Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 15.5s

Alternatives: 19

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate--l+ 99.6%

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

   2. +-commutative 99.6%

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

   3. associate-+l+ 99.6%

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

   4. +-commutative 99.6%

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

   5. fma-define 99.6%

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

   6. sub-neg 99.6%

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

   7. metadata-eval 99.6%

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

3. Simplified 99.6%

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

Alternative 2: 99.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. 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. Add Preprocessing

Alternative 3: 77.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.112 \lor \neg \left(a \leq 3.6\right):\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + \log \left(y \cdot {t}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -0.112) (not (<= a 3.6)))
   (- (* a (log t)) t)
   (+ (- (log z) t) (log (* y (pow t -0.5))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -0.112) || !(a <= 3.6)) {
		tmp = (a * log(t)) - t;
	} else {
		tmp = (log(z) - t) + log((y * pow(t, -0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((a <= (-0.112d0)) .or. (.not. (a <= 3.6d0))) then
        tmp = (a * log(t)) - t
    else
        tmp = (log(z) - t) + log((y * (t ** (-0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -0.112) || !(a <= 3.6)) {
		tmp = (a * Math.log(t)) - t;
	} else {
		tmp = (Math.log(z) - t) + Math.log((y * Math.pow(t, -0.5)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -0.112) or not (a <= 3.6):
		tmp = (a * math.log(t)) - t
	else:
		tmp = (math.log(z) - t) + math.log((y * math.pow(t, -0.5)))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -0.112) || !(a <= 3.6))
		tmp = Float64(Float64(a * log(t)) - t);
	else
		tmp = Float64(Float64(log(z) - t) + log(Float64(y * (t ^ -0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -0.112) || ~((a <= 3.6)))
		tmp = (a * log(t)) - t;
	else
		tmp = (log(z) - t) + log((y * (t ^ -0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -0.112], N[Not[LessEqual[a, 3.6]], $MachinePrecision]], N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] + N[Log[N[(y * N[Power[t, -0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -0.112000000000000002 or 3.60000000000000009 < a

   1. Initial program 99.6%

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

      1. associate--l+ 99.6%

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

      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 99.7%

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

      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 72.3%

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

   6. Taylor expanded in a around inf 98.1%

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

      1. *-commutative 98.1%

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

   8. Simplified 98.1%

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

   if -0.112000000000000002 < a < 3.60000000000000009

   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 66.9%

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

   6. Taylor expanded in a around 0 66.5%

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

      1. *-commutative 66.5%

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

   8. Simplified 66.5%

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

      1. *-un-lft-identity 66.5%

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

      2. add-log-exp 66.5%

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

      3. sum-log 55.7%

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

      4. exp-to-pow 55.7%

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

   10. Applied egg-rr 55.7%

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

       1. *-lft-identity 55.7%

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

   12. Simplified 55.7%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.112 \lor \neg \left(a \leq 3.6\right):\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + \log \left(y \cdot {t}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log z \leq 140:\\ \;\;\;\;\left(\log t \cdot \left(a - 0.5\right) + \log \left(z \cdot y\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + a \cdot \log t\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (log z) 140.0)
   (- (+ (* (log t) (- a 0.5)) (log (* z y))) t)
   (- (+ (log z) (* a (log t))) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (log(z) <= 140.0) {
		tmp = ((log(t) * (a - 0.5)) + log((z * y))) - t;
	} else {
		tmp = (log(z) + (a * log(t))) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (log(z) <= 140.0d0) then
        tmp = ((log(t) * (a - 0.5d0)) + log((z * y))) - t
    else
        tmp = (log(z) + (a * log(t))) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (Math.log(z) <= 140.0) {
		tmp = ((Math.log(t) * (a - 0.5)) + Math.log((z * y))) - t;
	} else {
		tmp = (Math.log(z) + (a * Math.log(t))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if math.log(z) <= 140.0:
		tmp = ((math.log(t) * (a - 0.5)) + math.log((z * y))) - t
	else:
		tmp = (math.log(z) + (a * math.log(t))) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (log(z) <= 140.0)
		tmp = Float64(Float64(Float64(log(t) * Float64(a - 0.5)) + log(Float64(z * y))) - t);
	else
		tmp = Float64(Float64(log(z) + Float64(a * log(t))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (log(z) <= 140.0)
		tmp = ((log(t) * (a - 0.5)) + log((z * y))) - t;
	else
		tmp = (log(z) + (a * log(t))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[Log[z], $MachinePrecision], 140.0], N[(N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision] + N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[z], $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (log.f64 z) < 140

   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. Step-by-step derivation

      1. add-cube-cbrt 98.1%

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

      2. pow3 98.1%

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

   6. Applied egg-rr 89.9%

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

   7. Taylor expanded in x around 0 57.4%

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

   if 140 < (log.f64 z)

   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+ 99.6%

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

      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 99.7%

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

      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 69.0%

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

   6. Taylor expanded in y around inf 69.0%

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

      1. +-commutative 69.0%

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

      2. mul-1-neg 69.0%

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

      3. sub-neg 69.0%

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

      4. metadata-eval 69.0%

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

      5. log-rec 69.0%

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

      6. remove-double-neg 69.0%

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

      7. fma-undefine 69.0%

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

   8. Simplified 69.0%

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

   9. Taylor expanded in a around inf 84.3%

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

       1. *-commutative 84.3%

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

   11. Simplified 84.3%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log z \leq 140:\\ \;\;\;\;\left(\log t \cdot \left(a - 0.5\right) + \log \left(z \cdot y\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + a \cdot \log t\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 112:\\ \;\;\;\;\log y + \left(\log z + \log t \cdot \left(a - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + a \cdot \log t\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 112.0)
   (+ (log y) (+ (log z) (* (log t) (- a 0.5))))
   (- (+ (log z) (* a (log t))) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 112.0) {
		tmp = log(y) + (log(z) + (log(t) * (a - 0.5)));
	} else {
		tmp = (log(z) + (a * log(t))) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 112.0d0) then
        tmp = log(y) + (log(z) + (log(t) * (a - 0.5d0)))
    else
        tmp = (log(z) + (a * log(t))) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 112.0) {
		tmp = Math.log(y) + (Math.log(z) + (Math.log(t) * (a - 0.5)));
	} else {
		tmp = (Math.log(z) + (a * Math.log(t))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 112.0:
		tmp = math.log(y) + (math.log(z) + (math.log(t) * (a - 0.5)))
	else:
		tmp = (math.log(z) + (a * math.log(t))) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 112.0)
		tmp = Float64(log(y) + Float64(log(z) + Float64(log(t) * Float64(a - 0.5))));
	else
		tmp = Float64(Float64(log(z) + Float64(a * log(t))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 112.0)
		tmp = log(y) + (log(z) + (log(t) * (a - 0.5)));
	else
		tmp = (log(z) + (a * log(t))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 112.0], N[(N[Log[y], $MachinePrecision] + N[(N[Log[z], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[z], $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 112

   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+ 99.4%

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

      4. +-commutative 99.4%

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

      5. fma-define 99.4%

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

      6. sub-neg 99.4%

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

      7. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 68.4%

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

   6. Taylor expanded in t around 0 67.8%

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

   if 112 < t

   1. Initial program 99.8%

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

      1. associate--l+ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+l+ 99.8%

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

      4. +-commutative 99.8%

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

      5. fma-define 99.8%

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

      6. sub-neg 99.8%

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

      7. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 71.4%

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

   6. Taylor expanded in y around inf 71.4%

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

      1. +-commutative 71.4%

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

      2. mul-1-neg 71.4%

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

      3. sub-neg 71.4%

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

      4. metadata-eval 71.4%

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

      5. log-rec 71.4%

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

      6. remove-double-neg 71.4%

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

      7. fma-undefine 71.4%

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

   8. Simplified 71.4%

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

   9. Taylor expanded in a around inf 98.1%

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

       1. *-commutative 98.1%

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

   11. Simplified 98.1%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 112:\\ \;\;\;\;\log y + \left(\log z + \log t \cdot \left(a - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + a \cdot \log t\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

Alternative 7: 70.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. add-cube-cbrt 98.0%

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

   2. pow3 98.0%

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

6. Applied egg-rr 70.0%

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

7. Taylor expanded in x around 0 49.1%

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

   1. *-commutative 49.1%

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

   2. log-prod 69.9%

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

9. Applied egg-rr 69.9%

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

10. Final simplification 69.9%

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

Alternative 8: 70.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate--l+ 99.6%

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

   2. +-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 69.9%

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

Alternative 9: 70.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

   \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
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 x around 0 69.9%

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

6. Final simplification 69.9%

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

Alternative 10: 70.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate--l+ 99.6%

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

   2. +-commutative 99.6%

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

   3. associate-+l+ 99.6%

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

   4. +-commutative 99.6%

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

   5. fma-define 99.6%

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

   6. sub-neg 99.6%

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

   7. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in x around 0 69.9%

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

Alternative 11: 75.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\log z + a \cdot \log t\right) - t\\ \mathbf{if}\;a \leq -3.95 \cdot 10^{-94}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq -2.8 \cdot 10^{-159}:\\ \;\;\;\;\log \left({t}^{-0.5} \cdot \left(z \cdot y\right)\right) - t\\ \mathbf{elif}\;a \leq 7 \cdot 10^{-203}:\\ \;\;\;\;\left(\log z - t\right) + \log \left(x + y\right)\\ \mathbf{elif}\;a \leq 1.05 \cdot 10^{-26}:\\ \;\;\;\;\log \left(z \cdot \left(y \cdot {t}^{-0.5}\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (+ (log z) (* a (log t))) t)))
   (if (<= a -3.95e-94)
     t_1
     (if (<= a -2.8e-159)
       (- (log (* (pow t -0.5) (* z y))) t)
       (if (<= a 7e-203)
         (+ (- (log z) t) (log (+ x y)))
         (if (<= a 1.05e-26) (- (log (* z (* y (pow t -0.5)))) t) t_1))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (log(z) + (a * log(t))) - t;
	double tmp;
	if (a <= -3.95e-94) {
		tmp = t_1;
	} else if (a <= -2.8e-159) {
		tmp = log((pow(t, -0.5) * (z * y))) - t;
	} else if (a <= 7e-203) {
		tmp = (log(z) - t) + log((x + y));
	} else if (a <= 1.05e-26) {
		tmp = log((z * (y * pow(t, -0.5)))) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (log(z) + (a * log(t))) - t
    if (a <= (-3.95d-94)) then
        tmp = t_1
    else if (a <= (-2.8d-159)) then
        tmp = log(((t ** (-0.5d0)) * (z * y))) - t
    else if (a <= 7d-203) then
        tmp = (log(z) - t) + log((x + y))
    else if (a <= 1.05d-26) then
        tmp = log((z * (y * (t ** (-0.5d0))))) - t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (Math.log(z) + (a * Math.log(t))) - t;
	double tmp;
	if (a <= -3.95e-94) {
		tmp = t_1;
	} else if (a <= -2.8e-159) {
		tmp = Math.log((Math.pow(t, -0.5) * (z * y))) - t;
	} else if (a <= 7e-203) {
		tmp = (Math.log(z) - t) + Math.log((x + y));
	} else if (a <= 1.05e-26) {
		tmp = Math.log((z * (y * Math.pow(t, -0.5)))) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (math.log(z) + (a * math.log(t))) - t
	tmp = 0
	if a <= -3.95e-94:
		tmp = t_1
	elif a <= -2.8e-159:
		tmp = math.log((math.pow(t, -0.5) * (z * y))) - t
	elif a <= 7e-203:
		tmp = (math.log(z) - t) + math.log((x + y))
	elif a <= 1.05e-26:
		tmp = math.log((z * (y * math.pow(t, -0.5)))) - t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(log(z) + Float64(a * log(t))) - t)
	tmp = 0.0
	if (a <= -3.95e-94)
		tmp = t_1;
	elseif (a <= -2.8e-159)
		tmp = Float64(log(Float64((t ^ -0.5) * Float64(z * y))) - t);
	elseif (a <= 7e-203)
		tmp = Float64(Float64(log(z) - t) + log(Float64(x + y)));
	elseif (a <= 1.05e-26)
		tmp = Float64(log(Float64(z * Float64(y * (t ^ -0.5)))) - t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (log(z) + (a * log(t))) - t;
	tmp = 0.0;
	if (a <= -3.95e-94)
		tmp = t_1;
	elseif (a <= -2.8e-159)
		tmp = log(((t ^ -0.5) * (z * y))) - t;
	elseif (a <= 7e-203)
		tmp = (log(z) - t) + log((x + y));
	elseif (a <= 1.05e-26)
		tmp = log((z * (y * (t ^ -0.5)))) - t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[Log[z], $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[a, -3.95e-94], t$95$1, If[LessEqual[a, -2.8e-159], N[(N[Log[N[(N[Power[t, -0.5], $MachinePrecision] * N[(z * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision], If[LessEqual[a, 7e-203], N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] + N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.05e-26], N[(N[Log[N[(z * N[(y * N[Power[t, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if a < -3.95e-94 or 1.05000000000000004e-26 < a

   1. Initial program 99.6%

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

      1. associate--l+ 99.6%

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

      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+ 99.7%

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

      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 71.1%

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

   6. Taylor expanded in y around inf 71.1%

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

      1. +-commutative 71.1%

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

      2. mul-1-neg 71.1%

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

      3. sub-neg 71.1%

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

      4. metadata-eval 71.1%

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

      5. log-rec 71.1%

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

      6. remove-double-neg 71.1%

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

      7. fma-undefine 71.1%

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

   8. Simplified 71.1%

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

   9. Taylor expanded in a around inf 93.3%

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

       1. *-commutative 93.3%

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

   11. Simplified 93.3%

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

   if -3.95e-94 < a < -2.8000000000000002e-159

   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+ 99.5%

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

      4. +-commutative 99.5%

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

      5. fma-define 99.5%

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

      6. sub-neg 99.5%

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

      7. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

      1. add-cube-cbrt 98.0%

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

      2. pow3 98.2%

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

   6. Applied egg-rr 87.2%

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

   7. Taylor expanded in x around 0 62.1%

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

   8. Taylor expanded in a around 0 62.1%

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

      1. +-commutative 62.1%

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

      2. *-commutative 62.1%

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

   10. Simplified 62.1%

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

       1. +-commutative 62.1%

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

       2. add-log-exp 62.1%

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

       3. sum-log 53.4%

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

       4. *-commutative 53.4%

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

       5. exp-to-pow 53.4%

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

   12. Applied egg-rr 53.4%

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

   if -2.8000000000000002e-159 < a < 7.0000000000000003e-203

   1. Initial program 99.7%

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

      1. associate-+l- 99.7%

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

      2. associate--l+ 99.8%

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

      3. sub-neg 99.8%

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

      4. +-commutative 99.8%

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

      5. *-commutative 99.8%

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

      6. distribute-rgt-neg-in 99.8%

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

      7. fma-undefine 99.8%

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

      8. sub-neg 99.8%

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

      9. +-commutative 99.8%

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

      10. distribute-neg-in 99.8%

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

      11. metadata-eval 99.8%

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

      12. metadata-eval 99.8%

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

      13. unsub-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t around inf 68.8%

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

   if 7.0000000000000003e-203 < a < 1.05000000000000004e-26

   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. Step-by-step derivation

      1. add-cube-cbrt 98.1%

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

      2. pow3 98.1%

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

   6. Applied egg-rr 75.1%

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

   7. Taylor expanded in x around 0 47.9%

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

   8. Taylor expanded in a around 0 47.9%

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

      1. +-commutative 47.9%

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

      2. *-commutative 47.9%

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

   10. Simplified 47.9%

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

       1. *-un-lft-identity 47.9%

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

       2. +-commutative 47.9%

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

       3. add-log-exp 47.9%

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

       4. sum-log 42.2%

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

       5. *-commutative 42.2%

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

       6. exp-to-pow 42.2%

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

   12. Applied egg-rr 42.2%

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

       1. *-lft-identity 42.2%

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

       2. associate-*l* 45.3%

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

   14. Simplified 45.3%

       \[\leadsto \color{blue}{\log \left(z \cdot \left(y \cdot {t}^{-0.5}\right)\right)} - t \]
3. Recombined 4 regimes into one program.

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.95 \cdot 10^{-94}:\\ \;\;\;\;\left(\log z + a \cdot \log t\right) - t\\ \mathbf{elif}\;a \leq -2.8 \cdot 10^{-159}:\\ \;\;\;\;\log \left({t}^{-0.5} \cdot \left(z \cdot y\right)\right) - t\\ \mathbf{elif}\;a \leq 7 \cdot 10^{-203}:\\ \;\;\;\;\left(\log z - t\right) + \log \left(x + y\right)\\ \mathbf{elif}\;a \leq 1.05 \cdot 10^{-26}:\\ \;\;\;\;\log \left(z \cdot \left(y \cdot {t}^{-0.5}\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + a \cdot \log t\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 81.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.4 \cdot 10^{-135}:\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t + \log \left(z \cdot \left(x + y\right)\right)\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-114} \lor \neg \left(t \leq 1.75 \cdot 10^{-22}\right):\\ \;\;\;\;\left(\log z + a \cdot \log t\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot \left(a - 0.5\right) + \log \left(z \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 1.4e-135)
   (+ (* (+ a -0.5) (log t)) (log (* z (+ x y))))
   (if (or (<= t 3.8e-114) (not (<= t 1.75e-22)))
     (- (+ (log z) (* a (log t))) t)
     (+ (* (log t) (- a 0.5)) (log (* z y))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 1.4e-135) {
		tmp = ((a + -0.5) * log(t)) + log((z * (x + y)));
	} else if ((t <= 3.8e-114) || !(t <= 1.75e-22)) {
		tmp = (log(z) + (a * log(t))) - t;
	} else {
		tmp = (log(t) * (a - 0.5)) + log((z * y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 1.4d-135) then
        tmp = ((a + (-0.5d0)) * log(t)) + log((z * (x + y)))
    else if ((t <= 3.8d-114) .or. (.not. (t <= 1.75d-22))) then
        tmp = (log(z) + (a * log(t))) - t
    else
        tmp = (log(t) * (a - 0.5d0)) + log((z * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 1.4e-135) {
		tmp = ((a + -0.5) * Math.log(t)) + Math.log((z * (x + y)));
	} else if ((t <= 3.8e-114) || !(t <= 1.75e-22)) {
		tmp = (Math.log(z) + (a * Math.log(t))) - t;
	} else {
		tmp = (Math.log(t) * (a - 0.5)) + Math.log((z * y));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 1.4e-135:
		tmp = ((a + -0.5) * math.log(t)) + math.log((z * (x + y)))
	elif (t <= 3.8e-114) or not (t <= 1.75e-22):
		tmp = (math.log(z) + (a * math.log(t))) - t
	else:
		tmp = (math.log(t) * (a - 0.5)) + math.log((z * y))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 1.4e-135)
		tmp = Float64(Float64(Float64(a + -0.5) * log(t)) + log(Float64(z * Float64(x + y))));
	elseif ((t <= 3.8e-114) || !(t <= 1.75e-22))
		tmp = Float64(Float64(log(z) + Float64(a * log(t))) - t);
	else
		tmp = Float64(Float64(log(t) * Float64(a - 0.5)) + log(Float64(z * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 1.4e-135)
		tmp = ((a + -0.5) * log(t)) + log((z * (x + y)));
	elseif ((t <= 3.8e-114) || ~((t <= 1.75e-22)))
		tmp = (log(z) + (a * log(t))) - t;
	else
		tmp = (log(t) * (a - 0.5)) + log((z * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 1.4e-135], N[(N[(N[(a + -0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision] + N[Log[N[(z * N[(x + y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t, 3.8e-114], N[Not[LessEqual[t, 1.75e-22]], $MachinePrecision]], N[(N[(N[Log[z], $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision] + N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.40000000000000012e-135

   1. Initial program 99.3%

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

      1. associate--l+ 99.3%

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

      2. +-commutative 99.3%

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

      3. associate-+l+ 99.4%

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

      4. +-commutative 99.4%

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

      5. fma-define 99.5%

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

      6. sub-neg 99.5%

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

      7. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

      1. add-cube-cbrt 97.9%

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

      2. pow3 98.0%

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

   6. Applied egg-rr 73.9%

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

   7. Taylor expanded in t around 0 75.0%

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

      1. +-commutative 75.0%

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

      2. sub-neg 75.0%

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

      3. metadata-eval 75.0%

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

      4. +-commutative 75.0%

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

      5. distribute-rgt-out 75.0%

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

      6. +-commutative 75.0%

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

      7. distribute-rgt-in 75.0%

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

      8. +-commutative 75.0%

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

   9. Simplified 75.0%

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

   if 1.40000000000000012e-135 < t < 3.7999999999999998e-114 or 1.75000000000000003e-22 < t

   1. Initial program 99.8%

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

      1. associate--l+ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+l+ 99.7%

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

      4. +-commutative 99.7%

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

      5. fma-define 99.8%

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

      6. sub-neg 99.8%

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

      7. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 72.1%

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

   6. Taylor expanded in y around inf 72.0%

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

      1. +-commutative 72.0%

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

      2. mul-1-neg 72.0%

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

      3. sub-neg 72.0%

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

      4. metadata-eval 72.0%

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

      5. log-rec 72.0%

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

      6. remove-double-neg 72.0%

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

      7. fma-undefine 72.0%

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

   8. Simplified 72.0%

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

   9. Taylor expanded in a around inf 95.6%

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

       1. *-commutative 95.6%

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

   11. Simplified 95.6%

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

   if 3.7999999999999998e-114 < t < 1.75000000000000003e-22

   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. Step-by-step derivation

      1. add-cube-cbrt 97.9%

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

      2. pow3 98.1%

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

   6. Applied egg-rr 86.3%

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

   7. Taylor expanded in x around 0 60.4%

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

   8. Taylor expanded in t around 0 60.4%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.4 \cdot 10^{-135}:\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t + \log \left(z \cdot \left(x + y\right)\right)\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-114} \lor \neg \left(t \leq 1.75 \cdot 10^{-22}\right):\\ \;\;\;\;\left(\log z + a \cdot \log t\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot \left(a - 0.5\right) + \log \left(z \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 74.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 8.4 \cdot 10^{-135} \lor \neg \left(t \leq 1.35 \cdot 10^{-112}\right) \land t \leq 1.5 \cdot 10^{-22}:\\ \;\;\;\;\log t \cdot \left(a - 0.5\right) + \log \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + a \cdot \log t\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t 8.4e-135) (and (not (<= t 1.35e-112)) (<= t 1.5e-22)))
   (+ (* (log t) (- a 0.5)) (log (* z y)))
   (- (+ (log z) (* a (log t))) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= 8.4e-135) || (!(t <= 1.35e-112) && (t <= 1.5e-22))) {
		tmp = (log(t) * (a - 0.5)) + log((z * y));
	} else {
		tmp = (log(z) + (a * log(t))) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((t <= 8.4d-135) .or. (.not. (t <= 1.35d-112)) .and. (t <= 1.5d-22)) then
        tmp = (log(t) * (a - 0.5d0)) + log((z * y))
    else
        tmp = (log(z) + (a * log(t))) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= 8.4e-135) || (!(t <= 1.35e-112) && (t <= 1.5e-22))) {
		tmp = (Math.log(t) * (a - 0.5)) + Math.log((z * y));
	} else {
		tmp = (Math.log(z) + (a * Math.log(t))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (t <= 8.4e-135) or (not (t <= 1.35e-112) and (t <= 1.5e-22)):
		tmp = (math.log(t) * (a - 0.5)) + math.log((z * y))
	else:
		tmp = (math.log(z) + (a * math.log(t))) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((t <= 8.4e-135) || (!(t <= 1.35e-112) && (t <= 1.5e-22)))
		tmp = Float64(Float64(log(t) * Float64(a - 0.5)) + log(Float64(z * y)));
	else
		tmp = Float64(Float64(log(z) + Float64(a * log(t))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((t <= 8.4e-135) || (~((t <= 1.35e-112)) && (t <= 1.5e-22)))
		tmp = (log(t) * (a - 0.5)) + log((z * y));
	else
		tmp = (log(z) + (a * log(t))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[t, 8.4e-135], And[N[Not[LessEqual[t, 1.35e-112]], $MachinePrecision], LessEqual[t, 1.5e-22]]], N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision] + N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[z], $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 8.4000000000000001e-135 or 1.35e-112 < t < 1.5e-22

   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+ 99.4%

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

      4. +-commutative 99.4%

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

      5. fma-define 99.4%

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

      6. sub-neg 99.4%

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

      7. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

      1. add-cube-cbrt 97.9%

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

      2. pow3 98.0%

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

   6. Applied egg-rr 78.6%

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

   7. Taylor expanded in x around 0 51.3%

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

   8. Taylor expanded in t around 0 51.3%

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

   if 8.4000000000000001e-135 < t < 1.35e-112 or 1.5e-22 < t

   1. Initial program 99.8%

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

      1. associate--l+ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+l+ 99.7%

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

      4. +-commutative 99.7%

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

      5. fma-define 99.8%

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

      6. sub-neg 99.8%

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

      7. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 72.1%

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

   6. Taylor expanded in y around inf 72.0%

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

      1. +-commutative 72.0%

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

      2. mul-1-neg 72.0%

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

      3. sub-neg 72.0%

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

      4. metadata-eval 72.0%

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

      5. log-rec 72.0%

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

      6. remove-double-neg 72.0%

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

      7. fma-undefine 72.0%

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

   8. Simplified 72.0%

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

   9. Taylor expanded in a around inf 95.6%

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

       1. *-commutative 95.6%

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

   11. Simplified 95.6%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8.4 \cdot 10^{-135} \lor \neg \left(t \leq 1.35 \cdot 10^{-112}\right) \land t \leq 1.5 \cdot 10^{-22}:\\ \;\;\;\;\log t \cdot \left(a - 0.5\right) + \log \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + a \cdot \log t\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 72.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.25 \cdot 10^{-129} \lor \neg \left(a \leq 2.3 \cdot 10^{-18}\right):\\ \;\;\;\;\left(\log z + a \cdot \log t\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log \left(z \cdot \left(y \cdot {t}^{-0.5}\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -2.25e-129) (not (<= a 2.3e-18)))
   (- (+ (log z) (* a (log t))) t)
   (- (log (* z (* y (pow t -0.5)))) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -2.25e-129) || !(a <= 2.3e-18)) {
		tmp = (log(z) + (a * log(t))) - t;
	} else {
		tmp = log((z * (y * pow(t, -0.5)))) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((a <= (-2.25d-129)) .or. (.not. (a <= 2.3d-18))) then
        tmp = (log(z) + (a * log(t))) - t
    else
        tmp = log((z * (y * (t ** (-0.5d0))))) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -2.25e-129) || !(a <= 2.3e-18)) {
		tmp = (Math.log(z) + (a * Math.log(t))) - t;
	} else {
		tmp = Math.log((z * (y * Math.pow(t, -0.5)))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -2.25e-129) or not (a <= 2.3e-18):
		tmp = (math.log(z) + (a * math.log(t))) - t
	else:
		tmp = math.log((z * (y * math.pow(t, -0.5)))) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -2.25e-129) || !(a <= 2.3e-18))
		tmp = Float64(Float64(log(z) + Float64(a * log(t))) - t);
	else
		tmp = Float64(log(Float64(z * Float64(y * (t ^ -0.5)))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -2.25e-129) || ~((a <= 2.3e-18)))
		tmp = (log(z) + (a * log(t))) - t;
	else
		tmp = log((z * (y * (t ^ -0.5)))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -2.25e-129], N[Not[LessEqual[a, 2.3e-18]], $MachinePrecision]], N[(N[(N[Log[z], $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[Log[N[(z * N[(y * N[Power[t, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.25000000000000015e-129 or 2.3000000000000001e-18 < a

   1. Initial program 99.6%

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

      1. associate--l+ 99.6%

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

      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+ 99.7%

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

      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 69.8%

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

   6. Taylor expanded in y around inf 69.8%

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

      1. +-commutative 69.8%

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

      2. mul-1-neg 69.8%

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

      3. sub-neg 69.8%

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

      4. metadata-eval 69.8%

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

      5. log-rec 69.8%

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

      6. remove-double-neg 69.8%

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

      7. fma-undefine 69.8%

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

   8. Simplified 69.8%

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

   9. Taylor expanded in a around inf 92.8%

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

       1. *-commutative 92.8%

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

   11. Simplified 92.8%

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

   if -2.25000000000000015e-129 < a < 2.3000000000000001e-18

   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+ 99.5%

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

      4. +-commutative 99.5%

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

      5. fma-define 99.5%

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

      6. sub-neg 99.5%

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

      7. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

      1. add-cube-cbrt 97.8%

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

      2. pow3 97.8%

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

   6. Applied egg-rr 65.7%

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

   7. Taylor expanded in x around 0 47.4%

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

   8. Taylor expanded in a around 0 47.4%

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

      1. +-commutative 47.4%

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

      2. *-commutative 47.4%

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

   10. Simplified 47.4%

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

       1. *-un-lft-identity 47.4%

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

       2. +-commutative 47.4%

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

       3. add-log-exp 47.4%

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

       4. sum-log 41.8%

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

       5. *-commutative 41.8%

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

       6. exp-to-pow 41.8%

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

   12. Applied egg-rr 41.8%

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

       1. *-lft-identity 41.8%

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

       2. associate-*l* 44.8%

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

   14. Simplified 44.8%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.25 \cdot 10^{-129} \lor \neg \left(a \leq 2.3 \cdot 10^{-18}\right):\\ \;\;\;\;\left(\log z + a \cdot \log t\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log \left(z \cdot \left(y \cdot {t}^{-0.5}\right)\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 78.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.49 \lor \neg \left(a \leq 1.65\right):\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + \log \left(x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -0.49) (not (<= a 1.65)))
   (- (* a (log t)) t)
   (+ (- (log z) t) (log (+ x y)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -0.49) || !(a <= 1.65)) {
		tmp = (a * log(t)) - t;
	} else {
		tmp = (log(z) - t) + log((x + y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((a <= (-0.49d0)) .or. (.not. (a <= 1.65d0))) then
        tmp = (a * log(t)) - t
    else
        tmp = (log(z) - t) + log((x + y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -0.49) || !(a <= 1.65)) {
		tmp = (a * Math.log(t)) - t;
	} else {
		tmp = (Math.log(z) - t) + Math.log((x + y));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -0.49) or not (a <= 1.65):
		tmp = (a * math.log(t)) - t
	else:
		tmp = (math.log(z) - t) + math.log((x + y))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -0.49) || !(a <= 1.65))
		tmp = Float64(Float64(a * log(t)) - t);
	else
		tmp = Float64(Float64(log(z) - t) + log(Float64(x + y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -0.49) || ~((a <= 1.65)))
		tmp = (a * log(t)) - t;
	else
		tmp = (log(z) - t) + log((x + y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -0.49], N[Not[LessEqual[a, 1.65]], $MachinePrecision]], N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] + N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -0.48999999999999999 or 1.6499999999999999 < a

   1. Initial program 99.6%

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

      1. associate--l+ 99.6%

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

      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 99.7%

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

      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 72.3%

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

   6. Taylor expanded in a around inf 98.1%

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

      1. *-commutative 98.1%

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

   8. Simplified 98.1%

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

   if -0.48999999999999999 < a < 1.6499999999999999

   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 59.0%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.49 \lor \neg \left(a \leq 1.65\right):\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + \log \left(x + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 77.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate--l+ 99.6%

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

   2. +-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 69.9%

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

6. Taylor expanded in y around inf 69.9%

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

   1. +-commutative 69.9%

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

   2. mul-1-neg 69.9%

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

   3. sub-neg 69.9%

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

   4. metadata-eval 69.9%

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

   5. log-rec 69.9%

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

   6. remove-double-neg 69.9%

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

   7. fma-undefine 69.9%

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

8. Simplified 69.9%

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

9. Taylor expanded in a around inf 79.5%

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

    1. *-commutative 79.5%

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

11. Simplified 79.5%

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

12. Final simplification 79.5%

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

Alternative 17: 63.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 9 \cdot 10^{+17}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (if (<= t 9e+17) (* a (log t)) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 9e+17) {
		tmp = a * log(t);
	} else {
		tmp = -t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 9d+17) then
        tmp = a * log(t)
    else
        tmp = -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 9e+17) {
		tmp = a * Math.log(t);
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 9e+17:
		tmp = a * math.log(t)
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 9e+17)
		tmp = Float64(a * log(t));
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 9e+17)
		tmp = a * log(t);
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 9e+17], N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if t < 9e17

   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+ 99.4%

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

      3. sub-neg 99.4%

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

      4. +-commutative 99.4%

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

      5. *-commutative 99.4%

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

      6. distribute-rgt-neg-in 99.4%

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

      7. fma-undefine 99.4%

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

      8. sub-neg 99.4%

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

      9. +-commutative 99.4%

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

      10. distribute-neg-in 99.4%

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

      11. metadata-eval 99.4%

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

      12. metadata-eval 99.4%

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

      13. unsub-neg 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in a around inf 56.8%

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

      1. *-commutative 56.8%

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

   7. Simplified 56.8%

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

   if 9e17 < t

   1. Initial program 99.8%

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

      1. associate-+l- 99.8%

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

      2. associate--l+ 99.8%

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

      3. sub-neg 99.8%

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

      4. +-commutative 99.8%

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

      5. *-commutative 99.8%

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

      6. distribute-rgt-neg-in 99.8%

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

      7. fma-undefine 99.8%

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

      8. sub-neg 99.8%

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

      9. +-commutative 99.8%

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

      10. distribute-neg-in 99.8%

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

      11. metadata-eval 99.8%

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

      12. metadata-eval 99.8%

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

      13. unsub-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t around inf 75.8%

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

      1. mul-1-neg 75.8%

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

   7. Simplified 75.8%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9 \cdot 10^{+17}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 74.8% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate--l+ 99.6%

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

   2. +-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 69.9%

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

6. Taylor expanded in a around inf 77.4%

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

   1. *-commutative 77.4%

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

8. Simplified 77.4%

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

9. Final simplification 77.4%

   \[\leadsto a \cdot \log t - t \]
10. Add Preprocessing

Alternative 19: 38.9% 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 37.5%

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

   1. mul-1-neg 37.5%

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

7. Simplified 37.5%

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

Developer target: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024089 
(FPCore (x y z t a)
  :name "Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2"
  :precision binary64

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

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