Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 15.8s

Alternatives: 16

Speedup: 1.0×

• Could not converge on a ground truth

  1. x = 1.4847440944883767e-283
  2. y = 1.2068276692430032e+298
  3. z = 1.6195148971989164e+29
  4. t = 2.707489982797899e-86
  5. a = -3.07600261207461e+189

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

\[\begin{array}{l} \\ \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \left(2 \cdot \log \left(\sqrt{t}\right)\right) \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (* 2.0 (log (sqrt t))))))

C

double code(double x, double y, double z, double t, double a) {
	return ((log((x + y)) + log(z)) - t) + ((a - 0.5) * (2.0 * log(sqrt(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) * (2.0d0 * log(sqrt(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) * (2.0 * Math.log(Math.sqrt(t))));
}

Python

def code(x, y, z, t, a):
	return ((math.log((x + y)) + math.log(z)) - t) + ((a - 0.5) * (2.0 * math.log(math.sqrt(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) * Float64(2.0 * log(sqrt(t)))))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = ((log((x + y)) + log(z)) - t) + ((a - 0.5) * (2.0 * log(sqrt(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[(2.0 * N[Log[N[Sqrt[t], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. add-sqr-sqrt 99.6%

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

   2. log-prod 99.6%

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

4. Applied egg-rr 99.6%

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

   1. count-2 99.6%

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

6. Simplified 99.6%

   \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\left(2 \cdot \log \left(\sqrt{t}\right)\right)} \]
7. Add Preprocessing

Alternative 2: 68.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 0.43:\\ \;\;\;\;\log z + \left(\log y + \left(a - 0.5\right) \cdot \log t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + \left(\log y + a \cdot \log t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 0.43)
   (+ (log z) (+ (log y) (* (- a 0.5) (log t))))
   (+ (- (log z) t) (+ (log y) (* a (log t))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 0.43) {
		tmp = log(z) + (log(y) + ((a - 0.5) * log(t)));
	} else {
		tmp = (log(z) - t) + (log(y) + (a * log(t)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 0.43d0) then
        tmp = log(z) + (log(y) + ((a - 0.5d0) * log(t)))
    else
        tmp = (log(z) - t) + (log(y) + (a * log(t)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 0.43) {
		tmp = Math.log(z) + (Math.log(y) + ((a - 0.5) * Math.log(t)));
	} else {
		tmp = (Math.log(z) - t) + (Math.log(y) + (a * Math.log(t)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 0.43:
		tmp = math.log(z) + (math.log(y) + ((a - 0.5) * math.log(t)))
	else:
		tmp = (math.log(z) - t) + (math.log(y) + (a * math.log(t)))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 0.43)
		tmp = Float64(log(z) + Float64(log(y) + Float64(Float64(a - 0.5) * log(t))));
	else
		tmp = Float64(Float64(log(z) - t) + Float64(log(y) + Float64(a * log(t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 0.43)
		tmp = log(z) + (log(y) + ((a - 0.5) * log(t)));
	else
		tmp = (log(z) - t) + (log(y) + (a * log(t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 0.43], N[(N[Log[z], $MachinePrecision] + N[(N[Log[y], $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] + N[(N[Log[y], $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 0.429999999999999993

   1. Initial program 99.3%

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

      1. associate--l+ 99.3%

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

      2. +-commutative 99.3%

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

      3. associate-+l+ 99.3%

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

      4. +-commutative 99.3%

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

      5. fma-define 99.3%

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

      6. sub-neg 99.3%

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

      7. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in x around 0 66.7%

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

   6. Taylor expanded in t around 0 66.6%

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

   if 0.429999999999999993 < 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+ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+l+ 99.8%

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

      4. +-commutative 99.8%

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

      5. fma-define 99.8%

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

      6. sub-neg 99.8%

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

      7. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 81.1%

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

   6. Taylor expanded in a around inf 80.6%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 0.43:\\ \;\;\;\;\log z + \left(\log y + \left(a - 0.5\right) \cdot \log t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + \left(\log y + a \cdot \log t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 380:\\ \;\;\;\;\log z + \left(\log y + \left(a - 0.5\right) \cdot \log t\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\log t \cdot \frac{a + -0.5}{t} + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 380.0)
   (+ (log z) (+ (log y) (* (- a 0.5) (log t))))
   (* t (+ (* (log t) (/ (+ a -0.5) t)) -1.0))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 380.0) {
		tmp = log(z) + (log(y) + ((a - 0.5) * log(t)));
	} else {
		tmp = t * ((log(t) * ((a + -0.5) / t)) + -1.0);
	}
	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 <= 380.0d0) then
        tmp = log(z) + (log(y) + ((a - 0.5d0) * log(t)))
    else
        tmp = t * ((log(t) * ((a + (-0.5d0)) / t)) + (-1.0d0))
    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 <= 380.0) {
		tmp = Math.log(z) + (Math.log(y) + ((a - 0.5) * Math.log(t)));
	} else {
		tmp = t * ((Math.log(t) * ((a + -0.5) / t)) + -1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 380.0:
		tmp = math.log(z) + (math.log(y) + ((a - 0.5) * math.log(t)))
	else:
		tmp = t * ((math.log(t) * ((a + -0.5) / t)) + -1.0)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 380.0)
		tmp = Float64(log(z) + Float64(log(y) + Float64(Float64(a - 0.5) * log(t))));
	else
		tmp = Float64(t * Float64(Float64(log(t) * Float64(Float64(a + -0.5) / t)) + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 380.0)
		tmp = log(z) + (log(y) + ((a - 0.5) * log(t)));
	else
		tmp = t * ((log(t) * ((a + -0.5) / t)) + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 380.0], N[(N[Log[z], $MachinePrecision] + N[(N[Log[y], $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t * N[(N[(N[Log[t], $MachinePrecision] * N[(N[(a + -0.5), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 380

   1. Initial program 99.3%

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

      1. associate--l+ 99.3%

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

      2. +-commutative 99.3%

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

      3. associate-+l+ 99.3%

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

      4. +-commutative 99.3%

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

      5. fma-define 99.3%

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

      6. sub-neg 99.3%

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

      7. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in x around 0 67.0%

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

   6. Taylor expanded in t around 0 66.3%

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

   if 380 < 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+ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+l+ 99.8%

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

      4. +-commutative 99.8%

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

      5. fma-define 99.8%

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

      6. sub-neg 99.8%

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

      7. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 81.0%

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

   6. Taylor expanded in t around inf 80.9%

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

      1. associate--l+ 80.9%

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

      2. mul-1-neg 80.9%

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

      3. associate-/l* 80.8%

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

      4. distribute-lft-neg-in 80.8%

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

      5. log-rec 80.8%

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

      6. remove-double-neg 80.8%

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

      7. sub-neg 80.8%

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

      8. metadata-eval 80.8%

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

      9. +-commutative 80.8%

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

      10. associate--l+ 80.9%

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

   8. Simplified 80.9%

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

   9. Taylor expanded in t around inf 99.1%

      \[\leadsto t \cdot \left(\log t \cdot \frac{a + -0.5}{t} + \color{blue}{-1}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 380:\\ \;\;\;\;\log z + \left(\log y + \left(a - 0.5\right) \cdot \log t\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\log t \cdot \frac{a + -0.5}{t} + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

Alternative 5: 68.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate--l+ 99.6%

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

   2. +-commutative 99.6%

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

   3. associate-+l+ 99.6%

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

   4. +-commutative 99.6%

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

   5. fma-define 99.6%

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

   6. sub-neg 99.6%

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

   7. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in x around 0 74.1%

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

6. Final simplification 74.1%

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

Alternative 6: 68.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate--l+ 99.6%

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

   2. +-commutative 99.6%

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

   3. associate-+l+ 99.6%

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

   4. +-commutative 99.6%

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

   5. fma-define 99.6%

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

   6. sub-neg 99.6%

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

   7. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in x around 0 74.1%

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

6. Final simplification 74.1%

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

Alternative 7: 73.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.05 \cdot 10^{-53}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{-19}:\\ \;\;\;\;\log \left({t}^{\left(a + -0.5\right)} \cdot \left(\left(x + y\right) \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\log t \cdot \frac{a + -0.5}{t} + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 1.05e-53)
   (* a (log t))
   (if (<= t 7.5e-19)
     (log (* (pow t (+ a -0.5)) (* (+ x y) z)))
     (* t (+ (* (log t) (/ (+ a -0.5) t)) -1.0)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 1.05e-53) {
		tmp = a * log(t);
	} else if (t <= 7.5e-19) {
		tmp = log((pow(t, (a + -0.5)) * ((x + y) * z)));
	} else {
		tmp = t * ((log(t) * ((a + -0.5) / t)) + -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 1.05d-53) then
        tmp = a * log(t)
    else if (t <= 7.5d-19) then
        tmp = log(((t ** (a + (-0.5d0))) * ((x + y) * z)))
    else
        tmp = t * ((log(t) * ((a + (-0.5d0)) / t)) + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 1.05e-53) {
		tmp = a * Math.log(t);
	} else if (t <= 7.5e-19) {
		tmp = Math.log((Math.pow(t, (a + -0.5)) * ((x + y) * z)));
	} else {
		tmp = t * ((Math.log(t) * ((a + -0.5) / t)) + -1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 1.05e-53:
		tmp = a * math.log(t)
	elif t <= 7.5e-19:
		tmp = math.log((math.pow(t, (a + -0.5)) * ((x + y) * z)))
	else:
		tmp = t * ((math.log(t) * ((a + -0.5) / t)) + -1.0)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 1.05e-53)
		tmp = Float64(a * log(t));
	elseif (t <= 7.5e-19)
		tmp = log(Float64((t ^ Float64(a + -0.5)) * Float64(Float64(x + y) * z)));
	else
		tmp = Float64(t * Float64(Float64(log(t) * Float64(Float64(a + -0.5) / t)) + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 1.05e-53)
		tmp = a * log(t);
	elseif (t <= 7.5e-19)
		tmp = log(((t ^ (a + -0.5)) * ((x + y) * z)));
	else
		tmp = t * ((log(t) * ((a + -0.5) / t)) + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 1.05e-53], N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 7.5e-19], N[Log[N[(N[Power[t, N[(a + -0.5), $MachinePrecision]], $MachinePrecision] * N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(t * N[(N[(N[Log[t], $MachinePrecision] * N[(N[(a + -0.5), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.04999999999999989e-53

   1. Initial program 99.3%

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

      1. associate-+l- 99.3%

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

      2. associate--l+ 99.4%

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

      3. sub-neg 99.4%

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

      4. +-commutative 99.4%

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

      5. *-commutative 99.4%

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

      6. distribute-rgt-neg-in 99.4%

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

      7. fma-undefine 99.4%

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

      8. sub-neg 99.4%

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

      9. +-commutative 99.4%

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

      10. distribute-neg-in 99.4%

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

      11. metadata-eval 99.4%

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

      12. metadata-eval 99.4%

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

      13. unsub-neg 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in a around inf 55.4%

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

      1. *-commutative 55.4%

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

   7. Simplified 55.4%

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

   if 1.04999999999999989e-53 < t < 7.49999999999999957e-19

   1. Initial program 99.1%

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

      1. associate--l+ 99.1%

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

      2. +-commutative 99.1%

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

      3. associate-+l+ 99.0%

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

      4. +-commutative 99.0%

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

      5. fma-define 99.0%

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

      6. sub-neg 99.0%

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

      7. metadata-eval 99.0%

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

   3. Simplified 99.0%

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

      1. add-log-exp 84.5%

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

      2. +-commutative 84.5%

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

      3. exp-sum 84.6%

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

      4. fma-undefine 84.6%

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

      5. metadata-eval 84.6%

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

      6. sub-neg 84.6%

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

      7. exp-sum 84.2%

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

      8. add-exp-log 84.7%

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

      9. sub-neg 84.7%

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

      10. metadata-eval 84.7%

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

      11. *-commutative 84.7%

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

      12. exp-to-pow 84.8%

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

      13. exp-diff 84.8%

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

      14. add-exp-log 85.4%

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

   6. Applied egg-rr 85.4%

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

      1. associate-*l* 85.0%

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

      2. associate-*r/ 85.0%

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

      3. *-commutative 85.0%

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

      4. +-commutative 85.0%

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

   8. Simplified 85.0%

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

   9. Taylor expanded in t around 0 85.0%

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

       1. +-commutative 85.0%

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

   11. Simplified 85.0%

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

   if 7.49999999999999957e-19 < t

   1. Initial program 99.8%

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

      1. associate--l+ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+l+ 99.8%

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

      4. +-commutative 99.8%

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

      5. fma-define 99.8%

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

      6. sub-neg 99.8%

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

      7. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 80.1%

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

   6. Taylor expanded in t around inf 80.0%

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

      1. associate--l+ 80.0%

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

      2. mul-1-neg 80.0%

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

      3. associate-/l* 80.0%

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

      4. distribute-lft-neg-in 80.0%

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

      5. log-rec 80.0%

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

      6. remove-double-neg 80.0%

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

      7. sub-neg 80.0%

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

      8. metadata-eval 80.0%

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

      9. +-commutative 80.0%

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

      10. associate--l+ 80.0%

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

   8. Simplified 80.0%

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

   9. Taylor expanded in t around inf 96.5%

      \[\leadsto t \cdot \left(\log t \cdot \frac{a + -0.5}{t} + \color{blue}{-1}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.05 \cdot 10^{-53}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{-19}:\\ \;\;\;\;\log \left({t}^{\left(a + -0.5\right)} \cdot \left(\left(x + y\right) \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\log t \cdot \frac{a + -0.5}{t} + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 2.25 \cdot 10^{-52}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{elif}\;t \leq 8.6 \cdot 10^{-19}:\\ \;\;\;\;\log \left(\left(y \cdot z\right) \cdot {t}^{\left(a + -0.5\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\log t \cdot \frac{a + -0.5}{t} + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 2.25e-52)
   (* a (log t))
   (if (<= t 8.6e-19)
     (log (* (* y z) (pow t (+ a -0.5))))
     (* t (+ (* (log t) (/ (+ a -0.5) t)) -1.0)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 2.25e-52) {
		tmp = a * log(t);
	} else if (t <= 8.6e-19) {
		tmp = log(((y * z) * pow(t, (a + -0.5))));
	} else {
		tmp = t * ((log(t) * ((a + -0.5) / t)) + -1.0);
	}
	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 <= 2.25d-52) then
        tmp = a * log(t)
    else if (t <= 8.6d-19) then
        tmp = log(((y * z) * (t ** (a + (-0.5d0)))))
    else
        tmp = t * ((log(t) * ((a + (-0.5d0)) / t)) + (-1.0d0))
    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 <= 2.25e-52) {
		tmp = a * Math.log(t);
	} else if (t <= 8.6e-19) {
		tmp = Math.log(((y * z) * Math.pow(t, (a + -0.5))));
	} else {
		tmp = t * ((Math.log(t) * ((a + -0.5) / t)) + -1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 2.25e-52:
		tmp = a * math.log(t)
	elif t <= 8.6e-19:
		tmp = math.log(((y * z) * math.pow(t, (a + -0.5))))
	else:
		tmp = t * ((math.log(t) * ((a + -0.5) / t)) + -1.0)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 2.25e-52)
		tmp = Float64(a * log(t));
	elseif (t <= 8.6e-19)
		tmp = log(Float64(Float64(y * z) * (t ^ Float64(a + -0.5))));
	else
		tmp = Float64(t * Float64(Float64(log(t) * Float64(Float64(a + -0.5) / t)) + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 2.25e-52)
		tmp = a * log(t);
	elseif (t <= 8.6e-19)
		tmp = log(((y * z) * (t ^ (a + -0.5))));
	else
		tmp = t * ((log(t) * ((a + -0.5) / t)) + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 2.25e-52], N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 8.6e-19], N[Log[N[(N[(y * z), $MachinePrecision] * N[Power[t, N[(a + -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(t * N[(N[(N[Log[t], $MachinePrecision] * N[(N[(a + -0.5), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 2.25e-52

   1. Initial program 99.3%

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

      1. associate-+l- 99.3%

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

      2. associate--l+ 99.4%

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

      3. sub-neg 99.4%

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

      4. +-commutative 99.4%

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

      5. *-commutative 99.4%

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

      6. distribute-rgt-neg-in 99.4%

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

      7. fma-undefine 99.4%

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

      8. sub-neg 99.4%

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

      9. +-commutative 99.4%

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

      10. distribute-neg-in 99.4%

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

      11. metadata-eval 99.4%

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

      12. metadata-eval 99.4%

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

      13. unsub-neg 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in a around inf 55.4%

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

      1. *-commutative 55.4%

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

   7. Simplified 55.4%

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

   if 2.25e-52 < t < 8.6e-19

   1. Initial program 99.1%

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

      1. associate--l+ 99.1%

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

      2. +-commutative 99.1%

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

      3. associate-+l+ 99.0%

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

      4. +-commutative 99.0%

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

      5. fma-define 99.0%

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

      6. sub-neg 99.0%

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

      7. metadata-eval 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in x around 0 70.6%

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

   6. Taylor expanded in t around 0 70.6%

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

      1. *-un-lft-identity 70.6%

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

      2. add-log-exp 56.0%

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

      3. sum-log 56.0%

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

      4. sub-neg 56.0%

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

      5. metadata-eval 56.0%

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

      6. exp-sum 55.9%

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

      7. add-exp-log 56.5%

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

      8. pow-to-exp 56.7%

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

   8. Applied egg-rr 56.7%

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

      1. *-lft-identity 56.7%

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

      2. associate-*r* 56.3%

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

      3. *-commutative 56.3%

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

      4. metadata-eval 56.3%

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

      5. sub-neg 56.3%

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

      6. exp-to-pow 56.1%

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

      7. exp-to-pow 56.3%

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

      8. sub-neg 56.3%

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

      9. metadata-eval 56.3%

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

      10. +-commutative 56.3%

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

   10. Simplified 56.3%

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

   if 8.6e-19 < t

   1. Initial program 99.8%

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

      1. associate--l+ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+l+ 99.8%

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

      4. +-commutative 99.8%

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

      5. fma-define 99.8%

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

      6. sub-neg 99.8%

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

      7. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 80.1%

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

   6. Taylor expanded in t around inf 80.0%

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

      1. associate--l+ 80.0%

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

      2. mul-1-neg 80.0%

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

      3. associate-/l* 80.0%

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

      4. distribute-lft-neg-in 80.0%

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

      5. log-rec 80.0%

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

      6. remove-double-neg 80.0%

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

      7. sub-neg 80.0%

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

      8. metadata-eval 80.0%

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

      9. +-commutative 80.0%

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

      10. associate--l+ 80.0%

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

   8. Simplified 80.0%

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

   9. Taylor expanded in t around inf 96.5%

      \[\leadsto t \cdot \left(\log t \cdot \frac{a + -0.5}{t} + \color{blue}{-1}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.25 \cdot 10^{-52}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{elif}\;t \leq 8.6 \cdot 10^{-19}:\\ \;\;\;\;\log \left(\left(y \cdot z\right) \cdot {t}^{\left(a + -0.5\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\log t \cdot \frac{a + -0.5}{t} + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 87.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 520:\\ \;\;\;\;\left(\left(a - 0.5\right) \cdot \log t + \log \left(\left(x + y\right) \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\log t \cdot \frac{a + -0.5}{t} + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 520.0)
   (- (+ (* (- a 0.5) (log t)) (log (* (+ x y) z))) t)
   (* t (+ (* (log t) (/ (+ a -0.5) t)) -1.0))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 520.0) {
		tmp = (((a - 0.5) * log(t)) + log(((x + y) * z))) - t;
	} else {
		tmp = t * ((log(t) * ((a + -0.5) / t)) + -1.0);
	}
	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 <= 520.0d0) then
        tmp = (((a - 0.5d0) * log(t)) + log(((x + y) * z))) - t
    else
        tmp = t * ((log(t) * ((a + (-0.5d0)) / t)) + (-1.0d0))
    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 <= 520.0) {
		tmp = (((a - 0.5) * Math.log(t)) + Math.log(((x + y) * z))) - t;
	} else {
		tmp = t * ((Math.log(t) * ((a + -0.5) / t)) + -1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 520.0:
		tmp = (((a - 0.5) * math.log(t)) + math.log(((x + y) * z))) - t
	else:
		tmp = t * ((math.log(t) * ((a + -0.5) / t)) + -1.0)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 520.0)
		tmp = Float64(Float64(Float64(Float64(a - 0.5) * log(t)) + log(Float64(Float64(x + y) * z))) - t);
	else
		tmp = Float64(t * Float64(Float64(log(t) * Float64(Float64(a + -0.5) / t)) + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 520.0)
		tmp = (((a - 0.5) * log(t)) + log(((x + y) * z))) - t;
	else
		tmp = t * ((log(t) * ((a + -0.5) / t)) + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 520.0], N[(N[(N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision] + N[Log[N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(t * N[(N[(N[Log[t], $MachinePrecision] * N[(N[(a + -0.5), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 520

   1. Initial program 99.3%

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

      1. associate-+l- 99.3%

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

      2. associate--l+ 99.3%

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

      3. sub-neg 99.3%

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

      4. +-commutative 99.3%

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

      5. *-commutative 99.3%

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

      6. distribute-rgt-neg-in 99.3%

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

      7. fma-undefine 99.3%

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

      8. sub-neg 99.3%

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

      9. +-commutative 99.3%

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

      10. distribute-neg-in 99.3%

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

      11. metadata-eval 99.3%

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

      12. metadata-eval 99.3%

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

      13. unsub-neg 99.3%

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

   3. Simplified 99.3%

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

      1. associate-+r- 99.3%

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

      2. fma-undefine 99.3%

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

      3. associate--r+ 99.3%

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

      4. sum-log 76.1%

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

   6. Applied egg-rr 76.1%

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

   if 520 < 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+ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+l+ 99.8%

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

      4. +-commutative 99.8%

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

      5. fma-define 99.8%

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

      6. sub-neg 99.8%

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

      7. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 81.0%

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

   6. Taylor expanded in t around inf 80.9%

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

      1. associate--l+ 80.9%

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

      2. mul-1-neg 80.9%

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

      3. associate-/l* 80.8%

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

      4. distribute-lft-neg-in 80.8%

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

      5. log-rec 80.8%

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

      6. remove-double-neg 80.8%

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

      7. sub-neg 80.8%

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

      8. metadata-eval 80.8%

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

      9. +-commutative 80.8%

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

      10. associate--l+ 80.9%

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

   8. Simplified 80.9%

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

   9. Taylor expanded in t around inf 99.1%

      \[\leadsto t \cdot \left(\log t \cdot \frac{a + -0.5}{t} + \color{blue}{-1}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 520:\\ \;\;\;\;\left(\left(a - 0.5\right) \cdot \log t + \log \left(\left(x + y\right) \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\log t \cdot \frac{a + -0.5}{t} + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 73.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 680:\\ \;\;\;\;\left(\left(a - 0.5\right) \cdot \log t + \log \left(y \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\log t \cdot \frac{a + -0.5}{t} + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 680.0)
   (- (+ (* (- a 0.5) (log t)) (log (* y z))) t)
   (* t (+ (* (log t) (/ (+ a -0.5) t)) -1.0))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 680.0) {
		tmp = (((a - 0.5) * log(t)) + log((y * z))) - t;
	} else {
		tmp = t * ((log(t) * ((a + -0.5) / t)) + -1.0);
	}
	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 <= 680.0d0) then
        tmp = (((a - 0.5d0) * log(t)) + log((y * z))) - t
    else
        tmp = t * ((log(t) * ((a + (-0.5d0)) / t)) + (-1.0d0))
    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 <= 680.0) {
		tmp = (((a - 0.5) * Math.log(t)) + Math.log((y * z))) - t;
	} else {
		tmp = t * ((Math.log(t) * ((a + -0.5) / t)) + -1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 680.0:
		tmp = (((a - 0.5) * math.log(t)) + math.log((y * z))) - t
	else:
		tmp = t * ((math.log(t) * ((a + -0.5) / t)) + -1.0)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 680.0)
		tmp = Float64(Float64(Float64(Float64(a - 0.5) * log(t)) + log(Float64(y * z))) - t);
	else
		tmp = Float64(t * Float64(Float64(log(t) * Float64(Float64(a + -0.5) / t)) + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 680.0)
		tmp = (((a - 0.5) * log(t)) + log((y * z))) - t;
	else
		tmp = t * ((log(t) * ((a + -0.5) / t)) + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 680.0], N[(N[(N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision] + N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(t * N[(N[(N[Log[t], $MachinePrecision] * N[(N[(a + -0.5), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 680

   1. Initial program 99.3%

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

      1. add-cbrt-cube 65.5%

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

      2. pow3 65.5%

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

      3. sub-neg 65.5%

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

      4. metadata-eval 65.5%

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

      5. *-commutative 65.5%

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

   4. Applied egg-rr 65.5%

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

      1. sum-log 51.9%

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

   6. Applied egg-rr 51.9%

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

   7. Taylor expanded in x around 0 52.8%

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

   if 680 < 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+ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+l+ 99.8%

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

      4. +-commutative 99.8%

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

      5. fma-define 99.8%

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

      6. sub-neg 99.8%

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

      7. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 81.0%

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

   6. Taylor expanded in t around inf 80.9%

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

      1. associate--l+ 80.9%

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

      2. mul-1-neg 80.9%

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

      3. associate-/l* 80.8%

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

      4. distribute-lft-neg-in 80.8%

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

      5. log-rec 80.8%

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

      6. remove-double-neg 80.8%

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

      7. sub-neg 80.8%

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

      8. metadata-eval 80.8%

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

      9. +-commutative 80.8%

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

      10. associate--l+ 80.9%

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

   8. Simplified 80.9%

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

   9. Taylor expanded in t around inf 99.1%

      \[\leadsto t \cdot \left(\log t \cdot \frac{a + -0.5}{t} + \color{blue}{-1}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 680:\\ \;\;\;\;\left(\left(a - 0.5\right) \cdot \log t + \log \left(y \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\log t \cdot \frac{a + -0.5}{t} + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 73.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 6.4 \cdot 10^{-55}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\log t \cdot \frac{a + -0.5}{t} + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 6.4e-55) (* a (log t)) (* t (+ (* (log t) (/ (+ a -0.5) t)) -1.0))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 6.4e-55) {
		tmp = a * log(t);
	} else {
		tmp = t * ((log(t) * ((a + -0.5) / t)) + -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 6.4d-55) then
        tmp = a * log(t)
    else
        tmp = t * ((log(t) * ((a + (-0.5d0)) / t)) + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 6.4e-55) {
		tmp = a * Math.log(t);
	} else {
		tmp = t * ((Math.log(t) * ((a + -0.5) / t)) + -1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 6.4e-55:
		tmp = a * math.log(t)
	else:
		tmp = t * ((math.log(t) * ((a + -0.5) / t)) + -1.0)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 6.4e-55)
		tmp = Float64(a * log(t));
	else
		tmp = Float64(t * Float64(Float64(log(t) * Float64(Float64(a + -0.5) / t)) + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 6.4e-55)
		tmp = a * log(t);
	else
		tmp = t * ((log(t) * ((a + -0.5) / t)) + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 6.4e-55], N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision], N[(t * N[(N[(N[Log[t], $MachinePrecision] * N[(N[(a + -0.5), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 6.4000000000000003e-55

   1. Initial program 99.3%

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

      1. associate-+l- 99.3%

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

      2. associate--l+ 99.4%

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

      3. sub-neg 99.4%

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

      4. +-commutative 99.4%

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

      5. *-commutative 99.4%

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

      6. distribute-rgt-neg-in 99.4%

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

      7. fma-undefine 99.4%

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

      8. sub-neg 99.4%

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

      9. +-commutative 99.4%

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

      10. distribute-neg-in 99.4%

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

      11. metadata-eval 99.4%

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

      12. metadata-eval 99.4%

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

      13. unsub-neg 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in a around inf 55.0%

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

      1. *-commutative 55.0%

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

   7. Simplified 55.0%

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

   if 6.4000000000000003e-55 < t

   1. Initial program 99.8%

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

      1. associate--l+ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+l+ 99.7%

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

      4. +-commutative 99.7%

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

      5. fma-define 99.7%

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

      6. sub-neg 99.7%

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

      7. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 79.4%

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

   6. Taylor expanded in t around inf 79.3%

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

      1. associate--l+ 79.3%

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

      2. mul-1-neg 79.3%

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

      3. associate-/l* 79.3%

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

      4. distribute-lft-neg-in 79.3%

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

      5. log-rec 79.3%

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

      6. remove-double-neg 79.3%

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

      7. sub-neg 79.3%

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

      8. metadata-eval 79.3%

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

      9. +-commutative 79.3%

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

      10. associate--l+ 79.3%

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

   8. Simplified 79.3%

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

   9. Taylor expanded in t around inf 90.7%

      \[\leadsto t \cdot \left(\log t \cdot \frac{a + -0.5}{t} + \color{blue}{-1}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.4 \cdot 10^{-55}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\log t \cdot \frac{a + -0.5}{t} + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 61.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.4 \cdot 10^{+39} \lor \neg \left(a \leq 5.5 \cdot 10^{+30}\right):\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-1 - t \cdot \left(\frac{-1}{t} - -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -3.4e+39) (not (<= a 5.5e+30)))
   (* a (log t))
   (- -1.0 (* t (- (/ -1.0 t) -1.0)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -3.4e+39) || !(a <= 5.5e+30)) {
		tmp = a * log(t);
	} else {
		tmp = -1.0 - (t * ((-1.0 / t) - -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((a <= (-3.4d+39)) .or. (.not. (a <= 5.5d+30))) then
        tmp = a * log(t)
    else
        tmp = (-1.0d0) - (t * (((-1.0d0) / t) - (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -3.4e+39) || !(a <= 5.5e+30)) {
		tmp = a * Math.log(t);
	} else {
		tmp = -1.0 - (t * ((-1.0 / t) - -1.0));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -3.4e+39) or not (a <= 5.5e+30):
		tmp = a * math.log(t)
	else:
		tmp = -1.0 - (t * ((-1.0 / t) - -1.0))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -3.4e+39) || !(a <= 5.5e+30))
		tmp = Float64(a * log(t));
	else
		tmp = Float64(-1.0 - Float64(t * Float64(Float64(-1.0 / t) - -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -3.4e+39) || ~((a <= 5.5e+30)))
		tmp = a * log(t);
	else
		tmp = -1.0 - (t * ((-1.0 / t) - -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -3.4e+39], N[Not[LessEqual[a, 5.5e+30]], $MachinePrecision]], N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision], N[(-1.0 - N[(t * N[(N[(-1.0 / t), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.3999999999999999e39 or 5.50000000000000025e30 < a

   1. Initial program 99.6%

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

      1. associate-+l- 99.6%

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

      2. associate--l+ 99.6%

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

      3. sub-neg 99.6%

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

      4. +-commutative 99.6%

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

      5. *-commutative 99.6%

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

      6. distribute-rgt-neg-in 99.6%

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

      7. fma-undefine 99.6%

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

      8. sub-neg 99.6%

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

      9. +-commutative 99.6%

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

      10. distribute-neg-in 99.6%

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

      11. metadata-eval 99.6%

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

      12. metadata-eval 99.6%

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

      13. unsub-neg 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in a around inf 78.5%

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

      1. *-commutative 78.5%

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

   7. Simplified 78.5%

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

   if -3.3999999999999999e39 < a < 5.50000000000000025e30

   1. Initial program 99.6%

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

      1. associate-+l- 99.6%

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

      2. associate--l+ 99.5%

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

      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-undefine 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 50.4%

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

      1. neg-mul-1 50.4%

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

   7. Simplified 50.4%

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

      1. expm1-log1p-u 1.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-t\right)\right)} \]

      2. expm1-undefine 1.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-t\right)} - 1} \]

   9. Applied egg-rr 1.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-t\right)} - 1} \]
   10. Step-by-step derivation

       1. sub-neg 1.6%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-t\right)} + \left(-1\right)} \]

       2. log1p-undefine 1.6%

          \[\leadsto e^{\color{blue}{\log \left(1 + \left(-t\right)\right)}} + \left(-1\right) \]

       3. rem-exp-log 50.6%

          \[\leadsto \color{blue}{\left(1 + \left(-t\right)\right)} + \left(-1\right) \]

       4. unsub-neg 50.6%

          \[\leadsto \color{blue}{\left(1 - t\right)} + \left(-1\right) \]

       5. metadata-eval 50.6%

          \[\leadsto \left(1 - t\right) + \color{blue}{-1} \]

   11. Simplified 50.6%

       \[\leadsto \color{blue}{\left(1 - t\right) + -1} \]

   12. Taylor expanded in t around inf 50.6%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.4 \cdot 10^{+39} \lor \neg \left(a \leq 5.5 \cdot 10^{+30}\right):\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-1 - t \cdot \left(\frac{-1}{t} - -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 37.3% accurate, 34.8× speedup

Math

\[\begin{array}{l} \\ -1 - t \cdot \left(\frac{-1}{t} - -1\right) \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (- -1.0 (* t (- (/ -1.0 t) -1.0))))

C

double code(double x, double y, double z, double t, double a) {
	return -1.0 - (t * ((-1.0 / t) - -1.0));
}

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 = (-1.0d0) - (t * (((-1.0d0) / t) - (-1.0d0)))
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return -1.0 - (t * ((-1.0 / t) - -1.0));
}

Python

def code(x, y, z, t, a):
	return -1.0 - (t * ((-1.0 / t) - -1.0))

Julia

function code(x, y, z, t, a)
	return Float64(-1.0 - Float64(t * Float64(Float64(-1.0 / t) - -1.0)))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = -1.0 - (t * ((-1.0 / t) - -1.0));
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(-1.0 - N[(t * N[(N[(-1.0 / t), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. associate-+l- 99.6%

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

   2. 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-undefine 99.6%

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

   8. sub-neg 99.6%

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

   9. +-commutative 99.6%

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

   10. distribute-neg-in 99.6%

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

   11. metadata-eval 99.6%

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

   12. metadata-eval 99.6%

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

   13. unsub-neg 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in t around inf 35.8%

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

   1. neg-mul-1 35.8%

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

7. Simplified 35.8%

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

   1. expm1-log1p-u 1.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-t\right)\right)} \]

   2. expm1-undefine 1.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-t\right)} - 1} \]

9. Applied egg-rr 1.4%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-t\right)} - 1} \]
10. Step-by-step derivation

    1. sub-neg 1.4%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-t\right)} + \left(-1\right)} \]

    2. log1p-undefine 1.4%

       \[\leadsto e^{\color{blue}{\log \left(1 + \left(-t\right)\right)}} + \left(-1\right) \]

    3. rem-exp-log 35.8%

       \[\leadsto \color{blue}{\left(1 + \left(-t\right)\right)} + \left(-1\right) \]

    4. unsub-neg 35.8%

       \[\leadsto \color{blue}{\left(1 - t\right)} + \left(-1\right) \]

    5. metadata-eval 35.8%

       \[\leadsto \left(1 - t\right) + \color{blue}{-1} \]

11. Simplified 35.8%

    \[\leadsto \color{blue}{\left(1 - t\right) + -1} \]

12. Taylor expanded in t around inf 35.9%

    \[\leadsto \color{blue}{t \cdot \left(\frac{1}{t} - 1\right)} + -1 \]

13. Final simplification 35.9%

    \[\leadsto -1 - t \cdot \left(\frac{-1}{t} - -1\right) \]
14. Add Preprocessing

Alternative 14: 37.2% accurate, 62.6× speedup

Math

\[\begin{array}{l} \\ -1 + \left(1 - t\right) \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (+ -1.0 (- 1.0 t)))

C

double code(double x, double y, double z, double t, double a) {
	return -1.0 + (1.0 - 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 = (-1.0d0) + (1.0d0 - t)
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return -1.0 + (1.0 - t);
}

Python

def code(x, y, z, t, a):
	return -1.0 + (1.0 - t)

Julia

function code(x, y, z, t, a)
	return Float64(-1.0 + Float64(1.0 - t))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = -1.0 + (1.0 - t);
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(-1.0 + N[(1.0 - t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. associate-+l- 99.6%

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

   2. 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-undefine 99.6%

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

   8. sub-neg 99.6%

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

   9. +-commutative 99.6%

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

   10. distribute-neg-in 99.6%

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

   11. metadata-eval 99.6%

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

   12. metadata-eval 99.6%

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

   13. unsub-neg 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in t around inf 35.8%

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

   1. neg-mul-1 35.8%

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

7. Simplified 35.8%

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

   1. expm1-log1p-u 1.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-t\right)\right)} \]

   2. expm1-undefine 1.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-t\right)} - 1} \]

9. Applied egg-rr 1.4%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-t\right)} - 1} \]
10. Step-by-step derivation

    1. sub-neg 1.4%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-t\right)} + \left(-1\right)} \]

    2. log1p-undefine 1.4%

       \[\leadsto e^{\color{blue}{\log \left(1 + \left(-t\right)\right)}} + \left(-1\right) \]

    3. rem-exp-log 35.8%

       \[\leadsto \color{blue}{\left(1 + \left(-t\right)\right)} + \left(-1\right) \]

    4. unsub-neg 35.8%

       \[\leadsto \color{blue}{\left(1 - t\right)} + \left(-1\right) \]

    5. metadata-eval 35.8%

       \[\leadsto \left(1 - t\right) + \color{blue}{-1} \]

11. Simplified 35.8%

    \[\leadsto \color{blue}{\left(1 - t\right) + -1} \]

12. Final simplification 35.8%

    \[\leadsto -1 + \left(1 - t\right) \]
13. Add Preprocessing

Alternative 15: 37.2% accurate, 156.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate-+l- 99.6%

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

   2. 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-undefine 99.6%

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

   8. sub-neg 99.6%

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

   9. +-commutative 99.6%

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

   10. distribute-neg-in 99.6%

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

   11. metadata-eval 99.6%

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

   12. metadata-eval 99.6%

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

   13. unsub-neg 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in t around inf 35.8%

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

   1. neg-mul-1 35.8%

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

7. Simplified 35.8%

   \[\leadsto \color{blue}{-t} \]
8. Add Preprocessing

Alternative 16: 2.4% accurate, 313.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 0.0)

C

double code(double x, double y, double z, double t, double a) {
	return 0.0;
}

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 = 0.0d0
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return 0.0;
}

Python

def code(x, y, z, t, a):
	return 0.0

Julia

function code(x, y, z, t, a)
	return 0.0
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = 0.0;
end

Wolfram

code[x_, y_, z_, t_, a_] := 0.0

Derivation

1. Initial program 99.6%

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

   1. associate-+l- 99.6%

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

   2. 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-undefine 99.6%

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

   8. sub-neg 99.6%

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

   9. +-commutative 99.6%

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

   10. distribute-neg-in 99.6%

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

   11. metadata-eval 99.6%

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

   12. metadata-eval 99.6%

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

   13. unsub-neg 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in t around inf 35.8%

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

   1. neg-mul-1 35.8%

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

7. Simplified 35.8%

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

   1. expm1-log1p-u 1.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-t\right)\right)} \]

   2. expm1-undefine 1.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-t\right)} - 1} \]

9. Applied egg-rr 1.4%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-t\right)} - 1} \]
10. Step-by-step derivation

    1. sub-neg 1.4%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-t\right)} + \left(-1\right)} \]

    2. log1p-undefine 1.4%

       \[\leadsto e^{\color{blue}{\log \left(1 + \left(-t\right)\right)}} + \left(-1\right) \]

    3. rem-exp-log 35.8%

       \[\leadsto \color{blue}{\left(1 + \left(-t\right)\right)} + \left(-1\right) \]

    4. unsub-neg 35.8%

       \[\leadsto \color{blue}{\left(1 - t\right)} + \left(-1\right) \]

    5. metadata-eval 35.8%

       \[\leadsto \left(1 - t\right) + \color{blue}{-1} \]

11. Simplified 35.8%

    \[\leadsto \color{blue}{\left(1 - t\right) + -1} \]

12. Taylor expanded in t around 0 2.4%

    \[\leadsto \color{blue}{1} + -1 \]
13. Step-by-step derivation

    1. metadata-eval 2.4%

       \[\leadsto \color{blue}{0} \]

14. Applied egg-rr 2.4%

    \[\leadsto \color{blue}{0} \]
15. Add Preprocessing

Developer Target 1: 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 2024170 
(FPCore (x y z t a)
  :name "Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (log (+ x y)) (+ (- (log z) t) (* (- a 1/2) (log t)))))

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