98.9%

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

      2. associate--l+ 99.2%

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

      3. sub-neg 99.2%

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

      4. +-commutative 99.2%

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

      5. *-commutative 99.2%

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

      6. distribute-rgt-neg-in 99.2%

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

      7. fma-udef 99.2%

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

      8. sub-neg 99.2%

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

      9. +-commutative 99.2%

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

      10. distribute-neg-in 99.2%

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

      11. metadata-eval 99.2%

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

      12. metadata-eval 99.2%

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

      13. unsub-neg 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in x around 0 90.1%

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

   6. Taylor expanded in t around 0 80.2%

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

      1. +-commutative 80.2%

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

      2. log-prod 61.1%

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

      3. add-log-exp 61.1%

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

      4. diff-log 61.1%

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

      5. exp-to-pow 61.1%

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

   8. Applied egg-rr 61.1%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.85:\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{elif}\;a \leq 6.5 \cdot 10^{-72}:\\ \;\;\;\;\log \left(x + y\right) + \left(\log z - t\right)\\ \mathbf{elif}\;a \leq 5.2 \cdot 10^{-33}:\\ \;\;\;\;\log \left(\frac{y \cdot z}{{t}^{\left(0.5 - a\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 86.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -880000000000 \lor \neg \left(a \leq 2.1 \cdot 10^{-32}\right):\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(\left(x + y\right) \cdot z\right) + \log t \cdot \left(a - 0.5\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -880000000000.0) (not (<= a 2.1e-32)))
   (- (* a (log t)) t)
   (- (+ (log (* (+ x y) z)) (* (log t) (- a 0.5))) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -880000000000.0) || !(a <= 2.1e-32)) {
		tmp = (a * log(t)) - t;
	} else {
		tmp = (log(((x + y) * z)) + (log(t) * (a - 0.5))) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((a <= (-880000000000.0d0)) .or. (.not. (a <= 2.1d-32))) then
        tmp = (a * log(t)) - t
    else
        tmp = (log(((x + y) * z)) + (log(t) * (a - 0.5d0))) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -880000000000.0) || !(a <= 2.1e-32)) {
		tmp = (a * Math.log(t)) - t;
	} else {
		tmp = (Math.log(((x + y) * z)) + (Math.log(t) * (a - 0.5))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -880000000000.0) or not (a <= 2.1e-32):
		tmp = (a * math.log(t)) - t
	else:
		tmp = (math.log(((x + y) * z)) + (math.log(t) * (a - 0.5))) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -880000000000.0) || !(a <= 2.1e-32))
		tmp = Float64(Float64(a * log(t)) - t);
	else
		tmp = Float64(Float64(log(Float64(Float64(x + y) * z)) + Float64(log(t) * Float64(a - 0.5))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -880000000000.0) || ~((a <= 2.1e-32)))
		tmp = (a * log(t)) - t;
	else
		tmp = (log(((x + y) * z)) + (log(t) * (a - 0.5))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -880000000000.0], N[Not[LessEqual[a, 2.1e-32]], $MachinePrecision]], N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], 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]]

Derivation

1. Split input into 2 regimes

2. if a < -8.8e11 or 2.0999999999999999e-32 < a

   1. Initial program 99.7%

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

      1. associate-+l- 99.7%

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

      2. associate--l+ 99.7%

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

      3. sub-neg 99.7%

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

      4. +-commutative 99.7%

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

      5. *-commutative 99.7%

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

      6. distribute-rgt-neg-in 99.7%

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

      7. fma-udef 99.7%

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

      8. sub-neg 99.7%

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

      9. +-commutative 99.7%

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

      10. distribute-neg-in 99.7%

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

      11. metadata-eval 99.7%

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

      12. metadata-eval 99.7%

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

      13. unsub-neg 99.7%

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

   3. Simplified 99.7%

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

      1. associate-+r- 99.7%

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

      2. fma-udef 99.7%

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

      3. associate--r+ 99.7%

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

      4. +-commutative 99.7%

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

      5. sum-log 80.7%

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

   6. Applied egg-rr 80.7%

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

   7. Taylor expanded in x around inf 72.3%

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

      1. mul-1-neg 72.3%

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

      2. log-rec 72.3%

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

      3. remove-double-neg 72.3%

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

      4. log-prod 56.2%

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

      5. *-commutative 56.2%

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

   9. Simplified 56.2%

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

   10. Taylor expanded in a around inf 98.9%

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

       1. *-commutative 98.9%

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

   12. Simplified 98.9%

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

   if -8.8e11 < a < 2.0999999999999999e-32

   1. Initial program 99.7%

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

      1. associate-+l- 99.7%

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

      2. associate--l+ 99.7%

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

      3. sub-neg 99.7%

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

      4. +-commutative 99.7%

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

      5. *-commutative 99.7%

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

      6. distribute-rgt-neg-in 99.7%

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

      7. fma-udef 99.7%

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

      8. sub-neg 99.7%

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

      9. +-commutative 99.7%

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

      10. distribute-neg-in 99.7%

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

      11. metadata-eval 99.7%

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

      12. metadata-eval 99.7%

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

      13. unsub-neg 99.7%

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

   3. Simplified 99.7%

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

      1. associate-+r- 99.7%

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

      2. fma-udef 99.7%

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

      3. associate--r+ 99.7%

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

      4. +-commutative 99.7%

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

      5. sum-log 80.1%

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

   6. Applied egg-rr 80.1%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -880000000000 \lor \neg \left(a \leq 2.1 \cdot 10^{-32}\right):\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(\left(x + y\right) \cdot z\right) + \log t \cdot \left(a - 0.5\right)\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \log t - t\\ \mathbf{if}\;a \leq -1.4:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 1.45 \cdot 10^{-78}:\\ \;\;\;\;\left(\log z + \log y\right) - t\\ \mathbf{elif}\;a \leq 6.2 \cdot 10^{-33}:\\ \;\;\;\;\log \left(y \cdot z\right) + -0.5 \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* a (log t)) t)))
   (if (<= a -1.4)
     t_1
     (if (<= a 1.45e-78)
       (- (+ (log z) (log y)) t)
       (if (<= a 6.2e-33) (+ (log (* y z)) (* -0.5 (log t))) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (a * log(t)) - t;
	double tmp;
	if (a <= -1.4) {
		tmp = t_1;
	} else if (a <= 1.45e-78) {
		tmp = (log(z) + log(y)) - t;
	} else if (a <= 6.2e-33) {
		tmp = log((y * z)) + (-0.5 * log(t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (a * log(t)) - t
    if (a <= (-1.4d0)) then
        tmp = t_1
    else if (a <= 1.45d-78) then
        tmp = (log(z) + log(y)) - t
    else if (a <= 6.2d-33) then
        tmp = log((y * z)) + ((-0.5d0) * log(t))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (a * Math.log(t)) - t;
	double tmp;
	if (a <= -1.4) {
		tmp = t_1;
	} else if (a <= 1.45e-78) {
		tmp = (Math.log(z) + Math.log(y)) - t;
	} else if (a <= 6.2e-33) {
		tmp = Math.log((y * z)) + (-0.5 * Math.log(t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (a * math.log(t)) - t
	tmp = 0
	if a <= -1.4:
		tmp = t_1
	elif a <= 1.45e-78:
		tmp = (math.log(z) + math.log(y)) - t
	elif a <= 6.2e-33:
		tmp = math.log((y * z)) + (-0.5 * math.log(t))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(a * log(t)) - t)
	tmp = 0.0
	if (a <= -1.4)
		tmp = t_1;
	elseif (a <= 1.45e-78)
		tmp = Float64(Float64(log(z) + log(y)) - t);
	elseif (a <= 6.2e-33)
		tmp = Float64(log(Float64(y * z)) + Float64(-0.5 * log(t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (a * log(t)) - t;
	tmp = 0.0;
	if (a <= -1.4)
		tmp = t_1;
	elseif (a <= 1.45e-78)
		tmp = (log(z) + log(y)) - t;
	elseif (a <= 6.2e-33)
		tmp = log((y * z)) + (-0.5 * log(t));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[a, -1.4], t$95$1, If[LessEqual[a, 1.45e-78], N[(N[(N[Log[z], $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[a, 6.2e-33], N[(N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision] + N[(-0.5 * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.3999999999999999 or 6.19999999999999994e-33 < a

   1. Initial program 99.7%

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

      1. associate-+l- 99.7%

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

      2. associate--l+ 99.6%

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

      3. sub-neg 99.6%

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

      4. +-commutative 99.6%

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

      5. *-commutative 99.6%

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

      6. distribute-rgt-neg-in 99.6%

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

      7. fma-udef 99.6%

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

      8. sub-neg 99.6%

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

      9. +-commutative 99.6%

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

      10. distribute-neg-in 99.6%

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

      11. metadata-eval 99.6%

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

      12. metadata-eval 99.6%

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

      13. unsub-neg 99.6%

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

   3. Simplified 99.6%

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

      1. associate-+r- 99.7%

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

      2. fma-udef 99.7%

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

      3. associate--r+ 99.7%

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

      4. +-commutative 99.7%

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

      5. sum-log 80.9%

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

   6. Applied egg-rr 80.9%

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

   7. Taylor expanded in x around inf 72.0%

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

      1. mul-1-neg 72.0%

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

      2. log-rec 72.0%

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

      3. remove-double-neg 72.0%

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

      4. log-prod 56.2%

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

      5. *-commutative 56.2%

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

   9. Simplified 56.2%

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

   10. Taylor expanded in a around inf 98.2%

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

       1. *-commutative 98.2%

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

   12. Simplified 98.2%

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

   if -1.3999999999999999 < a < 1.45e-78

   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%

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

      2. associate--l+ 99.8%

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

      3. sub-neg 99.8%

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

      4. +-commutative 99.8%

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

      5. *-commutative 99.8%

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

      6. distribute-rgt-neg-in 99.8%

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

      7. fma-udef 99.8%

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

      8. sub-neg 99.8%

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

      9. +-commutative 99.8%

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

      10. distribute-neg-in 99.8%

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

      11. metadata-eval 99.8%

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

      12. metadata-eval 99.8%

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

      13. unsub-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 64.4%

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

   6. Taylor expanded in t around inf 43.9%

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

   if 1.45e-78 < a < 6.19999999999999994e-33

   1. Initial program 98.9%

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

      1. associate-+l- 98.9%

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

      2. associate--l+ 99.2%

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

      3. sub-neg 99.2%

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

      4. +-commutative 99.2%

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

      5. *-commutative 99.2%

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

      6. distribute-rgt-neg-in 99.2%

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

      7. fma-udef 99.2%

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

      8. sub-neg 99.2%

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

      9. +-commutative 99.2%

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

      10. distribute-neg-in 99.2%

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

      11. metadata-eval 99.2%

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

      12. metadata-eval 99.2%

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

      13. unsub-neg 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in x around 0 90.1%

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

   6. Taylor expanded in t around 0 80.2%

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

   7. Taylor expanded in a around 0 80.2%

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

      1. cancel-sign-sub-inv 80.2%

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

      2. log-prod 61.1%

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

      3. metadata-eval 61.1%

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

      4. *-commutative 61.1%

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

   9. Simplified 61.1%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.4:\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{elif}\;a \leq 1.45 \cdot 10^{-78}:\\ \;\;\;\;\left(\log z + \log y\right) - t\\ \mathbf{elif}\;a \leq 6.2 \cdot 10^{-33}:\\ \;\;\;\;\log \left(y \cdot z\right) + -0.5 \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 77.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \log t - t\\ \mathbf{if}\;a \leq -1.4:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 1.05 \cdot 10^{-73}:\\ \;\;\;\;\log \left(x + y\right) + \left(\log z - t\right)\\ \mathbf{elif}\;a \leq 5.3 \cdot 10^{-33}:\\ \;\;\;\;\log \left(y \cdot z\right) + -0.5 \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* a (log t)) t)))
   (if (<= a -1.4)
     t_1
     (if (<= a 1.05e-73)
       (+ (log (+ x y)) (- (log z) t))
       (if (<= a 5.3e-33) (+ (log (* y z)) (* -0.5 (log t))) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (a * log(t)) - t;
	double tmp;
	if (a <= -1.4) {
		tmp = t_1;
	} else if (a <= 1.05e-73) {
		tmp = log((x + y)) + (log(z) - t);
	} else if (a <= 5.3e-33) {
		tmp = log((y * z)) + (-0.5 * log(t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (a * log(t)) - t
    if (a <= (-1.4d0)) then
        tmp = t_1
    else if (a <= 1.05d-73) then
        tmp = log((x + y)) + (log(z) - t)
    else if (a <= 5.3d-33) then
        tmp = log((y * z)) + ((-0.5d0) * log(t))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (a * Math.log(t)) - t;
	double tmp;
	if (a <= -1.4) {
		tmp = t_1;
	} else if (a <= 1.05e-73) {
		tmp = Math.log((x + y)) + (Math.log(z) - t);
	} else if (a <= 5.3e-33) {
		tmp = Math.log((y * z)) + (-0.5 * Math.log(t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (a * math.log(t)) - t
	tmp = 0
	if a <= -1.4:
		tmp = t_1
	elif a <= 1.05e-73:
		tmp = math.log((x + y)) + (math.log(z) - t)
	elif a <= 5.3e-33:
		tmp = math.log((y * z)) + (-0.5 * math.log(t))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(a * log(t)) - t)
	tmp = 0.0
	if (a <= -1.4)
		tmp = t_1;
	elseif (a <= 1.05e-73)
		tmp = Float64(log(Float64(x + y)) + Float64(log(z) - t));
	elseif (a <= 5.3e-33)
		tmp = Float64(log(Float64(y * z)) + Float64(-0.5 * log(t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (a * log(t)) - t;
	tmp = 0.0;
	if (a <= -1.4)
		tmp = t_1;
	elseif (a <= 1.05e-73)
		tmp = log((x + y)) + (log(z) - t);
	elseif (a <= 5.3e-33)
		tmp = log((y * z)) + (-0.5 * log(t));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[a, -1.4], t$95$1, If[LessEqual[a, 1.05e-73], N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 5.3e-33], N[(N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision] + N[(-0.5 * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.3999999999999999 or 5.29999999999999968e-33 < a

   1. Initial program 99.7%

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

      1. associate-+l- 99.7%

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

      2. associate--l+ 99.6%

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

      3. sub-neg 99.6%

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

      4. +-commutative 99.6%

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

      5. *-commutative 99.6%

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

      6. distribute-rgt-neg-in 99.6%

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

      7. fma-udef 99.6%

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

      8. sub-neg 99.6%

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

      9. +-commutative 99.6%

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

      10. distribute-neg-in 99.6%

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

      11. metadata-eval 99.6%

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

      12. metadata-eval 99.6%

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

      13. unsub-neg 99.6%

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

   3. Simplified 99.6%

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

      1. associate-+r- 99.7%

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

      2. fma-udef 99.7%

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

      3. associate--r+ 99.7%

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

      4. +-commutative 99.7%

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

      5. sum-log 80.9%

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

   6. Applied egg-rr 80.9%

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

   7. Taylor expanded in x around inf 72.0%

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

      1. mul-1-neg 72.0%

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

      2. log-rec 72.0%

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

      3. remove-double-neg 72.0%

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

      4. log-prod 56.2%

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

      5. *-commutative 56.2%

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

   9. Simplified 56.2%

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

   10. Taylor expanded in a around inf 98.2%

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

       1. *-commutative 98.2%

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

   12. Simplified 98.2%

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

   if -1.3999999999999999 < a < 1.0499999999999999e-73

   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%

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

      2. associate--l+ 99.8%

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

      3. sub-neg 99.8%

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

      4. +-commutative 99.8%

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

      5. *-commutative 99.8%

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

      6. distribute-rgt-neg-in 99.8%

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

      7. fma-udef 99.8%

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

      8. sub-neg 99.8%

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

      9. +-commutative 99.8%

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

      10. distribute-neg-in 99.8%

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

      11. metadata-eval 99.8%

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

      12. metadata-eval 99.8%

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

      13. unsub-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t around inf 60.0%

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

   if 1.0499999999999999e-73 < a < 5.29999999999999968e-33

   1. Initial program 98.9%

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

      1. associate-+l- 98.9%

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

      2. associate--l+ 99.2%

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

      3. sub-neg 99.2%

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

      4. +-commutative 99.2%

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

      5. *-commutative 99.2%

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

      6. distribute-rgt-neg-in 99.2%

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

      7. fma-udef 99.2%

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

      8. sub-neg 99.2%

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

      9. +-commutative 99.2%

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

      10. distribute-neg-in 99.2%

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

      11. metadata-eval 99.2%

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

      12. metadata-eval 99.2%

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

      13. unsub-neg 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in x around 0 90.1%

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

   6. Taylor expanded in t around 0 80.2%

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

   7. Taylor expanded in a around 0 80.2%

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

      1. cancel-sign-sub-inv 80.2%

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

      2. log-prod 61.1%

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

      3. metadata-eval 61.1%

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

      4. *-commutative 61.1%

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

   9. Simplified 61.1%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.4:\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{elif}\;a \leq 1.05 \cdot 10^{-73}:\\ \;\;\;\;\log \left(x + y\right) + \left(\log z - t\right)\\ \mathbf{elif}\;a \leq 5.3 \cdot 10^{-33}:\\ \;\;\;\;\log \left(y \cdot z\right) + -0.5 \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 73.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -220000000000 \lor \neg \left(a \leq 2.1 \cdot 10^{-32}\right):\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\log \left(y \cdot z\right) + \left(\log t \cdot \left(a - 0.5\right) - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -220000000000.0) (not (<= a 2.1e-32)))
   (- (* a (log t)) t)
   (+ (log (* y z)) (- (* (log t) (- a 0.5)) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -220000000000.0) || !(a <= 2.1e-32)) {
		tmp = (a * log(t)) - t;
	} else {
		tmp = log((y * z)) + ((log(t) * (a - 0.5)) - t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((a <= (-220000000000.0d0)) .or. (.not. (a <= 2.1d-32))) then
        tmp = (a * log(t)) - t
    else
        tmp = log((y * z)) + ((log(t) * (a - 0.5d0)) - t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -220000000000.0) || !(a <= 2.1e-32)) {
		tmp = (a * Math.log(t)) - t;
	} else {
		tmp = Math.log((y * z)) + ((Math.log(t) * (a - 0.5)) - t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -220000000000.0) or not (a <= 2.1e-32):
		tmp = (a * math.log(t)) - t
	else:
		tmp = math.log((y * z)) + ((math.log(t) * (a - 0.5)) - t)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -220000000000.0) || !(a <= 2.1e-32))
		tmp = Float64(Float64(a * log(t)) - t);
	else
		tmp = Float64(log(Float64(y * z)) + Float64(Float64(log(t) * Float64(a - 0.5)) - t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -220000000000.0) || ~((a <= 2.1e-32)))
		tmp = (a * log(t)) - t;
	else
		tmp = log((y * z)) + ((log(t) * (a - 0.5)) - t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -220000000000.0], N[Not[LessEqual[a, 2.1e-32]], $MachinePrecision]], N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision] + N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.2e11 or 2.0999999999999999e-32 < a

   1. Initial program 99.7%

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

      1. associate-+l- 99.7%

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

      2. associate--l+ 99.7%

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

      3. sub-neg 99.7%

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

      4. +-commutative 99.7%

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

      5. *-commutative 99.7%

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

      6. distribute-rgt-neg-in 99.7%

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

      7. fma-udef 99.7%

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

      8. sub-neg 99.7%

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

      9. +-commutative 99.7%

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

      10. distribute-neg-in 99.7%

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

      11. metadata-eval 99.7%

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

      12. metadata-eval 99.7%

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

      13. unsub-neg 99.7%

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

   3. Simplified 99.7%

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

      1. associate-+r- 99.7%

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

      2. fma-udef 99.7%

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

      3. associate--r+ 99.7%

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

      4. +-commutative 99.7%

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

      5. sum-log 80.7%

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

   6. Applied egg-rr 80.7%

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

   7. Taylor expanded in x around inf 72.3%

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

      1. mul-1-neg 72.3%

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

      2. log-rec 72.3%

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

      3. remove-double-neg 72.3%

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

      4. log-prod 56.2%

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

      5. *-commutative 56.2%

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

   9. Simplified 56.2%

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

   10. Taylor expanded in a around inf 98.9%

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

       1. *-commutative 98.9%

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

   12. Simplified 98.9%

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

   if -2.2e11 < a < 2.0999999999999999e-32

   1. Initial program 99.7%

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

      1. associate-+l- 99.7%

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

      2. associate--l+ 99.7%

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

      3. sub-neg 99.7%

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

      4. +-commutative 99.7%

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

      5. *-commutative 99.7%

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

      6. distribute-rgt-neg-in 99.7%

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

      7. fma-udef 99.7%

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

      8. sub-neg 99.7%

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

      9. +-commutative 99.7%

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

      10. distribute-neg-in 99.7%

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

      11. metadata-eval 99.7%

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

      12. metadata-eval 99.7%

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

      13. unsub-neg 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 66.7%

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

   6. Taylor expanded in y around inf 66.7%

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

      1. mul-1-neg 66.7%

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

      2. log-rec 66.7%

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

      3. remove-double-neg 66.7%

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

      4. log-prod 52.7%

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

   8. Simplified 52.7%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -220000000000 \lor \neg \left(a \leq 2.1 \cdot 10^{-32}\right):\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\log \left(y \cdot z\right) + \left(\log t \cdot \left(a - 0.5\right) - t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 74.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{-9} \lor \neg \left(a \leq 1.22 \cdot 10^{-32}\right):\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\log \left(y \cdot z\right) - \left(t + \log t \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -2e-9) (not (<= a 1.22e-32)))
   (- (* a (log t)) t)
   (- (log (* y z)) (+ t (* (log t) 0.5)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -2e-9) || !(a <= 1.22e-32)) {
		tmp = (a * log(t)) - t;
	} else {
		tmp = log((y * z)) - (t + (log(t) * 0.5));
	}
	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 <= (-2d-9)) .or. (.not. (a <= 1.22d-32))) then
        tmp = (a * log(t)) - t
    else
        tmp = log((y * z)) - (t + (log(t) * 0.5d0))
    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 <= -2e-9) || !(a <= 1.22e-32)) {
		tmp = (a * Math.log(t)) - t;
	} else {
		tmp = Math.log((y * z)) - (t + (Math.log(t) * 0.5));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -2e-9) or not (a <= 1.22e-32):
		tmp = (a * math.log(t)) - t
	else:
		tmp = math.log((y * z)) - (t + (math.log(t) * 0.5))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -2e-9) || !(a <= 1.22e-32))
		tmp = Float64(Float64(a * log(t)) - t);
	else
		tmp = Float64(log(Float64(y * z)) - Float64(t + Float64(log(t) * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -2e-9) || ~((a <= 1.22e-32)))
		tmp = (a * log(t)) - t;
	else
		tmp = log((y * z)) - (t + (log(t) * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -2e-9], N[Not[LessEqual[a, 1.22e-32]], $MachinePrecision]], N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision] - N[(t + N[(N[Log[t], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.00000000000000012e-9 or 1.22e-32 < a

   1. Initial program 99.7%

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

      1. associate-+l- 99.7%

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

      2. associate--l+ 99.7%

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

      3. sub-neg 99.7%

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

      4. +-commutative 99.7%

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

      5. *-commutative 99.7%

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

      6. distribute-rgt-neg-in 99.7%

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

      7. fma-udef 99.7%

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

      8. sub-neg 99.7%

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

      9. +-commutative 99.7%

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

      10. distribute-neg-in 99.7%

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

      11. metadata-eval 99.7%

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

      12. metadata-eval 99.7%

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

      13. unsub-neg 99.7%

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

   3. Simplified 99.7%

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

      1. associate-+r- 99.7%

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

      2. fma-udef 99.7%

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

      3. associate--r+ 99.7%

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

      4. +-commutative 99.7%

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

      5. sum-log 80.5%

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

   6. Applied egg-rr 80.5%

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

   7. Taylor expanded in x around inf 71.2%

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

      1. mul-1-neg 71.2%

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

      2. log-rec 71.2%

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

      3. remove-double-neg 71.2%

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

      4. log-prod 55.5%

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

      5. *-commutative 55.5%

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

   9. Simplified 55.5%

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

   10. Taylor expanded in a around inf 98.1%

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

       1. *-commutative 98.1%

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

   12. Simplified 98.1%

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

   if -2.00000000000000012e-9 < a < 1.22e-32

   1. Initial program 99.7%

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

      1. associate-+l- 99.7%

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

      2. associate--l+ 99.7%

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

      3. sub-neg 99.7%

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

      4. +-commutative 99.7%

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

      5. *-commutative 99.7%

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

      6. distribute-rgt-neg-in 99.7%

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

      7. fma-udef 99.7%

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

      8. sub-neg 99.7%

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

      9. +-commutative 99.7%

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

      10. distribute-neg-in 99.7%

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

      11. metadata-eval 99.7%

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

      12. metadata-eval 99.7%

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

      13. unsub-neg 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 65.5%

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

      1. add-cube-cbrt 65.3%

         \[\leadsto \left(\log y + \log z\right) - \left(t + \color{blue}{\left(\sqrt[3]{\log t \cdot \left(0.5 - a\right)} \cdot \sqrt[3]{\log t \cdot \left(0.5 - a\right)}\right) \cdot \sqrt[3]{\log t \cdot \left(0.5 - a\right)}}\right) \]

      2. pow3 65.3%

         \[\leadsto \left(\log y + \log z\right) - \left(t + \color{blue}{{\left(\sqrt[3]{\log t \cdot \left(0.5 - a\right)}\right)}^{3}}\right) \]

   7. Applied egg-rr 65.3%

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

      1. rem-cube-cbrt 65.5%

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

      2. *-commutative 65.5%

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

      3. flip-- 65.5%

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

      4. associate-*l/ 65.5%

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

      5. metadata-eval 65.5%

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

      6. pow2 65.5%

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

   9. Applied egg-rr 65.5%

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

   10. Taylor expanded in a around 0 65.5%

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

       1. log-prod 51.9%

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

       2. *-commutative 51.9%

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

   12. Simplified 51.9%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{-9} \lor \neg \left(a \leq 1.22 \cdot 10^{-32}\right):\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\log \left(y \cdot z\right) - \left(t + \log t \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 71.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.6 \lor \neg \left(a \leq 2.05\right):\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + \log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -1.6) (not (<= a 2.05)))
   (- (* a (log t)) t)
   (- (+ (log z) (log y)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -1.6) || !(a <= 2.05)) {
		tmp = (a * log(t)) - t;
	} else {
		tmp = (log(z) + log(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 <= (-1.6d0)) .or. (.not. (a <= 2.05d0))) then
        tmp = (a * log(t)) - t
    else
        tmp = (log(z) + log(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 <= -1.6) || !(a <= 2.05)) {
		tmp = (a * Math.log(t)) - t;
	} else {
		tmp = (Math.log(z) + Math.log(y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -1.6) or not (a <= 2.05):
		tmp = (a * math.log(t)) - t
	else:
		tmp = (math.log(z) + math.log(y)) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -1.6) || !(a <= 2.05))
		tmp = Float64(Float64(a * log(t)) - t);
	else
		tmp = Float64(Float64(log(z) + log(y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -1.6) || ~((a <= 2.05)))
		tmp = (a * log(t)) - t;
	else
		tmp = (log(z) + log(y)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -1.6], N[Not[LessEqual[a, 2.05]], $MachinePrecision]], N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[z], $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.6000000000000001 or 2.0499999999999998 < a

   1. Initial program 99.7%

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

      1. associate-+l- 99.7%

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

      2. associate--l+ 99.7%

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

      3. sub-neg 99.7%

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

      4. +-commutative 99.7%

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

      5. *-commutative 99.7%

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

      6. distribute-rgt-neg-in 99.7%

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

      7. fma-udef 99.7%

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

      8. sub-neg 99.7%

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

      9. +-commutative 99.7%

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

      10. distribute-neg-in 99.7%

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

      11. metadata-eval 99.7%

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

      12. metadata-eval 99.7%

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

      13. unsub-neg 99.7%

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

   3. Simplified 99.7%

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

      1. associate-+r- 99.7%

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

      2. fma-udef 99.7%

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

      3. associate--r+ 99.7%

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

      4. +-commutative 99.7%

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

      5. sum-log 81.2%

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

   6. Applied egg-rr 81.2%

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

   7. Taylor expanded in x around inf 71.7%

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

      1. mul-1-neg 71.7%

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

      2. log-rec 71.7%

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

      3. remove-double-neg 71.7%

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

      4. log-prod 56.2%

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

      5. *-commutative 56.2%

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

   9. Simplified 56.2%

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

   10. Taylor expanded in a around inf 99.3%

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

       1. *-commutative 99.3%

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

   12. Simplified 99.3%

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

   if -1.6000000000000001 < a < 2.0499999999999998

   1. Initial program 99.7%

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

      1. associate-+l- 99.7%

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

      2. associate--l+ 99.7%

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

      3. sub-neg 99.7%

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

      4. +-commutative 99.7%

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

      5. *-commutative 99.7%

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

      6. distribute-rgt-neg-in 99.7%

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

      7. fma-udef 99.7%

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

      8. sub-neg 99.7%

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

      9. +-commutative 99.7%

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

      10. distribute-neg-in 99.7%

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

      11. metadata-eval 99.7%

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

      12. metadata-eval 99.7%

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

      13. unsub-neg 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 66.4%

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

   6. Taylor expanded in t around inf 42.4%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.6 \lor \neg \left(a \leq 2.05\right):\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + \log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 57.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.1 \cdot 10^{+69} \lor \neg \left(a \leq 3.4 \cdot 10^{+16}\right):\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\log y - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -4.1e+69) (not (<= a 3.4e+16))) (* a (log t)) (- (log y) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -4.1e+69) || !(a <= 3.4e+16)) {
		tmp = a * log(t);
	} else {
		tmp = log(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 <= (-4.1d+69)) .or. (.not. (a <= 3.4d+16))) then
        tmp = a * log(t)
    else
        tmp = log(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 <= -4.1e+69) || !(a <= 3.4e+16)) {
		tmp = a * Math.log(t);
	} else {
		tmp = Math.log(y) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -4.1e+69) or not (a <= 3.4e+16):
		tmp = a * math.log(t)
	else:
		tmp = math.log(y) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -4.1e+69) || !(a <= 3.4e+16))
		tmp = Float64(a * log(t));
	else
		tmp = Float64(log(y) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -4.1e+69) || ~((a <= 3.4e+16)))
		tmp = a * log(t);
	else
		tmp = log(y) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -4.1e+69], N[Not[LessEqual[a, 3.4e+16]], $MachinePrecision]], N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision], N[(N[Log[y], $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4.0999999999999999e69 or 3.4e16 < a

   1. Initial program 99.7%

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

      1. associate-+l- 99.7%

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

      2. associate--l+ 99.7%

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

      3. sub-neg 99.7%

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

      4. +-commutative 99.7%

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

      5. *-commutative 99.7%

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

      6. distribute-rgt-neg-in 99.7%

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

      7. fma-udef 99.7%

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

      8. sub-neg 99.7%

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

      9. +-commutative 99.7%

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

      10. distribute-neg-in 99.7%

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

      11. metadata-eval 99.7%

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

      12. metadata-eval 99.7%

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

      13. unsub-neg 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 76.3%

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

   6. Taylor expanded in a around inf 81.5%

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

   if -4.0999999999999999e69 < a < 3.4e16

   1. Initial program 99.7%

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

      1. associate-+l- 99.7%

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

      2. associate--l+ 99.7%

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

      3. sub-neg 99.7%

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

      4. +-commutative 99.7%

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

      5. *-commutative 99.7%

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

      6. distribute-rgt-neg-in 99.7%

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

      7. fma-udef 99.7%

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

      8. sub-neg 99.7%

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

      9. +-commutative 99.7%

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

      10. distribute-neg-in 99.7%

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

      11. metadata-eval 99.7%

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

      12. metadata-eval 99.7%

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

      13. unsub-neg 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in t around inf 55.4%

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

      1. neg-mul-1 55.4%

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

   7. Simplified 55.4%

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

   8. Taylor expanded in x around 0 41.9%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.1 \cdot 10^{+69} \lor \neg \left(a \leq 3.4 \cdot 10^{+16}\right):\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\log y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 41.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 246:\\ \;\;\;\;\log \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (if (<= t 246.0) (log (+ x y)) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 246.0) {
		tmp = log((x + y));
	} 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 <= 246.0d0) then
        tmp = log((x + y))
    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 <= 246.0) {
		tmp = Math.log((x + y));
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 246.0:
		tmp = math.log((x + y))
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 246.0)
		tmp = log(Float64(x + y));
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 246.0)
		tmp = log((x + y));
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 246.0], N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if t < 246

   1. Initial program 99.5%

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

      1. associate-+l- 99.5%

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

      2. associate--l+ 99.5%

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

      3. sub-neg 99.5%

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

      4. +-commutative 99.5%

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

      5. *-commutative 99.5%

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

      6. distribute-rgt-neg-in 99.5%

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

      7. fma-udef 99.5%

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

      8. sub-neg 99.5%

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

      9. +-commutative 99.5%

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

      10. distribute-neg-in 99.5%

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

      11. metadata-eval 99.5%

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

      12. metadata-eval 99.5%

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

      13. unsub-neg 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in t around inf 7.9%

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

      1. neg-mul-1 7.9%

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

   7. Simplified 7.9%

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

   8. Taylor expanded in t around 0 7.9%

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

      1. +-commutative 7.9%

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

   10. Simplified 7.9%

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

   if 246 < 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. associate--l+ 99.9%

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

      3. sub-neg 99.9%

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

      4. +-commutative 99.9%

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

      5. *-commutative 99.9%

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

      6. distribute-rgt-neg-in 99.9%

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

      7. fma-udef 99.9%

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

      8. sub-neg 99.9%

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

      9. +-commutative 99.9%

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

      10. distribute-neg-in 99.9%

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

      11. metadata-eval 99.9%

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

      12. metadata-eval 99.9%

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

      13. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 74.3%

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

   6. Taylor expanded in t around inf 68.5%

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

      1. mul-1-neg 68.5%

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

   8. Simplified 68.5%

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

4. Final simplification 36.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 246:\\ \;\;\;\;\log \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 62.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 2.6 \cdot 10^{+58}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 2.6e+58) (* a (log t)) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 2.6e+58) {
		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 <= 2.6d+58) 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 <= 2.6e+58) {
		tmp = a * Math.log(t);
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 2.6e+58:
		tmp = a * math.log(t)
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 2.6e+58)
		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 <= 2.6e+58)
		tmp = a * log(t);
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 2.6e+58], N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if t < 2.59999999999999988e58

   1. Initial program 99.5%

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

      1. associate-+l- 99.5%

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

      2. associate--l+ 99.5%

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

      3. sub-neg 99.5%

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

      4. +-commutative 99.5%

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

      5. *-commutative 99.5%

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

      6. distribute-rgt-neg-in 99.5%

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

      7. fma-udef 99.5%

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

      8. sub-neg 99.5%

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

      9. +-commutative 99.5%

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

      10. distribute-neg-in 99.5%

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

      11. metadata-eval 99.5%

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

      12. metadata-eval 99.5%

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

      13. unsub-neg 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around 0 71.5%

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

   6. Taylor expanded in a around inf 60.4%

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

   if 2.59999999999999988e58 < 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. associate--l+ 99.9%

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

      3. sub-neg 99.9%

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

      4. +-commutative 99.9%

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

      5. *-commutative 99.9%

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

      6. distribute-rgt-neg-in 99.9%

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

      7. fma-udef 99.9%

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

      8. sub-neg 99.9%

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

      9. +-commutative 99.9%

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

      10. distribute-neg-in 99.9%

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

      11. metadata-eval 99.9%

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

      12. metadata-eval 99.9%

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

      13. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 73.9%

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

   6. Taylor expanded in t around inf 77.1%

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

      1. mul-1-neg 77.1%

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

   8. Simplified 77.1%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.6 \cdot 10^{+58}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 75.1% 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%

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

   2. associate--l+ 99.7%

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

   3. sub-neg 99.7%

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

   4. +-commutative 99.7%

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

   5. *-commutative 99.7%

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

   6. distribute-rgt-neg-in 99.7%

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

   7. fma-udef 99.7%

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

   8. sub-neg 99.7%

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

   9. +-commutative 99.7%

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

   10. distribute-neg-in 99.7%

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

   11. metadata-eval 99.7%

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

   12. metadata-eval 99.7%

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

   13. unsub-neg 99.7%

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

3. Simplified 99.7%

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

   1. associate-+r- 99.7%

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

   2. fma-udef 99.7%

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

   3. associate--r+ 99.7%

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

   4. +-commutative 99.7%

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

   5. sum-log 80.4%

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

6. Applied egg-rr 80.4%

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

7. Taylor expanded in x around inf 67.3%

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

   1. mul-1-neg 67.3%

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

   2. log-rec 67.3%

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

   3. remove-double-neg 67.3%

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

   4. log-prod 53.0%

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

   5. *-commutative 53.0%

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

9. Simplified 53.0%

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

10. Taylor expanded in a around inf 77.6%

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

    1. *-commutative 77.6%

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

12. Simplified 77.6%

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

13. Final simplification 77.6%

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

Alternative 18: 38.1% accurate, 156.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. associate-+l- 99.7%

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

   2. associate--l+ 99.7%

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

   3. sub-neg 99.7%

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

   4. +-commutative 99.7%

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

   5. *-commutative 99.7%

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

   6. distribute-rgt-neg-in 99.7%

      \[\leadsto \log \left(x + y\right) + \left(\log z - \left(\color{blue}{\log t \cdot \left(-\left(a - 0.5\right)\right)} + t\right)\right) \]

   7. fma-udef 99.7%

      \[\leadsto \log \left(x + y\right) + \left(\log z - \color{blue}{\mathsf{fma}\left(\log t, -\left(a - 0.5\right), t\right)}\right) \]

   8. sub-neg 99.7%

      \[\leadsto \log \left(x + y\right) + \left(\log z - \mathsf{fma}\left(\log t, -\color{blue}{\left(a + \left(-0.5\right)\right)}, t\right)\right) \]

   9. +-commutative 99.7%

      \[\leadsto \log \left(x + y\right) + \left(\log z - \mathsf{fma}\left(\log t, -\color{blue}{\left(\left(-0.5\right) + a\right)}, t\right)\right) \]

   10. distribute-neg-in 99.7%

       \[\leadsto \log \left(x + y\right) + \left(\log z - \mathsf{fma}\left(\log t, \color{blue}{\left(-\left(-0.5\right)\right) + \left(-a\right)}, t\right)\right) \]

   11. metadata-eval 99.7%

       \[\leadsto \log \left(x + y\right) + \left(\log z - \mathsf{fma}\left(\log t, \left(-\color{blue}{-0.5}\right) + \left(-a\right), t\right)\right) \]

   12. metadata-eval 99.7%

       \[\leadsto \log \left(x + y\right) + \left(\log z - \mathsf{fma}\left(\log t, \color{blue}{0.5} + \left(-a\right), t\right)\right) \]

   13. unsub-neg 99.7%

       \[\leadsto \log \left(x + y\right) + \left(\log z - \mathsf{fma}\left(\log t, \color{blue}{0.5 - a}, t\right)\right) \]

3. Simplified 99.7%

   \[\leadsto \color{blue}{\log \left(x + y\right) + \left(\log z - \mathsf{fma}\left(\log t, 0.5 - a, t\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 72.4%

   \[\leadsto \color{blue}{\left(\log y + \log z\right) - \left(t + \log t \cdot \left(0.5 - a\right)\right)} \]

6. Taylor expanded in t around inf 33.5%

   \[\leadsto \color{blue}{-1 \cdot t} \]
7. Step-by-step derivation

   1. mul-1-neg 33.5%

      \[\leadsto \color{blue}{-t} \]

8. Simplified 33.5%

   \[\leadsto \color{blue}{-t} \]

9. Final simplification 33.5%

   \[\leadsto -t \]
10. Add Preprocessing

Developer target: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \log \left(x + y\right) + \left(\left(\log z - t\right) + \left(a - 0.5\right) \cdot \log t\right) \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (+ (log (+ x y)) (+ (- (log z) t) (* (- a 0.5) (log t)))))

C

double code(double x, double y, double z, double t, double a) {
	return log((x + y)) + ((log(z) - t) + ((a - 0.5) * log(t)));
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = log((x + y)) + ((log(z) - t) + ((a - 0.5d0) * log(t)))
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return Math.log((x + y)) + ((Math.log(z) - t) + ((a - 0.5) * Math.log(t)));
}

Python

def code(x, y, z, t, a):
	return math.log((x + y)) + ((math.log(z) - t) + ((a - 0.5) * math.log(t)))

Julia

function code(x, y, z, t, a)
	return Float64(log(Float64(x + y)) + Float64(Float64(log(z) - t) + Float64(Float64(a - 0.5) * log(t))))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = log((x + y)) + ((log(z) - t) + ((a - 0.5) * log(t)));
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024031 
(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))))