Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 19.9s

Alternatives: 14

Speedup: 0.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. associate-+l+ 99.6%

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

   3. +-commutative 99.6%

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

   4. associate-+r- 99.6%

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

   5. fma-def 99.6%

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

   6. sub-neg 99.6%

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

   7. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 2: 86.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -100 \lor \neg \left(a - 0.5 \leq -0.4\right):\\ \;\;\;\;\left(\log y + a \cdot \log t\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + \left(\log \left(x + y\right) + -0.5 \cdot \log t\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= (- a 0.5) -100.0) (not (<= (- a 0.5) -0.4)))
   (- (+ (log y) (* a (log t))) t)
   (- (+ (log z) (+ (log (+ x y)) (* -0.5 (log t)))) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((a - 0.5) <= -100.0) || !((a - 0.5) <= -0.4)) {
		tmp = (log(y) + (a * log(t))) - t;
	} else {
		tmp = (log(z) + (log((x + y)) + (-0.5 * log(t)))) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (((a - 0.5d0) <= (-100.0d0)) .or. (.not. ((a - 0.5d0) <= (-0.4d0)))) then
        tmp = (log(y) + (a * log(t))) - t
    else
        tmp = (log(z) + (log((x + y)) + ((-0.5d0) * log(t)))) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((a - 0.5) <= -100.0) || !((a - 0.5) <= -0.4)) {
		tmp = (Math.log(y) + (a * Math.log(t))) - t;
	} else {
		tmp = (Math.log(z) + (Math.log((x + y)) + (-0.5 * Math.log(t)))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if ((a - 0.5) <= -100.0) or not ((a - 0.5) <= -0.4):
		tmp = (math.log(y) + (a * math.log(t))) - t
	else:
		tmp = (math.log(z) + (math.log((x + y)) + (-0.5 * math.log(t)))) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((Float64(a - 0.5) <= -100.0) || !(Float64(a - 0.5) <= -0.4))
		tmp = Float64(Float64(log(y) + Float64(a * log(t))) - t);
	else
		tmp = Float64(Float64(log(z) + Float64(log(Float64(x + y)) + Float64(-0.5 * log(t)))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (((a - 0.5) <= -100.0) || ~(((a - 0.5) <= -0.4)))
		tmp = (log(y) + (a * log(t))) - t;
	else
		tmp = (log(z) + (log((x + y)) + (-0.5 * log(t)))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[N[(a - 0.5), $MachinePrecision], -100.0], N[Not[LessEqual[N[(a - 0.5), $MachinePrecision], -0.4]], $MachinePrecision]], N[(N[(N[Log[y], $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[z], $MachinePrecision] + N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[(-0.5 * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 a 1/2) < -100 or -0.40000000000000002 < (-.f64 a 1/2)

   1. Initial program 99.7%

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

      1. associate--l+ 99.7%

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

      2. associate-+l+ 99.7%

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

      3. +-commutative 99.7%

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

      4. associate-+r- 99.7%

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

      5. fma-def 99.7%

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

      6. sub-neg 99.7%

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

      7. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in a around inf 99.2%

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

      1. *-commutative 99.2%

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

   6. Simplified 99.2%

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

   7. Taylor expanded in x around 0 75.2%

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

   if -100 < (-.f64 a 1/2) < -0.40000000000000002

   1. Initial program 99.4%

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

      1. associate--l+ 99.4%

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

      2. associate-+l+ 99.5%

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

      3. +-commutative 99.5%

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

      4. associate-+r- 99.5%

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

      5. fma-def 99.5%

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

      6. sub-neg 99.5%

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

      7. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in a around 0 97.1%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -100 \lor \neg \left(a - 0.5 \leq -0.4\right):\\ \;\;\;\;\left(\log y + a \cdot \log t\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + \left(\log \left(x + y\right) + -0.5 \cdot \log t\right)\right) - t\\ \end{array} \]

Alternative 3: 69.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -100 \lor \neg \left(a - 0.5 \leq -0.4\right):\\ \;\;\;\;\left(\log y + a \cdot \log t\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log y + \left(\log z + -0.5 \cdot \log t\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= (- a 0.5) -100.0) (not (<= (- a 0.5) -0.4)))
   (- (+ (log y) (* a (log t))) t)
   (- (+ (log y) (+ (log z) (* -0.5 (log t)))) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((a - 0.5) <= -100.0) || !((a - 0.5) <= -0.4)) {
		tmp = (log(y) + (a * log(t))) - t;
	} else {
		tmp = (log(y) + (log(z) + (-0.5 * log(t)))) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (((a - 0.5d0) <= (-100.0d0)) .or. (.not. ((a - 0.5d0) <= (-0.4d0)))) then
        tmp = (log(y) + (a * log(t))) - t
    else
        tmp = (log(y) + (log(z) + ((-0.5d0) * log(t)))) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((a - 0.5) <= -100.0) || !((a - 0.5) <= -0.4)) {
		tmp = (Math.log(y) + (a * Math.log(t))) - t;
	} else {
		tmp = (Math.log(y) + (Math.log(z) + (-0.5 * Math.log(t)))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if ((a - 0.5) <= -100.0) or not ((a - 0.5) <= -0.4):
		tmp = (math.log(y) + (a * math.log(t))) - t
	else:
		tmp = (math.log(y) + (math.log(z) + (-0.5 * math.log(t)))) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((Float64(a - 0.5) <= -100.0) || !(Float64(a - 0.5) <= -0.4))
		tmp = Float64(Float64(log(y) + Float64(a * log(t))) - t);
	else
		tmp = Float64(Float64(log(y) + Float64(log(z) + Float64(-0.5 * log(t)))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (((a - 0.5) <= -100.0) || ~(((a - 0.5) <= -0.4)))
		tmp = (log(y) + (a * log(t))) - t;
	else
		tmp = (log(y) + (log(z) + (-0.5 * log(t)))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[N[(a - 0.5), $MachinePrecision], -100.0], N[Not[LessEqual[N[(a - 0.5), $MachinePrecision], -0.4]], $MachinePrecision]], N[(N[(N[Log[y], $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] + N[(N[Log[z], $MachinePrecision] + N[(-0.5 * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 a 1/2) < -100 or -0.40000000000000002 < (-.f64 a 1/2)

   1. Initial program 99.7%

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

      1. associate--l+ 99.7%

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

      2. associate-+l+ 99.7%

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

      3. +-commutative 99.7%

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

      4. associate-+r- 99.7%

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

      5. fma-def 99.7%

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

      6. sub-neg 99.7%

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

      7. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in a around inf 99.2%

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

      1. *-commutative 99.2%

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

   6. Simplified 99.2%

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

   7. Taylor expanded in x around 0 75.2%

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

   if -100 < (-.f64 a 1/2) < -0.40000000000000002

   1. Initial program 99.4%

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

      1. associate--l+ 99.4%

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

      2. associate-+l+ 99.5%

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

      3. +-commutative 99.5%

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

      4. associate-+r- 99.5%

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

      5. fma-def 99.5%

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

      6. sub-neg 99.5%

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

      7. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in a around 0 97.1%

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

   5. Taylor expanded in x around 0 62.6%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -100 \lor \neg \left(a - 0.5 \leq -0.4\right):\\ \;\;\;\;\left(\log y + a \cdot \log t\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log y + \left(\log z + -0.5 \cdot \log t\right)\right) - t\\ \end{array} \]

Alternative 4: 69.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -100 \lor \neg \left(a - 0.5 \leq -0.4\right):\\ \;\;\;\;\left(\log y + a \cdot \log t\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + \left(\log y + -0.5 \cdot \log t\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= (- a 0.5) -100.0) (not (<= (- a 0.5) -0.4)))
   (- (+ (log y) (* a (log t))) t)
   (- (+ (log z) (+ (log y) (* -0.5 (log t)))) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((a - 0.5) <= -100.0) || !((a - 0.5) <= -0.4)) {
		tmp = (log(y) + (a * log(t))) - t;
	} else {
		tmp = (log(z) + (log(y) + (-0.5 * log(t)))) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (((a - 0.5d0) <= (-100.0d0)) .or. (.not. ((a - 0.5d0) <= (-0.4d0)))) then
        tmp = (log(y) + (a * log(t))) - t
    else
        tmp = (log(z) + (log(y) + ((-0.5d0) * log(t)))) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((a - 0.5) <= -100.0) || !((a - 0.5) <= -0.4)) {
		tmp = (Math.log(y) + (a * Math.log(t))) - t;
	} else {
		tmp = (Math.log(z) + (Math.log(y) + (-0.5 * Math.log(t)))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if ((a - 0.5) <= -100.0) or not ((a - 0.5) <= -0.4):
		tmp = (math.log(y) + (a * math.log(t))) - t
	else:
		tmp = (math.log(z) + (math.log(y) + (-0.5 * math.log(t)))) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((Float64(a - 0.5) <= -100.0) || !(Float64(a - 0.5) <= -0.4))
		tmp = Float64(Float64(log(y) + Float64(a * log(t))) - t);
	else
		tmp = Float64(Float64(log(z) + Float64(log(y) + Float64(-0.5 * log(t)))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (((a - 0.5) <= -100.0) || ~(((a - 0.5) <= -0.4)))
		tmp = (log(y) + (a * log(t))) - t;
	else
		tmp = (log(z) + (log(y) + (-0.5 * log(t)))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[N[(a - 0.5), $MachinePrecision], -100.0], N[Not[LessEqual[N[(a - 0.5), $MachinePrecision], -0.4]], $MachinePrecision]], N[(N[(N[Log[y], $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[z], $MachinePrecision] + N[(N[Log[y], $MachinePrecision] + N[(-0.5 * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 a 1/2) < -100 or -0.40000000000000002 < (-.f64 a 1/2)

   1. Initial program 99.7%

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

      1. associate--l+ 99.7%

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

      2. associate-+l+ 99.7%

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

      3. +-commutative 99.7%

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

      4. associate-+r- 99.7%

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

      5. fma-def 99.7%

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

      6. sub-neg 99.7%

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

      7. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in a around inf 99.2%

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

      1. *-commutative 99.2%

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

   6. Simplified 99.2%

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

   7. Taylor expanded in x around 0 75.2%

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

   if -100 < (-.f64 a 1/2) < -0.40000000000000002

   1. Initial program 99.4%

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

      1. associate--l+ 99.4%

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

      2. associate-+l+ 99.5%

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

      3. +-commutative 99.5%

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

      4. associate-+r- 99.5%

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

      5. fma-def 99.5%

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

      6. sub-neg 99.5%

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

      7. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in a around 0 97.1%

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

   5. Taylor expanded in x around 0 62.6%

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

      1. +-commutative 62.6%

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

   7. Simplified 62.6%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -100 \lor \neg \left(a - 0.5 \leq -0.4\right):\\ \;\;\;\;\left(\log y + a \cdot \log t\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + \left(\log y + -0.5 \cdot \log t\right)\right) - t\\ \end{array} \]

Alternative 5: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. cancel-sign-sub 99.6%

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

   2. cancel-sign-sub-inv 99.6%

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

   3. associate--l+ 99.6%

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

   4. remove-double-neg 99.6%

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

   5. sub-neg 99.6%

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

   6. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 6: 65.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.65 \lor \neg \left(a \leq 4.5 \cdot 10^{-5}\right):\\ \;\;\;\;\left(\log y + a \cdot \log t\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + \log \left(y \cdot {t}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -1.65) (not (<= a 4.5e-5)))
   (- (+ (log y) (* a (log t))) t)
   (+ (- (log z) t) (log (* y (pow t -0.5))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -1.65) || !(a <= 4.5e-5)) {
		tmp = (log(y) + (a * log(t))) - t;
	} else {
		tmp = (log(z) - t) + log((y * pow(t, -0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((a <= (-1.65d0)) .or. (.not. (a <= 4.5d-5))) then
        tmp = (log(y) + (a * log(t))) - t
    else
        tmp = (log(z) - t) + log((y * (t ** (-0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -1.65) || !(a <= 4.5e-5)) {
		tmp = (Math.log(y) + (a * Math.log(t))) - t;
	} else {
		tmp = (Math.log(z) - t) + Math.log((y * Math.pow(t, -0.5)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -1.65) or not (a <= 4.5e-5):
		tmp = (math.log(y) + (a * math.log(t))) - t
	else:
		tmp = (math.log(z) - t) + math.log((y * math.pow(t, -0.5)))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -1.65) || !(a <= 4.5e-5))
		tmp = Float64(Float64(log(y) + Float64(a * log(t))) - t);
	else
		tmp = Float64(Float64(log(z) - t) + log(Float64(y * (t ^ -0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -1.65) || ~((a <= 4.5e-5)))
		tmp = (log(y) + (a * log(t))) - t;
	else
		tmp = (log(z) - t) + log((y * (t ^ -0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -1.65], N[Not[LessEqual[a, 4.5e-5]], $MachinePrecision]], N[(N[(N[Log[y], $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] + N[Log[N[(y * N[Power[t, -0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.6499999999999999 or 4.50000000000000028e-5 < a

   1. Initial program 99.7%

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

      1. associate--l+ 99.7%

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

      2. associate-+l+ 99.7%

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

      3. +-commutative 99.7%

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

      4. associate-+r- 99.7%

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

      5. fma-def 99.7%

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

      6. sub-neg 99.7%

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

      7. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in a around inf 98.2%

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

      1. *-commutative 98.2%

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

   6. Simplified 98.2%

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

   7. Taylor expanded in x around 0 73.8%

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

   if -1.6499999999999999 < a < 4.50000000000000028e-5

   1. Initial program 99.4%

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

      1. associate--l+ 99.4%

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

      2. associate-+l+ 99.4%

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

      3. +-commutative 99.4%

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

      4. associate-+r- 99.4%

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

      5. fma-def 99.4%

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

      6. sub-neg 99.4%

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

      7. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in a around 0 98.2%

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

   5. Taylor expanded in x around 0 64.2%

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

      1. +-commutative 64.2%

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

   7. Simplified 64.2%

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

   8. Taylor expanded in z around 0 64.3%

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

      1. sub-neg 64.3%

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

      2. associate-+l+ 64.3%

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

      3. associate-+r+ 64.3%

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

      4. +-commutative 64.3%

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

      5. neg-mul-1 64.3%

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

      6. +-commutative 64.3%

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

      7. neg-mul-1 64.3%

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

      8. associate-+r+ 64.3%

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

      9. sub-neg 64.3%

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

      10. associate-+l+ 64.3%

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

      11. log-pow 64.3%

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

      12. exp-to-pow 64.3%

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

      13. log-prod 58.8%

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

      14. *-commutative 58.8%

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

   10. Simplified 58.8%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.65 \lor \neg \left(a \leq 4.5 \cdot 10^{-5}\right):\\ \;\;\;\;\left(\log y + a \cdot \log t\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + \log \left(y \cdot {t}^{-0.5}\right)\\ \end{array} \]

Alternative 7: 60.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.55 \cdot 10^{-163} \lor \neg \left(a \leq 7.6\right):\\ \;\;\;\;\left(\log y + a \cdot \log t\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a + -0.5\right) \cdot \log t + \log \left(y \cdot z\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -1.55e-163) (not (<= a 7.6)))
   (- (+ (log y) (* a (log t))) t)
   (- (+ (* (+ a -0.5) (log t)) (log (* y z))) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -1.55e-163) || !(a <= 7.6)) {
		tmp = (log(y) + (a * log(t))) - t;
	} else {
		tmp = (((a + -0.5) * log(t)) + log((y * z))) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((a <= (-1.55d-163)) .or. (.not. (a <= 7.6d0))) then
        tmp = (log(y) + (a * log(t))) - t
    else
        tmp = (((a + (-0.5d0)) * log(t)) + log((y * z))) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -1.55e-163) || !(a <= 7.6)) {
		tmp = (Math.log(y) + (a * Math.log(t))) - t;
	} else {
		tmp = (((a + -0.5) * Math.log(t)) + Math.log((y * z))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -1.55e-163) or not (a <= 7.6):
		tmp = (math.log(y) + (a * math.log(t))) - t
	else:
		tmp = (((a + -0.5) * math.log(t)) + math.log((y * z))) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -1.55e-163) || !(a <= 7.6))
		tmp = Float64(Float64(log(y) + Float64(a * log(t))) - t);
	else
		tmp = Float64(Float64(Float64(Float64(a + -0.5) * log(t)) + log(Float64(y * z))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -1.55e-163) || ~((a <= 7.6)))
		tmp = (log(y) + (a * log(t))) - t;
	else
		tmp = (((a + -0.5) * log(t)) + log((y * z))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -1.55e-163], N[Not[LessEqual[a, 7.6]], $MachinePrecision]], N[(N[(N[Log[y], $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[(N[(a + -0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision] + N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.54999999999999987e-163 or 7.5999999999999996 < a

   1. Initial program 99.7%

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

      1. associate--l+ 99.7%

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

      2. associate-+l+ 99.7%

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

      3. +-commutative 99.7%

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

      4. associate-+r- 99.7%

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

      5. fma-def 99.7%

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

      6. sub-neg 99.7%

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

      7. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in a around inf 96.7%

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

      1. *-commutative 96.7%

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

   6. Simplified 96.7%

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

   7. Taylor expanded in x around 0 73.5%

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

   if -1.54999999999999987e-163 < a < 7.5999999999999996

   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. cancel-sign-sub 99.3%

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

      2. cancel-sign-sub-inv 99.3%

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

      3. associate--l+ 99.4%

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

      4. remove-double-neg 99.4%

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

      5. sub-neg 99.4%

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

      6. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

      1. add-sqr-sqrt 55.9%

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

      2. pow2 55.9%

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

   5. Applied egg-rr 55.9%

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

   6. Taylor expanded in x around 0 62.1%

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

      1. associate-+r+ 62.1%

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

      2. log-prod 44.9%

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

      3. associate--l+ 44.9%

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

      4. log-prod 62.1%

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

      5. remove-double-neg 62.1%

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

      6. log-rec 62.1%

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

      7. mul-1-neg 62.1%

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

      8. +-commutative 62.1%

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

      9. associate--l+ 62.1%

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

   8. Simplified 44.9%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.55 \cdot 10^{-163} \lor \neg \left(a \leq 7.6\right):\\ \;\;\;\;\left(\log y + a \cdot \log t\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a + -0.5\right) \cdot \log t + \log \left(y \cdot z\right)\right) - t\\ \end{array} \]

Alternative 8: 60.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.38 \cdot 10^{-163} \lor \neg \left(a \leq 0.00125\right):\\ \;\;\;\;\left(\log y + a \cdot \log t\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(-0.5 \cdot \log t + \log \left(y \cdot z\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -1.38e-163) (not (<= a 0.00125)))
   (- (+ (log y) (* a (log t))) t)
   (- (+ (* -0.5 (log t)) (log (* y z))) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -1.38e-163) || !(a <= 0.00125)) {
		tmp = (log(y) + (a * log(t))) - t;
	} else {
		tmp = ((-0.5 * log(t)) + log((y * z))) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((a <= (-1.38d-163)) .or. (.not. (a <= 0.00125d0))) then
        tmp = (log(y) + (a * log(t))) - t
    else
        tmp = (((-0.5d0) * log(t)) + log((y * z))) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -1.38e-163) || !(a <= 0.00125)) {
		tmp = (Math.log(y) + (a * Math.log(t))) - t;
	} else {
		tmp = ((-0.5 * Math.log(t)) + Math.log((y * z))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -1.38e-163) or not (a <= 0.00125):
		tmp = (math.log(y) + (a * math.log(t))) - t
	else:
		tmp = ((-0.5 * math.log(t)) + math.log((y * z))) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -1.38e-163) || !(a <= 0.00125))
		tmp = Float64(Float64(log(y) + Float64(a * log(t))) - t);
	else
		tmp = Float64(Float64(Float64(-0.5 * log(t)) + log(Float64(y * z))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -1.38e-163) || ~((a <= 0.00125)))
		tmp = (log(y) + (a * log(t))) - t;
	else
		tmp = ((-0.5 * log(t)) + log((y * z))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -1.38e-163], N[Not[LessEqual[a, 0.00125]], $MachinePrecision]], N[(N[(N[Log[y], $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[(-0.5 * N[Log[t], $MachinePrecision]), $MachinePrecision] + N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.37999999999999999e-163 or 0.00125000000000000003 < a

   1. Initial program 99.7%

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

      1. associate--l+ 99.7%

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

      2. associate-+l+ 99.7%

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

      3. +-commutative 99.7%

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

      4. associate-+r- 99.7%

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

      5. fma-def 99.7%

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

      6. sub-neg 99.7%

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

      7. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in a around inf 96.7%

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

      1. *-commutative 96.7%

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

   6. Simplified 96.7%

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

   7. Taylor expanded in x around 0 73.5%

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

   if -1.37999999999999999e-163 < a < 0.00125000000000000003

   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+ 99.4%

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

      2. associate-+l+ 99.4%

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

      3. +-commutative 99.4%

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

      4. associate-+r- 99.4%

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

      5. fma-def 99.4%

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

      6. sub-neg 99.4%

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

      7. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in a around 0 96.7%

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

      1. associate-+r+ 96.6%

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

      2. log-prod 74.8%

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

      3. +-commutative 74.8%

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

   6. Simplified 74.8%

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

   7. Taylor expanded in y around inf 61.2%

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

      1. mul-1-neg 29.1%

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

      2. log-rec 29.1%

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

      3. remove-double-neg 29.1%

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

      4. log-prod 20.3%

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

   9. Simplified 43.9%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.38 \cdot 10^{-163} \lor \neg \left(a \leq 0.00125\right):\\ \;\;\;\;\left(\log y + a \cdot \log t\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(-0.5 \cdot \log t + \log \left(y \cdot z\right)\right) - t\\ \end{array} \]

Alternative 9: 60.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 8 \cdot 10^{-81}:\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t + \log \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log y + a \cdot \log t\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 8e-81)
   (+ (* (+ a -0.5) (log t)) (log (* y z)))
   (- (+ (log y) (* a (log t))) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 8e-81) {
		tmp = ((a + -0.5) * log(t)) + log((y * z));
	} else {
		tmp = (log(y) + (a * log(t))) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 8d-81) then
        tmp = ((a + (-0.5d0)) * log(t)) + log((y * z))
    else
        tmp = (log(y) + (a * log(t))) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 8e-81) {
		tmp = ((a + -0.5) * Math.log(t)) + Math.log((y * z));
	} else {
		tmp = (Math.log(y) + (a * Math.log(t))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 8e-81:
		tmp = ((a + -0.5) * math.log(t)) + math.log((y * z))
	else:
		tmp = (math.log(y) + (a * math.log(t))) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 8e-81)
		tmp = Float64(Float64(Float64(a + -0.5) * log(t)) + log(Float64(y * z)));
	else
		tmp = Float64(Float64(log(y) + Float64(a * log(t))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 8e-81)
		tmp = ((a + -0.5) * log(t)) + log((y * z));
	else
		tmp = (log(y) + (a * log(t))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 8e-81], N[(N[(N[(a + -0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision] + N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 7.9999999999999997e-81

   1. Initial program 99.2%

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

      1. cancel-sign-sub 99.2%

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

      2. cancel-sign-sub-inv 99.2%

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

      3. associate--l+ 99.2%

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

      4. remove-double-neg 99.2%

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

      5. sub-neg 99.2%

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

      6. metadata-eval 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in t around 0 99.2%

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

      1. log-prod 77.0%

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

      2. +-commutative 77.0%

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

   6. Simplified 77.0%

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

   7. Taylor expanded in y around inf 60.1%

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

      1. mul-1-neg 60.1%

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

      2. log-rec 60.1%

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

      3. remove-double-neg 60.1%

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

      4. log-prod 46.6%

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

   9. Simplified 46.6%

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

   if 7.9999999999999997e-81 < 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. associate-+l+ 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+r- 99.8%

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

      5. fma-def 99.8%

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

      6. sub-neg 99.8%

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

      7. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in a around inf 91.0%

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

      1. *-commutative 91.0%

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

   6. Simplified 91.0%

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

   7. Taylor expanded in x around 0 70.5%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8 \cdot 10^{-81}:\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t + \log \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log y + a \cdot \log t\right) - t\\ \end{array} \]

Alternative 10: 65.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 3.4 \cdot 10^{+30}:\\ \;\;\;\;\log \left(x + y\right) + a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 3.4e+30) (+ (log (+ x y)) (* a (log t))) (- t)))

C

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

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 3.4d+30) then
        tmp = log((x + y)) + (a * log(t))
    else
        tmp = -t
    end if
    code = tmp
end function

Java

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

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 3.4e+30:
		tmp = math.log((x + y)) + (a * math.log(t))
	else:
		tmp = -t
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 3.4e+30], N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if t < 3.4000000000000002e30

   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. associate-+l+ 99.3%

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

      3. +-commutative 99.3%

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

      4. associate-+r- 99.3%

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

      5. fma-def 99.3%

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

      6. sub-neg 99.3%

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

      7. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in a around inf 61.9%

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

      1. *-commutative 61.9%

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

   6. Simplified 61.9%

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

   7. Taylor expanded in t around 0 61.3%

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

      1. +-commutative 61.3%

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

   9. Simplified 61.3%

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

   if 3.4000000000000002e30 < t

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. associate-+l+ 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+r- 99.9%

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

      5. fma-def 99.9%

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

      6. sub-neg 99.9%

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

      7. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t around inf 74.3%

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

      1. neg-mul-1 74.3%

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

   6. Simplified 74.3%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.4 \cdot 10^{+30}:\\ \;\;\;\;\log \left(x + y\right) + a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 11: 58.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[Log[y], $MachinePrecision] + N[(a * N[Log[t], $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. associate-+l+ 99.6%

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

   3. +-commutative 99.6%

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

   4. associate-+r- 99.6%

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

   5. fma-def 99.6%

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

   6. sub-neg 99.6%

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

   7. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in a around inf 79.8%

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

   1. *-commutative 79.8%

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

6. Simplified 79.8%

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

7. Taylor expanded in x around 0 61.0%

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

8. Final simplification 61.0%

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

Alternative 12: 64.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -5 \cdot 10^{+56} \lor \neg \left(a - 0.5 \leq 10^{+52}\right):\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= (- a 0.5) -5e+56) (not (<= (- a 0.5) 1e+52)))
   (* a (log t))
   (- (log (+ x y)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((a - 0.5) <= -5e+56) || !((a - 0.5) <= 1e+52)) {
		tmp = a * log(t);
	} else {
		tmp = log((x + y)) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (((a - 0.5d0) <= (-5d+56)) .or. (.not. ((a - 0.5d0) <= 1d+52))) then
        tmp = a * log(t)
    else
        tmp = log((x + y)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((a - 0.5) <= -5e+56) || !((a - 0.5) <= 1e+52)) {
		tmp = a * Math.log(t);
	} else {
		tmp = Math.log((x + y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if ((a - 0.5) <= -5e+56) or not ((a - 0.5) <= 1e+52):
		tmp = a * math.log(t)
	else:
		tmp = math.log((x + y)) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((Float64(a - 0.5) <= -5e+56) || !(Float64(a - 0.5) <= 1e+52))
		tmp = Float64(a * log(t));
	else
		tmp = Float64(log(Float64(x + y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (((a - 0.5) <= -5e+56) || ~(((a - 0.5) <= 1e+52)))
		tmp = a * log(t);
	else
		tmp = log((x + y)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[N[(a - 0.5), $MachinePrecision], -5e+56], N[Not[LessEqual[N[(a - 0.5), $MachinePrecision], 1e+52]], $MachinePrecision]], N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision], N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 a 1/2) < -5.00000000000000024e56 or 9.9999999999999999e51 < (-.f64 a 1/2)

   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. cancel-sign-sub 99.6%

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

      2. cancel-sign-sub-inv 99.6%

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

      3. associate--l+ 99.6%

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

      4. remove-double-neg 99.6%

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

      5. sub-neg 99.6%

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

      6. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

      1. add-cube-cbrt 99.3%

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

      2. pow3 99.3%

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

   5. Applied egg-rr 99.3%

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

   6. Taylor expanded in t around inf 80.7%

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

      1. +-commutative 80.7%

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

   8. Simplified 80.7%

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

   9. Taylor expanded in a around inf 80.7%

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

   if -5.00000000000000024e56 < (-.f64 a 1/2) < 9.9999999999999999e51

   1. Initial program 99.5%

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

      1. associate--l+ 99.5%

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

      2. associate-+l+ 99.5%

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

      3. +-commutative 99.5%

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

      4. associate-+r- 99.5%

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

      5. fma-def 99.5%

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

      6. sub-neg 99.5%

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

      7. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in a around inf 61.2%

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

      1. *-commutative 61.2%

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

   6. Simplified 61.2%

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

   7. Taylor expanded in a around 0 56.8%

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

      1. +-commutative 56.8%

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

   9. Simplified 56.8%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -5 \cdot 10^{+56} \lor \neg \left(a - 0.5 \leq 10^{+52}\right):\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + y\right) - t\\ \end{array} \]

Alternative 13: 62.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 2.9 \cdot 10^{+30}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 2.9e+30) (* a (log t)) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 2.9e+30) {
		tmp = a * log(t);
	} else {
		tmp = -t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 2.9d+30) then
        tmp = a * log(t)
    else
        tmp = -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 2.9e+30) {
		tmp = a * Math.log(t);
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 2.9e+30:
		tmp = a * math.log(t)
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 2.9e+30)
		tmp = Float64(a * log(t));
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 2.9e+30)
		tmp = a * log(t);
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 2.9e+30], N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if t < 2.8999999999999998e30

   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. cancel-sign-sub 99.3%

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

      2. cancel-sign-sub-inv 99.3%

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

      3. associate--l+ 99.3%

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

      4. remove-double-neg 99.3%

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

      5. sub-neg 99.3%

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

      6. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

      1. add-cube-cbrt 98.8%

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

      2. pow3 98.8%

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

   5. Applied egg-rr 98.8%

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

   6. Taylor expanded in t around inf 61.8%

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

      1. +-commutative 61.8%

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

   8. Simplified 61.8%

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

   9. Taylor expanded in a around inf 56.3%

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

   if 2.8999999999999998e30 < t

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. associate-+l+ 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+r- 99.9%

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

      5. fma-def 99.9%

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

      6. sub-neg 99.9%

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

      7. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t around inf 74.3%

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

      1. neg-mul-1 74.3%

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

   6. Simplified 74.3%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.9 \cdot 10^{+30}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 14: 38.3% accurate, 156.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

   \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
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. associate-+l+ 99.6%

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

   3. +-commutative 99.6%

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

   4. associate-+r- 99.6%

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

   5. fma-def 99.6%

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

   6. sub-neg 99.6%

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

   7. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in t around inf 37.0%

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

   1. neg-mul-1 37.0%

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

6. Simplified 37.0%

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

7. Final simplification 37.0%

   \[\leadsto -t \]

Developer target: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023297 
(FPCore (x y z t a)
  :name "Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2"
  :precision binary64

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

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