Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 19.6s

Alternatives: 17

Speedup: 0.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. associate--l+ 99.5%

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

   2. associate-+l+ 99.5%

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

   3. +-commutative 99.5%

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

   4. associate-+r- 99.5%

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

   5. fma-def 99.5%

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

   6. sub-neg 99.5%

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

   7. metadata-eval 99.5%

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

3. Simplified 99.5%

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

4. Final simplification 99.5%

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

Alternative 2: 74.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \log t\\ \mathbf{if}\;a - 0.5 \leq -2000000:\\ \;\;\;\;\left(\log y + t_1\right) - t\\ \mathbf{elif}\;a - 0.5 \leq -0.4:\\ \;\;\;\;\left(\log y + \left(\log z + -0.5 \cdot \log t\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;t_1 - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* a (log t))))
   (if (<= (- a 0.5) -2000000.0)
     (- (+ (log y) t_1) t)
     (if (<= (- a 0.5) -0.4)
       (- (+ (log y) (+ (log z) (* -0.5 (log t)))) t)
       (- t_1 t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = a * log(t);
	double tmp;
	if ((a - 0.5) <= -2000000.0) {
		tmp = (log(y) + t_1) - t;
	} else if ((a - 0.5) <= -0.4) {
		tmp = (log(y) + (log(z) + (-0.5 * log(t)))) - t;
	} else {
		tmp = t_1 - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = a * log(t)
    if ((a - 0.5d0) <= (-2000000.0d0)) then
        tmp = (log(y) + t_1) - t
    else if ((a - 0.5d0) <= (-0.4d0)) then
        tmp = (log(y) + (log(z) + ((-0.5d0) * log(t)))) - t
    else
        tmp = t_1 - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = a * Math.log(t);
	double tmp;
	if ((a - 0.5) <= -2000000.0) {
		tmp = (Math.log(y) + t_1) - t;
	} else if ((a - 0.5) <= -0.4) {
		tmp = (Math.log(y) + (Math.log(z) + (-0.5 * Math.log(t)))) - t;
	} else {
		tmp = t_1 - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = a * math.log(t)
	tmp = 0
	if (a - 0.5) <= -2000000.0:
		tmp = (math.log(y) + t_1) - t
	elif (a - 0.5) <= -0.4:
		tmp = (math.log(y) + (math.log(z) + (-0.5 * math.log(t)))) - t
	else:
		tmp = t_1 - t
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(a * log(t))
	tmp = 0.0
	if (Float64(a - 0.5) <= -2000000.0)
		tmp = Float64(Float64(log(y) + t_1) - t);
	elseif (Float64(a - 0.5) <= -0.4)
		tmp = Float64(Float64(log(y) + Float64(log(z) + Float64(-0.5 * log(t)))) - t);
	else
		tmp = Float64(t_1 - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = a * log(t);
	tmp = 0.0;
	if ((a - 0.5) <= -2000000.0)
		tmp = (log(y) + t_1) - t;
	elseif ((a - 0.5) <= -0.4)
		tmp = (log(y) + (log(z) + (-0.5 * log(t)))) - t;
	else
		tmp = t_1 - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a - 0.5), $MachinePrecision], -2000000.0], N[(N[(N[Log[y], $MachinePrecision] + t$95$1), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[N[(a - 0.5), $MachinePrecision], -0.4], N[(N[(N[Log[y], $MachinePrecision] + N[(N[Log[z], $MachinePrecision] + N[(-0.5 * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(t$95$1 - t), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.7%

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

      1. associate--l+ 99.7%

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

      2. associate-+l+ 99.7%

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

      3. +-commutative 99.7%

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

      4. associate-+r- 99.7%

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

      5. fma-def 99.7%

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

      6. sub-neg 99.7%

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

      7. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in a around inf 98.9%

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

      1. *-commutative 98.9%

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

   6. Simplified 98.9%

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

   7. Taylor expanded in x around 0 72.8%

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

      1. log-pow 5.6%

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

      2. associate--l+ 5.6%

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

      3. remove-double-neg 5.6%

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

      4. log-rec 5.6%

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

      5. mul-1-neg 5.6%

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

      6. associate--l+ 5.6%

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

      7. log-pow 72.8%

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

      8. mul-1-neg 72.8%

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

      9. log-rec 72.8%

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

      10. remove-double-neg 72.8%

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

      11. *-commutative 72.8%

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

   9. Simplified 72.8%

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

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

   1. Initial program 99.3%

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

      1. associate--l+ 99.3%

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

      2. associate-+l+ 99.3%

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

      3. +-commutative 99.3%

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

      4. associate-+r- 99.3%

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

      5. fma-def 99.3%

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

      6. sub-neg 99.3%

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

      7. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in x around 0 61.6%

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

   5. Taylor expanded in a around 0 61.4%

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

      1. *-commutative 61.4%

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

   7. Simplified 61.4%

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

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

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 65.9%

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

      1. remove-double-neg 65.9%

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

      2. log-rec 65.9%

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

      3. mul-1-neg 65.9%

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

      4. +-commutative 65.9%

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

      5. associate--l+ 65.9%

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

      6. mul-1-neg 65.9%

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

      7. log-rec 65.9%

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

      8. remove-double-neg 65.9%

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

   4. Simplified 65.9%

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

   5. Taylor expanded in t around inf 99.6%

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

      1. neg-mul-1 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in t around 0 99.6%

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

      1. mul-1-neg 99.6%

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

      2. +-commutative 99.6%

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

      3. sub-neg 99.6%

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

      4. metadata-eval 99.6%

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

      5. +-commutative 99.6%

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

      6. distribute-rgt-out 99.6%

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

      7. sub-neg 99.6%

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

      8. distribute-rgt-out 99.6%

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

      9. +-commutative 99.6%

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

   10. Simplified 99.6%

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

   11. Taylor expanded in a around inf 99.6%

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

       1. *-commutative 99.6%

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

   13. Simplified 99.6%

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

4. Final simplification 74.2%

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

Alternative 3: 74.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \log t\\ \mathbf{if}\;a \leq -14000:\\ \;\;\;\;\left(\log y + t_1\right) - t\\ \mathbf{elif}\;a \leq 0.96:\\ \;\;\;\;\left(\log z + \left(\log y - t\right)\right) + -0.5 \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;t_1 - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* a (log t))))
   (if (<= a -14000.0)
     (- (+ (log y) t_1) t)
     (if (<= a 0.96)
       (+ (+ (log z) (- (log y) t)) (* -0.5 (log t)))
       (- t_1 t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = a * log(t);
	double tmp;
	if (a <= -14000.0) {
		tmp = (log(y) + t_1) - t;
	} else if (a <= 0.96) {
		tmp = (log(z) + (log(y) - t)) + (-0.5 * log(t));
	} else {
		tmp = t_1 - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = a * log(t)
    if (a <= (-14000.0d0)) then
        tmp = (log(y) + t_1) - t
    else if (a <= 0.96d0) then
        tmp = (log(z) + (log(y) - t)) + ((-0.5d0) * log(t))
    else
        tmp = t_1 - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = a * Math.log(t);
	double tmp;
	if (a <= -14000.0) {
		tmp = (Math.log(y) + t_1) - t;
	} else if (a <= 0.96) {
		tmp = (Math.log(z) + (Math.log(y) - t)) + (-0.5 * Math.log(t));
	} else {
		tmp = t_1 - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = a * math.log(t)
	tmp = 0
	if a <= -14000.0:
		tmp = (math.log(y) + t_1) - t
	elif a <= 0.96:
		tmp = (math.log(z) + (math.log(y) - t)) + (-0.5 * math.log(t))
	else:
		tmp = t_1 - t
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(a * log(t))
	tmp = 0.0
	if (a <= -14000.0)
		tmp = Float64(Float64(log(y) + t_1) - t);
	elseif (a <= 0.96)
		tmp = Float64(Float64(log(z) + Float64(log(y) - t)) + Float64(-0.5 * log(t)));
	else
		tmp = Float64(t_1 - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = a * log(t);
	tmp = 0.0;
	if (a <= -14000.0)
		tmp = (log(y) + t_1) - t;
	elseif (a <= 0.96)
		tmp = (log(z) + (log(y) - t)) + (-0.5 * log(t));
	else
		tmp = t_1 - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -14000.0], N[(N[(N[Log[y], $MachinePrecision] + t$95$1), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[a, 0.96], N[(N[(N[Log[z], $MachinePrecision] + N[(N[Log[y], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 - t), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -14000

   1. Initial program 99.7%

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

      1. associate--l+ 99.7%

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

      2. associate-+l+ 99.7%

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

      3. +-commutative 99.7%

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

      4. associate-+r- 99.7%

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

      5. fma-def 99.7%

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

      6. sub-neg 99.7%

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

      7. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in a around inf 98.9%

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

      1. *-commutative 98.9%

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

   6. Simplified 98.9%

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

   7. Taylor expanded in x around 0 72.8%

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

      1. log-pow 5.6%

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

      2. associate--l+ 5.6%

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

      3. remove-double-neg 5.6%

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

      4. log-rec 5.6%

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

      5. mul-1-neg 5.6%

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

      6. associate--l+ 5.6%

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

      7. log-pow 72.8%

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

      8. mul-1-neg 72.8%

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

      9. log-rec 72.8%

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

      10. remove-double-neg 72.8%

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

      11. *-commutative 72.8%

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

   9. Simplified 72.8%

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

   if -14000 < a < 0.95999999999999996

   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. Taylor expanded in x around 0 61.6%

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

      1. remove-double-neg 61.6%

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

      2. log-rec 61.6%

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

      3. mul-1-neg 61.6%

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

      4. +-commutative 61.6%

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

      5. associate--l+ 61.6%

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

      6. mul-1-neg 61.6%

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

      7. log-rec 61.6%

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

      8. remove-double-neg 61.6%

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

   4. Simplified 61.6%

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

   5. Taylor expanded in a around 0 61.4%

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

   if 0.95999999999999996 < 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. Taylor expanded in x around 0 65.9%

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

      1. remove-double-neg 65.9%

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

      2. log-rec 65.9%

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

      3. mul-1-neg 65.9%

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

      4. +-commutative 65.9%

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

      5. associate--l+ 65.9%

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

      6. mul-1-neg 65.9%

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

      7. log-rec 65.9%

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

      8. remove-double-neg 65.9%

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

   4. Simplified 65.9%

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

   5. Taylor expanded in t around inf 99.6%

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

      1. neg-mul-1 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in t around 0 99.6%

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

      1. mul-1-neg 99.6%

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

      2. +-commutative 99.6%

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

      3. sub-neg 99.6%

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

      4. metadata-eval 99.6%

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

      5. +-commutative 99.6%

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

      6. distribute-rgt-out 99.6%

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

      7. sub-neg 99.6%

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

      8. distribute-rgt-out 99.6%

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

      9. +-commutative 99.6%

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

   10. Simplified 99.6%

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

   11. Taylor expanded in a around inf 99.6%

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

       1. *-commutative 99.6%

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

   13. Simplified 99.6%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -14000:\\ \;\;\;\;\left(\log y + a \cdot \log t\right) - t\\ \mathbf{elif}\;a \leq 0.96:\\ \;\;\;\;\left(\log z + \left(\log y - t\right)\right) + -0.5 \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \]

Alternative 4: 71.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \log t\\ \mathbf{if}\;a \leq -14000:\\ \;\;\;\;\left(\log y + t_1\right) - t\\ \mathbf{elif}\;a \leq 0.006:\\ \;\;\;\;\left(\log y + \log \left(z \cdot {t}^{-0.5}\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;t_1 - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* a (log t))))
   (if (<= a -14000.0)
     (- (+ (log y) t_1) t)
     (if (<= a 0.006) (- (+ (log y) (log (* z (pow t -0.5)))) t) (- t_1 t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = a * log(t);
	double tmp;
	if (a <= -14000.0) {
		tmp = (log(y) + t_1) - t;
	} else if (a <= 0.006) {
		tmp = (log(y) + log((z * pow(t, -0.5)))) - t;
	} else {
		tmp = t_1 - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = a * log(t)
    if (a <= (-14000.0d0)) then
        tmp = (log(y) + t_1) - t
    else if (a <= 0.006d0) then
        tmp = (log(y) + log((z * (t ** (-0.5d0))))) - t
    else
        tmp = t_1 - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = a * Math.log(t);
	double tmp;
	if (a <= -14000.0) {
		tmp = (Math.log(y) + t_1) - t;
	} else if (a <= 0.006) {
		tmp = (Math.log(y) + Math.log((z * Math.pow(t, -0.5)))) - t;
	} else {
		tmp = t_1 - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = a * math.log(t)
	tmp = 0
	if a <= -14000.0:
		tmp = (math.log(y) + t_1) - t
	elif a <= 0.006:
		tmp = (math.log(y) + math.log((z * math.pow(t, -0.5)))) - t
	else:
		tmp = t_1 - t
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(a * log(t))
	tmp = 0.0
	if (a <= -14000.0)
		tmp = Float64(Float64(log(y) + t_1) - t);
	elseif (a <= 0.006)
		tmp = Float64(Float64(log(y) + log(Float64(z * (t ^ -0.5)))) - t);
	else
		tmp = Float64(t_1 - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = a * log(t);
	tmp = 0.0;
	if (a <= -14000.0)
		tmp = (log(y) + t_1) - t;
	elseif (a <= 0.006)
		tmp = (log(y) + log((z * (t ^ -0.5)))) - t;
	else
		tmp = t_1 - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -14000.0], N[(N[(N[Log[y], $MachinePrecision] + t$95$1), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[a, 0.006], N[(N[(N[Log[y], $MachinePrecision] + N[Log[N[(z * N[Power[t, -0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(t$95$1 - t), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -14000

   1. Initial program 99.7%

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

      1. associate--l+ 99.7%

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

      2. associate-+l+ 99.7%

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

      3. +-commutative 99.7%

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

      4. associate-+r- 99.7%

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

      5. fma-def 99.7%

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

      6. sub-neg 99.7%

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

      7. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in a around inf 98.9%

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

      1. *-commutative 98.9%

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

   6. Simplified 98.9%

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

   7. Taylor expanded in x around 0 72.8%

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

      1. log-pow 5.6%

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

      2. associate--l+ 5.6%

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

      3. remove-double-neg 5.6%

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

      4. log-rec 5.6%

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

      5. mul-1-neg 5.6%

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

      6. associate--l+ 5.6%

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

      7. log-pow 72.8%

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

      8. mul-1-neg 72.8%

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

      9. log-rec 72.8%

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

      10. remove-double-neg 72.8%

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

      11. *-commutative 72.8%

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

   9. Simplified 72.8%

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

   if -14000 < a < 0.0060000000000000001

   1. Initial program 99.3%

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

      1. associate--l+ 99.3%

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

      2. associate-+l+ 99.3%

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

      3. +-commutative 99.3%

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

      4. associate-+r- 99.3%

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

      5. fma-def 99.3%

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

      6. sub-neg 99.3%

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

      7. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in x around 0 61.6%

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

   5. Taylor expanded in a around 0 61.4%

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

      1. *-commutative 61.4%

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

   7. Simplified 61.4%

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

      1. add-log-exp 61.4%

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

      2. sum-log 56.3%

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

      3. exp-to-pow 56.3%

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

   9. Applied egg-rr 56.3%

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

   if 0.0060000000000000001 < 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. Taylor expanded in x around 0 65.9%

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

      1. remove-double-neg 65.9%

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

      2. log-rec 65.9%

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

      3. mul-1-neg 65.9%

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

      4. +-commutative 65.9%

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

      5. associate--l+ 65.9%

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

      6. mul-1-neg 65.9%

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

      7. log-rec 65.9%

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

      8. remove-double-neg 65.9%

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

   4. Simplified 65.9%

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

   5. Taylor expanded in t around inf 99.6%

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

      1. neg-mul-1 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in t around 0 99.6%

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

      1. mul-1-neg 99.6%

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

      2. +-commutative 99.6%

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

      3. sub-neg 99.6%

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

      4. metadata-eval 99.6%

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

      5. +-commutative 99.6%

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

      6. distribute-rgt-out 99.6%

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

      7. sub-neg 99.6%

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

      8. distribute-rgt-out 99.6%

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

      9. +-commutative 99.6%

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

   10. Simplified 99.6%

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

   11. Taylor expanded in a around inf 99.6%

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

       1. *-commutative 99.6%

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

   13. Simplified 99.6%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -14000:\\ \;\;\;\;\left(\log y + a \cdot \log t\right) - t\\ \mathbf{elif}\;a \leq 0.006:\\ \;\;\;\;\left(\log y + \log \left(z \cdot {t}^{-0.5}\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \]

Alternative 5: 69.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in x around 0 66.4%

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

   1. remove-double-neg 66.4%

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

   2. log-rec 66.4%

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

   3. mul-1-neg 66.4%

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

   4. +-commutative 66.4%

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

   5. associate--l+ 66.4%

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

   6. mul-1-neg 66.4%

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

   7. log-rec 66.4%

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

   8. remove-double-neg 66.4%

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

4. Simplified 66.4%

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

5. Final simplification 66.4%

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

Alternative 6: 65.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \log t\\ \mathbf{if}\;a - 0.5 \leq -2000000:\\ \;\;\;\;\left(\log y + t_1\right) - t\\ \mathbf{elif}\;a - 0.5 \leq -0.4:\\ \;\;\;\;\log \left(y \cdot \left(z \cdot {t}^{-0.5}\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;t_1 - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* a (log t))))
   (if (<= (- a 0.5) -2000000.0)
     (- (+ (log y) t_1) t)
     (if (<= (- a 0.5) -0.4) (- (log (* y (* z (pow t -0.5)))) t) (- t_1 t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = a * log(t);
	double tmp;
	if ((a - 0.5) <= -2000000.0) {
		tmp = (log(y) + t_1) - t;
	} else if ((a - 0.5) <= -0.4) {
		tmp = log((y * (z * pow(t, -0.5)))) - t;
	} else {
		tmp = t_1 - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = a * log(t)
    if ((a - 0.5d0) <= (-2000000.0d0)) then
        tmp = (log(y) + t_1) - t
    else if ((a - 0.5d0) <= (-0.4d0)) then
        tmp = log((y * (z * (t ** (-0.5d0))))) - t
    else
        tmp = t_1 - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = a * Math.log(t);
	double tmp;
	if ((a - 0.5) <= -2000000.0) {
		tmp = (Math.log(y) + t_1) - t;
	} else if ((a - 0.5) <= -0.4) {
		tmp = Math.log((y * (z * Math.pow(t, -0.5)))) - t;
	} else {
		tmp = t_1 - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = a * math.log(t)
	tmp = 0
	if (a - 0.5) <= -2000000.0:
		tmp = (math.log(y) + t_1) - t
	elif (a - 0.5) <= -0.4:
		tmp = math.log((y * (z * math.pow(t, -0.5)))) - t
	else:
		tmp = t_1 - t
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(a * log(t))
	tmp = 0.0
	if (Float64(a - 0.5) <= -2000000.0)
		tmp = Float64(Float64(log(y) + t_1) - t);
	elseif (Float64(a - 0.5) <= -0.4)
		tmp = Float64(log(Float64(y * Float64(z * (t ^ -0.5)))) - t);
	else
		tmp = Float64(t_1 - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = a * log(t);
	tmp = 0.0;
	if ((a - 0.5) <= -2000000.0)
		tmp = (log(y) + t_1) - t;
	elseif ((a - 0.5) <= -0.4)
		tmp = log((y * (z * (t ^ -0.5)))) - t;
	else
		tmp = t_1 - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a - 0.5), $MachinePrecision], -2000000.0], N[(N[(N[Log[y], $MachinePrecision] + t$95$1), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[N[(a - 0.5), $MachinePrecision], -0.4], N[(N[Log[N[(y * N[(z * N[Power[t, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision], N[(t$95$1 - t), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.7%

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

      1. associate--l+ 99.7%

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

      2. associate-+l+ 99.7%

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

      3. +-commutative 99.7%

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

      4. associate-+r- 99.7%

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

      5. fma-def 99.7%

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

      6. sub-neg 99.7%

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

      7. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in a around inf 98.9%

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

      1. *-commutative 98.9%

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

   6. Simplified 98.9%

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

   7. Taylor expanded in x around 0 72.8%

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

      1. log-pow 5.6%

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

      2. associate--l+ 5.6%

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

      3. remove-double-neg 5.6%

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

      4. log-rec 5.6%

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

      5. mul-1-neg 5.6%

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

      6. associate--l+ 5.6%

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

      7. log-pow 72.8%

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

      8. mul-1-neg 72.8%

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

      9. log-rec 72.8%

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

      10. remove-double-neg 72.8%

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

      11. *-commutative 72.8%

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

   9. Simplified 72.8%

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

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

   1. Initial program 99.3%

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

      1. associate--l+ 99.3%

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

      2. associate-+l+ 99.3%

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

      3. +-commutative 99.3%

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

      4. associate-+r- 99.3%

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

      5. fma-def 99.3%

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

      6. sub-neg 99.3%

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

      7. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in a around 0 98.3%

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

      1. associate-+r+ 98.4%

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

      2. log-prod 80.7%

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

      3. +-commutative 80.7%

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

   6. Simplified 80.7%

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

   7. Taylor expanded in y around inf 61.3%

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

      1. associate-+r+ 61.4%

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

      2. +-commutative 61.4%

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

      3. mul-1-neg 61.4%

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

      4. log-rec 61.4%

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

      5. remove-double-neg 61.4%

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

      6. log-prod 46.2%

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

      7. +-commutative 46.2%

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

      8. *-commutative 46.2%

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

   9. Simplified 46.2%

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

       1. +-commutative 46.2%

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

       2. log-prod 61.4%

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

       3. associate-+r+ 61.4%

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

       4. add-log-exp 56.2%

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

       5. sum-log 43.8%

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

       6. exp-sum 43.9%

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

       7. add-exp-log 44.0%

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

       8. exp-to-pow 44.1%

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

   11. Applied egg-rr 44.1%

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

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

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 65.9%

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

      1. remove-double-neg 65.9%

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

      2. log-rec 65.9%

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

      3. mul-1-neg 65.9%

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

      4. +-commutative 65.9%

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

      5. associate--l+ 65.9%

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

      6. mul-1-neg 65.9%

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

      7. log-rec 65.9%

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

      8. remove-double-neg 65.9%

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

   4. Simplified 65.9%

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

   5. Taylor expanded in t around inf 99.6%

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

      1. neg-mul-1 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in t around 0 99.6%

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

      1. mul-1-neg 99.6%

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

      2. +-commutative 99.6%

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

      3. sub-neg 99.6%

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

      4. metadata-eval 99.6%

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

      5. +-commutative 99.6%

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

      6. distribute-rgt-out 99.6%

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

      7. sub-neg 99.6%

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

      8. distribute-rgt-out 99.6%

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

      9. +-commutative 99.6%

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

   10. Simplified 99.6%

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

   11. Taylor expanded in a around inf 99.6%

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

       1. *-commutative 99.6%

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

   13. Simplified 99.6%

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

4. Final simplification 66.5%

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

Alternative 7: 87.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a + -0.5\right) \cdot \log t - t\\ \mathbf{if}\;z \leq 6.5 \cdot 10^{+81}:\\ \;\;\;\;\log \left(\left(x + y\right) \cdot z\right) + t_1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* (+ a -0.5) (log t)) t)))
   (if (<= z 6.5e+81) (+ (log (* (+ x y) z)) t_1) t_1)))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((a + -0.5) * log(t)) - t;
	double tmp;
	if (z <= 6.5e+81) {
		tmp = log(((x + y) * z)) + t_1;
	} 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 = ((a + (-0.5d0)) * log(t)) - t
    if (z <= 6.5d+81) then
        tmp = log(((x + y) * z)) + t_1
    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 = ((a + -0.5) * Math.log(t)) - t;
	double tmp;
	if (z <= 6.5e+81) {
		tmp = Math.log(((x + y) * z)) + t_1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = ((a + -0.5) * math.log(t)) - t
	tmp = 0
	if z <= 6.5e+81:
		tmp = math.log(((x + y) * z)) + t_1
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(a + -0.5) * log(t)) - t)
	tmp = 0.0
	if (z <= 6.5e+81)
		tmp = Float64(log(Float64(Float64(x + y) * z)) + t_1);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = ((a + -0.5) * log(t)) - t;
	tmp = 0.0;
	if (z <= 6.5e+81)
		tmp = log(((x + y) * z)) + t_1;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(a + -0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[z, 6.5e+81], N[(N[Log[N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision] + t$95$1), $MachinePrecision], t$95$1]]

Derivation

1. Split input into 2 regimes

2. if z < 6.4999999999999996e81

   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. +-commutative 99.4%

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

      3. sum-log 95.1%

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

      4. sub-neg 95.1%

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

      5. metadata-eval 95.1%

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

   3. Applied egg-rr 95.1%

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

   if 6.4999999999999996e81 < 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. Taylor expanded in x around 0 65.9%

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

      1. remove-double-neg 65.9%

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

      2. log-rec 65.9%

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

      3. mul-1-neg 65.9%

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

      4. +-commutative 65.9%

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

      5. associate--l+ 65.9%

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

      6. mul-1-neg 65.9%

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

      7. log-rec 65.9%

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

      8. remove-double-neg 65.9%

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

   4. Simplified 65.9%

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

   5. Taylor expanded in t around inf 85.5%

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

      1. neg-mul-1 85.5%

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

   7. Simplified 85.5%

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

   8. Taylor expanded in t around 0 85.5%

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

      1. mul-1-neg 85.5%

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

      2. +-commutative 85.5%

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

      3. sub-neg 85.5%

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

      4. metadata-eval 85.5%

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

      5. +-commutative 85.5%

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

      6. distribute-rgt-out 85.5%

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

      7. sub-neg 85.5%

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

      8. distribute-rgt-out 85.5%

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

      9. +-commutative 85.5%

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

   10. Simplified 85.5%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 6.5 \cdot 10^{+81}:\\ \;\;\;\;\log \left(\left(x + y\right) \cdot z\right) + \left(\left(a + -0.5\right) \cdot \log t - t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t - t\\ \end{array} \]

Alternative 8: 61.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 0.0115:\\ \;\;\;\;\log \left(y \cdot z\right) + \log t \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log y + a \cdot \log t\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 0.0115)
   (+ (log (* y z)) (* (log t) (- a 0.5)))
   (- (+ (log y) (* a (log t))) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 0.0115) {
		tmp = log((y * z)) + (log(t) * (a - 0.5));
	} else {
		tmp = (log(y) + (a * log(t))) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 0.0115d0) then
        tmp = log((y * z)) + (log(t) * (a - 0.5d0))
    else
        tmp = (log(y) + (a * log(t))) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 0.0115) {
		tmp = Math.log((y * z)) + (Math.log(t) * (a - 0.5));
	} else {
		tmp = (Math.log(y) + (a * Math.log(t))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 0.0115:
		tmp = math.log((y * z)) + (math.log(t) * (a - 0.5))
	else:
		tmp = (math.log(y) + (a * math.log(t))) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 0.0115)
		tmp = Float64(log(Float64(y * z)) + Float64(log(t) * Float64(a - 0.5)));
	else
		tmp = Float64(Float64(log(y) + Float64(a * log(t))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 0.0115)
		tmp = log((y * z)) + (log(t) * (a - 0.5));
	else
		tmp = (log(y) + (a * log(t))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 0.0115], N[(N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 0.0115

   1. Initial program 99.1%

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

   2. Taylor expanded in x around 0 62.0%

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

      1. remove-double-neg 62.0%

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

      2. log-rec 62.0%

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

      3. mul-1-neg 62.0%

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

      4. +-commutative 62.0%

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

      5. associate--l+ 62.0%

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

      6. mul-1-neg 62.0%

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

      7. log-rec 62.0%

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

      8. remove-double-neg 62.0%

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

   4. Simplified 62.0%

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

   5. Taylor expanded in t around 0 61.7%

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

      1. log-prod 49.8%

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

   7. Simplified 49.8%

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

   if 0.0115 < t

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. associate-+l+ 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+r- 99.9%

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

      5. fma-def 99.9%

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

      6. sub-neg 99.9%

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

      7. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in a around inf 99.5%

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

      1. *-commutative 99.5%

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

   6. Simplified 99.5%

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

   7. Taylor expanded in x around 0 70.4%

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

      1. log-pow 32.4%

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

      2. associate--l+ 32.4%

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

      3. remove-double-neg 32.4%

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

      4. log-rec 32.4%

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

      5. mul-1-neg 32.4%

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

      6. associate--l+ 32.4%

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

      7. log-pow 70.4%

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

      8. mul-1-neg 70.4%

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

      9. log-rec 70.4%

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

      10. remove-double-neg 70.4%

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

      11. *-commutative 70.4%

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

   9. Simplified 70.4%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 0.0115:\\ \;\;\;\;\log \left(y \cdot z\right) + \log t \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log y + a \cdot \log t\right) - t\\ \end{array} \]

Alternative 9: 73.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 3.4 \cdot 10^{-5}:\\ \;\;\;\;\log \left(y \cdot z\right) + \log t \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + y\right) + \left(a \cdot \log t - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 3.4e-5)
   (+ (log (* y z)) (* (log t) (- a 0.5)))
   (+ (log (+ x y)) (- (* a (log t)) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 3.4e-5) {
		tmp = log((y * z)) + (log(t) * (a - 0.5));
	} else {
		tmp = log((x + y)) + ((a * log(t)) - t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 3.4d-5) then
        tmp = log((y * z)) + (log(t) * (a - 0.5d0))
    else
        tmp = log((x + y)) + ((a * log(t)) - t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 3.4e-5) {
		tmp = Math.log((y * z)) + (Math.log(t) * (a - 0.5));
	} else {
		tmp = Math.log((x + y)) + ((a * Math.log(t)) - t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 3.4e-5:
		tmp = math.log((y * z)) + (math.log(t) * (a - 0.5))
	else:
		tmp = math.log((x + y)) + ((a * math.log(t)) - t)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 3.4e-5)
		tmp = Float64(log(Float64(y * z)) + Float64(log(t) * Float64(a - 0.5)));
	else
		tmp = Float64(log(Float64(x + y)) + Float64(Float64(a * log(t)) - t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 3.4e-5)
		tmp = log((y * z)) + (log(t) * (a - 0.5));
	else
		tmp = log((x + y)) + ((a * log(t)) - t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 3.4e-5], N[(N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 3.4e-5

   1. Initial program 99.1%

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

   2. Taylor expanded in x around 0 62.0%

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

      1. remove-double-neg 62.0%

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

      2. log-rec 62.0%

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

      3. mul-1-neg 62.0%

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

      4. +-commutative 62.0%

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

      5. associate--l+ 62.0%

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

      6. mul-1-neg 62.0%

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

      7. log-rec 62.0%

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

      8. remove-double-neg 62.0%

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

   4. Simplified 62.0%

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

   5. Taylor expanded in t around 0 61.7%

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

      1. log-prod 49.8%

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

   7. Simplified 49.8%

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

   if 3.4e-5 < t

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. associate-+l+ 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+r- 99.9%

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

      5. fma-def 99.9%

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

      6. sub-neg 99.9%

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

      7. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in a around inf 99.5%

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

      1. *-commutative 99.5%

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

   6. Simplified 99.5%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.4 \cdot 10^{-5}:\\ \;\;\;\;\log \left(y \cdot z\right) + \log t \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + y\right) + \left(a \cdot \log t - t\right)\\ \end{array} \]

Alternative 10: 64.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.12 \cdot 10^{+20} \lor \neg \left(t \leq 3 \cdot 10^{+63}\right) \land t \leq 1.22 \cdot 10^{+100}:\\ \;\;\;\;\log t \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t 1.12e+20) (and (not (<= t 3e+63)) (<= t 1.22e+100)))
   (* (log t) (- a 0.5))
   (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= 1.12e+20) || (!(t <= 3e+63) && (t <= 1.22e+100))) {
		tmp = log(t) * (a - 0.5);
	} 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 <= 1.12d+20) .or. (.not. (t <= 3d+63)) .and. (t <= 1.22d+100)) then
        tmp = log(t) * (a - 0.5d0)
    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 <= 1.12e+20) || (!(t <= 3e+63) && (t <= 1.22e+100))) {
		tmp = Math.log(t) * (a - 0.5);
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (t <= 1.12e+20) or (not (t <= 3e+63) and (t <= 1.22e+100)):
		tmp = math.log(t) * (a - 0.5)
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((t <= 1.12e+20) || (!(t <= 3e+63) && (t <= 1.22e+100)))
		tmp = Float64(log(t) * Float64(a - 0.5));
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((t <= 1.12e+20) || (~((t <= 3e+63)) && (t <= 1.22e+100)))
		tmp = log(t) * (a - 0.5);
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[t, 1.12e+20], And[N[Not[LessEqual[t, 3e+63]], $MachinePrecision], LessEqual[t, 1.22e+100]]], N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if t < 1.12e20 or 2.99999999999999999e63 < t < 1.21999999999999995e100

   1. Initial program 99.2%

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

   2. Taylor expanded in x around 0 64.7%

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

      1. remove-double-neg 64.7%

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

      2. log-rec 64.7%

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

      3. mul-1-neg 64.7%

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

      4. +-commutative 64.7%

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

      5. associate--l+ 64.7%

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

      6. mul-1-neg 64.7%

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

      7. log-rec 64.7%

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

      8. remove-double-neg 64.7%

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

   4. Simplified 64.7%

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

   5. Taylor expanded in t around inf 65.9%

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

      1. neg-mul-1 65.9%

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

   7. Simplified 65.9%

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

   8. Taylor expanded in t around 0 62.0%

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

   if 1.12e20 < t < 2.99999999999999999e63 or 1.21999999999999995e100 < t

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. associate-+l+ 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+r- 99.9%

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

      5. fma-def 99.9%

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

      6. sub-neg 99.9%

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

      7. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t around inf 77.2%

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

      1. neg-mul-1 77.2%

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

   6. Simplified 77.2%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.12 \cdot 10^{+20} \lor \neg \left(t \leq 3 \cdot 10^{+63}\right) \land t \leq 1.22 \cdot 10^{+100}:\\ \;\;\;\;\log t \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 11: 66.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2000000 \lor \neg \left(a \leq 10^{+37}\right):\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -2000000.0) (not (<= a 1e+37)))
   (* a (log t))
   (- (log (+ x y)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -2000000.0) || !(a <= 1e+37)) {
		tmp = a * log(t);
	} else {
		tmp = log((x + y)) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((a <= (-2000000.0d0)) .or. (.not. (a <= 1d+37))) then
        tmp = a * log(t)
    else
        tmp = log((x + y)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -2000000.0) || !(a <= 1e+37)) {
		tmp = a * Math.log(t);
	} else {
		tmp = Math.log((x + y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -2000000.0) or not (a <= 1e+37):
		tmp = a * math.log(t)
	else:
		tmp = math.log((x + y)) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -2000000.0) || !(a <= 1e+37))
		tmp = Float64(a * log(t));
	else
		tmp = Float64(log(Float64(x + y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -2000000.0) || ~((a <= 1e+37)))
		tmp = a * log(t);
	else
		tmp = log((x + y)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -2000000.0], N[Not[LessEqual[a, 1e+37]], $MachinePrecision]], N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision], N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2e6 or 9.99999999999999954e36 < a

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 69.9%

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

      1. remove-double-neg 69.9%

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

      2. log-rec 69.9%

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

      3. mul-1-neg 69.9%

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

      4. +-commutative 69.9%

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

      5. associate--l+ 69.9%

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

      6. mul-1-neg 69.9%

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

      7. log-rec 69.9%

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

      8. remove-double-neg 69.9%

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

   4. Simplified 69.9%

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

   5. Taylor expanded in a around inf 79.2%

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

      1. *-commutative 79.2%

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

   7. Simplified 79.2%

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

   if -2e6 < a < 9.99999999999999954e36

   1. Initial program 99.3%

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

      1. associate--l+ 99.3%

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

      2. associate-+l+ 99.3%

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

      3. +-commutative 99.3%

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

      4. associate-+r- 99.3%

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

      5. fma-def 99.3%

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

      6. sub-neg 99.3%

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

      7. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in a around inf 57.9%

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

      1. *-commutative 57.9%

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

   6. Simplified 57.9%

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

   7. Taylor expanded in a around 0 55.6%

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

      1. +-commutative 55.6%

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

   9. Simplified 55.6%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2000000 \lor \neg \left(a \leq 10^{+37}\right):\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + y\right) - t\\ \end{array} \]

Alternative 12: 77.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 0.0195:\\ \;\;\;\;t + \left(a + -0.5\right) \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 0.0195) (+ t (* (+ a -0.5) (log t))) (- (* a (log t)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 0.0195) {
		tmp = t + ((a + -0.5) * log(t));
	} else {
		tmp = (a * log(t)) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 0.0195d0) then
        tmp = t + ((a + (-0.5d0)) * log(t))
    else
        tmp = (a * log(t)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 0.0195) {
		tmp = t + ((a + -0.5) * Math.log(t));
	} else {
		tmp = (a * Math.log(t)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 0.0195:
		tmp = t + ((a + -0.5) * math.log(t))
	else:
		tmp = (a * math.log(t)) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 0.0195)
		tmp = Float64(t + Float64(Float64(a + -0.5) * log(t)));
	else
		tmp = Float64(Float64(a * log(t)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 0.0195)
		tmp = t + ((a + -0.5) * log(t));
	else
		tmp = (a * log(t)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 0.0195], N[(t + N[(N[(a + -0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 0.0195

   1. Initial program 99.1%

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

   2. Taylor expanded in x around 0 62.0%

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

      1. remove-double-neg 62.0%

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

      2. log-rec 62.0%

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

      3. mul-1-neg 62.0%

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

      4. +-commutative 62.0%

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

      5. associate--l+ 62.0%

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

      6. mul-1-neg 62.0%

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

      7. log-rec 62.0%

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

      8. remove-double-neg 62.0%

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

   4. Simplified 62.0%

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

   5. Taylor expanded in t around inf 60.0%

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

      1. neg-mul-1 60.0%

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

   7. Simplified 60.0%

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

   8. Taylor expanded in t around 0 60.0%

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

      1. mul-1-neg 60.0%

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

      2. +-commutative 60.0%

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

      3. sub-neg 60.0%

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

      4. metadata-eval 60.0%

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

      5. +-commutative 60.0%

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

      6. distribute-rgt-out 60.0%

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

      7. sub-neg 60.0%

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

      8. distribute-rgt-out 60.0%

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

      9. +-commutative 60.0%

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

   10. Simplified 60.0%

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

       1. sub-neg 60.0%

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

       2. add-sqr-sqrt 0.0%

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

       3. sqrt-unprod 60.0%

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

       4. sqr-neg 60.0%

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

       5. sqrt-unprod 60.0%

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

       6. add-sqr-sqrt 60.0%

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

   12. Applied egg-rr 60.0%

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

   if 0.0195 < t

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 70.8%

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

      1. remove-double-neg 70.8%

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

      2. log-rec 70.8%

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

      3. mul-1-neg 70.8%

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

      4. +-commutative 70.8%

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

      5. associate--l+ 70.8%

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

      6. mul-1-neg 70.8%

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

      7. log-rec 70.8%

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

      8. remove-double-neg 70.8%

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

   4. Simplified 70.8%

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

   5. Taylor expanded in t around inf 99.5%

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

      1. neg-mul-1 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in t around 0 99.5%

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

      1. mul-1-neg 99.5%

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

      2. +-commutative 99.5%

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

      3. sub-neg 99.5%

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

      4. metadata-eval 99.5%

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

      5. +-commutative 99.5%

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

      6. distribute-rgt-out 99.5%

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

      7. sub-neg 99.5%

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

      8. distribute-rgt-out 99.5%

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

      9. +-commutative 99.5%

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

   10. Simplified 99.5%

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

   11. Taylor expanded in a around inf 99.5%

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

       1. *-commutative 99.5%

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

   13. Simplified 99.5%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 0.0195:\\ \;\;\;\;t + \left(a + -0.5\right) \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \]

Alternative 13: 63.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2000000 \lor \neg \left(a \leq 10^{+37}\right):\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -2000000.0) (not (<= a 1e+37))) (* a (log t)) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -2000000.0) || !(a <= 1e+37)) {
		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 ((a <= (-2000000.0d0)) .or. (.not. (a <= 1d+37))) 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 ((a <= -2000000.0) || !(a <= 1e+37)) {
		tmp = a * Math.log(t);
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -2000000.0) or not (a <= 1e+37):
		tmp = a * math.log(t)
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -2000000.0) || !(a <= 1e+37))
		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 ((a <= -2000000.0) || ~((a <= 1e+37)))
		tmp = a * log(t);
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -2000000.0], N[Not[LessEqual[a, 1e+37]], $MachinePrecision]], N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if a < -2e6 or 9.99999999999999954e36 < a

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 69.9%

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

      1. remove-double-neg 69.9%

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

      2. log-rec 69.9%

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

      3. mul-1-neg 69.9%

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

      4. +-commutative 69.9%

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

      5. associate--l+ 69.9%

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

      6. mul-1-neg 69.9%

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

      7. log-rec 69.9%

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

      8. remove-double-neg 69.9%

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

   4. Simplified 69.9%

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

   5. Taylor expanded in a around inf 79.2%

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

      1. *-commutative 79.2%

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

   7. Simplified 79.2%

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

   if -2e6 < a < 9.99999999999999954e36

   1. Initial program 99.3%

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

      1. associate--l+ 99.3%

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

      2. associate-+l+ 99.3%

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

      3. +-commutative 99.3%

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

      4. associate-+r- 99.3%

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

      5. fma-def 99.3%

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

      6. sub-neg 99.3%

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

      7. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in t around inf 50.3%

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

      1. neg-mul-1 50.3%

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

   6. Simplified 50.3%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2000000 \lor \neg \left(a \leq 10^{+37}\right):\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 14: 77.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 0.00045:\\ \;\;\;\;\log t \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 0.00045) (* (log t) (- a 0.5)) (- (* a (log t)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 0.00045) {
		tmp = log(t) * (a - 0.5);
	} else {
		tmp = (a * log(t)) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 0.00045d0) then
        tmp = log(t) * (a - 0.5d0)
    else
        tmp = (a * log(t)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 0.00045) {
		tmp = Math.log(t) * (a - 0.5);
	} else {
		tmp = (a * Math.log(t)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 0.00045:
		tmp = math.log(t) * (a - 0.5)
	else:
		tmp = (a * math.log(t)) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 0.00045)
		tmp = Float64(log(t) * Float64(a - 0.5));
	else
		tmp = Float64(Float64(a * log(t)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 0.00045)
		tmp = log(t) * (a - 0.5);
	else
		tmp = (a * log(t)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 0.00045], N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 4.4999999999999999e-4

   1. Initial program 99.1%

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

   2. Taylor expanded in x around 0 62.0%

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

      1. remove-double-neg 62.0%

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

      2. log-rec 62.0%

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

      3. mul-1-neg 62.0%

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

      4. +-commutative 62.0%

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

      5. associate--l+ 62.0%

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

      6. mul-1-neg 62.0%

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

      7. log-rec 62.0%

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

      8. remove-double-neg 62.0%

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

   4. Simplified 62.0%

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

   5. Taylor expanded in t around inf 60.0%

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

      1. neg-mul-1 60.0%

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

   7. Simplified 60.0%

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

   8. Taylor expanded in t around 0 60.0%

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

   if 4.4999999999999999e-4 < t

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 70.8%

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

      1. remove-double-neg 70.8%

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

      2. log-rec 70.8%

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

      3. mul-1-neg 70.8%

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

      4. +-commutative 70.8%

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

      5. associate--l+ 70.8%

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

      6. mul-1-neg 70.8%

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

      7. log-rec 70.8%

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

      8. remove-double-neg 70.8%

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

   4. Simplified 70.8%

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

   5. Taylor expanded in t around inf 99.5%

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

      1. neg-mul-1 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in t around 0 99.5%

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

      1. mul-1-neg 99.5%

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

      2. +-commutative 99.5%

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

      3. sub-neg 99.5%

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

      4. metadata-eval 99.5%

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

      5. +-commutative 99.5%

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

      6. distribute-rgt-out 99.5%

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

      7. sub-neg 99.5%

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

      8. distribute-rgt-out 99.5%

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

      9. +-commutative 99.5%

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

   10. Simplified 99.5%

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

   11. Taylor expanded in a around inf 99.5%

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

       1. *-commutative 99.5%

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

   13. Simplified 99.5%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 0.00045:\\ \;\;\;\;\log t \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \]

Alternative 15: 77.6% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in x around 0 66.4%

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

   1. remove-double-neg 66.4%

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

   2. log-rec 66.4%

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

   3. mul-1-neg 66.4%

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

   4. +-commutative 66.4%

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

   5. associate--l+ 66.4%

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

   6. mul-1-neg 66.4%

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

   7. log-rec 66.4%

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

   8. remove-double-neg 66.4%

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

4. Simplified 66.4%

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

5. Taylor expanded in t around inf 79.6%

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

   1. neg-mul-1 79.6%

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

7. Simplified 79.6%

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

8. Taylor expanded in t around 0 79.6%

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

   1. mul-1-neg 79.6%

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

   2. +-commutative 79.6%

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

   3. sub-neg 79.6%

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

   4. metadata-eval 79.6%

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

   5. +-commutative 79.6%

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

   6. distribute-rgt-out 79.6%

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

   7. sub-neg 79.6%

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

   8. distribute-rgt-out 79.6%

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

   9. +-commutative 79.6%

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

10. Simplified 79.6%

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

11. Final simplification 79.6%

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

Alternative 16: 40.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 0.92:\\ \;\;\;\;-0.5 \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 0.92) (* -0.5 (log t)) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 0.92) {
		tmp = -0.5 * 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 <= 0.92d0) then
        tmp = (-0.5d0) * 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 <= 0.92) {
		tmp = -0.5 * Math.log(t);
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 0.92:
		tmp = -0.5 * math.log(t)
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 0.92)
		tmp = Float64(-0.5 * 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 <= 0.92)
		tmp = -0.5 * log(t);
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 0.92], N[(-0.5 * N[Log[t], $MachinePrecision]), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if t < 0.92000000000000004

   1. Initial program 99.1%

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

   2. Taylor expanded in x around 0 62.3%

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

      1. remove-double-neg 62.3%

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

      2. log-rec 62.3%

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

      3. mul-1-neg 62.3%

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

      4. +-commutative 62.3%

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

      5. associate--l+ 62.3%

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

      6. mul-1-neg 62.3%

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

      7. log-rec 62.3%

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

      8. remove-double-neg 62.3%

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

   4. Simplified 62.3%

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

   5. Taylor expanded in t around inf 60.3%

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

      1. neg-mul-1 60.3%

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

   7. Simplified 60.3%

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

   8. Taylor expanded in t around 0 60.3%

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

   9. Taylor expanded in a around 0 8.4%

      \[\leadsto \color{blue}{-0.5 \cdot \log t} \]
   10. Step-by-step derivation

       1. *-commutative 8.4%

          \[\leadsto \color{blue}{\log t \cdot -0.5} \]

   11. Simplified 8.4%

       \[\leadsto \color{blue}{\log t \cdot -0.5} \]

   if 0.92000000000000004 < t

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. associate-+l+ 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+r- 99.9%

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

      5. fma-def 99.9%

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

      6. sub-neg 99.9%

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

      7. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t around inf 68.9%

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

      1. neg-mul-1 68.9%

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

   6. Simplified 68.9%

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

4. Final simplification 38.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 0.92:\\ \;\;\;\;-0.5 \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 17: 37.5% 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 99.5%

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

   1. associate--l+ 99.5%

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

   2. associate-+l+ 99.5%

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

   3. +-commutative 99.5%

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

   4. associate-+r- 99.5%

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

   5. fma-def 99.5%

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

   6. sub-neg 99.5%

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

   7. metadata-eval 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in t around inf 35.3%

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

   1. neg-mul-1 35.3%

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

6. Simplified 35.3%

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

7. Final simplification 35.3%

   \[\leadsto -t \]

Developer target: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023271 
(FPCore (x y z t a)
  :name "Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2"
  :precision binary64

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

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