Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 23.1s

Alternatives: 19

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. associate-+l- 99.5%

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

   2. associate--l+ 99.6%

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

   3. sub-neg 99.6%

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

   4. +-commutative 99.6%

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

   5. *-commutative 99.6%

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

   6. distribute-rgt-neg-in 99.6%

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

   7. fma-undefine 99.6%

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

   8. sub-neg 99.6%

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

   9. +-commutative 99.6%

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

   10. distribute-neg-in 99.6%

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

   11. metadata-eval 99.6%

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

   12. metadata-eval 99.6%

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

   13. unsub-neg 99.6%

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

3. Simplified 99.6%

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

Alternative 2: 96.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log t \cdot \left(a - 0.5\right)\\ \mathbf{if}\;t \leq 1.8 \cdot 10^{-17}:\\ \;\;\;\;\log \left(x + y\right) + \left(\log z + t\_1\right)\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{+22}:\\ \;\;\;\;\left(\log \left(y \cdot z\right) + t\_1\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log t, a, -t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (log t) (- a 0.5))))
   (if (<= t 1.8e-17)
     (+ (log (+ x y)) (+ (log z) t_1))
     (if (<= t 5.5e+22) (- (+ (log (* y z)) t_1) t) (fma (log t) a (- t))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log(t) * (a - 0.5);
	double tmp;
	if (t <= 1.8e-17) {
		tmp = log((x + y)) + (log(z) + t_1);
	} else if (t <= 5.5e+22) {
		tmp = (log((y * z)) + t_1) - t;
	} else {
		tmp = fma(log(t), a, -t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(t) * Float64(a - 0.5))
	tmp = 0.0
	if (t <= 1.8e-17)
		tmp = Float64(log(Float64(x + y)) + Float64(log(z) + t_1));
	elseif (t <= 5.5e+22)
		tmp = Float64(Float64(log(Float64(y * z)) + t_1) - t);
	else
		tmp = fma(log(t), a, Float64(-t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, 1.8e-17], N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[(N[Log[z], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.5e+22], N[(N[(N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision] + t$95$1), $MachinePrecision] - t), $MachinePrecision], N[(N[Log[t], $MachinePrecision] * a + (-t)), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.79999999999999997e-17

   1. Initial program 99.2%

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

      1. associate-+l- 99.2%

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

      2. associate--l+ 99.2%

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

      3. sub-neg 99.2%

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

      4. +-commutative 99.2%

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

      5. *-commutative 99.2%

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

      6. distribute-rgt-neg-in 99.2%

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

      7. fma-undefine 99.2%

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

      8. sub-neg 99.2%

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

      9. +-commutative 99.2%

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

      10. distribute-neg-in 99.2%

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

      11. metadata-eval 99.2%

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

      12. metadata-eval 99.2%

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

      13. unsub-neg 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in t around 0 99.2%

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

   if 1.79999999999999997e-17 < t < 5.50000000000000021e22

   1. Initial program 99.9%

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

      1. add-cbrt-cube 76.6%

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

      2. pow1/3 33.7%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \color{blue}{{\left(\left(\left(\left(a - 0.5\right) \cdot \log t\right) \cdot \left(\left(a - 0.5\right) \cdot \log t\right)\right) \cdot \left(\left(a - 0.5\right) \cdot \log t\right)\right)}^{0.3333333333333333}} \]

      3. pow3 33.7%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + {\color{blue}{\left({\left(\left(a - 0.5\right) \cdot \log t\right)}^{3}\right)}}^{0.3333333333333333} \]

      4. sub-neg 33.7%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + {\left({\left(\color{blue}{\left(a + \left(-0.5\right)\right)} \cdot \log t\right)}^{3}\right)}^{0.3333333333333333} \]

      5. metadata-eval 33.7%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + {\left({\left(\left(a + \color{blue}{-0.5}\right) \cdot \log t\right)}^{3}\right)}^{0.3333333333333333} \]

      6. *-commutative 33.7%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + {\left({\color{blue}{\left(\log t \cdot \left(a + -0.5\right)\right)}}^{3}\right)}^{0.3333333333333333} \]

   4. Applied egg-rr 33.7%

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \color{blue}{{\left({\left(\log t \cdot \left(a + -0.5\right)\right)}^{3}\right)}^{0.3333333333333333}} \]
   5. Step-by-step derivation

      1. pow-pow 99.9%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \color{blue}{{\left(\log t \cdot \left(a + -0.5\right)\right)}^{\left(3 \cdot 0.3333333333333333\right)}} \]

      2. metadata-eval 99.9%

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

      3. pow1 99.9%

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

      4. associate-+l- 99.9%

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

      5. +-commutative 99.9%

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

      6. sum-log 91.8%

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

      7. +-commutative 91.8%

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

   6. Applied egg-rr 91.8%

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

   7. Taylor expanded in x around 0 60.3%

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

   if 5.50000000000000021e22 < t

   1. Initial program 99.9%

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

      1. add-cbrt-cube 71.7%

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

      2. pow1/3 7.6%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \color{blue}{{\left(\left(\left(\left(a - 0.5\right) \cdot \log t\right) \cdot \left(\left(a - 0.5\right) \cdot \log t\right)\right) \cdot \left(\left(a - 0.5\right) \cdot \log t\right)\right)}^{0.3333333333333333}} \]

      3. pow3 7.6%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + {\color{blue}{\left({\left(\left(a - 0.5\right) \cdot \log t\right)}^{3}\right)}}^{0.3333333333333333} \]

      4. sub-neg 7.6%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + {\left({\left(\color{blue}{\left(a + \left(-0.5\right)\right)} \cdot \log t\right)}^{3}\right)}^{0.3333333333333333} \]

      5. metadata-eval 7.6%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + {\left({\left(\left(a + \color{blue}{-0.5}\right) \cdot \log t\right)}^{3}\right)}^{0.3333333333333333} \]

      6. *-commutative 7.6%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + {\left({\color{blue}{\left(\log t \cdot \left(a + -0.5\right)\right)}}^{3}\right)}^{0.3333333333333333} \]

   4. Applied egg-rr 7.6%

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \color{blue}{{\left({\left(\log t \cdot \left(a + -0.5\right)\right)}^{3}\right)}^{0.3333333333333333}} \]
   5. Step-by-step derivation

      1. pow-pow 99.9%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \color{blue}{{\left(\log t \cdot \left(a + -0.5\right)\right)}^{\left(3 \cdot 0.3333333333333333\right)}} \]

      2. metadata-eval 99.9%

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

      3. pow1 99.9%

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

      4. associate-+l- 99.9%

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

      5. +-commutative 99.9%

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

      6. sum-log 81.5%

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

      7. +-commutative 81.5%

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

   6. Applied egg-rr 81.5%

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

   7. Taylor expanded in x around 0 64.5%

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

   8. Taylor expanded in a around inf 99.9%

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

      1. *-commutative 99.9%

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

   10. Simplified 99.9%

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

       1. fma-neg 99.9%

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

   12. Applied egg-rr 99.9%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.8 \cdot 10^{-17}:\\ \;\;\;\;\log \left(x + y\right) + \left(\log z + \log t \cdot \left(a - 0.5\right)\right)\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{+22}:\\ \;\;\;\;\left(\log \left(y \cdot z\right) + \log t \cdot \left(a - 0.5\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log t, a, -t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 4.6 \cdot 10^{-30}:\\ \;\;\;\;\log z + \left(\log y + \log t \cdot \left(a - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log t, a, -t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 4.6e-30)
   (+ (log z) (+ (log y) (* (log t) (- a 0.5))))
   (fma (log t) a (- t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 4.6e-30) {
		tmp = log(z) + (log(y) + (log(t) * (a - 0.5)));
	} else {
		tmp = fma(log(t), a, -t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 4.6e-30)
		tmp = Float64(log(z) + Float64(log(y) + Float64(log(t) * Float64(a - 0.5))));
	else
		tmp = fma(log(t), a, Float64(-t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 4.6e-30], N[(N[Log[z], $MachinePrecision] + N[(N[Log[y], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Log[t], $MachinePrecision] * a + (-t)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 4.59999999999999968e-30

   1. Initial program 99.2%

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

      1. associate--l+ 99.2%

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

      2. +-commutative 99.2%

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

      3. associate-+l+ 99.2%

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

      4. +-commutative 99.2%

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

      5. fma-define 99.2%

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

      6. sub-neg 99.2%

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

      7. metadata-eval 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in x around 0 63.6%

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

   6. Taylor expanded in t around 0 63.6%

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

   if 4.59999999999999968e-30 < t

   1. Initial program 99.9%

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

      1. add-cbrt-cube 70.6%

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

      2. pow1/3 11.0%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \color{blue}{{\left(\left(\left(\left(a - 0.5\right) \cdot \log t\right) \cdot \left(\left(a - 0.5\right) \cdot \log t\right)\right) \cdot \left(\left(a - 0.5\right) \cdot \log t\right)\right)}^{0.3333333333333333}} \]

      3. pow3 11.0%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + {\color{blue}{\left({\left(\left(a - 0.5\right) \cdot \log t\right)}^{3}\right)}}^{0.3333333333333333} \]

      4. sub-neg 11.0%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + {\left({\left(\color{blue}{\left(a + \left(-0.5\right)\right)} \cdot \log t\right)}^{3}\right)}^{0.3333333333333333} \]

      5. metadata-eval 11.0%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + {\left({\left(\left(a + \color{blue}{-0.5}\right) \cdot \log t\right)}^{3}\right)}^{0.3333333333333333} \]

      6. *-commutative 11.0%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + {\left({\color{blue}{\left(\log t \cdot \left(a + -0.5\right)\right)}}^{3}\right)}^{0.3333333333333333} \]

   4. Applied egg-rr 11.0%

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \color{blue}{{\left({\left(\log t \cdot \left(a + -0.5\right)\right)}^{3}\right)}^{0.3333333333333333}} \]
   5. Step-by-step derivation

      1. pow-pow 99.9%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \color{blue}{{\left(\log t \cdot \left(a + -0.5\right)\right)}^{\left(3 \cdot 0.3333333333333333\right)}} \]

      2. metadata-eval 99.9%

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

      3. pow1 99.9%

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

      4. associate-+l- 99.9%

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

      5. +-commutative 99.9%

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

      6. sum-log 81.8%

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

      7. +-commutative 81.8%

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

   6. Applied egg-rr 81.8%

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

   7. Taylor expanded in x around 0 65.0%

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

   8. Taylor expanded in a around inf 97.4%

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

      1. *-commutative 97.4%

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

   10. Simplified 97.4%

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

       1. fma-neg 97.4%

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

   12. Applied egg-rr 97.4%

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

Alternative 4: 80.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 4.6 \cdot 10^{-30}:\\ \;\;\;\;\log y + \left(\log z + \log t \cdot \left(a - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log t, a, -t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 4.6e-30)
   (+ (log y) (+ (log z) (* (log t) (- a 0.5))))
   (fma (log t) a (- t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 4.6e-30) {
		tmp = log(y) + (log(z) + (log(t) * (a - 0.5)));
	} else {
		tmp = fma(log(t), a, -t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 4.6e-30)
		tmp = Float64(log(y) + Float64(log(z) + Float64(log(t) * Float64(a - 0.5))));
	else
		tmp = fma(log(t), a, Float64(-t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 4.6e-30], N[(N[Log[y], $MachinePrecision] + N[(N[Log[z], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Log[t], $MachinePrecision] * a + (-t)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 4.59999999999999968e-30

   1. Initial program 99.2%

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

      1. associate-+l- 99.2%

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

      2. associate--l+ 99.2%

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

      3. sub-neg 99.2%

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

      4. +-commutative 99.2%

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

      5. *-commutative 99.2%

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

      6. distribute-rgt-neg-in 99.2%

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

      7. fma-undefine 99.2%

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

      8. sub-neg 99.2%

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

      9. +-commutative 99.2%

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

      10. distribute-neg-in 99.2%

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

      11. metadata-eval 99.2%

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

      12. metadata-eval 99.2%

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

      13. unsub-neg 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in t around 0 99.2%

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

   6. Taylor expanded in x around 0 63.6%

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

      1. associate--l+ 63.6%

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

   8. Simplified 63.6%

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

   if 4.59999999999999968e-30 < t

   1. Initial program 99.9%

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

      1. add-cbrt-cube 70.6%

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

      2. pow1/3 11.0%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \color{blue}{{\left(\left(\left(\left(a - 0.5\right) \cdot \log t\right) \cdot \left(\left(a - 0.5\right) \cdot \log t\right)\right) \cdot \left(\left(a - 0.5\right) \cdot \log t\right)\right)}^{0.3333333333333333}} \]

      3. pow3 11.0%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + {\color{blue}{\left({\left(\left(a - 0.5\right) \cdot \log t\right)}^{3}\right)}}^{0.3333333333333333} \]

      4. sub-neg 11.0%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + {\left({\left(\color{blue}{\left(a + \left(-0.5\right)\right)} \cdot \log t\right)}^{3}\right)}^{0.3333333333333333} \]

      5. metadata-eval 11.0%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + {\left({\left(\left(a + \color{blue}{-0.5}\right) \cdot \log t\right)}^{3}\right)}^{0.3333333333333333} \]

      6. *-commutative 11.0%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + {\left({\color{blue}{\left(\log t \cdot \left(a + -0.5\right)\right)}}^{3}\right)}^{0.3333333333333333} \]

   4. Applied egg-rr 11.0%

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \color{blue}{{\left({\left(\log t \cdot \left(a + -0.5\right)\right)}^{3}\right)}^{0.3333333333333333}} \]
   5. Step-by-step derivation

      1. pow-pow 99.9%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \color{blue}{{\left(\log t \cdot \left(a + -0.5\right)\right)}^{\left(3 \cdot 0.3333333333333333\right)}} \]

      2. metadata-eval 99.9%

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

      3. pow1 99.9%

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

      4. associate-+l- 99.9%

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

      5. +-commutative 99.9%

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

      6. sum-log 81.8%

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

      7. +-commutative 81.8%

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

   6. Applied egg-rr 81.8%

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

   7. Taylor expanded in x around 0 65.0%

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

   8. Taylor expanded in a around inf 97.4%

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

      1. *-commutative 97.4%

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

   10. Simplified 97.4%

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

       1. fma-neg 97.4%

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

   12. Applied egg-rr 97.4%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.6 \cdot 10^{-30}:\\ \;\;\;\;\log y + \left(\log z + \log t \cdot \left(a - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log t, a, -t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Final simplification 99.5%

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

Alternative 6: 69.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

   \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
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. +-commutative 99.5%

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

   3. associate-+l+ 99.6%

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

   4. +-commutative 99.6%

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

   5. fma-define 99.5%

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

   6. sub-neg 99.5%

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

   7. metadata-eval 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in x around 0 70.7%

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

Alternative 7: 87.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -7 \cdot 10^{+27} \lor \neg \left(a \leq 5\right):\\ \;\;\;\;\mathsf{fma}\left(\log t, a, -t\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(z \cdot \left(x + y\right)\right) + \left(\log t \cdot \left(a + -0.5\right) - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -7e+27) (not (<= a 5.0)))
   (fma (log t) a (- t))
   (+ (log (* z (+ x y))) (- (* (log t) (+ a -0.5)) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -7e+27) || !(a <= 5.0)) {
		tmp = fma(log(t), a, -t);
	} else {
		tmp = log((z * (x + y))) + ((log(t) * (a + -0.5)) - t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -7e+27) || !(a <= 5.0))
		tmp = fma(log(t), a, Float64(-t));
	else
		tmp = Float64(log(Float64(z * Float64(x + y))) + Float64(Float64(log(t) * Float64(a + -0.5)) - t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -7e+27], N[Not[LessEqual[a, 5.0]], $MachinePrecision]], N[(N[Log[t], $MachinePrecision] * a + (-t)), $MachinePrecision], N[(N[Log[N[(z * N[(x + y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -7.0000000000000004e27 or 5 < a

   1. Initial program 99.7%

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

      1. add-cbrt-cube 33.2%

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

      2. pow1/3 17.9%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \color{blue}{{\left(\left(\left(\left(a - 0.5\right) \cdot \log t\right) \cdot \left(\left(a - 0.5\right) \cdot \log t\right)\right) \cdot \left(\left(a - 0.5\right) \cdot \log t\right)\right)}^{0.3333333333333333}} \]

      3. pow3 17.9%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + {\color{blue}{\left({\left(\left(a - 0.5\right) \cdot \log t\right)}^{3}\right)}}^{0.3333333333333333} \]

      4. sub-neg 17.9%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + {\left({\left(\color{blue}{\left(a + \left(-0.5\right)\right)} \cdot \log t\right)}^{3}\right)}^{0.3333333333333333} \]

      5. metadata-eval 17.9%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + {\left({\left(\left(a + \color{blue}{-0.5}\right) \cdot \log t\right)}^{3}\right)}^{0.3333333333333333} \]

      6. *-commutative 17.9%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + {\left({\color{blue}{\left(\log t \cdot \left(a + -0.5\right)\right)}}^{3}\right)}^{0.3333333333333333} \]

   4. Applied egg-rr 17.9%

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \color{blue}{{\left({\left(\log t \cdot \left(a + -0.5\right)\right)}^{3}\right)}^{0.3333333333333333}} \]
   5. Step-by-step derivation

      1. pow-pow 99.7%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \color{blue}{{\left(\log t \cdot \left(a + -0.5\right)\right)}^{\left(3 \cdot 0.3333333333333333\right)}} \]

      2. metadata-eval 99.7%

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

      3. pow1 99.7%

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

      4. associate-+l- 99.7%

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

      5. +-commutative 99.7%

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

      6. sum-log 78.7%

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

      7. +-commutative 78.7%

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

   6. Applied egg-rr 78.7%

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

   7. Taylor expanded in x around 0 60.0%

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

   8. Taylor expanded in a around inf 99.0%

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

      1. *-commutative 99.0%

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

   10. Simplified 99.0%

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

       1. fma-neg 99.1%

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

   12. Applied egg-rr 99.1%

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

   if -7.0000000000000004e27 < a < 5

   1. Initial program 99.4%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   2. Add Preprocessing
   3. 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 83.3%

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

      4. sub-neg 83.3%

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

      5. metadata-eval 83.3%

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

      6. *-commutative 83.3%

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

   4. Applied egg-rr 83.3%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -7 \cdot 10^{+27} \lor \neg \left(a \leq 5\right):\\ \;\;\;\;\mathsf{fma}\left(\log t, a, -t\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(z \cdot \left(x + y\right)\right) + \left(\log t \cdot \left(a + -0.5\right) - t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -0.500005 \lor \neg \left(a - 0.5 \leq -0.49999999996\right):\\ \;\;\;\;\mathsf{fma}\left(\log t, a, -t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(y \cdot z\right) + \log t \cdot -0.5\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= (- a 0.5) -0.500005) (not (<= (- a 0.5) -0.49999999996)))
   (fma (log t) a (- t))
   (- (+ (log (* y z)) (* (log t) -0.5)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((a - 0.5) <= -0.500005) || !((a - 0.5) <= -0.49999999996)) {
		tmp = fma(log(t), a, -t);
	} else {
		tmp = (log((y * z)) + (log(t) * -0.5)) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((Float64(a - 0.5) <= -0.500005) || !(Float64(a - 0.5) <= -0.49999999996))
		tmp = fma(log(t), a, Float64(-t));
	else
		tmp = Float64(Float64(log(Float64(y * z)) + Float64(log(t) * -0.5)) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[N[(a - 0.5), $MachinePrecision], -0.500005], N[Not[LessEqual[N[(a - 0.5), $MachinePrecision], -0.49999999996]], $MachinePrecision]], N[(N[Log[t], $MachinePrecision] * a + (-t)), $MachinePrecision], N[(N[(N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 a #s(literal 1/2 binary64)) < -0.50000500000000003 or -0.49999999996 < (-.f64 a #s(literal 1/2 binary64))

   1. Initial program 99.6%

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

      1. add-cbrt-cube 38.4%

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

      2. pow1/3 20.2%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \color{blue}{{\left(\left(\left(\left(a - 0.5\right) \cdot \log t\right) \cdot \left(\left(a - 0.5\right) \cdot \log t\right)\right) \cdot \left(\left(a - 0.5\right) \cdot \log t\right)\right)}^{0.3333333333333333}} \]

      3. pow3 20.2%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + {\color{blue}{\left({\left(\left(a - 0.5\right) \cdot \log t\right)}^{3}\right)}}^{0.3333333333333333} \]

      4. sub-neg 20.2%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + {\left({\left(\color{blue}{\left(a + \left(-0.5\right)\right)} \cdot \log t\right)}^{3}\right)}^{0.3333333333333333} \]

      5. metadata-eval 20.2%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + {\left({\left(\left(a + \color{blue}{-0.5}\right) \cdot \log t\right)}^{3}\right)}^{0.3333333333333333} \]

      6. *-commutative 20.2%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + {\left({\color{blue}{\left(\log t \cdot \left(a + -0.5\right)\right)}}^{3}\right)}^{0.3333333333333333} \]

   4. Applied egg-rr 20.2%

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \color{blue}{{\left({\left(\log t \cdot \left(a + -0.5\right)\right)}^{3}\right)}^{0.3333333333333333}} \]
   5. Step-by-step derivation

      1. pow-pow 99.6%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \color{blue}{{\left(\log t \cdot \left(a + -0.5\right)\right)}^{\left(3 \cdot 0.3333333333333333\right)}} \]

      2. metadata-eval 99.6%

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

      3. pow1 99.6%

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

      4. associate-+l- 99.6%

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

      5. +-commutative 99.6%

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

      6. sum-log 79.2%

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

      7. +-commutative 79.2%

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

   6. Applied egg-rr 79.2%

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

   7. Taylor expanded in x around 0 59.6%

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

   8. Taylor expanded in a around inf 97.5%

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

      1. *-commutative 97.5%

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

   10. Simplified 97.5%

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

       1. fma-neg 97.5%

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

   12. Applied egg-rr 97.5%

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

   if -0.50000500000000003 < (-.f64 a #s(literal 1/2 binary64)) < -0.49999999996

   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. +-commutative 99.5%

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

      2. fma-define 99.5%

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

      3. sub-neg 99.5%

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

      4. metadata-eval 99.5%

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

      5. associate--l+ 99.5%

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

   3. Simplified 99.5%

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

      1. associate-+r- 99.5%

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

      2. +-commutative 99.5%

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

      3. sum-log 83.2%

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

   6. Applied egg-rr 83.2%

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

      1. +-commutative 83.2%

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

   8. Simplified 83.2%

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

   9. Taylor expanded in x around 0 55.7%

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

   10. Taylor expanded in a around 0 55.6%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -0.500005 \lor \neg \left(a - 0.5 \leq -0.49999999996\right):\\ \;\;\;\;\mathsf{fma}\left(\log t, a, -t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(y \cdot z\right) + \log t \cdot -0.5\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -6.5 \cdot 10^{+27} \lor \neg \left(a \leq 4.2\right):\\ \;\;\;\;\mathsf{fma}\left(\log t, a, -t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(y \cdot z\right) + \log t \cdot \left(a - 0.5\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -6.5e+27) (not (<= a 4.2)))
   (fma (log t) a (- t))
   (- (+ (log (* y z)) (* (log t) (- a 0.5))) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -6.5e+27) || !(a <= 4.2)) {
		tmp = fma(log(t), a, -t);
	} else {
		tmp = (log((y * z)) + (log(t) * (a - 0.5))) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -6.5e+27) || !(a <= 4.2))
		tmp = fma(log(t), a, Float64(-t));
	else
		tmp = Float64(Float64(log(Float64(y * z)) + Float64(log(t) * Float64(a - 0.5))) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -6.5e+27], N[Not[LessEqual[a, 4.2]], $MachinePrecision]], N[(N[Log[t], $MachinePrecision] * a + (-t)), $MachinePrecision], N[(N[(N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -6.5000000000000005e27 or 4.20000000000000018 < a

   1. Initial program 99.7%

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

      1. add-cbrt-cube 33.2%

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

      2. pow1/3 17.9%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \color{blue}{{\left(\left(\left(\left(a - 0.5\right) \cdot \log t\right) \cdot \left(\left(a - 0.5\right) \cdot \log t\right)\right) \cdot \left(\left(a - 0.5\right) \cdot \log t\right)\right)}^{0.3333333333333333}} \]

      3. pow3 17.9%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + {\color{blue}{\left({\left(\left(a - 0.5\right) \cdot \log t\right)}^{3}\right)}}^{0.3333333333333333} \]

      4. sub-neg 17.9%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + {\left({\left(\color{blue}{\left(a + \left(-0.5\right)\right)} \cdot \log t\right)}^{3}\right)}^{0.3333333333333333} \]

      5. metadata-eval 17.9%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + {\left({\left(\left(a + \color{blue}{-0.5}\right) \cdot \log t\right)}^{3}\right)}^{0.3333333333333333} \]

      6. *-commutative 17.9%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + {\left({\color{blue}{\left(\log t \cdot \left(a + -0.5\right)\right)}}^{3}\right)}^{0.3333333333333333} \]

   4. Applied egg-rr 17.9%

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \color{blue}{{\left({\left(\log t \cdot \left(a + -0.5\right)\right)}^{3}\right)}^{0.3333333333333333}} \]
   5. Step-by-step derivation

      1. pow-pow 99.7%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \color{blue}{{\left(\log t \cdot \left(a + -0.5\right)\right)}^{\left(3 \cdot 0.3333333333333333\right)}} \]

      2. metadata-eval 99.7%

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

      3. pow1 99.7%

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

      4. associate-+l- 99.7%

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

      5. +-commutative 99.7%

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

      6. sum-log 78.7%

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

      7. +-commutative 78.7%

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

   6. Applied egg-rr 78.7%

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

   7. Taylor expanded in x around 0 60.0%

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

   8. Taylor expanded in a around inf 99.0%

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

      1. *-commutative 99.0%

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

   10. Simplified 99.0%

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

       1. fma-neg 99.1%

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

   12. Applied egg-rr 99.1%

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

   if -6.5000000000000005e27 < a < 4.20000000000000018

   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. Add Preprocessing
   3. Step-by-step derivation

      1. add-cbrt-cube 99.4%

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

      2. pow1/3 46.9%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \color{blue}{{\left(\left(\left(\left(a - 0.5\right) \cdot \log t\right) \cdot \left(\left(a - 0.5\right) \cdot \log t\right)\right) \cdot \left(\left(a - 0.5\right) \cdot \log t\right)\right)}^{0.3333333333333333}} \]

      3. pow3 46.8%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + {\color{blue}{\left({\left(\left(a - 0.5\right) \cdot \log t\right)}^{3}\right)}}^{0.3333333333333333} \]

      4. sub-neg 46.8%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + {\left({\left(\color{blue}{\left(a + \left(-0.5\right)\right)} \cdot \log t\right)}^{3}\right)}^{0.3333333333333333} \]

      5. metadata-eval 46.8%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + {\left({\left(\left(a + \color{blue}{-0.5}\right) \cdot \log t\right)}^{3}\right)}^{0.3333333333333333} \]

      6. *-commutative 46.8%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + {\left({\color{blue}{\left(\log t \cdot \left(a + -0.5\right)\right)}}^{3}\right)}^{0.3333333333333333} \]

   4. Applied egg-rr 46.8%

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \color{blue}{{\left({\left(\log t \cdot \left(a + -0.5\right)\right)}^{3}\right)}^{0.3333333333333333}} \]
   5. Step-by-step derivation

      1. pow-pow 99.4%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \color{blue}{{\left(\log t \cdot \left(a + -0.5\right)\right)}^{\left(3 \cdot 0.3333333333333333\right)}} \]

      2. metadata-eval 99.4%

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

      3. pow1 99.4%

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

      4. associate-+l- 99.4%

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

      5. +-commutative 99.4%

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

      6. sum-log 83.3%

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

      7. +-commutative 83.3%

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

   6. Applied egg-rr 83.3%

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

   7. Taylor expanded in x around 0 55.7%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6.5 \cdot 10^{+27} \lor \neg \left(a \leq 4.2\right):\\ \;\;\;\;\mathsf{fma}\left(\log t, a, -t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(y \cdot z\right) + \log t \cdot \left(a - 0.5\right)\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 76.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 3.7 \cdot 10^{-117}:\\ \;\;\;\;\log \left(x + y\right) + \log t \cdot a\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{-91}:\\ \;\;\;\;\log \left(z \cdot \frac{x + y}{\sqrt{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log t, a, -t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 3.7e-117)
   (+ (log (+ x y)) (* (log t) a))
   (if (<= t 1.4e-91) (log (* z (/ (+ x y) (sqrt t)))) (fma (log t) a (- t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 3.7e-117) {
		tmp = log((x + y)) + (log(t) * a);
	} else if (t <= 1.4e-91) {
		tmp = log((z * ((x + y) / sqrt(t))));
	} else {
		tmp = fma(log(t), a, -t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 3.7e-117)
		tmp = Float64(log(Float64(x + y)) + Float64(log(t) * a));
	elseif (t <= 1.4e-91)
		tmp = log(Float64(z * Float64(Float64(x + y) / sqrt(t))));
	else
		tmp = fma(log(t), a, Float64(-t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 3.7e-117], N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.4e-91], N[Log[N[(z * N[(N[(x + y), $MachinePrecision] / N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Log[t], $MachinePrecision] * a + (-t)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 3.7000000000000002e-117

   1. Initial program 99.0%

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

      1. associate-+l- 99.0%

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

      2. associate--l+ 99.1%

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

      3. sub-neg 99.1%

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

      4. +-commutative 99.1%

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

      5. *-commutative 99.1%

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

      6. distribute-rgt-neg-in 99.1%

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

      7. fma-undefine 99.1%

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

      8. sub-neg 99.1%

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

      9. +-commutative 99.1%

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

      10. distribute-neg-in 99.1%

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

      11. metadata-eval 99.1%

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

      12. metadata-eval 99.1%

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

      13. unsub-neg 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in a around inf 59.4%

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

      1. *-commutative 59.4%

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

   7. Simplified 59.4%

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

   if 3.7000000000000002e-117 < t < 1.4e-91

   1. Initial program 99.0%

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

      1. associate-+l- 99.0%

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

      2. associate--l+ 99.4%

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

      3. sub-neg 99.4%

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

      4. +-commutative 99.4%

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

      5. *-commutative 99.4%

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

      6. distribute-rgt-neg-in 99.4%

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

      7. fma-undefine 99.4%

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

      8. sub-neg 99.4%

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

      9. +-commutative 99.4%

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

      10. distribute-neg-in 99.4%

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

      11. metadata-eval 99.4%

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

      12. metadata-eval 99.4%

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

      13. unsub-neg 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in t around 0 99.4%

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

      1. add-sqr-sqrt 9.8%

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

      2. pow2 9.8%

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

   7. Applied egg-rr 9.8%

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

   8. Taylor expanded in a around 0 80.8%

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

      1. *-commutative 80.8%

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

   10. Simplified 80.8%

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

       1. +-commutative 80.8%

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

       2. *-un-lft-identity 80.8%

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

       3. fma-define 80.8%

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

       4. add-log-exp 80.8%

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

       5. diff-log 61.6%

          \[\leadsto \mathsf{fma}\left(1, \color{blue}{\log \left(\frac{z}{e^{\log t \cdot 0.5}}\right)}, \log \left(x + y\right)\right) \]

       6. exp-to-pow 61.6%

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

       7. pow1/2 61.6%

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

       8. +-commutative 61.6%

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

   12. Applied egg-rr 61.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(1, \log \left(\frac{z}{\sqrt{t}}\right), \log \left(y + x\right)\right)} \]
   13. Step-by-step derivation

       1. fma-undefine 61.6%

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

       2. *-lft-identity 61.6%

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

       3. +-commutative 61.6%

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

   14. Simplified 61.6%

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

       1. *-un-lft-identity 61.6%

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

       2. sum-log 52.2%

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

       3. +-commutative 52.2%

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

   16. Applied egg-rr 52.2%

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

       1. *-lft-identity 52.2%

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

       2. associate-*l/ 71.6%

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

       3. associate-/l* 71.7%

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

   18. Simplified 71.7%

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

   if 1.4e-91 < t

   1. Initial program 99.8%

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

      1. add-cbrt-cube 70.5%

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

      2. pow1/3 20.5%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \color{blue}{{\left(\left(\left(\left(a - 0.5\right) \cdot \log t\right) \cdot \left(\left(a - 0.5\right) \cdot \log t\right)\right) \cdot \left(\left(a - 0.5\right) \cdot \log t\right)\right)}^{0.3333333333333333}} \]

      3. pow3 20.5%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + {\color{blue}{\left({\left(\left(a - 0.5\right) \cdot \log t\right)}^{3}\right)}}^{0.3333333333333333} \]

      4. sub-neg 20.5%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + {\left({\left(\color{blue}{\left(a + \left(-0.5\right)\right)} \cdot \log t\right)}^{3}\right)}^{0.3333333333333333} \]

      5. metadata-eval 20.5%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + {\left({\left(\left(a + \color{blue}{-0.5}\right) \cdot \log t\right)}^{3}\right)}^{0.3333333333333333} \]

      6. *-commutative 20.5%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + {\left({\color{blue}{\left(\log t \cdot \left(a + -0.5\right)\right)}}^{3}\right)}^{0.3333333333333333} \]

   4. Applied egg-rr 20.5%

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \color{blue}{{\left({\left(\log t \cdot \left(a + -0.5\right)\right)}^{3}\right)}^{0.3333333333333333}} \]
   5. Step-by-step derivation

      1. pow-pow 99.8%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \color{blue}{{\left(\log t \cdot \left(a + -0.5\right)\right)}^{\left(3 \cdot 0.3333333333333333\right)}} \]

      2. metadata-eval 99.8%

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

      3. pow1 99.8%

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

      4. associate-+l- 99.8%

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

      5. +-commutative 99.8%

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

      6. sum-log 82.6%

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

      7. +-commutative 82.6%

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

   6. Applied egg-rr 82.6%

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

   7. Taylor expanded in x around 0 62.3%

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

   8. Taylor expanded in a around inf 88.4%

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

      1. *-commutative 88.4%

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

   10. Simplified 88.4%

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

       1. fma-neg 88.5%

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

   12. Applied egg-rr 88.5%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.7 \cdot 10^{-117}:\\ \;\;\;\;\log \left(x + y\right) + \log t \cdot a\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{-91}:\\ \;\;\;\;\log \left(z \cdot \frac{x + y}{\sqrt{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log t, a, -t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 85.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 4.6 \cdot 10^{-30}:\\ \;\;\;\;\log \left(z \cdot \left(x + y\right)\right) + \log t \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log t, a, -t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 4.6e-30)
   (+ (log (* z (+ x y))) (* (log t) (- a 0.5)))
   (fma (log t) a (- t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 4.6e-30) {
		tmp = log((z * (x + y))) + (log(t) * (a - 0.5));
	} else {
		tmp = fma(log(t), a, -t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 4.6e-30)
		tmp = Float64(log(Float64(z * Float64(x + y))) + Float64(log(t) * Float64(a - 0.5)));
	else
		tmp = fma(log(t), a, Float64(-t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 4.6e-30], N[(N[Log[N[(z * N[(x + y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Log[t], $MachinePrecision] * a + (-t)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 4.59999999999999968e-30

   1. Initial program 99.2%

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

      1. associate-+l- 99.2%

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

      2. associate--l+ 99.2%

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

      3. sub-neg 99.2%

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

      4. +-commutative 99.2%

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

      5. *-commutative 99.2%

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

      6. distribute-rgt-neg-in 99.2%

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

      7. fma-undefine 99.2%

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

      8. sub-neg 99.2%

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

      9. +-commutative 99.2%

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

      10. distribute-neg-in 99.2%

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

      11. metadata-eval 99.2%

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

      12. metadata-eval 99.2%

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

      13. unsub-neg 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in t around 0 99.2%

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

      1. associate-+r- 99.2%

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

      2. +-commutative 99.2%

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

      3. sum-log 80.6%

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

      4. +-commutative 80.6%

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

   7. Applied egg-rr 80.6%

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

   if 4.59999999999999968e-30 < t

   1. Initial program 99.9%

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

      1. add-cbrt-cube 70.6%

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

      2. pow1/3 11.0%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \color{blue}{{\left(\left(\left(\left(a - 0.5\right) \cdot \log t\right) \cdot \left(\left(a - 0.5\right) \cdot \log t\right)\right) \cdot \left(\left(a - 0.5\right) \cdot \log t\right)\right)}^{0.3333333333333333}} \]

      3. pow3 11.0%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + {\color{blue}{\left({\left(\left(a - 0.5\right) \cdot \log t\right)}^{3}\right)}}^{0.3333333333333333} \]

      4. sub-neg 11.0%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + {\left({\left(\color{blue}{\left(a + \left(-0.5\right)\right)} \cdot \log t\right)}^{3}\right)}^{0.3333333333333333} \]

      5. metadata-eval 11.0%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + {\left({\left(\left(a + \color{blue}{-0.5}\right) \cdot \log t\right)}^{3}\right)}^{0.3333333333333333} \]

      6. *-commutative 11.0%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + {\left({\color{blue}{\left(\log t \cdot \left(a + -0.5\right)\right)}}^{3}\right)}^{0.3333333333333333} \]

   4. Applied egg-rr 11.0%

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \color{blue}{{\left({\left(\log t \cdot \left(a + -0.5\right)\right)}^{3}\right)}^{0.3333333333333333}} \]
   5. Step-by-step derivation

      1. pow-pow 99.9%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \color{blue}{{\left(\log t \cdot \left(a + -0.5\right)\right)}^{\left(3 \cdot 0.3333333333333333\right)}} \]

      2. metadata-eval 99.9%

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

      3. pow1 99.9%

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

      4. associate-+l- 99.9%

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

      5. +-commutative 99.9%

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

      6. sum-log 81.8%

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

      7. +-commutative 81.8%

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

   6. Applied egg-rr 81.8%

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

   7. Taylor expanded in x around 0 65.0%

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

   8. Taylor expanded in a around inf 97.4%

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

      1. *-commutative 97.4%

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

   10. Simplified 97.4%

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

       1. fma-neg 97.4%

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

   12. Applied egg-rr 97.4%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.6 \cdot 10^{-30}:\\ \;\;\;\;\log \left(z \cdot \left(x + y\right)\right) + \log t \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log t, a, -t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 71.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1800000 \lor \neg \left(a \leq 3.2 \cdot 10^{-41}\right):\\ \;\;\;\;\mathsf{fma}\left(\log t, a, -t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + \log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -1800000.0) (not (<= a 3.2e-41)))
   (fma (log t) a (- t))
   (- (+ (log z) (log y)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -1800000.0) || !(a <= 3.2e-41)) {
		tmp = fma(log(t), a, -t);
	} else {
		tmp = (log(z) + log(y)) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -1800000.0) || !(a <= 3.2e-41))
		tmp = fma(log(t), a, Float64(-t));
	else
		tmp = Float64(Float64(log(z) + log(y)) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -1800000.0], N[Not[LessEqual[a, 3.2e-41]], $MachinePrecision]], N[(N[Log[t], $MachinePrecision] * a + (-t)), $MachinePrecision], N[(N[(N[Log[z], $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.8e6 or 3.20000000000000012e-41 < a

   1. Initial program 99.7%

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

      1. add-cbrt-cube 40.8%

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

      2. pow1/3 23.2%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \color{blue}{{\left(\left(\left(\left(a - 0.5\right) \cdot \log t\right) \cdot \left(\left(a - 0.5\right) \cdot \log t\right)\right) \cdot \left(\left(a - 0.5\right) \cdot \log t\right)\right)}^{0.3333333333333333}} \]

      3. pow3 23.2%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + {\color{blue}{\left({\left(\left(a - 0.5\right) \cdot \log t\right)}^{3}\right)}}^{0.3333333333333333} \]

      4. sub-neg 23.2%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + {\left({\left(\color{blue}{\left(a + \left(-0.5\right)\right)} \cdot \log t\right)}^{3}\right)}^{0.3333333333333333} \]

      5. metadata-eval 23.2%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + {\left({\left(\left(a + \color{blue}{-0.5}\right) \cdot \log t\right)}^{3}\right)}^{0.3333333333333333} \]

      6. *-commutative 23.2%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + {\left({\color{blue}{\left(\log t \cdot \left(a + -0.5\right)\right)}}^{3}\right)}^{0.3333333333333333} \]

   4. Applied egg-rr 23.2%

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \color{blue}{{\left({\left(\log t \cdot \left(a + -0.5\right)\right)}^{3}\right)}^{0.3333333333333333}} \]
   5. Step-by-step derivation

      1. pow-pow 99.7%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \color{blue}{{\left(\log t \cdot \left(a + -0.5\right)\right)}^{\left(3 \cdot 0.3333333333333333\right)}} \]

      2. metadata-eval 99.7%

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

      3. pow1 99.7%

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

      4. associate-+l- 99.7%

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

      5. +-commutative 99.7%

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

      6. sum-log 79.6%

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

      7. +-commutative 79.6%

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

   6. Applied egg-rr 79.6%

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

   7. Taylor expanded in x around 0 60.0%

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

   8. Taylor expanded in a around inf 93.9%

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

      1. *-commutative 93.9%

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

   10. Simplified 93.9%

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

       1. fma-neg 94.0%

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

   12. Applied egg-rr 94.0%

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

   if -1.8e6 < a < 3.20000000000000012e-41

   1. Initial program 99.4%

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

      1. associate-+l- 99.4%

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

      2. associate--l+ 99.5%

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

      3. sub-neg 99.5%

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

      4. +-commutative 99.5%

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

      5. *-commutative 99.5%

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

      6. distribute-rgt-neg-in 99.5%

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

      7. fma-undefine 99.5%

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

      8. sub-neg 99.5%

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

      9. +-commutative 99.5%

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

      10. distribute-neg-in 99.5%

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

      11. metadata-eval 99.5%

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

      12. metadata-eval 99.5%

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

      13. unsub-neg 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in t around inf 61.8%

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

   6. Taylor expanded in x around 0 46.2%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1800000 \lor \neg \left(a \leq 3.2 \cdot 10^{-41}\right):\\ \;\;\;\;\mathsf{fma}\left(\log t, a, -t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + \log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 73.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 6.1 \cdot 10^{-61}:\\ \;\;\;\;\log t \cdot \left(a - 0.5\right) + \log \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log t, a, -t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 6.1e-61)
   (+ (* (log t) (- a 0.5)) (log (* y z)))
   (fma (log t) a (- t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 6.1e-61) {
		tmp = (log(t) * (a - 0.5)) + log((y * z));
	} else {
		tmp = fma(log(t), a, -t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 6.1e-61)
		tmp = Float64(Float64(log(t) * Float64(a - 0.5)) + log(Float64(y * z)));
	else
		tmp = fma(log(t), a, Float64(-t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 6.1e-61], N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision] + N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Log[t], $MachinePrecision] * a + (-t)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 6.1000000000000001e-61

   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. Step-by-step derivation

      1. +-commutative 99.1%

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

      2. fma-define 99.2%

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

      3. sub-neg 99.2%

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

      4. metadata-eval 99.2%

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

      5. associate--l+ 99.2%

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

   3. Simplified 99.2%

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

      1. associate-+r- 99.2%

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

      2. +-commutative 99.2%

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

      3. sum-log 81.2%

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

   6. Applied egg-rr 81.2%

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

      1. +-commutative 81.2%

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

   8. Simplified 81.2%

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

   9. Taylor expanded in x around 0 50.0%

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

   10. Taylor expanded in t around 0 50.0%

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

   if 6.1000000000000001e-61 < t

   1. Initial program 99.8%

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

      1. add-cbrt-cube 70.4%

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

      2. pow1/3 15.5%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \color{blue}{{\left(\left(\left(\left(a - 0.5\right) \cdot \log t\right) \cdot \left(\left(a - 0.5\right) \cdot \log t\right)\right) \cdot \left(\left(a - 0.5\right) \cdot \log t\right)\right)}^{0.3333333333333333}} \]

      3. pow3 15.5%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + {\color{blue}{\left({\left(\left(a - 0.5\right) \cdot \log t\right)}^{3}\right)}}^{0.3333333333333333} \]

      4. sub-neg 15.5%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + {\left({\left(\color{blue}{\left(a + \left(-0.5\right)\right)} \cdot \log t\right)}^{3}\right)}^{0.3333333333333333} \]

      5. metadata-eval 15.5%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + {\left({\left(\left(a + \color{blue}{-0.5}\right) \cdot \log t\right)}^{3}\right)}^{0.3333333333333333} \]

      6. *-commutative 15.5%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + {\left({\color{blue}{\left(\log t \cdot \left(a + -0.5\right)\right)}}^{3}\right)}^{0.3333333333333333} \]

   4. Applied egg-rr 15.5%

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \color{blue}{{\left({\left(\log t \cdot \left(a + -0.5\right)\right)}^{3}\right)}^{0.3333333333333333}} \]
   5. Step-by-step derivation

      1. pow-pow 99.8%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \color{blue}{{\left(\log t \cdot \left(a + -0.5\right)\right)}^{\left(3 \cdot 0.3333333333333333\right)}} \]

      2. metadata-eval 99.8%

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

      3. pow1 99.8%

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

      4. associate-+l- 99.8%

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

      5. +-commutative 99.8%

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

      6. sum-log 81.3%

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

      7. +-commutative 81.3%

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

   6. Applied egg-rr 81.3%

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

   7. Taylor expanded in x around 0 63.3%

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

   8. Taylor expanded in a around inf 93.0%

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

      1. *-commutative 93.0%

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

   10. Simplified 93.0%

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

       1. fma-neg 93.0%

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

   12. Applied egg-rr 93.0%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.1 \cdot 10^{-61}:\\ \;\;\;\;\log t \cdot \left(a - 0.5\right) + \log \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log t, a, -t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 74.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[Log[t], $MachinePrecision] * a + (-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. Add Preprocessing
3. Step-by-step derivation

   1. add-cbrt-cube 69.4%

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

   2. pow1/3 33.7%

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \color{blue}{{\left(\left(\left(\left(a - 0.5\right) \cdot \log t\right) \cdot \left(\left(a - 0.5\right) \cdot \log t\right)\right) \cdot \left(\left(a - 0.5\right) \cdot \log t\right)\right)}^{0.3333333333333333}} \]

   3. pow3 33.7%

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + {\color{blue}{\left({\left(\left(a - 0.5\right) \cdot \log t\right)}^{3}\right)}}^{0.3333333333333333} \]

   4. sub-neg 33.7%

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + {\left({\left(\color{blue}{\left(a + \left(-0.5\right)\right)} \cdot \log t\right)}^{3}\right)}^{0.3333333333333333} \]

   5. metadata-eval 33.7%

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + {\left({\left(\left(a + \color{blue}{-0.5}\right) \cdot \log t\right)}^{3}\right)}^{0.3333333333333333} \]

   6. *-commutative 33.7%

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + {\left({\color{blue}{\left(\log t \cdot \left(a + -0.5\right)\right)}}^{3}\right)}^{0.3333333333333333} \]

4. Applied egg-rr 33.7%

   \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \color{blue}{{\left({\left(\log t \cdot \left(a + -0.5\right)\right)}^{3}\right)}^{0.3333333333333333}} \]
5. Step-by-step derivation

   1. pow-pow 99.5%

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \color{blue}{{\left(\log t \cdot \left(a + -0.5\right)\right)}^{\left(3 \cdot 0.3333333333333333\right)}} \]

   2. metadata-eval 99.5%

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

   3. pow1 99.5%

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

   4. associate-+l- 99.5%

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

   5. +-commutative 99.5%

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

   6. sum-log 81.2%

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

   7. +-commutative 81.2%

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

6. Applied egg-rr 81.2%

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

7. Taylor expanded in x around 0 57.6%

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

8. Taylor expanded in a around inf 74.9%

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

   1. *-commutative 74.9%

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

10. Simplified 74.9%

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

    1. fma-neg 74.9%

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

12. Applied egg-rr 74.9%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\log t, a, -t\right)} \]
13. Add Preprocessing

Alternative 15: 58.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.85 \cdot 10^{+19} \lor \neg \left(a \leq 70000\right):\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;\log y - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -1.85e+19) (not (<= a 70000.0))) (* (log t) a) (- (log y) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -1.85e+19) || !(a <= 70000.0)) {
		tmp = log(t) * a;
	} else {
		tmp = log(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 <= (-1.85d+19)) .or. (.not. (a <= 70000.0d0))) then
        tmp = log(t) * a
    else
        tmp = log(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 <= -1.85e+19) || !(a <= 70000.0)) {
		tmp = Math.log(t) * a;
	} else {
		tmp = Math.log(y) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -1.85e+19) or not (a <= 70000.0):
		tmp = math.log(t) * a
	else:
		tmp = math.log(y) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -1.85e+19) || !(a <= 70000.0))
		tmp = Float64(log(t) * a);
	else
		tmp = Float64(log(y) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -1.85e+19) || ~((a <= 70000.0)))
		tmp = log(t) * a;
	else
		tmp = log(y) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -1.85e+19], N[Not[LessEqual[a, 70000.0]], $MachinePrecision]], N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision], N[(N[Log[y], $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.85e19 or 7e4 < a

   1. Initial program 99.7%

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

      1. associate--l+ 99.7%

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

      2. +-commutative 99.7%

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

      3. associate-+l+ 99.7%

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

      4. +-commutative 99.7%

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

      5. fma-define 99.7%

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

      6. sub-neg 99.7%

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

      7. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 75.4%

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

   6. Taylor expanded in a around inf 82.7%

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

      1. *-commutative 82.7%

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

   8. Simplified 82.7%

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

   if -1.85e19 < a < 7e4

   1. Initial program 99.4%

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

      1. associate-+l- 99.4%

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

      2. associate--l+ 99.5%

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

      3. sub-neg 99.5%

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

      4. +-commutative 99.5%

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

      5. *-commutative 99.5%

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

      6. distribute-rgt-neg-in 99.5%

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

      7. fma-undefine 99.5%

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

      8. sub-neg 99.5%

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

      9. +-commutative 99.5%

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

      10. distribute-neg-in 99.5%

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

      11. metadata-eval 99.5%

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

      12. metadata-eval 99.5%

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

      13. unsub-neg 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in t around inf 58.6%

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

      1. neg-mul-1 58.6%

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

   7. Simplified 58.6%

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

   8. Taylor expanded in x around 0 42.7%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.85 \cdot 10^{+19} \lor \neg \left(a \leq 70000\right):\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;\log y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 62.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -7.8 \cdot 10^{+27} \lor \neg \left(a \leq 0.00044\right):\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -7.8e+27) (not (<= a 0.00044))) (* (log t) a) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -7.8e+27) || !(a <= 0.00044)) {
		tmp = log(t) * a;
	} 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 <= (-7.8d+27)) .or. (.not. (a <= 0.00044d0))) then
        tmp = log(t) * a
    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 <= -7.8e+27) || !(a <= 0.00044)) {
		tmp = Math.log(t) * a;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -7.8e+27) or not (a <= 0.00044):
		tmp = math.log(t) * a
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -7.8e+27) || !(a <= 0.00044))
		tmp = Float64(log(t) * a);
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -7.8e+27) || ~((a <= 0.00044)))
		tmp = log(t) * a;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -7.8e+27], N[Not[LessEqual[a, 0.00044]], $MachinePrecision]], N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if a < -7.7999999999999997e27 or 4.40000000000000016e-4 < a

   1. Initial program 99.7%

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

      1. associate--l+ 99.7%

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

      2. +-commutative 99.7%

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

      3. associate-+l+ 99.7%

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

      4. +-commutative 99.7%

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

      5. fma-define 99.7%

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

      6. sub-neg 99.7%

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

      7. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 73.6%

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

   6. Taylor expanded in a around inf 80.8%

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

      1. *-commutative 80.8%

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

   8. Simplified 80.8%

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

   if -7.7999999999999997e27 < a < 4.40000000000000016e-4

   1. Initial program 99.4%

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

      1. associate--l+ 99.4%

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

      2. +-commutative 99.4%

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

      3. associate-+l+ 99.5%

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

      4. +-commutative 99.5%

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

      5. fma-define 99.5%

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

      6. sub-neg 99.5%

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

      7. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around 0 68.3%

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

   6. Taylor expanded in t around inf 52.8%

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

      1. neg-mul-1 52.8%

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

   8. Simplified 52.8%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -7.8 \cdot 10^{+27} \lor \neg \left(a \leq 0.00044\right):\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 40.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\log \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (if (<= t 5e-7) (log (+ x y)) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 5e-7) {
		tmp = log((x + y));
	} 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 <= 5d-7) then
        tmp = log((x + y))
    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 <= 5e-7) {
		tmp = Math.log((x + y));
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 5e-7:
		tmp = math.log((x + y))
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 5e-7)
		tmp = log(Float64(x + y));
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 5e-7)
		tmp = log((x + y));
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 5e-7], N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if t < 4.99999999999999977e-7

   1. Initial program 99.2%

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

      1. associate-+l- 99.2%

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

      2. associate--l+ 99.2%

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

      3. sub-neg 99.2%

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

      4. +-commutative 99.2%

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

      5. *-commutative 99.2%

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

      6. distribute-rgt-neg-in 99.2%

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

      7. fma-undefine 99.2%

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

      8. sub-neg 99.2%

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

      9. +-commutative 99.2%

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

      10. distribute-neg-in 99.2%

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

      11. metadata-eval 99.2%

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

      12. metadata-eval 99.2%

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

      13. unsub-neg 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in t around inf 9.0%

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

      1. neg-mul-1 9.0%

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

   7. Simplified 9.0%

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

   8. Taylor expanded in t around 0 9.0%

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

      1. +-commutative 9.0%

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

   10. Simplified 9.0%

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

   if 4.99999999999999977e-7 < 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. +-commutative 99.9%

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

      3. associate-+l+ 99.9%

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

      4. +-commutative 99.9%

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

      5. fma-define 99.9%

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

      6. sub-neg 99.9%

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

      7. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 76.4%

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

   6. Taylor expanded in t around inf 72.5%

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

      1. neg-mul-1 72.5%

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

   8. Simplified 72.5%

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

4. Final simplification 40.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\log \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 74.9% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[Log[t], $MachinePrecision] * a), $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. Add Preprocessing
3. Step-by-step derivation

   1. add-cbrt-cube 69.4%

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

   2. pow1/3 33.7%

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \color{blue}{{\left(\left(\left(\left(a - 0.5\right) \cdot \log t\right) \cdot \left(\left(a - 0.5\right) \cdot \log t\right)\right) \cdot \left(\left(a - 0.5\right) \cdot \log t\right)\right)}^{0.3333333333333333}} \]

   3. pow3 33.7%

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + {\color{blue}{\left({\left(\left(a - 0.5\right) \cdot \log t\right)}^{3}\right)}}^{0.3333333333333333} \]

   4. sub-neg 33.7%

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + {\left({\left(\color{blue}{\left(a + \left(-0.5\right)\right)} \cdot \log t\right)}^{3}\right)}^{0.3333333333333333} \]

   5. metadata-eval 33.7%

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + {\left({\left(\left(a + \color{blue}{-0.5}\right) \cdot \log t\right)}^{3}\right)}^{0.3333333333333333} \]

   6. *-commutative 33.7%

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + {\left({\color{blue}{\left(\log t \cdot \left(a + -0.5\right)\right)}}^{3}\right)}^{0.3333333333333333} \]

4. Applied egg-rr 33.7%

   \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \color{blue}{{\left({\left(\log t \cdot \left(a + -0.5\right)\right)}^{3}\right)}^{0.3333333333333333}} \]
5. Step-by-step derivation

   1. pow-pow 99.5%

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \color{blue}{{\left(\log t \cdot \left(a + -0.5\right)\right)}^{\left(3 \cdot 0.3333333333333333\right)}} \]

   2. metadata-eval 99.5%

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

   3. pow1 99.5%

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

   4. associate-+l- 99.5%

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

   5. +-commutative 99.5%

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

   6. sum-log 81.2%

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

   7. +-commutative 81.2%

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

6. Applied egg-rr 81.2%

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

7. Taylor expanded in x around 0 57.6%

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

8. Taylor expanded in a around inf 74.9%

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

   1. *-commutative 74.9%

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

10. Simplified 74.9%

    \[\leadsto \color{blue}{\log t \cdot a} - t \]
11. Add Preprocessing

Alternative 19: 37.7% 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. +-commutative 99.5%

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

   3. associate-+l+ 99.6%

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

   4. +-commutative 99.6%

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

   5. fma-define 99.5%

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

   6. sub-neg 99.5%

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

   7. metadata-eval 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in x around 0 70.7%

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

6. Taylor expanded in t around inf 37.3%

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

   1. neg-mul-1 37.3%

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

8. Simplified 37.3%

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

Developer target: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024107 
(FPCore (x y z t a)
  :name "Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2"
  :precision binary64

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

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