Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 30.0s

Alternatives: 14

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. +-commutative 99.6%

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

   2. fma-def 99.7%

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

   3. sub-neg 99.7%

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

   4. metadata-eval 99.7%

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

   5. associate--l+ 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 2: 80.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -2000000 \lor \neg \left(a - 0.5 \leq 0.4\right):\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log y + \left(\log z + -0.5 \cdot \log t\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= (- a 0.5) -2000000.0) (not (<= (- a 0.5) 0.4)))
   (- (* 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) <= -2000000.0) || !((a - 0.5) <= 0.4)) {
		tmp = (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) <= (-2000000.0d0)) .or. (.not. ((a - 0.5d0) <= 0.4d0))) then
        tmp = (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) <= -2000000.0) || !((a - 0.5) <= 0.4)) {
		tmp = (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) <= -2000000.0) or not ((a - 0.5) <= 0.4):
		tmp = (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) <= -2000000.0) || !(Float64(a - 0.5) <= 0.4))
		tmp = Float64(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) <= -2000000.0) || ~(((a - 0.5) <= 0.4)))
		tmp = (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], -2000000.0], N[Not[LessEqual[N[(a - 0.5), $MachinePrecision], 0.4]], $MachinePrecision]], N[(N[(a * N[Log[t], $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) < -2e6 or 0.40000000000000002 < (-.f64 a 1/2)

   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 x around 0 70.6%

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

   5. Taylor expanded in a around inf 98.3%

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

      1. *-commutative 98.3%

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

   7. Simplified 98.3%

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

   if -2e6 < (-.f64 a 1/2) < 0.40000000000000002

   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 x around 0 61.9%

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

   5. Taylor expanded in a around 0 61.6%

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

      1. *-commutative 61.6%

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

   7. Simplified 61.6%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -2000000 \lor \neg \left(a - 0.5 \leq 0.4\right):\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log y + \left(\log z + -0.5 \cdot \log t\right)\right) - t\\ \end{array} \]

Alternative 3: 76.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -2000000 \lor \neg \left(a - 0.5 \leq 0.4\right):\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + \log \left(y \cdot {t}^{-0.5}\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= (- a 0.5) -2000000.0) (not (<= (- a 0.5) 0.4)))
   (- (* a (log t)) t)
   (- (+ (log z) (log (* y (pow t -0.5)))) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((a - 0.5) <= -2000000.0) || !((a - 0.5) <= 0.4)) {
		tmp = (a * log(t)) - t;
	} else {
		tmp = (log(z) + log((y * pow(t, -0.5)))) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (((a - 0.5d0) <= (-2000000.0d0)) .or. (.not. ((a - 0.5d0) <= 0.4d0))) then
        tmp = (a * log(t)) - t
    else
        tmp = (log(z) + log((y * (t ** (-0.5d0))))) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((a - 0.5) <= -2000000.0) || !((a - 0.5) <= 0.4)) {
		tmp = (a * Math.log(t)) - t;
	} else {
		tmp = (Math.log(z) + Math.log((y * Math.pow(t, -0.5)))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if ((a - 0.5) <= -2000000.0) or not ((a - 0.5) <= 0.4):
		tmp = (a * math.log(t)) - t
	else:
		tmp = (math.log(z) + math.log((y * math.pow(t, -0.5)))) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((Float64(a - 0.5) <= -2000000.0) || !(Float64(a - 0.5) <= 0.4))
		tmp = Float64(Float64(a * log(t)) - t);
	else
		tmp = Float64(Float64(log(z) + log(Float64(y * (t ^ -0.5)))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (((a - 0.5) <= -2000000.0) || ~(((a - 0.5) <= 0.4)))
		tmp = (a * log(t)) - t;
	else
		tmp = (log(z) + log((y * (t ^ -0.5)))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[N[(a - 0.5), $MachinePrecision], -2000000.0], N[Not[LessEqual[N[(a - 0.5), $MachinePrecision], 0.4]], $MachinePrecision]], N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[z], $MachinePrecision] + N[Log[N[(y * N[Power[t, -0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 a 1/2) < -2e6 or 0.40000000000000002 < (-.f64 a 1/2)

   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 x around 0 70.6%

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

   5. Taylor expanded in a around inf 98.3%

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

      1. *-commutative 98.3%

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

   7. Simplified 98.3%

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

   if -2e6 < (-.f64 a 1/2) < 0.40000000000000002

   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 0 98.5%

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

   5. Taylor expanded in z around inf 98.5%

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

      1. associate-+r+ 98.6%

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

      2. +-commutative 98.6%

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

      3. mul-1-neg 98.6%

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

      4. log-rec 98.6%

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

      5. remove-double-neg 98.6%

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

      6. log-prod 76.6%

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

      7. +-commutative 76.6%

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

      8. log-pow 76.6%

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

      9. log-prod 69.8%

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

      10. *-commutative 69.8%

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

   7. Simplified 69.8%

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

   8. Taylor expanded in y around inf 43.1%

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

      1. associate-*r* 42.5%

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

      2. log-prod 51.8%

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

   10. Applied egg-rr 51.8%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -2000000 \lor \neg \left(a - 0.5 \leq 0.4\right):\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + \log \left(y \cdot {t}^{-0.5}\right)\right) - t\\ \end{array} \]

Alternative 4: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

2. Final simplification 99.6%

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

Alternative 5: 69.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[Log[y], $MachinePrecision] + N[(N[Log[z], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $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. 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 x around 0 66.5%

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

5. Final simplification 66.5%

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

Alternative 6: 85.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 3.1 \cdot 10^{+97}:\\ \;\;\;\;\log \left(\left(x + y\right) \cdot z\right) + \left(\left(a + -0.5\right) \cdot \log t - t\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + y\right) + \mathsf{fma}\left(\log t, a, -t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z 3.1e+97)
   (+ (log (* (+ x y) z)) (- (* (+ a -0.5) (log t)) t))
   (+ (log (+ x y)) (fma (log t) a (- t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= 3.1e+97) {
		tmp = log(((x + y) * z)) + (((a + -0.5) * log(t)) - t);
	} else {
		tmp = log((x + y)) + fma(log(t), a, -t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= 3.1e+97)
		tmp = Float64(log(Float64(Float64(x + y) * z)) + Float64(Float64(Float64(a + -0.5) * log(t)) - t));
	else
		tmp = Float64(log(Float64(x + y)) + fma(log(t), a, Float64(-t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, 3.1e+97], N[(N[Log[N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision] + N[(N[(N[(a + -0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * a + (-t)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 3.09999999999999981e97

   1. Initial program 99.5%

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

      1. associate-+l- 99.5%

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

      2. +-commutative 99.5%

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

      3. sum-log 91.1%

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

      4. sub-neg 91.1%

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

      5. metadata-eval 91.1%

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

   3. Applied egg-rr 91.1%

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

   if 3.09999999999999981e97 < z

   1. Initial program 99.8%

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

      1. associate--l+ 99.8%

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

      2. 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 82.9%

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

      1. *-commutative 82.9%

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

   6. Simplified 82.9%

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

   7. Taylor expanded in t around 0 82.9%

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

      1. neg-mul-1 82.9%

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

      2. +-commutative 82.9%

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

      3. +-commutative 82.9%

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

      4. *-commutative 82.9%

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

      5. fma-udef 82.9%

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

   9. Simplified 82.9%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.1 \cdot 10^{+97}:\\ \;\;\;\;\log \left(\left(x + y\right) \cdot z\right) + \left(\left(a + -0.5\right) \cdot \log t - t\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + y\right) + \mathsf{fma}\left(\log t, a, -t\right)\\ \end{array} \]

Alternative 7: 79.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 2.2 \cdot 10^{+97}:\\ \;\;\;\;\log \left(\left(x + y\right) \cdot z\right) + \left(\left(a + -0.5\right) \cdot \log t - t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log y + a \cdot \log t\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z 2.2e+97)
   (+ (log (* (+ x y) z)) (- (* (+ a -0.5) (log t)) t))
   (- (+ (log y) (* a (log t))) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= 2.2e+97) {
		tmp = log(((x + y) * z)) + (((a + -0.5) * log(t)) - t);
	} 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 (z <= 2.2d+97) then
        tmp = log(((x + y) * z)) + (((a + (-0.5d0)) * log(t)) - t)
    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 (z <= 2.2e+97) {
		tmp = Math.log(((x + y) * z)) + (((a + -0.5) * Math.log(t)) - t);
	} else {
		tmp = (Math.log(y) + (a * Math.log(t))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= 2.2e+97:
		tmp = math.log(((x + y) * z)) + (((a + -0.5) * math.log(t)) - t)
	else:
		tmp = (math.log(y) + (a * math.log(t))) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= 2.2e+97)
		tmp = Float64(log(Float64(Float64(x + y) * z)) + Float64(Float64(Float64(a + -0.5) * log(t)) - t));
	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 (z <= 2.2e+97)
		tmp = log(((x + y) * z)) + (((a + -0.5) * log(t)) - t);
	else
		tmp = (log(y) + (a * log(t))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, 2.2e+97], N[(N[Log[N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision] + N[(N[(N[(a + -0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $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 z < 2.2000000000000001e97

   1. Initial program 99.5%

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

      1. associate-+l- 99.5%

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

      2. +-commutative 99.5%

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

      3. sum-log 91.1%

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

      4. sub-neg 91.1%

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

      5. metadata-eval 91.1%

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

   3. Applied egg-rr 91.1%

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

   if 2.2000000000000001e97 < z

   1. Initial program 99.8%

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

      1. associate--l+ 99.8%

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

      2. 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 82.9%

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

      1. *-commutative 82.9%

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

   6. Simplified 82.9%

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

   7. Taylor expanded in x around 0 62.0%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.2 \cdot 10^{+97}:\\ \;\;\;\;\log \left(\left(x + y\right) \cdot z\right) + \left(\left(a + -0.5\right) \cdot \log t - t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log y + a \cdot \log t\right) - t\\ \end{array} \]

Alternative 8: 67.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \log t\\ \mathbf{if}\;a \leq -0.00088:\\ \;\;\;\;t_1 - t\\ \mathbf{elif}\;a \leq 6 \cdot 10^{-45}:\\ \;\;\;\;\left(-0.5 \cdot \log t + \log \left(y \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log y + t_1\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* a (log t))))
   (if (<= a -0.00088)
     (- t_1 t)
     (if (<= a 6e-45)
       (- (+ (* -0.5 (log t)) (log (* y z))) t)
       (- (+ (log y) t_1) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = a * log(t);
	double tmp;
	if (a <= -0.00088) {
		tmp = t_1 - t;
	} else if (a <= 6e-45) {
		tmp = ((-0.5 * log(t)) + log((y * z))) - t;
	} else {
		tmp = (log(y) + t_1) - 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) :: t_1
    real(8) :: tmp
    t_1 = a * log(t)
    if (a <= (-0.00088d0)) then
        tmp = t_1 - t
    else if (a <= 6d-45) then
        tmp = (((-0.5d0) * log(t)) + log((y * z))) - t
    else
        tmp = (log(y) + t_1) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = a * Math.log(t);
	double tmp;
	if (a <= -0.00088) {
		tmp = t_1 - t;
	} else if (a <= 6e-45) {
		tmp = ((-0.5 * Math.log(t)) + Math.log((y * z))) - t;
	} else {
		tmp = (Math.log(y) + t_1) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = a * math.log(t)
	tmp = 0
	if a <= -0.00088:
		tmp = t_1 - t
	elif a <= 6e-45:
		tmp = ((-0.5 * math.log(t)) + math.log((y * z))) - t
	else:
		tmp = (math.log(y) + t_1) - t
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(a * log(t))
	tmp = 0.0
	if (a <= -0.00088)
		tmp = Float64(t_1 - t);
	elseif (a <= 6e-45)
		tmp = Float64(Float64(Float64(-0.5 * log(t)) + log(Float64(y * z))) - t);
	else
		tmp = Float64(Float64(log(y) + t_1) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = a * log(t);
	tmp = 0.0;
	if (a <= -0.00088)
		tmp = t_1 - t;
	elseif (a <= 6e-45)
		tmp = ((-0.5 * log(t)) + log((y * z))) - t;
	else
		tmp = (log(y) + t_1) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -0.00088], N[(t$95$1 - t), $MachinePrecision], If[LessEqual[a, 6e-45], N[(N[(N[(-0.5 * N[Log[t], $MachinePrecision]), $MachinePrecision] + N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] + t$95$1), $MachinePrecision] - t), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -8.80000000000000031e-4

   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 x around 0 62.2%

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

   5. Taylor expanded in a around inf 99.1%

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

      1. *-commutative 99.1%

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

   7. Simplified 99.1%

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

   if -8.80000000000000031e-4 < a < 6.00000000000000022e-45

   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 x around 0 60.9%

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

   5. Taylor expanded in a around 0 60.9%

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

      1. associate-+r+ 60.8%

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

      2. log-prod 50.4%

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

      3. *-commutative 50.4%

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

      4. *-commutative 50.4%

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

   7. Simplified 50.4%

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

   if 6.00000000000000022e-45 < 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 92.6%

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

      1. *-commutative 92.6%

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

   6. Simplified 92.6%

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

   7. Taylor expanded in x around 0 73.3%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.00088:\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{elif}\;a \leq 6 \cdot 10^{-45}:\\ \;\;\;\;\left(-0.5 \cdot \log t + \log \left(y \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log y + a \cdot \log t\right) - t\\ \end{array} \]

Alternative 9: 74.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 0.0013:\\ \;\;\;\;\log t \cdot \left(a - 0.5\right) + \log \left(\left(x + y\right) \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log y + a \cdot \log t\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 0.0013)
   (+ (* (log t) (- a 0.5)) (log (* (+ x 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 <= 0.0013) {
		tmp = (log(t) * (a - 0.5)) + log(((x + 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 <= 0.0013d0) then
        tmp = (log(t) * (a - 0.5d0)) + log(((x + 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 <= 0.0013) {
		tmp = (Math.log(t) * (a - 0.5)) + Math.log(((x + 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 <= 0.0013:
		tmp = (math.log(t) * (a - 0.5)) + math.log(((x + 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 <= 0.0013)
		tmp = Float64(Float64(log(t) * Float64(a - 0.5)) + log(Float64(Float64(x + 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 <= 0.0013)
		tmp = (log(t) * (a - 0.5)) + log(((x + 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, 0.0013], N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision] + N[Log[N[(N[(x + y), $MachinePrecision] * 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 < 0.0012999999999999999

   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. add-cbrt-cube 70.4%

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

      2. pow3 70.5%

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

      3. +-commutative 70.5%

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

      4. sub-neg 70.5%

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

      5. metadata-eval 70.5%

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

      6. fma-def 70.5%

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

      7. +-commutative 70.5%

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

      8. sum-log 55.3%

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

   3. Applied egg-rr 55.3%

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

   4. Taylor expanded in t around 0 75.7%

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

   if 0.0012999999999999999 < 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 a around inf 99.4%

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

      1. *-commutative 99.4%

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

   6. Simplified 99.4%

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

   7. Taylor expanded in x around 0 70.1%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 0.0013:\\ \;\;\;\;\log t \cdot \left(a - 0.5\right) + \log \left(\left(x + y\right) \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log y + a \cdot \log t\right) - t\\ \end{array} \]

Alternative 10: 65.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \log t\\ \mathbf{if}\;a \leq -3.5 \cdot 10^{-20}:\\ \;\;\;\;\left(\log y + t_1\right) - t\\ \mathbf{elif}\;a \leq 0.00055:\\ \;\;\;\;\log \left(y \cdot \left(z \cdot {t}^{-0.5}\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;t_1 - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* a (log t))))
   (if (<= a -3.5e-20)
     (- (+ (log y) t_1) t)
     (if (<= a 0.00055) (- (log (* y (* z (pow t -0.5)))) t) (- t_1 t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = a * log(t);
	double tmp;
	if (a <= -3.5e-20) {
		tmp = (log(y) + t_1) - t;
	} else if (a <= 0.00055) {
		tmp = log((y * (z * pow(t, -0.5)))) - t;
	} else {
		tmp = t_1 - 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) :: t_1
    real(8) :: tmp
    t_1 = a * log(t)
    if (a <= (-3.5d-20)) then
        tmp = (log(y) + t_1) - t
    else if (a <= 0.00055d0) then
        tmp = log((y * (z * (t ** (-0.5d0))))) - t
    else
        tmp = t_1 - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = a * Math.log(t);
	double tmp;
	if (a <= -3.5e-20) {
		tmp = (Math.log(y) + t_1) - t;
	} else if (a <= 0.00055) {
		tmp = Math.log((y * (z * Math.pow(t, -0.5)))) - t;
	} else {
		tmp = t_1 - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = a * math.log(t)
	tmp = 0
	if a <= -3.5e-20:
		tmp = (math.log(y) + t_1) - t
	elif a <= 0.00055:
		tmp = math.log((y * (z * math.pow(t, -0.5)))) - t
	else:
		tmp = t_1 - t
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(a * log(t))
	tmp = 0.0
	if (a <= -3.5e-20)
		tmp = Float64(Float64(log(y) + t_1) - t);
	elseif (a <= 0.00055)
		tmp = Float64(log(Float64(y * Float64(z * (t ^ -0.5)))) - t);
	else
		tmp = Float64(t_1 - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = a * log(t);
	tmp = 0.0;
	if (a <= -3.5e-20)
		tmp = (log(y) + t_1) - t;
	elseif (a <= 0.00055)
		tmp = log((y * (z * (t ^ -0.5)))) - t;
	else
		tmp = t_1 - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -3.5e-20], N[(N[(N[Log[y], $MachinePrecision] + t$95$1), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[a, 0.00055], N[(N[Log[N[(y * N[(z * N[Power[t, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision], N[(t$95$1 - t), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.50000000000000003e-20

   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 97.9%

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

      1. *-commutative 97.9%

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

   6. Simplified 97.9%

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

   7. Taylor expanded in x around 0 61.6%

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

   if -3.50000000000000003e-20 < a < 5.50000000000000033e-4

   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 0 98.6%

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

   5. Taylor expanded in z around inf 98.7%

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

      1. associate-+r+ 98.7%

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

      2. +-commutative 98.7%

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

      3. mul-1-neg 98.7%

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

      4. log-rec 98.7%

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

      5. remove-double-neg 98.7%

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

      6. log-prod 77.6%

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

      7. +-commutative 77.6%

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

      8. log-pow 77.6%

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

      9. log-prod 71.4%

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

      10. *-commutative 71.4%

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

   7. Simplified 71.4%

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

   8. Taylor expanded in y around inf 44.5%

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

   9. Taylor expanded in y around 0 57.5%

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

       1. log-prod 61.0%

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

       2. unpow1/2 61.0%

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

       3. log-pow 61.0%

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

       4. *-commutative 61.0%

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

       5. log-rec 61.0%

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

       6. distribute-lft-neg-out 61.0%

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

       7. distribute-rgt-neg-in 61.0%

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

       8. metadata-eval 61.0%

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

       9. *-commutative 61.0%

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

       10. log-pow 61.0%

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

       11. log-prod 57.5%

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

       12. log-prod 45.3%

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

       13. *-commutative 45.3%

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

   11. Simplified 45.3%

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

   if 5.50000000000000033e-4 < a

   1. Initial program 99.8%

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

      1. associate--l+ 99.7%

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

      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 x around 0 80.2%

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

   5. Taylor expanded in a around inf 97.4%

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

      1. *-commutative 97.4%

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

   7. Simplified 97.4%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.5 \cdot 10^{-20}:\\ \;\;\;\;\left(\log y + a \cdot \log t\right) - t\\ \mathbf{elif}\;a \leq 0.00055:\\ \;\;\;\;\log \left(y \cdot \left(z \cdot {t}^{-0.5}\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \]

Alternative 11: 58.0% 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 77.3%

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

   1. *-commutative 77.3%

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

6. Simplified 77.3%

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

7. Taylor expanded in x around 0 55.1%

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

8. Final simplification 55.1%

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

Alternative 12: 41.4% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[Log[N[(x + y), $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 77.3%

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

   1. *-commutative 77.3%

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

6. Simplified 77.3%

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

7. Taylor expanded in a around 0 39.0%

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

   1. +-commutative 39.0%

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

9. Simplified 39.0%

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

10. Final simplification 39.0%

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

Alternative 13: 74.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ a \cdot \log t - t \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

   \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
2. 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 x around 0 66.5%

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

5. Taylor expanded in a around inf 74.5%

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

   1. *-commutative 74.5%

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

7. Simplified 74.5%

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

8. Final simplification 74.5%

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

Alternative 14: 38.1% accurate, 156.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

   \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
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 35.8%

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

   1. neg-mul-1 35.8%

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

6. Simplified 35.8%

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

7. Final simplification 35.8%

   \[\leadsto -t \]

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 2023275 
(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))))