Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 15.8s

Alternatives: 18

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

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

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. Add Preprocessing

3. Final simplification 99.6%

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

Alternative 2: 73.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.82:\\ \;\;\;\;a \cdot \left(\log \left(\sqrt[3]{t}\right) \cdot 3\right) - t\\ \mathbf{elif}\;a \leq 8.8 \cdot 10^{-20}:\\ \;\;\;\;\left(\left(\log z + \log y\right) + \log t \cdot -0.5\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log y + \left(\log z + a \cdot \log t\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -0.82)
   (- (* a (* (log (cbrt t)) 3.0)) t)
   (if (<= a 8.8e-20)
     (- (+ (+ (log z) (log y)) (* (log t) -0.5)) t)
     (- (+ (log y) (+ (log z) (* a (log t)))) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -0.82) {
		tmp = (a * (log(cbrt(t)) * 3.0)) - t;
	} else if (a <= 8.8e-20) {
		tmp = ((log(z) + log(y)) + (log(t) * -0.5)) - t;
	} else {
		tmp = (log(y) + (log(z) + (a * log(t)))) - t;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -0.82) {
		tmp = (a * (Math.log(Math.cbrt(t)) * 3.0)) - t;
	} else if (a <= 8.8e-20) {
		tmp = ((Math.log(z) + Math.log(y)) + (Math.log(t) * -0.5)) - t;
	} else {
		tmp = (Math.log(y) + (Math.log(z) + (a * Math.log(t)))) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -0.82)
		tmp = Float64(Float64(a * Float64(log(cbrt(t)) * 3.0)) - t);
	elseif (a <= 8.8e-20)
		tmp = Float64(Float64(Float64(log(z) + log(y)) + Float64(log(t) * -0.5)) - t);
	else
		tmp = Float64(Float64(log(y) + Float64(log(z) + Float64(a * log(t)))) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -0.82], N[(N[(a * N[(N[Log[N[Power[t, 1/3], $MachinePrecision]], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[a, 8.8e-20], N[(N[(N[(N[Log[z], $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] + N[(N[Log[z], $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -0.819999999999999951

   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. Add Preprocessing
   3. Step-by-step derivation

      1. add-exp-log 56.9%

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

   4. Applied egg-rr 56.9%

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

      1. rem-exp-log 99.7%

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

      2. add-cube-cbrt 99.7%

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

      3. unpow2 99.7%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log \left(\color{blue}{{\left(\sqrt[3]{t}\right)}^{2}} \cdot \sqrt[3]{t}\right) \]

      4. sum-log 99.6%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\left(\log \left({\left(\sqrt[3]{t}\right)}^{2}\right) + \log \left(\sqrt[3]{t}\right)\right)} \]

      5. log-pow 99.6%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \left(\color{blue}{2 \cdot \log \left(\sqrt[3]{t}\right)} + \log \left(\sqrt[3]{t}\right)\right) \]

   6. Applied egg-rr 99.6%

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\left(2 \cdot \log \left(\sqrt[3]{t}\right) + \log \left(\sqrt[3]{t}\right)\right)} \]
   7. Step-by-step derivation

      1. distribute-lft1-in 99.6%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\left(\left(2 + 1\right) \cdot \log \left(\sqrt[3]{t}\right)\right)} \]

      2. metadata-eval 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in x around 0 83.4%

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

   10. Taylor expanded in a around inf 98.4%

       \[\leadsto \color{blue}{3 \cdot \left(a \cdot \log \left(\sqrt[3]{t}\right)\right)} - t \]
   11. Step-by-step derivation

       1. *-commutative 98.4%

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

       2. associate-*r* 98.2%

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

   12. Simplified 98.2%

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

   if -0.819999999999999951 < a < 8.79999999999999964e-20

   1. Initial program 99.6%

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

      1. associate--l+ 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-+l+ 99.6%

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

      4. +-commutative 99.6%

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

      5. fma-define 99.6%

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

      6. sub-neg 99.6%

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

      7. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 64.4%

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

   6. Taylor expanded in a around 0 64.4%

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

      1. +-commutative 64.4%

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

      2. +-commutative 64.4%

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

      3. associate-+l+ 64.4%

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

   8. Simplified 64.4%

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

   if 8.79999999999999964e-20 < 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. +-commutative 99.7%

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

      3. associate-+l+ 99.7%

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

      4. +-commutative 99.7%

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

      5. fma-define 99.7%

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

      6. sub-neg 99.7%

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

      7. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 61.6%

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

   6. Taylor expanded in a around inf 61.6%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.82:\\ \;\;\;\;a \cdot \left(\log \left(\sqrt[3]{t}\right) \cdot 3\right) - t\\ \mathbf{elif}\;a \leq 8.8 \cdot 10^{-20}:\\ \;\;\;\;\left(\left(\log z + \log y\right) + \log t \cdot -0.5\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log y + \left(\log z + a \cdot \log t\right)\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.62:\\ \;\;\;\;a \cdot \left(\log \left(\sqrt[3]{t}\right) \cdot 3\right) - t\\ \mathbf{elif}\;a \leq 8.8 \cdot 10^{-20}:\\ \;\;\;\;\left(\log y + \left(\log z + \log t \cdot -0.5\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log y + \left(\log z + a \cdot \log t\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -0.62)
   (- (* a (* (log (cbrt t)) 3.0)) t)
   (if (<= a 8.8e-20)
     (- (+ (log y) (+ (log z) (* (log t) -0.5))) t)
     (- (+ (log y) (+ (log z) (* a (log t)))) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -0.62) {
		tmp = (a * (log(cbrt(t)) * 3.0)) - t;
	} else if (a <= 8.8e-20) {
		tmp = (log(y) + (log(z) + (log(t) * -0.5))) - t;
	} else {
		tmp = (log(y) + (log(z) + (a * log(t)))) - t;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -0.62) {
		tmp = (a * (Math.log(Math.cbrt(t)) * 3.0)) - t;
	} else if (a <= 8.8e-20) {
		tmp = (Math.log(y) + (Math.log(z) + (Math.log(t) * -0.5))) - t;
	} else {
		tmp = (Math.log(y) + (Math.log(z) + (a * Math.log(t)))) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -0.62)
		tmp = Float64(Float64(a * Float64(log(cbrt(t)) * 3.0)) - t);
	elseif (a <= 8.8e-20)
		tmp = Float64(Float64(log(y) + Float64(log(z) + Float64(log(t) * -0.5))) - t);
	else
		tmp = Float64(Float64(log(y) + Float64(log(z) + Float64(a * log(t)))) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -0.62], N[(N[(a * N[(N[Log[N[Power[t, 1/3], $MachinePrecision]], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[a, 8.8e-20], N[(N[(N[Log[y], $MachinePrecision] + N[(N[Log[z], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] + N[(N[Log[z], $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -0.619999999999999996

   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. Add Preprocessing
   3. Step-by-step derivation

      1. add-exp-log 56.9%

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

   4. Applied egg-rr 56.9%

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

      1. rem-exp-log 99.7%

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

      2. add-cube-cbrt 99.7%

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

      3. unpow2 99.7%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log \left(\color{blue}{{\left(\sqrt[3]{t}\right)}^{2}} \cdot \sqrt[3]{t}\right) \]

      4. sum-log 99.6%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\left(\log \left({\left(\sqrt[3]{t}\right)}^{2}\right) + \log \left(\sqrt[3]{t}\right)\right)} \]

      5. log-pow 99.6%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \left(\color{blue}{2 \cdot \log \left(\sqrt[3]{t}\right)} + \log \left(\sqrt[3]{t}\right)\right) \]

   6. Applied egg-rr 99.6%

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\left(2 \cdot \log \left(\sqrt[3]{t}\right) + \log \left(\sqrt[3]{t}\right)\right)} \]
   7. Step-by-step derivation

      1. distribute-lft1-in 99.6%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\left(\left(2 + 1\right) \cdot \log \left(\sqrt[3]{t}\right)\right)} \]

      2. metadata-eval 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in x around 0 83.4%

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

   10. Taylor expanded in a around inf 98.4%

       \[\leadsto \color{blue}{3 \cdot \left(a \cdot \log \left(\sqrt[3]{t}\right)\right)} - t \]
   11. Step-by-step derivation

       1. *-commutative 98.4%

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

       2. associate-*r* 98.2%

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

   12. Simplified 98.2%

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

   if -0.619999999999999996 < a < 8.79999999999999964e-20

   1. Initial program 99.6%

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

      1. associate--l+ 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-+l+ 99.6%

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

      4. +-commutative 99.6%

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

      5. fma-define 99.6%

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

      6. sub-neg 99.6%

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

      7. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 64.4%

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

   6. Taylor expanded in a around 0 64.4%

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

      1. +-commutative 64.4%

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

   8. Simplified 64.4%

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

   if 8.79999999999999964e-20 < 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. +-commutative 99.7%

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

      3. associate-+l+ 99.7%

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

      4. +-commutative 99.7%

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

      5. fma-define 99.7%

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

      6. sub-neg 99.7%

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

      7. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 61.6%

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

   6. Taylor expanded in a around inf 61.6%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.62:\\ \;\;\;\;a \cdot \left(\log \left(\sqrt[3]{t}\right) \cdot 3\right) - t\\ \mathbf{elif}\;a \leq 8.8 \cdot 10^{-20}:\\ \;\;\;\;\left(\log y + \left(\log z + \log t \cdot -0.5\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log y + \left(\log z + a \cdot \log t\right)\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.216:\\ \;\;\;\;a \cdot \left(\log \left(\sqrt[3]{t}\right) \cdot 3\right) - t\\ \mathbf{elif}\;a \leq 8.8 \cdot 10^{-20}:\\ \;\;\;\;\left(\log y + \log \left(z \cdot {t}^{-0.5}\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log y + \left(\log z + a \cdot \log t\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -0.216)
   (- (* a (* (log (cbrt t)) 3.0)) t)
   (if (<= a 8.8e-20)
     (- (+ (log y) (log (* z (pow t -0.5)))) t)
     (- (+ (log y) (+ (log z) (* a (log t)))) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -0.216) {
		tmp = (a * (log(cbrt(t)) * 3.0)) - t;
	} else if (a <= 8.8e-20) {
		tmp = (log(y) + log((z * pow(t, -0.5)))) - t;
	} else {
		tmp = (log(y) + (log(z) + (a * log(t)))) - t;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -0.216) {
		tmp = (a * (Math.log(Math.cbrt(t)) * 3.0)) - t;
	} else if (a <= 8.8e-20) {
		tmp = (Math.log(y) + Math.log((z * Math.pow(t, -0.5)))) - t;
	} else {
		tmp = (Math.log(y) + (Math.log(z) + (a * Math.log(t)))) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -0.216)
		tmp = Float64(Float64(a * Float64(log(cbrt(t)) * 3.0)) - t);
	elseif (a <= 8.8e-20)
		tmp = Float64(Float64(log(y) + log(Float64(z * (t ^ -0.5)))) - t);
	else
		tmp = Float64(Float64(log(y) + Float64(log(z) + Float64(a * log(t)))) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -0.216], N[(N[(a * N[(N[Log[N[Power[t, 1/3], $MachinePrecision]], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[a, 8.8e-20], N[(N[(N[Log[y], $MachinePrecision] + N[Log[N[(z * N[Power[t, -0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] + N[(N[Log[z], $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -0.215999999999999998

   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. Add Preprocessing
   3. Step-by-step derivation

      1. add-exp-log 56.9%

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

   4. Applied egg-rr 56.9%

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

      1. rem-exp-log 99.7%

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

      2. add-cube-cbrt 99.7%

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

      3. unpow2 99.7%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log \left(\color{blue}{{\left(\sqrt[3]{t}\right)}^{2}} \cdot \sqrt[3]{t}\right) \]

      4. sum-log 99.6%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\left(\log \left({\left(\sqrt[3]{t}\right)}^{2}\right) + \log \left(\sqrt[3]{t}\right)\right)} \]

      5. log-pow 99.6%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \left(\color{blue}{2 \cdot \log \left(\sqrt[3]{t}\right)} + \log \left(\sqrt[3]{t}\right)\right) \]

   6. Applied egg-rr 99.6%

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\left(2 \cdot \log \left(\sqrt[3]{t}\right) + \log \left(\sqrt[3]{t}\right)\right)} \]
   7. Step-by-step derivation

      1. distribute-lft1-in 99.6%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\left(\left(2 + 1\right) \cdot \log \left(\sqrt[3]{t}\right)\right)} \]

      2. metadata-eval 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in x around 0 83.4%

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

   10. Taylor expanded in a around inf 98.4%

       \[\leadsto \color{blue}{3 \cdot \left(a \cdot \log \left(\sqrt[3]{t}\right)\right)} - t \]
   11. Step-by-step derivation

       1. *-commutative 98.4%

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

       2. associate-*r* 98.2%

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

   12. Simplified 98.2%

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

   if -0.215999999999999998 < a < 8.79999999999999964e-20

   1. Initial program 99.6%

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

      1. associate--l+ 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-+l+ 99.6%

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

      4. +-commutative 99.6%

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

      5. fma-define 99.6%

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

      6. sub-neg 99.6%

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

      7. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 64.4%

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

   6. Taylor expanded in a around 0 64.4%

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

      1. +-commutative 64.4%

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

      2. +-commutative 64.4%

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

      3. associate-+l+ 64.4%

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

   8. Simplified 64.4%

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

      1. *-un-lft-identity 64.4%

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

      2. +-commutative 64.4%

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

      3. sum-log 48.2%

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

      4. add-log-exp 48.2%

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

      5. sum-log 44.7%

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

      6. add-exp-log 44.6%

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

      7. sum-log 44.3%

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

      8. +-commutative 44.3%

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

      9. sum-log 44.6%

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

      10. add-exp-log 44.7%

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

      11. *-commutative 44.7%

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

      12. pow-to-exp 44.8%

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

   10. Applied egg-rr 44.8%

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

       1. *-lft-identity 44.8%

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

   12. Simplified 44.8%

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

       1. associate-*l* 46.5%

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

       2. log-prod 57.4%

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

   14. Applied egg-rr 57.4%

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

   if 8.79999999999999964e-20 < 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. +-commutative 99.7%

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

      3. associate-+l+ 99.7%

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

      4. +-commutative 99.7%

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

      5. fma-define 99.7%

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

      6. sub-neg 99.7%

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

      7. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 61.6%

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

   6. Taylor expanded in a around inf 61.6%

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

Alternative 5: 76.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.124 \lor \neg \left(a \leq 8.8 \cdot 10^{-20}\right):\\ \;\;\;\;a \cdot \left(\log \left(\sqrt[3]{t}\right) \cdot 3\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log y + \log \left(z \cdot {t}^{-0.5}\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -0.124) (not (<= a 8.8e-20)))
   (- (* a (* (log (cbrt t)) 3.0)) t)
   (- (+ (log y) (log (* z (pow t -0.5)))) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -0.124) || !(a <= 8.8e-20)) {
		tmp = (a * (log(cbrt(t)) * 3.0)) - t;
	} else {
		tmp = (log(y) + log((z * pow(t, -0.5)))) - t;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -0.124) || !(a <= 8.8e-20)) {
		tmp = (a * (Math.log(Math.cbrt(t)) * 3.0)) - t;
	} else {
		tmp = (Math.log(y) + Math.log((z * Math.pow(t, -0.5)))) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -0.124) || !(a <= 8.8e-20))
		tmp = Float64(Float64(a * Float64(log(cbrt(t)) * 3.0)) - t);
	else
		tmp = Float64(Float64(log(y) + log(Float64(z * (t ^ -0.5)))) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -0.124], N[Not[LessEqual[a, 8.8e-20]], $MachinePrecision]], N[(N[(a * N[(N[Log[N[Power[t, 1/3], $MachinePrecision]], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] + N[Log[N[(z * N[Power[t, -0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -0.124 or 8.79999999999999964e-20 < 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. Add Preprocessing
   3. Step-by-step derivation

      1. add-exp-log 52.5%

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

   4. Applied egg-rr 52.5%

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

      1. rem-exp-log 99.7%

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

      2. add-cube-cbrt 99.7%

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

      3. unpow2 99.7%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log \left(\color{blue}{{\left(\sqrt[3]{t}\right)}^{2}} \cdot \sqrt[3]{t}\right) \]

      4. sum-log 99.6%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\left(\log \left({\left(\sqrt[3]{t}\right)}^{2}\right) + \log \left(\sqrt[3]{t}\right)\right)} \]

      5. log-pow 99.6%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \left(\color{blue}{2 \cdot \log \left(\sqrt[3]{t}\right)} + \log \left(\sqrt[3]{t}\right)\right) \]

   6. Applied egg-rr 99.6%

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\left(2 \cdot \log \left(\sqrt[3]{t}\right) + \log \left(\sqrt[3]{t}\right)\right)} \]
   7. Step-by-step derivation

      1. distribute-lft1-in 99.6%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\left(\left(2 + 1\right) \cdot \log \left(\sqrt[3]{t}\right)\right)} \]

      2. metadata-eval 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in x around 0 72.5%

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

   10. Taylor expanded in a around inf 98.9%

       \[\leadsto \color{blue}{3 \cdot \left(a \cdot \log \left(\sqrt[3]{t}\right)\right)} - t \]
   11. Step-by-step derivation

       1. *-commutative 98.9%

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

       2. associate-*r* 98.8%

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

   12. Simplified 98.8%

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

   if -0.124 < a < 8.79999999999999964e-20

   1. Initial program 99.6%

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

      1. associate--l+ 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-+l+ 99.6%

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

      4. +-commutative 99.6%

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

      5. fma-define 99.6%

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

      6. sub-neg 99.6%

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

      7. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 64.4%

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

   6. Taylor expanded in a around 0 64.4%

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

      1. +-commutative 64.4%

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

      2. +-commutative 64.4%

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

      3. associate-+l+ 64.4%

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

   8. Simplified 64.4%

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

      1. *-un-lft-identity 64.4%

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

      2. +-commutative 64.4%

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

      3. sum-log 48.2%

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

      4. add-log-exp 48.2%

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

      5. sum-log 44.7%

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

      6. add-exp-log 44.6%

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

      7. sum-log 44.3%

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

      8. +-commutative 44.3%

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

      9. sum-log 44.6%

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

      10. add-exp-log 44.7%

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

      11. *-commutative 44.7%

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

      12. pow-to-exp 44.8%

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

   10. Applied egg-rr 44.8%

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

       1. *-lft-identity 44.8%

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

   12. Simplified 44.8%

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

       1. associate-*l* 46.5%

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

       2. log-prod 57.4%

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

   14. Applied egg-rr 57.4%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.124 \lor \neg \left(a \leq 8.8 \cdot 10^{-20}\right):\\ \;\;\;\;a \cdot \left(\log \left(\sqrt[3]{t}\right) \cdot 3\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log y + \log \left(z \cdot {t}^{-0.5}\right)\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate--l+ 99.6%

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

   2. +-commutative 99.6%

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

   3. associate-+l+ 99.6%

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

   4. +-commutative 99.6%

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

   5. fma-define 99.6%

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

   6. sub-neg 99.6%

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

   7. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in x around 0 68.2%

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

   1. associate-+r+ 68.2%

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

   2. sub-neg 68.2%

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

   3. metadata-eval 68.2%

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

7. Simplified 68.2%

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

8. Final simplification 68.2%

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

Alternative 7: 67.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate--l+ 99.6%

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

   2. +-commutative 99.6%

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

   3. associate-+l+ 99.6%

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

   4. +-commutative 99.6%

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

   5. fma-define 99.6%

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

   6. sub-neg 99.6%

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

   7. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in x around 0 68.2%

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

6. Final simplification 68.2%

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

Alternative 8: 72.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.05 \cdot 10^{-5} \lor \neg \left(a \leq 4.2 \cdot 10^{-36}\right):\\ \;\;\;\;a \cdot \left(\log \left(\sqrt[3]{t}\right) \cdot 3\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log t \cdot -0.5 + \log \left(y \cdot z\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -3.05e-5) (not (<= a 4.2e-36)))
   (- (* a (* (log (cbrt t)) 3.0)) t)
   (- (+ (* (log t) -0.5) (log (* y z))) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -3.05e-5) || !(a <= 4.2e-36)) {
		tmp = (a * (log(cbrt(t)) * 3.0)) - t;
	} else {
		tmp = ((log(t) * -0.5) + log((y * z))) - t;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -3.05e-5) || !(a <= 4.2e-36)) {
		tmp = (a * (Math.log(Math.cbrt(t)) * 3.0)) - t;
	} else {
		tmp = ((Math.log(t) * -0.5) + Math.log((y * z))) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -3.05e-5) || !(a <= 4.2e-36))
		tmp = Float64(Float64(a * Float64(log(cbrt(t)) * 3.0)) - t);
	else
		tmp = Float64(Float64(Float64(log(t) * -0.5) + log(Float64(y * z))) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -3.05e-5], N[Not[LessEqual[a, 4.2e-36]], $MachinePrecision]], N[(N[(a * N[(N[Log[N[Power[t, 1/3], $MachinePrecision]], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[(N[Log[t], $MachinePrecision] * -0.5), $MachinePrecision] + N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.04999999999999994e-5 or 4.19999999999999982e-36 < 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. Add Preprocessing
   3. Step-by-step derivation

      1. add-exp-log 52.9%

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

   4. Applied egg-rr 52.9%

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

      1. rem-exp-log 99.7%

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

      2. add-cube-cbrt 99.7%

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

      3. unpow2 99.7%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log \left(\color{blue}{{\left(\sqrt[3]{t}\right)}^{2}} \cdot \sqrt[3]{t}\right) \]

      4. sum-log 99.6%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\left(\log \left({\left(\sqrt[3]{t}\right)}^{2}\right) + \log \left(\sqrt[3]{t}\right)\right)} \]

      5. log-pow 99.6%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \left(\color{blue}{2 \cdot \log \left(\sqrt[3]{t}\right)} + \log \left(\sqrt[3]{t}\right)\right) \]

   6. Applied egg-rr 99.6%

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\left(2 \cdot \log \left(\sqrt[3]{t}\right) + \log \left(\sqrt[3]{t}\right)\right)} \]
   7. Step-by-step derivation

      1. distribute-lft1-in 99.6%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\left(\left(2 + 1\right) \cdot \log \left(\sqrt[3]{t}\right)\right)} \]

      2. metadata-eval 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in x around 0 72.4%

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

   10. Taylor expanded in a around inf 98.2%

       \[\leadsto \color{blue}{3 \cdot \left(a \cdot \log \left(\sqrt[3]{t}\right)\right)} - t \]
   11. Step-by-step derivation

       1. *-commutative 98.2%

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

       2. associate-*r* 98.1%

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

   12. Simplified 98.1%

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

   if -3.04999999999999994e-5 < a < 4.19999999999999982e-36

   1. Initial program 99.6%

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

      1. associate--l+ 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-+l+ 99.6%

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

      4. +-commutative 99.6%

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

      5. fma-define 99.6%

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

      6. sub-neg 99.6%

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

      7. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 64.4%

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

   6. Taylor expanded in a around 0 64.4%

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

      1. +-commutative 64.4%

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

      2. +-commutative 64.4%

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

      3. associate-+l+ 64.4%

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

   8. Simplified 64.4%

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

      1. *-un-lft-identity 64.4%

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

      2. +-commutative 64.4%

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

      3. sum-log 48.6%

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

   10. Applied egg-rr 48.6%

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

       1. *-lft-identity 48.6%

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

   12. Simplified 48.6%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.05 \cdot 10^{-5} \lor \neg \left(a \leq 4.2 \cdot 10^{-36}\right):\\ \;\;\;\;a \cdot \left(\log \left(\sqrt[3]{t}\right) \cdot 3\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log t \cdot -0.5 + \log \left(y \cdot z\right)\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 71.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.00017 \lor \neg \left(a \leq 9.5 \cdot 10^{-36}\right):\\ \;\;\;\;a \cdot \left(\log \left(\sqrt[3]{t}\right) \cdot 3\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log \left({t}^{-0.5} \cdot \left(y \cdot z\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -0.00017) (not (<= a 9.5e-36)))
   (- (* a (* (log (cbrt t)) 3.0)) t)
   (- (log (* (pow t -0.5) (* y z))) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -0.00017) || !(a <= 9.5e-36)) {
		tmp = (a * (log(cbrt(t)) * 3.0)) - t;
	} else {
		tmp = log((pow(t, -0.5) * (y * z))) - t;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -0.00017) || !(a <= 9.5e-36)) {
		tmp = (a * (Math.log(Math.cbrt(t)) * 3.0)) - t;
	} else {
		tmp = Math.log((Math.pow(t, -0.5) * (y * z))) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -0.00017) || !(a <= 9.5e-36))
		tmp = Float64(Float64(a * Float64(log(cbrt(t)) * 3.0)) - t);
	else
		tmp = Float64(log(Float64((t ^ -0.5) * Float64(y * z))) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -0.00017], N[Not[LessEqual[a, 9.5e-36]], $MachinePrecision]], N[(N[(a * N[(N[Log[N[Power[t, 1/3], $MachinePrecision]], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[Log[N[(N[Power[t, -0.5], $MachinePrecision] * N[(y * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.7e-4 or 9.5000000000000003e-36 < 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. Add Preprocessing
   3. Step-by-step derivation

      1. add-exp-log 52.9%

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

   4. Applied egg-rr 52.9%

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

      1. rem-exp-log 99.7%

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

      2. add-cube-cbrt 99.7%

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

      3. unpow2 99.7%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log \left(\color{blue}{{\left(\sqrt[3]{t}\right)}^{2}} \cdot \sqrt[3]{t}\right) \]

      4. sum-log 99.6%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\left(\log \left({\left(\sqrt[3]{t}\right)}^{2}\right) + \log \left(\sqrt[3]{t}\right)\right)} \]

      5. log-pow 99.6%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \left(\color{blue}{2 \cdot \log \left(\sqrt[3]{t}\right)} + \log \left(\sqrt[3]{t}\right)\right) \]

   6. Applied egg-rr 99.6%

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\left(2 \cdot \log \left(\sqrt[3]{t}\right) + \log \left(\sqrt[3]{t}\right)\right)} \]
   7. Step-by-step derivation

      1. distribute-lft1-in 99.6%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\left(\left(2 + 1\right) \cdot \log \left(\sqrt[3]{t}\right)\right)} \]

      2. metadata-eval 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in x around 0 72.4%

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

   10. Taylor expanded in a around inf 98.2%

       \[\leadsto \color{blue}{3 \cdot \left(a \cdot \log \left(\sqrt[3]{t}\right)\right)} - t \]
   11. Step-by-step derivation

       1. *-commutative 98.2%

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

       2. associate-*r* 98.1%

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

   12. Simplified 98.1%

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

   if -1.7e-4 < a < 9.5000000000000003e-36

   1. Initial program 99.6%

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

      1. associate--l+ 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-+l+ 99.6%

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

      4. +-commutative 99.6%

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

      5. fma-define 99.6%

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

      6. sub-neg 99.6%

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

      7. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 64.4%

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

   6. Taylor expanded in a around 0 64.4%

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

      1. +-commutative 64.4%

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

      2. +-commutative 64.4%

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

      3. associate-+l+ 64.4%

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

   8. Simplified 64.4%

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

      1. *-un-lft-identity 64.4%

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

      2. +-commutative 64.4%

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

      3. sum-log 48.6%

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

      4. add-log-exp 48.6%

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

      5. sum-log 45.0%

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

      6. add-exp-log 44.8%

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

      7. sum-log 44.6%

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

      8. +-commutative 44.6%

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

      9. sum-log 44.8%

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

      10. add-exp-log 45.0%

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

      11. *-commutative 45.0%

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

      12. pow-to-exp 45.1%

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

   10. Applied egg-rr 45.1%

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

       1. *-lft-identity 45.1%

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

   12. Simplified 45.1%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.00017 \lor \neg \left(a \leq 9.5 \cdot 10^{-36}\right):\\ \;\;\;\;a \cdot \left(\log \left(\sqrt[3]{t}\right) \cdot 3\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log \left({t}^{-0.5} \cdot \left(y \cdot z\right)\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 71.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.4 \cdot 10^{-5} \lor \neg \left(a \leq 6 \cdot 10^{-36}\right):\\ \;\;\;\;a \cdot \left(\log \left(\sqrt[3]{t}\right) \cdot 3\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log \left(y \cdot \left(z \cdot {t}^{-0.5}\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -1.4e-5) (not (<= a 6e-36)))
   (- (* a (* (log (cbrt t)) 3.0)) t)
   (- (log (* y (* z (pow t -0.5)))) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -1.4e-5) || !(a <= 6e-36)) {
		tmp = (a * (log(cbrt(t)) * 3.0)) - t;
	} else {
		tmp = log((y * (z * pow(t, -0.5)))) - t;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -1.4e-5) || !(a <= 6e-36)) {
		tmp = (a * (Math.log(Math.cbrt(t)) * 3.0)) - t;
	} else {
		tmp = Math.log((y * (z * Math.pow(t, -0.5)))) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -1.4e-5) || !(a <= 6e-36))
		tmp = Float64(Float64(a * Float64(log(cbrt(t)) * 3.0)) - t);
	else
		tmp = Float64(log(Float64(y * Float64(z * (t ^ -0.5)))) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -1.4e-5], N[Not[LessEqual[a, 6e-36]], $MachinePrecision]], N[(N[(a * N[(N[Log[N[Power[t, 1/3], $MachinePrecision]], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[Log[N[(y * N[(z * N[Power[t, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.39999999999999998e-5 or 6.0000000000000003e-36 < 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. Add Preprocessing
   3. Step-by-step derivation

      1. add-exp-log 52.9%

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

   4. Applied egg-rr 52.9%

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

      1. rem-exp-log 99.7%

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

      2. add-cube-cbrt 99.7%

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

      3. unpow2 99.7%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log \left(\color{blue}{{\left(\sqrt[3]{t}\right)}^{2}} \cdot \sqrt[3]{t}\right) \]

      4. sum-log 99.6%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\left(\log \left({\left(\sqrt[3]{t}\right)}^{2}\right) + \log \left(\sqrt[3]{t}\right)\right)} \]

      5. log-pow 99.6%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \left(\color{blue}{2 \cdot \log \left(\sqrt[3]{t}\right)} + \log \left(\sqrt[3]{t}\right)\right) \]

   6. Applied egg-rr 99.6%

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\left(2 \cdot \log \left(\sqrt[3]{t}\right) + \log \left(\sqrt[3]{t}\right)\right)} \]
   7. Step-by-step derivation

      1. distribute-lft1-in 99.6%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\left(\left(2 + 1\right) \cdot \log \left(\sqrt[3]{t}\right)\right)} \]

      2. metadata-eval 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in x around 0 72.4%

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

   10. Taylor expanded in a around inf 98.2%

       \[\leadsto \color{blue}{3 \cdot \left(a \cdot \log \left(\sqrt[3]{t}\right)\right)} - t \]
   11. Step-by-step derivation

       1. *-commutative 98.2%

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

       2. associate-*r* 98.1%

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

   12. Simplified 98.1%

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

   if -1.39999999999999998e-5 < a < 6.0000000000000003e-36

   1. Initial program 99.6%

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

      1. associate--l+ 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-+l+ 99.6%

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

      4. +-commutative 99.6%

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

      5. fma-define 99.6%

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

      6. sub-neg 99.6%

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

      7. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 64.4%

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

   6. Taylor expanded in a around 0 64.4%

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

      1. +-commutative 64.4%

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

      2. +-commutative 64.4%

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

      3. associate-+l+ 64.4%

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

   8. Simplified 64.4%

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

      1. *-un-lft-identity 64.4%

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

      2. +-commutative 64.4%

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

      3. sum-log 48.6%

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

      4. add-log-exp 48.6%

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

      5. sum-log 45.0%

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

      6. add-exp-log 44.8%

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

      7. sum-log 44.6%

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

      8. +-commutative 44.6%

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

      9. sum-log 44.8%

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

      10. add-exp-log 45.0%

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

      11. *-commutative 45.0%

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

      12. pow-to-exp 45.1%

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

   10. Applied egg-rr 45.1%

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

       1. *-lft-identity 45.1%

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

   12. Simplified 45.1%

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

       1. add-sqr-sqrt 22.0%

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

       2. add-sqr-sqrt 21.9%

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

       3. difference-of-squares 21.9%

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

       4. associate-*l* 21.2%

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

       5. associate-*l* 24.2%

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

   14. Applied egg-rr 24.2%

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

       1. difference-of-squares 24.2%

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

       2. rem-square-sqrt 46.5%

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

       3. rem-square-sqrt 46.8%

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

   16. Simplified 46.8%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.4 \cdot 10^{-5} \lor \neg \left(a \leq 6 \cdot 10^{-36}\right):\\ \;\;\;\;a \cdot \left(\log \left(\sqrt[3]{t}\right) \cdot 3\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log \left(y \cdot \left(z \cdot {t}^{-0.5}\right)\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 86.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 165000000000:\\ \;\;\;\;\left(\log \left(\left(x + y\right) \cdot z\right) + \log t \cdot \left(a - 0.5\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\log \left(\sqrt[3]{t}\right) \cdot 3\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 165000000000.0)
   (- (+ (log (* (+ x y) z)) (* (log t) (- a 0.5))) t)
   (- (* a (* (log (cbrt t)) 3.0)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 165000000000.0) {
		tmp = (log(((x + y) * z)) + (log(t) * (a - 0.5))) - t;
	} else {
		tmp = (a * (log(cbrt(t)) * 3.0)) - t;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 165000000000.0) {
		tmp = (Math.log(((x + y) * z)) + (Math.log(t) * (a - 0.5))) - t;
	} else {
		tmp = (a * (Math.log(Math.cbrt(t)) * 3.0)) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 165000000000.0)
		tmp = Float64(Float64(log(Float64(Float64(x + y) * z)) + Float64(log(t) * Float64(a - 0.5))) - t);
	else
		tmp = Float64(Float64(a * Float64(log(cbrt(t)) * 3.0)) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 165000000000.0], N[(N[(N[Log[N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(a * N[(N[Log[N[Power[t, 1/3], $MachinePrecision]], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.65e11

   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%

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

      2. associate--l+ 99.3%

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

      3. sub-neg 99.3%

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

      4. +-commutative 99.3%

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

      5. *-commutative 99.3%

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

      6. distribute-rgt-neg-in 99.3%

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

      7. fma-undefine 99.3%

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

      8. sub-neg 99.3%

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

      9. +-commutative 99.3%

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

      10. distribute-neg-in 99.3%

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

      11. metadata-eval 99.3%

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

      12. metadata-eval 99.3%

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

      13. unsub-neg 99.3%

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

   3. Simplified 99.3%

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

      1. associate-+r- 99.4%

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

      2. fma-undefine 99.4%

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

      3. associate--r+ 99.4%

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

      4. sum-log 75.7%

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

   6. Applied egg-rr 75.7%

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

   if 1.65e11 < 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. Add Preprocessing
   3. Step-by-step derivation

      1. add-exp-log 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. rem-exp-log 99.9%

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

      2. add-cube-cbrt 99.9%

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

      3. unpow2 99.9%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log \left(\color{blue}{{\left(\sqrt[3]{t}\right)}^{2}} \cdot \sqrt[3]{t}\right) \]

      4. sum-log 99.9%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\left(\log \left({\left(\sqrt[3]{t}\right)}^{2}\right) + \log \left(\sqrt[3]{t}\right)\right)} \]

      5. log-pow 99.9%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \left(\color{blue}{2 \cdot \log \left(\sqrt[3]{t}\right)} + \log \left(\sqrt[3]{t}\right)\right) \]

   6. Applied egg-rr 99.9%

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\left(2 \cdot \log \left(\sqrt[3]{t}\right) + \log \left(\sqrt[3]{t}\right)\right)} \]
   7. Step-by-step derivation

      1. distribute-lft1-in 99.9%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\left(\left(2 + 1\right) \cdot \log \left(\sqrt[3]{t}\right)\right)} \]

      2. metadata-eval 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in x around 0 77.7%

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

   10. Taylor expanded in a around inf 99.7%

       \[\leadsto \color{blue}{3 \cdot \left(a \cdot \log \left(\sqrt[3]{t}\right)\right)} - t \]
   11. Step-by-step derivation

       1. *-commutative 99.7%

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

       2. associate-*r* 99.7%

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

   12. Simplified 99.7%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 165000000000:\\ \;\;\;\;\left(\log \left(\left(x + y\right) \cdot z\right) + \log t \cdot \left(a - 0.5\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\log \left(\sqrt[3]{t}\right) \cdot 3\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 69.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1 \lor \neg \left(a \leq 9.5 \cdot 10^{-36}\right):\\ \;\;\;\;a \cdot \left(\log \left(\sqrt[3]{t}\right) \cdot 3\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log y + \left(\log z - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -1.0) (not (<= a 9.5e-36)))
   (- (* a (* (log (cbrt t)) 3.0)) t)
   (+ (log y) (- (log z) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -1.0) || !(a <= 9.5e-36)) {
		tmp = (a * (log(cbrt(t)) * 3.0)) - t;
	} else {
		tmp = log(y) + (log(z) - t);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -1.0) || !(a <= 9.5e-36)) {
		tmp = (a * (Math.log(Math.cbrt(t)) * 3.0)) - t;
	} else {
		tmp = Math.log(y) + (Math.log(z) - t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -1.0) || !(a <= 9.5e-36))
		tmp = Float64(Float64(a * Float64(log(cbrt(t)) * 3.0)) - t);
	else
		tmp = Float64(log(y) + Float64(log(z) - t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -1.0], N[Not[LessEqual[a, 9.5e-36]], $MachinePrecision]], N[(N[(a * N[(N[Log[N[Power[t, 1/3], $MachinePrecision]], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[Log[y], $MachinePrecision] + N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1 or 9.5000000000000003e-36 < 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. Add Preprocessing
   3. Step-by-step derivation

      1. add-exp-log 52.9%

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

   4. Applied egg-rr 52.9%

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

      1. rem-exp-log 99.7%

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

      2. add-cube-cbrt 99.7%

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

      3. unpow2 99.7%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log \left(\color{blue}{{\left(\sqrt[3]{t}\right)}^{2}} \cdot \sqrt[3]{t}\right) \]

      4. sum-log 99.6%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\left(\log \left({\left(\sqrt[3]{t}\right)}^{2}\right) + \log \left(\sqrt[3]{t}\right)\right)} \]

      5. log-pow 99.6%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \left(\color{blue}{2 \cdot \log \left(\sqrt[3]{t}\right)} + \log \left(\sqrt[3]{t}\right)\right) \]

   6. Applied egg-rr 99.6%

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\left(2 \cdot \log \left(\sqrt[3]{t}\right) + \log \left(\sqrt[3]{t}\right)\right)} \]
   7. Step-by-step derivation

      1. distribute-lft1-in 99.6%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\left(\left(2 + 1\right) \cdot \log \left(\sqrt[3]{t}\right)\right)} \]

      2. metadata-eval 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in x around 0 72.4%

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

   10. Taylor expanded in a around inf 98.2%

       \[\leadsto \color{blue}{3 \cdot \left(a \cdot \log \left(\sqrt[3]{t}\right)\right)} - t \]
   11. Step-by-step derivation

       1. *-commutative 98.2%

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

       2. associate-*r* 98.1%

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

   12. Simplified 98.1%

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

   if -1 < a < 9.5000000000000003e-36

   1. Initial program 99.6%

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

      1. associate--l+ 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-+l+ 99.6%

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

      4. +-commutative 99.6%

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

      5. fma-define 99.6%

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

      6. sub-neg 99.6%

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

      7. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 64.4%

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

   6. Taylor expanded in a around inf 42.6%

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

   7. Taylor expanded in a around 0 42.6%

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

      1. associate--l+ 42.6%

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

   9. Simplified 42.6%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1 \lor \neg \left(a \leq 9.5 \cdot 10^{-36}\right):\\ \;\;\;\;a \cdot \left(\log \left(\sqrt[3]{t}\right) \cdot 3\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log y + \left(\log z - t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 72.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 195000000000:\\ \;\;\;\;\left(\log t \cdot \left(a + -0.5\right) + \log \left(y \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\log \left(\sqrt[3]{t}\right) \cdot 3\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 195000000000.0)
   (- (+ (* (log t) (+ a -0.5)) (log (* y z))) t)
   (- (* a (* (log (cbrt t)) 3.0)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 195000000000.0) {
		tmp = ((log(t) * (a + -0.5)) + log((y * z))) - t;
	} else {
		tmp = (a * (log(cbrt(t)) * 3.0)) - t;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 195000000000.0) {
		tmp = ((Math.log(t) * (a + -0.5)) + Math.log((y * z))) - t;
	} else {
		tmp = (a * (Math.log(Math.cbrt(t)) * 3.0)) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 195000000000.0)
		tmp = Float64(Float64(Float64(log(t) * Float64(a + -0.5)) + log(Float64(y * z))) - t);
	else
		tmp = Float64(Float64(a * Float64(log(cbrt(t)) * 3.0)) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 195000000000.0], N[(N[(N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision]), $MachinePrecision] + N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(a * N[(N[Log[N[Power[t, 1/3], $MachinePrecision]], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.95e11

   1. Initial program 99.4%

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

      1. associate--l+ 99.4%

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

      2. +-commutative 99.4%

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

      3. associate-+l+ 99.3%

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

      4. +-commutative 99.3%

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

      5. fma-define 99.3%

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

      6. sub-neg 99.3%

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

      7. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in x around 0 59.1%

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

      1. *-un-lft-identity 59.1%

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

      2. add-log-exp 27.8%

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

      3. sum-log 22.7%

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

      4. exp-sum 22.7%

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

      5. add-exp-log 22.8%

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

      6. exp-to-pow 22.9%

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

      7. sub-neg 22.9%

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

      8. metadata-eval 22.9%

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

   7. Applied egg-rr 22.9%

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

      1. *-lft-identity 22.9%

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

   9. Simplified 22.9%

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

       1. associate-*r* 23.3%

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

       2. log-prod 24.3%

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

       3. log-pow 44.0%

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

   11. Applied egg-rr 44.0%

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

   if 1.95e11 < 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. Add Preprocessing
   3. Step-by-step derivation

      1. add-exp-log 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. rem-exp-log 99.9%

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

      2. add-cube-cbrt 99.9%

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

      3. unpow2 99.9%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log \left(\color{blue}{{\left(\sqrt[3]{t}\right)}^{2}} \cdot \sqrt[3]{t}\right) \]

      4. sum-log 99.9%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\left(\log \left({\left(\sqrt[3]{t}\right)}^{2}\right) + \log \left(\sqrt[3]{t}\right)\right)} \]

      5. log-pow 99.9%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \left(\color{blue}{2 \cdot \log \left(\sqrt[3]{t}\right)} + \log \left(\sqrt[3]{t}\right)\right) \]

   6. Applied egg-rr 99.9%

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\left(2 \cdot \log \left(\sqrt[3]{t}\right) + \log \left(\sqrt[3]{t}\right)\right)} \]
   7. Step-by-step derivation

      1. distribute-lft1-in 99.9%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\left(\left(2 + 1\right) \cdot \log \left(\sqrt[3]{t}\right)\right)} \]

      2. metadata-eval 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in x around 0 77.7%

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

   10. Taylor expanded in a around inf 99.7%

       \[\leadsto \color{blue}{3 \cdot \left(a \cdot \log \left(\sqrt[3]{t}\right)\right)} - t \]
   11. Step-by-step derivation

       1. *-commutative 99.7%

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

       2. associate-*r* 99.7%

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

   12. Simplified 99.7%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 195000000000:\\ \;\;\;\;\left(\log t \cdot \left(a + -0.5\right) + \log \left(y \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\log \left(\sqrt[3]{t}\right) \cdot 3\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 61.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 5.8 \cdot 10^{+20}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 5.8e+20) (* a (log t)) (+ -1.0 (- 1.0 t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 5.8e+20) {
		tmp = a * log(t);
	} else {
		tmp = -1.0 + (1.0 - 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 <= 5.8d+20) then
        tmp = a * log(t)
    else
        tmp = (-1.0d0) + (1.0d0 - 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 <= 5.8e+20) {
		tmp = a * Math.log(t);
	} else {
		tmp = -1.0 + (1.0 - t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 5.8e+20:
		tmp = a * math.log(t)
	else:
		tmp = -1.0 + (1.0 - t)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 5.8e+20)
		tmp = Float64(a * log(t));
	else
		tmp = Float64(-1.0 + Float64(1.0 - t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 5.8e+20)
		tmp = a * log(t);
	else
		tmp = -1.0 + (1.0 - t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 5.8e+20], N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(1.0 - t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 5.8e20

   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%

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

      2. associate--l+ 99.3%

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

      3. sub-neg 99.3%

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

      4. +-commutative 99.3%

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

      5. *-commutative 99.3%

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

      6. distribute-rgt-neg-in 99.3%

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

      7. fma-undefine 99.3%

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

      8. sub-neg 99.3%

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

      9. +-commutative 99.3%

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

      10. distribute-neg-in 99.3%

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

      11. metadata-eval 99.3%

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

      12. metadata-eval 99.3%

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

      13. unsub-neg 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in a around inf 48.6%

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

      1. *-commutative 48.6%

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

   7. Simplified 48.6%

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

   if 5.8e20 < t

   1. Initial program 99.9%

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

      1. associate-+l- 99.9%

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

      2. associate--l+ 99.9%

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

      3. sub-neg 99.9%

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

      4. +-commutative 99.9%

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

      5. *-commutative 99.9%

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

      6. distribute-rgt-neg-in 99.9%

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

      7. fma-undefine 99.9%

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

      8. sub-neg 99.9%

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

      9. +-commutative 99.9%

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

      10. distribute-neg-in 99.9%

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

      11. metadata-eval 99.9%

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

      12. metadata-eval 99.9%

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

      13. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t around inf 84.1%

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

      1. neg-mul-1 84.1%

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

   7. Simplified 84.1%

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

      1. expm1-log1p-u 0.0%

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

      2. expm1-undefine 0.0%

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

   9. Applied egg-rr 0.0%

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

       1. sub-neg 0.0%

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

       2. log1p-undefine 0.0%

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

       3. rem-exp-log 84.1%

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

       4. unsub-neg 84.1%

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

       5. metadata-eval 84.1%

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

   11. Simplified 84.1%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.8 \cdot 10^{+20}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 - t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 36.9% accurate, 34.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = (-1.0d0) + (t * ((-1.0d0) + (1.0d0 / t)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate-+l- 99.6%

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

   2. associate--l+ 99.6%

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

   3. sub-neg 99.6%

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

   4. +-commutative 99.6%

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

   5. *-commutative 99.6%

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

   6. distribute-rgt-neg-in 99.6%

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

   7. fma-undefine 99.6%

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

   8. sub-neg 99.6%

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

   9. +-commutative 99.6%

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

   10. distribute-neg-in 99.6%

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

   11. metadata-eval 99.6%

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

   12. metadata-eval 99.6%

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

   13. unsub-neg 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in t around inf 41.9%

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

   1. neg-mul-1 41.9%

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

7. Simplified 41.9%

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

   1. expm1-log1p-u 1.3%

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

   2. expm1-undefine 1.3%

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

9. Applied egg-rr 1.3%

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

    1. sub-neg 1.3%

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

    2. log1p-undefine 1.3%

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

    3. rem-exp-log 41.9%

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

    4. unsub-neg 41.9%

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

    5. metadata-eval 41.9%

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

11. Simplified 41.9%

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

12. Taylor expanded in t around inf 41.9%

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

13. Final simplification 41.9%

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

Alternative 16: 36.8% accurate, 62.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = (-1.0d0) + (1.0d0 - t)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate-+l- 99.6%

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

   2. associate--l+ 99.6%

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

   3. sub-neg 99.6%

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

   4. +-commutative 99.6%

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

   5. *-commutative 99.6%

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

   6. distribute-rgt-neg-in 99.6%

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

   7. fma-undefine 99.6%

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

   8. sub-neg 99.6%

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

   9. +-commutative 99.6%

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

   10. distribute-neg-in 99.6%

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

   11. metadata-eval 99.6%

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

   12. metadata-eval 99.6%

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

   13. unsub-neg 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in t around inf 41.9%

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

   1. neg-mul-1 41.9%

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

7. Simplified 41.9%

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

   1. expm1-log1p-u 1.3%

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

   2. expm1-undefine 1.3%

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

9. Applied egg-rr 1.3%

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

    1. sub-neg 1.3%

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

    2. log1p-undefine 1.3%

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

    3. rem-exp-log 41.9%

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

    4. unsub-neg 41.9%

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

    5. metadata-eval 41.9%

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

11. Simplified 41.9%

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

12. Final simplification 41.9%

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

Alternative 17: 36.9% accurate, 156.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate-+l- 99.6%

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

   2. associate--l+ 99.6%

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

   3. sub-neg 99.6%

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

   4. +-commutative 99.6%

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

   5. *-commutative 99.6%

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

   6. distribute-rgt-neg-in 99.6%

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

   7. fma-undefine 99.6%

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

   8. sub-neg 99.6%

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

   9. +-commutative 99.6%

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

   10. distribute-neg-in 99.6%

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

   11. metadata-eval 99.6%

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

   12. metadata-eval 99.6%

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

   13. unsub-neg 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in t around inf 41.9%

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

   1. neg-mul-1 41.9%

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

7. Simplified 41.9%

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

Alternative 18: 2.4% accurate, 313.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = 0.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate-+l- 99.6%

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

   2. associate--l+ 99.6%

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

   3. sub-neg 99.6%

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

   4. +-commutative 99.6%

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

   5. *-commutative 99.6%

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

   6. distribute-rgt-neg-in 99.6%

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

   7. fma-undefine 99.6%

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

   8. sub-neg 99.6%

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

   9. +-commutative 99.6%

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

   10. distribute-neg-in 99.6%

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

   11. metadata-eval 99.6%

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

   12. metadata-eval 99.6%

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

   13. unsub-neg 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in t around inf 41.9%

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

   1. neg-mul-1 41.9%

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

7. Simplified 41.9%

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

   1. expm1-log1p-u 1.3%

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

   2. expm1-undefine 1.3%

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

9. Applied egg-rr 1.3%

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

    1. sub-neg 1.3%

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

    2. log1p-undefine 1.3%

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

    3. rem-exp-log 41.9%

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

    4. unsub-neg 41.9%

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

    5. metadata-eval 41.9%

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

11. Simplified 41.9%

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

12. Taylor expanded in t around 0 2.4%

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

    1. metadata-eval 2.4%

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

14. Applied egg-rr 2.4%

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

Developer Target 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024165 
(FPCore (x y z t a)
  :name "Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2"
  :precision binary64

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

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