Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 12.5s

Alternatives: 14

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t, double a) {
	return ((log((y + x)) + log(z)) - t) - ((0.5 - a) * 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((y + x)) + log(z)) - t) - ((0.5d0 - a) * log(t))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Final simplification 99.6%

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

Alternative 2: 97.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(0.5 - a\right) \cdot \log t\\ t_2 := \log \left(y + x\right)\\ t_3 := \left(\left(t\_2 + \log z\right) - t\right) - t\_1\\ \mathbf{if}\;t\_3 \leq -200000:\\ \;\;\;\;\left(-t\right) - t\_1\\ \mathbf{elif}\;t\_3 \leq 1800:\\ \;\;\;\;\log z + \mathsf{fma}\left(-0.5, \log t, t\_2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, {\left(\frac{-1}{t}\right)}^{-1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (- 0.5 a) (log t)))
        (t_2 (log (+ y x)))
        (t_3 (- (- (+ t_2 (log z)) t) t_1)))
   (if (<= t_3 -200000.0)
     (- (- t) t_1)
     (if (<= t_3 1800.0)
       (+ (log z) (fma -0.5 (log t) t_2))
       (fma (- a 0.5) (log t) (pow (/ -1.0 t) -1.0))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (0.5 - a) * log(t);
	double t_2 = log((y + x));
	double t_3 = ((t_2 + log(z)) - t) - t_1;
	double tmp;
	if (t_3 <= -200000.0) {
		tmp = -t - t_1;
	} else if (t_3 <= 1800.0) {
		tmp = log(z) + fma(-0.5, log(t), t_2);
	} else {
		tmp = fma((a - 0.5), log(t), pow((-1.0 / t), -1.0));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(0.5 - a) * log(t))
	t_2 = log(Float64(y + x))
	t_3 = Float64(Float64(Float64(t_2 + log(z)) - t) - t_1)
	tmp = 0.0
	if (t_3 <= -200000.0)
		tmp = Float64(Float64(-t) - t_1);
	elseif (t_3 <= 1800.0)
		tmp = Float64(log(z) + fma(-0.5, log(t), t_2));
	else
		tmp = fma(Float64(a - 0.5), log(t), (Float64(-1.0 / t) ^ -1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(0.5 - a), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t$95$2 + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] - t$95$1), $MachinePrecision]}, If[LessEqual[t$95$3, -200000.0], N[((-t) - t$95$1), $MachinePrecision], If[LessEqual[t$95$3, 1800.0], N[(N[Log[z], $MachinePrecision] + N[(-0.5 * N[Log[t], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[Power[N[(-1.0 / t), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < -2e5

   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. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) - 1\right)} + \left(a - \frac{1}{2}\right) \cdot \log t \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto t \cdot \color{blue}{\left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)} + \left(a - \frac{1}{2}\right) \cdot \log t \]

      2. metadata-eval N/A

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

      3. distribute-rgt-in N/A

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

      4. mul-1-neg N/A

         \[\leadsto \left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) \cdot t + \color{blue}{\left(\mathsf{neg}\left(t\right)\right)}\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}, t, \mathsf{neg}\left(t\right)\right)} + \left(a - \frac{1}{2}\right) \cdot \log t \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\log \left(x + y\right)}{t} + \frac{\log z}{t}}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      7. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\log \left(x + y\right)}{t} + \frac{\log z}{t}}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\log \left(x + y\right)}{t}} + \frac{\log z}{t}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      9. lower-log.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\log \left(x + y\right)}}{t} + \frac{\log z}{t}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\log \color{blue}{\left(y + x\right)}}{t} + \frac{\log z}{t}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      11. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\log \color{blue}{\left(y + x\right)}}{t} + \frac{\log z}{t}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\log \left(y + x\right)}{t} + \color{blue}{\frac{\log z}{t}}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      13. lower-log.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\log \left(y + x\right)}{t} + \frac{\color{blue}{\log z}}{t}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      14. lower-neg.f64 99.2

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in t around inf

      \[\leadsto -1 \cdot \color{blue}{t} + \left(a - \frac{1}{2}\right) \cdot \log t \]
   7. Step-by-step derivation

   8. Applied rewrites 98.6%

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

   if -2e5 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 1800

   1. Initial program 99.3%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 99.2

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

   4. Applied rewrites 76.7%

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

   5. Taylor expanded in a around 0

      \[\leadsto \frac{1}{\frac{1}{\log \left(z \cdot \left(y + x\right)\right) - t}} + \color{blue}{\frac{-1}{2}} \cdot \log t \]
   6. Step-by-step derivation

   7. Applied rewrites 75.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. remove-double-div N/A

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

      6. lift--.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. log-prod N/A

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

      12. lift-log.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. lift-log.f64 N/A

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

      17. +-commutative N/A

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

      18. associate--l+ N/A

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

   9. Applied rewrites 97.6%

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

   10. Taylor expanded in t around 0

       \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \log t, \log \left(y + x\right)\right) + \color{blue}{\log z} \]
   11. Step-by-step derivation

       1. lower-log.f64 96.5

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

   12. Applied rewrites 96.5%

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

   if 1800 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t)))

   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. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 99.4

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

   4. Applied rewrites 86.1%

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

   5. Taylor expanded in t around inf

      \[\leadsto \frac{1}{\color{blue}{\frac{-1}{t}}} + \left(a - \frac{1}{2}\right) \cdot \log t \]
   6. Step-by-step derivation

      1. lower-/.f64 97.6

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

   7. Applied rewrites 97.6%

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

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{-1}{t}} + \left(a - \frac{1}{2}\right) \cdot \log t} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(a - \frac{1}{2}\right) \cdot \log t + \frac{1}{\frac{-1}{t}}} \]

      3. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(a - \frac{1}{2}\right)} \cdot \log t + \frac{1}{\frac{-1}{t}} \]

      4. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(a - \frac{1}{2}\right) \cdot \log t} + \frac{1}{\frac{-1}{t}} \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a - \frac{1}{2}, \log t, \frac{1}{\frac{-1}{t}}\right)} \]

      6. lift--.f64 97.6

         \[\leadsto \mathsf{fma}\left(\color{blue}{a - 0.5}, \log t, \frac{1}{\frac{-1}{t}}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, \log t, \color{blue}{\frac{1}{\frac{-1}{t}}}\right) \]

      8. inv-pow N/A

         \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, \log t, \color{blue}{{\left(\frac{-1}{t}\right)}^{-1}}\right) \]

      9. lower-pow.f64 97.6

         \[\leadsto \mathsf{fma}\left(a - 0.5, \log t, \color{blue}{{\left(\frac{-1}{t}\right)}^{-1}}\right) \]

   9. Applied rewrites 97.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a - 0.5, \log t, {\left(\frac{-1}{t}\right)}^{-1}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\log \left(y + x\right) + \log z\right) - t\right) - \left(0.5 - a\right) \cdot \log t \leq -200000:\\ \;\;\;\;\left(-t\right) - \left(0.5 - a\right) \cdot \log t\\ \mathbf{elif}\;\left(\left(\log \left(y + x\right) + \log z\right) - t\right) - \left(0.5 - a\right) \cdot \log t \leq 1800:\\ \;\;\;\;\log z + \mathsf{fma}\left(-0.5, \log t, \log \left(y + x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, {\left(\frac{-1}{t}\right)}^{-1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(0.5 - a\right) \cdot \log t\\ t_2 := \left(\left(\log \left(y + x\right) + \log z\right) - t\right) - t\_1\\ t_3 := \left(-t\right) - t\_1\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+18}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 1000:\\ \;\;\;\;-0.5 \cdot \log t + \left(\log \left(z \cdot y\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (- 0.5 a) (log t)))
        (t_2 (- (- (+ (log (+ y x)) (log z)) t) t_1))
        (t_3 (- (- t) t_1)))
   (if (<= t_2 -1e+18)
     t_3
     (if (<= t_2 1000.0) (+ (* -0.5 (log t)) (- (log (* z y)) t)) t_3))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (0.5 - a) * log(t);
	double t_2 = ((log((y + x)) + log(z)) - t) - t_1;
	double t_3 = -t - t_1;
	double tmp;
	if (t_2 <= -1e+18) {
		tmp = t_3;
	} else if (t_2 <= 1000.0) {
		tmp = (-0.5 * log(t)) + (log((z * y)) - t);
	} else {
		tmp = t_3;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = (0.5d0 - a) * log(t)
    t_2 = ((log((y + x)) + log(z)) - t) - t_1
    t_3 = -t - t_1
    if (t_2 <= (-1d+18)) then
        tmp = t_3
    else if (t_2 <= 1000.0d0) then
        tmp = ((-0.5d0) * log(t)) + (log((z * y)) - t)
    else
        tmp = t_3
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (0.5 - a) * Math.log(t);
	double t_2 = ((Math.log((y + x)) + Math.log(z)) - t) - t_1;
	double t_3 = -t - t_1;
	double tmp;
	if (t_2 <= -1e+18) {
		tmp = t_3;
	} else if (t_2 <= 1000.0) {
		tmp = (-0.5 * Math.log(t)) + (Math.log((z * y)) - t);
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (0.5 - a) * math.log(t)
	t_2 = ((math.log((y + x)) + math.log(z)) - t) - t_1
	t_3 = -t - t_1
	tmp = 0
	if t_2 <= -1e+18:
		tmp = t_3
	elif t_2 <= 1000.0:
		tmp = (-0.5 * math.log(t)) + (math.log((z * y)) - t)
	else:
		tmp = t_3
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(0.5 - a) * log(t))
	t_2 = Float64(Float64(Float64(log(Float64(y + x)) + log(z)) - t) - t_1)
	t_3 = Float64(Float64(-t) - t_1)
	tmp = 0.0
	if (t_2 <= -1e+18)
		tmp = t_3;
	elseif (t_2 <= 1000.0)
		tmp = Float64(Float64(-0.5 * log(t)) + Float64(log(Float64(z * y)) - t));
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (0.5 - a) * log(t);
	t_2 = ((log((y + x)) + log(z)) - t) - t_1;
	t_3 = -t - t_1;
	tmp = 0.0;
	if (t_2 <= -1e+18)
		tmp = t_3;
	elseif (t_2 <= 1000.0)
		tmp = (-0.5 * log(t)) + (log((z * y)) - t);
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(0.5 - a), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] - t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[((-t) - t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+18], t$95$3, If[LessEqual[t$95$2, 1000.0], N[(N[(-0.5 * N[Log[t], $MachinePrecision]), $MachinePrecision] + N[(N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < -1e18 or 1e3 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 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. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) - 1\right)} + \left(a - \frac{1}{2}\right) \cdot \log t \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto t \cdot \color{blue}{\left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)} + \left(a - \frac{1}{2}\right) \cdot \log t \]

      2. metadata-eval N/A

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

      3. distribute-rgt-in N/A

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

      4. mul-1-neg N/A

         \[\leadsto \left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) \cdot t + \color{blue}{\left(\mathsf{neg}\left(t\right)\right)}\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}, t, \mathsf{neg}\left(t\right)\right)} + \left(a - \frac{1}{2}\right) \cdot \log t \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\log \left(x + y\right)}{t} + \frac{\log z}{t}}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      7. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\log \left(x + y\right)}{t} + \frac{\log z}{t}}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\log \left(x + y\right)}{t}} + \frac{\log z}{t}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      9. lower-log.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\log \left(x + y\right)}}{t} + \frac{\log z}{t}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\log \color{blue}{\left(y + x\right)}}{t} + \frac{\log z}{t}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      11. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\log \color{blue}{\left(y + x\right)}}{t} + \frac{\log z}{t}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\log \left(y + x\right)}{t} + \color{blue}{\frac{\log z}{t}}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      13. lower-log.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\log \left(y + x\right)}{t} + \frac{\color{blue}{\log z}}{t}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      14. lower-neg.f64 98.7

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in t around inf

      \[\leadsto -1 \cdot \color{blue}{t} + \left(a - \frac{1}{2}\right) \cdot \log t \]
   7. Step-by-step derivation

   8. Applied rewrites 94.2%

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

   if -1e18 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 1e3

   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. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 99.2

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

   4. Applied rewrites 92.8%

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

   5. Taylor expanded in a around 0

      \[\leadsto \frac{1}{\frac{1}{\log \left(z \cdot \left(y + x\right)\right) - t}} + \color{blue}{\frac{-1}{2}} \cdot \log t \]
   6. Step-by-step derivation

   7. Applied rewrites 89.3%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\log \left(y \cdot z\right) - t\right)} + \frac{-1}{2} \cdot \log t \]
   9. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\log \left(y \cdot z\right) - t\right)} + \frac{-1}{2} \cdot \log t \]

      2. lower-log.f64 N/A

         \[\leadsto \left(\color{blue}{\log \left(y \cdot z\right)} - t\right) + \frac{-1}{2} \cdot \log t \]

      3. *-commutative N/A

         \[\leadsto \left(\log \color{blue}{\left(z \cdot y\right)} - t\right) + \frac{-1}{2} \cdot \log t \]

      4. lower-*.f64 53.3

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

   10. Applied rewrites 53.3%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\log \left(y + x\right) + \log z\right) - t\right) - \left(0.5 - a\right) \cdot \log t \leq -1 \cdot 10^{+18}:\\ \;\;\;\;\left(-t\right) - \left(0.5 - a\right) \cdot \log t\\ \mathbf{elif}\;\left(\left(\log \left(y + x\right) + \log z\right) - t\right) - \left(0.5 - a\right) \cdot \log t \leq 1000:\\ \;\;\;\;-0.5 \cdot \log t + \left(\log \left(z \cdot y\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - \left(0.5 - a\right) \cdot \log t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(0.5 - a\right) \cdot \log t\\ t_2 := \left(\left(\log \left(y + x\right) + \log z\right) - t\right) - t\_1\\ t_3 := \left(-t\right) - t\_1\\ \mathbf{if}\;t\_2 \leq -1000000000:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 1000:\\ \;\;\;\;\mathsf{fma}\left(\log t, a - 0.5, \log \left(z \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (- 0.5 a) (log t)))
        (t_2 (- (- (+ (log (+ y x)) (log z)) t) t_1))
        (t_3 (- (- t) t_1)))
   (if (<= t_2 -1000000000.0)
     t_3
     (if (<= t_2 1000.0) (fma (log t) (- a 0.5) (log (* z y))) t_3))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (0.5 - a) * log(t);
	double t_2 = ((log((y + x)) + log(z)) - t) - t_1;
	double t_3 = -t - t_1;
	double tmp;
	if (t_2 <= -1000000000.0) {
		tmp = t_3;
	} else if (t_2 <= 1000.0) {
		tmp = fma(log(t), (a - 0.5), log((z * y)));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(0.5 - a) * log(t))
	t_2 = Float64(Float64(Float64(log(Float64(y + x)) + log(z)) - t) - t_1)
	t_3 = Float64(Float64(-t) - t_1)
	tmp = 0.0
	if (t_2 <= -1000000000.0)
		tmp = t_3;
	elseif (t_2 <= 1000.0)
		tmp = fma(log(t), Float64(a - 0.5), log(Float64(z * y)));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(0.5 - a), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] - t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[((-t) - t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -1000000000.0], t$95$3, If[LessEqual[t$95$2, 1000.0], N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision] + N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$3]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < -1e9 or 1e3 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 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. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) - 1\right)} + \left(a - \frac{1}{2}\right) \cdot \log t \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto t \cdot \color{blue}{\left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)} + \left(a - \frac{1}{2}\right) \cdot \log t \]

      2. metadata-eval N/A

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

      3. distribute-rgt-in N/A

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

      4. mul-1-neg N/A

         \[\leadsto \left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) \cdot t + \color{blue}{\left(\mathsf{neg}\left(t\right)\right)}\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}, t, \mathsf{neg}\left(t\right)\right)} + \left(a - \frac{1}{2}\right) \cdot \log t \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\log \left(x + y\right)}{t} + \frac{\log z}{t}}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      7. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\log \left(x + y\right)}{t} + \frac{\log z}{t}}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\log \left(x + y\right)}{t}} + \frac{\log z}{t}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      9. lower-log.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\log \left(x + y\right)}}{t} + \frac{\log z}{t}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\log \color{blue}{\left(y + x\right)}}{t} + \frac{\log z}{t}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      11. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\log \color{blue}{\left(y + x\right)}}{t} + \frac{\log z}{t}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\log \left(y + x\right)}{t} + \color{blue}{\frac{\log z}{t}}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      13. lower-log.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\log \left(y + x\right)}{t} + \frac{\color{blue}{\log z}}{t}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      14. lower-neg.f64 98.7

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in t around inf

      \[\leadsto -1 \cdot \color{blue}{t} + \left(a - \frac{1}{2}\right) \cdot \log t \]
   7. Step-by-step derivation

   8. Applied rewrites 94.0%

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

   if -1e9 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 1e3

   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. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 99.1

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

   4. Applied rewrites 92.7%

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

   5. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\log \left(z \cdot \left(x + y\right)\right) + \log t \cdot \left(a - \frac{1}{2}\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\log t, a - \frac{1}{2}, \log \left(z \cdot \left(x + y\right)\right)\right)} \]

      3. lower-log.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\log t}, a - \frac{1}{2}, \log \left(z \cdot \left(x + y\right)\right)\right) \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\log t, \color{blue}{a - \frac{1}{2}}, \log \left(z \cdot \left(x + y\right)\right)\right) \]

      5. lower-log.f64 N/A

         \[\leadsto \mathsf{fma}\left(\log t, a - \frac{1}{2}, \color{blue}{\log \left(z \cdot \left(x + y\right)\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\log t, a - \frac{1}{2}, \log \color{blue}{\left(\left(x + y\right) \cdot z\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\log t, a - \frac{1}{2}, \log \color{blue}{\left(\left(x + y\right) \cdot z\right)}\right) \]

      8. lower-+.f64 89.6

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

   7. Applied rewrites 89.6%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\log t, a - \frac{1}{2}, \log \left(y \cdot z\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 51.3%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\log \left(y + x\right) + \log z\right) - t\right) - \left(0.5 - a\right) \cdot \log t \leq -1000000000:\\ \;\;\;\;\left(-t\right) - \left(0.5 - a\right) \cdot \log t\\ \mathbf{elif}\;\left(\left(\log \left(y + x\right) + \log z\right) - t\right) - \left(0.5 - a\right) \cdot \log t \leq 1000:\\ \;\;\;\;\mathsf{fma}\left(\log t, a - 0.5, \log \left(z \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - \left(0.5 - a\right) \cdot \log t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(0.5 - a\right) \cdot \log t\\ t_2 := \left(\left(\log \left(y + x\right) + \log z\right) - t\right) - t\_1\\ t_3 := \left(-t\right) - t\_1\\ \mathbf{if}\;t\_2 \leq -200000:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 1000:\\ \;\;\;\;\mathsf{fma}\left(\log t, -0.5, \log \left(z \cdot \left(y + x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (- 0.5 a) (log t)))
        (t_2 (- (- (+ (log (+ y x)) (log z)) t) t_1))
        (t_3 (- (- t) t_1)))
   (if (<= t_2 -200000.0)
     t_3
     (if (<= t_2 1000.0) (fma (log t) -0.5 (log (* z (+ y x)))) t_3))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (0.5 - a) * log(t);
	double t_2 = ((log((y + x)) + log(z)) - t) - t_1;
	double t_3 = -t - t_1;
	double tmp;
	if (t_2 <= -200000.0) {
		tmp = t_3;
	} else if (t_2 <= 1000.0) {
		tmp = fma(log(t), -0.5, log((z * (y + x))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(0.5 - a) * log(t))
	t_2 = Float64(Float64(Float64(log(Float64(y + x)) + log(z)) - t) - t_1)
	t_3 = Float64(Float64(-t) - t_1)
	tmp = 0.0
	if (t_2 <= -200000.0)
		tmp = t_3;
	elseif (t_2 <= 1000.0)
		tmp = fma(log(t), -0.5, log(Float64(z * Float64(y + x))));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(0.5 - a), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] - t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[((-t) - t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -200000.0], t$95$3, If[LessEqual[t$95$2, 1000.0], N[(N[Log[t], $MachinePrecision] * -0.5 + N[Log[N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$3]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < -2e5 or 1e3 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 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. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) - 1\right)} + \left(a - \frac{1}{2}\right) \cdot \log t \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto t \cdot \color{blue}{\left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)} + \left(a - \frac{1}{2}\right) \cdot \log t \]

      2. metadata-eval N/A

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

      3. distribute-rgt-in N/A

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

      4. mul-1-neg N/A

         \[\leadsto \left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) \cdot t + \color{blue}{\left(\mathsf{neg}\left(t\right)\right)}\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}, t, \mathsf{neg}\left(t\right)\right)} + \left(a - \frac{1}{2}\right) \cdot \log t \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\log \left(x + y\right)}{t} + \frac{\log z}{t}}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      7. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\log \left(x + y\right)}{t} + \frac{\log z}{t}}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\log \left(x + y\right)}{t}} + \frac{\log z}{t}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      9. lower-log.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\log \left(x + y\right)}}{t} + \frac{\log z}{t}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\log \color{blue}{\left(y + x\right)}}{t} + \frac{\log z}{t}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      11. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\log \color{blue}{\left(y + x\right)}}{t} + \frac{\log z}{t}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\log \left(y + x\right)}{t} + \color{blue}{\frac{\log z}{t}}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      13. lower-log.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\log \left(y + x\right)}{t} + \frac{\color{blue}{\log z}}{t}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      14. lower-neg.f64 98.7

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in t around inf

      \[\leadsto -1 \cdot \color{blue}{t} + \left(a - \frac{1}{2}\right) \cdot \log t \]
   7. Step-by-step derivation

   8. Applied rewrites 93.4%

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

   if -2e5 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 1e3

   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. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 99.1

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

   4. Applied rewrites 92.4%

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

   5. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\log \left(z \cdot \left(x + y\right)\right) + \log t \cdot \left(a - \frac{1}{2}\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\log t, a - \frac{1}{2}, \log \left(z \cdot \left(x + y\right)\right)\right)} \]

      3. lower-log.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\log t}, a - \frac{1}{2}, \log \left(z \cdot \left(x + y\right)\right)\right) \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\log t, \color{blue}{a - \frac{1}{2}}, \log \left(z \cdot \left(x + y\right)\right)\right) \]

      5. lower-log.f64 N/A

         \[\leadsto \mathsf{fma}\left(\log t, a - \frac{1}{2}, \color{blue}{\log \left(z \cdot \left(x + y\right)\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\log t, a - \frac{1}{2}, \log \color{blue}{\left(\left(x + y\right) \cdot z\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\log t, a - \frac{1}{2}, \log \color{blue}{\left(\left(x + y\right) \cdot z\right)}\right) \]

      8. lower-+.f64 91.0

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

   7. Applied rewrites 91.0%

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

   8. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma}\left(\log t, \frac{-1}{2}, \log \left(\left(x + y\right) \cdot z\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 89.1%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\log \left(y + x\right) + \log z\right) - t\right) - \left(0.5 - a\right) \cdot \log t \leq -200000:\\ \;\;\;\;\left(-t\right) - \left(0.5 - a\right) \cdot \log t\\ \mathbf{elif}\;\left(\left(\log \left(y + x\right) + \log z\right) - t\right) - \left(0.5 - a\right) \cdot \log t \leq 1000:\\ \;\;\;\;\mathsf{fma}\left(\log t, -0.5, \log \left(z \cdot \left(y + x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - \left(0.5 - a\right) \cdot \log t\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(y + x\right) + \log z\\ t_2 := \left(-t\right) - \left(0.5 - a\right) \cdot \log t\\ \mathbf{if}\;t\_1 \leq -750:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 710:\\ \;\;\;\;\log \left(z \cdot \left(y + x\right)\right) - \left(t - \log t \cdot \left(a - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log (+ y x)) (log z))) (t_2 (- (- t) (* (- 0.5 a) (log t)))))
   (if (<= t_1 -750.0)
     t_2
     (if (<= t_1 710.0)
       (- (log (* z (+ y x))) (- t (* (log t) (- a 0.5))))
       t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((y + x)) + log(z);
	double t_2 = -t - ((0.5 - a) * log(t));
	double tmp;
	if (t_1 <= -750.0) {
		tmp = t_2;
	} else if (t_1 <= 710.0) {
		tmp = log((z * (y + x))) - (t - (log(t) * (a - 0.5)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = log((y + x)) + log(z)
    t_2 = -t - ((0.5d0 - a) * log(t))
    if (t_1 <= (-750.0d0)) then
        tmp = t_2
    else if (t_1 <= 710.0d0) then
        tmp = log((z * (y + x))) - (t - (log(t) * (a - 0.5d0)))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = Math.log((y + x)) + Math.log(z);
	double t_2 = -t - ((0.5 - a) * Math.log(t));
	double tmp;
	if (t_1 <= -750.0) {
		tmp = t_2;
	} else if (t_1 <= 710.0) {
		tmp = Math.log((z * (y + x))) - (t - (Math.log(t) * (a - 0.5)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = math.log((y + x)) + math.log(z)
	t_2 = -t - ((0.5 - a) * math.log(t))
	tmp = 0
	if t_1 <= -750.0:
		tmp = t_2
	elif t_1 <= 710.0:
		tmp = math.log((z * (y + x))) - (t - (math.log(t) * (a - 0.5)))
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(Float64(y + x)) + log(z))
	t_2 = Float64(Float64(-t) - Float64(Float64(0.5 - a) * log(t)))
	tmp = 0.0
	if (t_1 <= -750.0)
		tmp = t_2;
	elseif (t_1 <= 710.0)
		tmp = Float64(log(Float64(z * Float64(y + x))) - Float64(t - Float64(log(t) * Float64(a - 0.5))));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = log((y + x)) + log(z);
	t_2 = -t - ((0.5 - a) * log(t));
	tmp = 0.0;
	if (t_1 <= -750.0)
		tmp = t_2;
	elseif (t_1 <= 710.0)
		tmp = log((z * (y + x))) - (t - (log(t) * (a - 0.5)));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[((-t) - N[(N[(0.5 - a), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -750.0], t$95$2, If[LessEqual[t$95$1, 710.0], N[(N[Log[N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(t - N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -750 or 710 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z))

   1. Initial program 99.7%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) - 1\right)} + \left(a - \frac{1}{2}\right) \cdot \log t \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto t \cdot \color{blue}{\left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)} + \left(a - \frac{1}{2}\right) \cdot \log t \]

      2. metadata-eval N/A

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

      3. distribute-rgt-in N/A

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

      4. mul-1-neg N/A

         \[\leadsto \left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) \cdot t + \color{blue}{\left(\mathsf{neg}\left(t\right)\right)}\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}, t, \mathsf{neg}\left(t\right)\right)} + \left(a - \frac{1}{2}\right) \cdot \log t \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\log \left(x + y\right)}{t} + \frac{\log z}{t}}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      7. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\log \left(x + y\right)}{t} + \frac{\log z}{t}}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\log \left(x + y\right)}{t}} + \frac{\log z}{t}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      9. lower-log.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\log \left(x + y\right)}}{t} + \frac{\log z}{t}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\log \color{blue}{\left(y + x\right)}}{t} + \frac{\log z}{t}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      11. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\log \color{blue}{\left(y + x\right)}}{t} + \frac{\log z}{t}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\log \left(y + x\right)}{t} + \color{blue}{\frac{\log z}{t}}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      13. lower-log.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\log \left(y + x\right)}{t} + \frac{\color{blue}{\log z}}{t}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      14. lower-neg.f64 96.1

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

   5. Applied rewrites 96.1%

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

   6. Taylor expanded in t around inf

      \[\leadsto -1 \cdot \color{blue}{t} + \left(a - \frac{1}{2}\right) \cdot \log t \]
   7. Step-by-step derivation

   8. Applied rewrites 75.2%

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

   if -750 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 710

   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. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. sum-log N/A

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

      9. lower-log.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 99.6

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(y + x\right) + \log z \leq -750:\\ \;\;\;\;\left(-t\right) - \left(0.5 - a\right) \cdot \log t\\ \mathbf{elif}\;\log \left(y + x\right) + \log z \leq 710:\\ \;\;\;\;\log \left(z \cdot \left(y + x\right)\right) - \left(t - \log t \cdot \left(a - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - \left(0.5 - a\right) \cdot \log t\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 94.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(y + x\right) + \log z\\ t_2 := \left(-t\right) - \left(0.5 - a\right) \cdot \log t\\ \mathbf{if}\;t\_1 \leq -750:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 710:\\ \;\;\;\;\mathsf{fma}\left(\log t, a - 0.5, \log \left(z \cdot \left(y + x\right)\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log (+ y x)) (log z))) (t_2 (- (- t) (* (- 0.5 a) (log t)))))
   (if (<= t_1 -750.0)
     t_2
     (if (<= t_1 710.0)
       (- (fma (log t) (- a 0.5) (log (* z (+ y x)))) t)
       t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((y + x)) + log(z);
	double t_2 = -t - ((0.5 - a) * log(t));
	double tmp;
	if (t_1 <= -750.0) {
		tmp = t_2;
	} else if (t_1 <= 710.0) {
		tmp = fma(log(t), (a - 0.5), log((z * (y + x)))) - t;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(Float64(y + x)) + log(z))
	t_2 = Float64(Float64(-t) - Float64(Float64(0.5 - a) * log(t)))
	tmp = 0.0
	if (t_1 <= -750.0)
		tmp = t_2;
	elseif (t_1 <= 710.0)
		tmp = Float64(fma(log(t), Float64(a - 0.5), log(Float64(z * Float64(y + x)))) - t);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[((-t) - N[(N[(0.5 - a), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -750.0], t$95$2, If[LessEqual[t$95$1, 710.0], N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision] + N[Log[N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -750 or 710 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z))

   1. Initial program 99.7%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) - 1\right)} + \left(a - \frac{1}{2}\right) \cdot \log t \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto t \cdot \color{blue}{\left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)} + \left(a - \frac{1}{2}\right) \cdot \log t \]

      2. metadata-eval N/A

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

      3. distribute-rgt-in N/A

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

      4. mul-1-neg N/A

         \[\leadsto \left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) \cdot t + \color{blue}{\left(\mathsf{neg}\left(t\right)\right)}\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}, t, \mathsf{neg}\left(t\right)\right)} + \left(a - \frac{1}{2}\right) \cdot \log t \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\log \left(x + y\right)}{t} + \frac{\log z}{t}}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      7. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\log \left(x + y\right)}{t} + \frac{\log z}{t}}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\log \left(x + y\right)}{t}} + \frac{\log z}{t}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      9. lower-log.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\log \left(x + y\right)}}{t} + \frac{\log z}{t}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\log \color{blue}{\left(y + x\right)}}{t} + \frac{\log z}{t}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      11. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\log \color{blue}{\left(y + x\right)}}{t} + \frac{\log z}{t}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\log \left(y + x\right)}{t} + \color{blue}{\frac{\log z}{t}}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      13. lower-log.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\log \left(y + x\right)}{t} + \frac{\color{blue}{\log z}}{t}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      14. lower-neg.f64 96.1

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

   5. Applied rewrites 96.1%

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

   6. Taylor expanded in t around inf

      \[\leadsto -1 \cdot \color{blue}{t} + \left(a - \frac{1}{2}\right) \cdot \log t \]
   7. Step-by-step derivation

   8. Applied rewrites 75.2%

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

   if -750 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 710

   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. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+r- N/A

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

      5. lower--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 99.6

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

      9. lift-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\log t, a - \frac{1}{2}, \color{blue}{\log \left(x + y\right) + \log z}\right) - t \]

      10. lift-log.f64 N/A

          \[\leadsto \mathsf{fma}\left(\log t, a - \frac{1}{2}, \color{blue}{\log \left(x + y\right)} + \log z\right) - t \]

      11. lift-log.f64 N/A

          \[\leadsto \mathsf{fma}\left(\log t, a - \frac{1}{2}, \log \left(x + y\right) + \color{blue}{\log z}\right) - t \]

      12. sum-log N/A

          \[\leadsto \mathsf{fma}\left(\log t, a - \frac{1}{2}, \color{blue}{\log \left(\left(x + y\right) \cdot z\right)}\right) - t \]

      13. lower-log.f64 N/A

          \[\leadsto \mathsf{fma}\left(\log t, a - \frac{1}{2}, \color{blue}{\log \left(\left(x + y\right) \cdot z\right)}\right) - t \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\log t, a - \frac{1}{2}, \log \color{blue}{\left(z \cdot \left(x + y\right)\right)}\right) - t \]

      15. lower-*.f64 99.6

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

      16. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\log t, a - \frac{1}{2}, \log \left(z \cdot \color{blue}{\left(x + y\right)}\right)\right) - t \]

      17. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\log t, a - \frac{1}{2}, \log \left(z \cdot \color{blue}{\left(y + x\right)}\right)\right) - t \]

      18. lower-+.f64 99.6

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

   4. Applied rewrites 99.6%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(y + x\right) + \log z \leq -750:\\ \;\;\;\;\left(-t\right) - \left(0.5 - a\right) \cdot \log t\\ \mathbf{elif}\;\log \left(y + x\right) + \log z \leq 710:\\ \;\;\;\;\mathsf{fma}\left(\log t, a - 0.5, \log \left(z \cdot \left(y + x\right)\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - \left(0.5 - a\right) \cdot \log t\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(y + x\right) + \log z\\ t_2 := \left(-t\right) - \left(0.5 - a\right) \cdot \log t\\ \mathbf{if}\;t\_1 \leq -750:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 710:\\ \;\;\;\;\mathsf{fma}\left(\log t, a - 0.5, \log \left(z \cdot y\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log (+ y x)) (log z))) (t_2 (- (- t) (* (- 0.5 a) (log t)))))
   (if (<= t_1 -750.0)
     t_2
     (if (<= t_1 710.0) (- (fma (log t) (- a 0.5) (log (* z y))) t) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((y + x)) + log(z);
	double t_2 = -t - ((0.5 - a) * log(t));
	double tmp;
	if (t_1 <= -750.0) {
		tmp = t_2;
	} else if (t_1 <= 710.0) {
		tmp = fma(log(t), (a - 0.5), log((z * y))) - t;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(Float64(y + x)) + log(z))
	t_2 = Float64(Float64(-t) - Float64(Float64(0.5 - a) * log(t)))
	tmp = 0.0
	if (t_1 <= -750.0)
		tmp = t_2;
	elseif (t_1 <= 710.0)
		tmp = Float64(fma(log(t), Float64(a - 0.5), log(Float64(z * y))) - t);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[((-t) - N[(N[(0.5 - a), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -750.0], t$95$2, If[LessEqual[t$95$1, 710.0], N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision] + N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -750 or 710 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z))

   1. Initial program 99.7%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) - 1\right)} + \left(a - \frac{1}{2}\right) \cdot \log t \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto t \cdot \color{blue}{\left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)} + \left(a - \frac{1}{2}\right) \cdot \log t \]

      2. metadata-eval N/A

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

      3. distribute-rgt-in N/A

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

      4. mul-1-neg N/A

         \[\leadsto \left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) \cdot t + \color{blue}{\left(\mathsf{neg}\left(t\right)\right)}\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}, t, \mathsf{neg}\left(t\right)\right)} + \left(a - \frac{1}{2}\right) \cdot \log t \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\log \left(x + y\right)}{t} + \frac{\log z}{t}}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      7. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\log \left(x + y\right)}{t} + \frac{\log z}{t}}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\log \left(x + y\right)}{t}} + \frac{\log z}{t}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      9. lower-log.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\log \left(x + y\right)}}{t} + \frac{\log z}{t}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\log \color{blue}{\left(y + x\right)}}{t} + \frac{\log z}{t}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      11. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\log \color{blue}{\left(y + x\right)}}{t} + \frac{\log z}{t}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\log \left(y + x\right)}{t} + \color{blue}{\frac{\log z}{t}}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      13. lower-log.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\log \left(y + x\right)}{t} + \frac{\color{blue}{\log z}}{t}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      14. lower-neg.f64 96.1

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

   5. Applied rewrites 96.1%

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

   6. Taylor expanded in t around inf

      \[\leadsto -1 \cdot \color{blue}{t} + \left(a - \frac{1}{2}\right) \cdot \log t \]
   7. Step-by-step derivation

   8. Applied rewrites 75.2%

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

   if -750 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 710

   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. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\log \left(y \cdot z\right) + \log t \cdot \left(a - \frac{1}{2}\right)\right) - t} \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\log t, a - \frac{1}{2}, \log \left(y \cdot z\right)\right)} - t \]

      4. lower-log.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\log t}, a - \frac{1}{2}, \log \left(y \cdot z\right)\right) - t \]

      5. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\log t, \color{blue}{a - \frac{1}{2}}, \log \left(y \cdot z\right)\right) - t \]

      6. lower-log.f64 N/A

         \[\leadsto \mathsf{fma}\left(\log t, a - \frac{1}{2}, \color{blue}{\log \left(y \cdot z\right)}\right) - t \]

      7. lower-*.f64 62.9

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

   7. Applied rewrites 62.9%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(y + x\right) + \log z \leq -750:\\ \;\;\;\;\left(-t\right) - \left(0.5 - a\right) \cdot \log t\\ \mathbf{elif}\;\log \left(y + x\right) + \log z \leq 710:\\ \;\;\;\;\mathsf{fma}\left(\log t, a - 0.5, \log \left(z \cdot y\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - \left(0.5 - a\right) \cdot \log t\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-t\right) - \left(0.5 - a\right) \cdot \log t\\ \mathbf{if}\;a \leq -0.92:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 1.65:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log t, \log \left(y + x\right)\right) - \left(t - \log z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (- t) (* (- 0.5 a) (log t)))))
   (if (<= a -0.92)
     t_1
     (if (<= a 1.65) (- (fma -0.5 (log t) (log (+ y x))) (- t (log z))) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = -t - ((0.5 - a) * log(t));
	double tmp;
	if (a <= -0.92) {
		tmp = t_1;
	} else if (a <= 1.65) {
		tmp = fma(-0.5, log(t), log((y + x))) - (t - log(z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(-t) - Float64(Float64(0.5 - a) * log(t)))
	tmp = 0.0
	if (a <= -0.92)
		tmp = t_1;
	elseif (a <= 1.65)
		tmp = Float64(fma(-0.5, log(t), log(Float64(y + x))) - Float64(t - log(z)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[((-t) - N[(N[(0.5 - a), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -0.92], t$95$1, If[LessEqual[a, 1.65], N[(N[(-0.5 * N[Log[t], $MachinePrecision] + N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(t - N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -0.92000000000000004 or 1.6499999999999999 < a

   1. Initial program 99.6%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) - 1\right)} + \left(a - \frac{1}{2}\right) \cdot \log t \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto t \cdot \color{blue}{\left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)} + \left(a - \frac{1}{2}\right) \cdot \log t \]

      2. metadata-eval N/A

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

      3. distribute-rgt-in N/A

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

      4. mul-1-neg N/A

         \[\leadsto \left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) \cdot t + \color{blue}{\left(\mathsf{neg}\left(t\right)\right)}\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}, t, \mathsf{neg}\left(t\right)\right)} + \left(a - \frac{1}{2}\right) \cdot \log t \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\log \left(x + y\right)}{t} + \frac{\log z}{t}}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      7. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\log \left(x + y\right)}{t} + \frac{\log z}{t}}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\log \left(x + y\right)}{t}} + \frac{\log z}{t}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      9. lower-log.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\log \left(x + y\right)}}{t} + \frac{\log z}{t}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\log \color{blue}{\left(y + x\right)}}{t} + \frac{\log z}{t}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      11. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\log \color{blue}{\left(y + x\right)}}{t} + \frac{\log z}{t}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\log \left(y + x\right)}{t} + \color{blue}{\frac{\log z}{t}}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      13. lower-log.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\log \left(y + x\right)}{t} + \frac{\color{blue}{\log z}}{t}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      14. lower-neg.f64 98.1

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

   5. Applied rewrites 98.1%

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

   6. Taylor expanded in t around inf

      \[\leadsto -1 \cdot \color{blue}{t} + \left(a - \frac{1}{2}\right) \cdot \log t \]
   7. Step-by-step derivation

   8. Applied rewrites 98.4%

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

   if -0.92000000000000004 < a < 1.6499999999999999

   1. Initial program 99.6%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(\log z + \left(\log \left(x + y\right) + \frac{-1}{2} \cdot \log t\right)\right) - t} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \log t, \log \left(x + y\right)\right)} - \left(t - \log z\right) \]

      7. lower-log.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\log t}, \log \left(x + y\right)\right) - \left(t - \log z\right) \]

      8. lower-log.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \log t, \color{blue}{\log \left(x + y\right)}\right) - \left(t - \log z\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \log t, \log \color{blue}{\left(y + x\right)}\right) - \left(t - \log z\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \log t, \log \color{blue}{\left(y + x\right)}\right) - \left(t - \log z\right) \]

      11. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \log t, \log \left(y + x\right)\right) - \color{blue}{\left(t - \log z\right)} \]

      12. lower-log.f64 98.7

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

   5. Applied rewrites 98.7%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.92:\\ \;\;\;\;\left(-t\right) - \left(0.5 - a\right) \cdot \log t\\ \mathbf{elif}\;a \leq 1.65:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log t, \log \left(y + x\right)\right) - \left(t - \log z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - \left(0.5 - a\right) \cdot \log t\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift--.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. associate--l+ N/A

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

   6. +-commutative N/A

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

   7. associate-+r+ N/A

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

   8. lower-+.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(\log t, a - \frac{1}{2}, \log z - t\right)} + \log \left(x + y\right) \]

   12. lower--.f64 99.6

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

   13. lift-+.f64 N/A

       \[\leadsto \mathsf{fma}\left(\log t, a - \frac{1}{2}, \log z - t\right) + \log \color{blue}{\left(x + y\right)} \]

   14. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(\log t, a - \frac{1}{2}, \log z - t\right) + \log \color{blue}{\left(y + x\right)} \]

   15. lower-+.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Final simplification 99.6%

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

Alternative 11: 68.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\left(\log y + \left(\log z + \log t \cdot \left(a - \frac{1}{2}\right)\right)\right) - t} \]
4. Step-by-step derivation

   1. associate--l+ N/A

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

   2. +-commutative N/A

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

   3. associate-+l- N/A

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

   4. lower--.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(a - \frac{1}{2}, \log t, \log z\right)} - \left(t - \log y\right) \]

   8. lower--.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{a - \frac{1}{2}}, \log t, \log z\right) - \left(t - \log y\right) \]

   9. lower-log.f64 N/A

      \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, \color{blue}{\log t}, \log z\right) - \left(t - \log y\right) \]

   10. lower-log.f64 N/A

       \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, \log t, \color{blue}{\log z}\right) - \left(t - \log y\right) \]

   11. lower--.f64 N/A

       \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, \log t, \log z\right) - \color{blue}{\left(t - \log y\right)} \]

   12. lower-log.f64 68.3

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

5. Applied rewrites 68.3%

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

Alternative 12: 77.3% accurate, 2.8× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t, double a) {
	return -t - ((0.5 - a) * 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 = -t - ((0.5d0 - a) * log(t))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in t around inf

   \[\leadsto \color{blue}{t \cdot \left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) - 1\right)} + \left(a - \frac{1}{2}\right) \cdot \log t \]
4. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto t \cdot \color{blue}{\left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)} + \left(a - \frac{1}{2}\right) \cdot \log t \]

   2. metadata-eval N/A

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

   3. distribute-rgt-in N/A

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

   4. mul-1-neg N/A

      \[\leadsto \left(\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}\right) \cdot t + \color{blue}{\left(\mathsf{neg}\left(t\right)\right)}\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

   5. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\log z}{t} + \frac{\log \left(x + y\right)}{t}, t, \mathsf{neg}\left(t\right)\right)} + \left(a - \frac{1}{2}\right) \cdot \log t \]

   6. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\log \left(x + y\right)}{t} + \frac{\log z}{t}}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

   7. lower-+.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\log \left(x + y\right)}{t} + \frac{\log z}{t}}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

   8. lower-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\log \left(x + y\right)}{t}} + \frac{\log z}{t}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

   9. lower-log.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\log \left(x + y\right)}}{t} + \frac{\log z}{t}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

   10. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(\frac{\log \color{blue}{\left(y + x\right)}}{t} + \frac{\log z}{t}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

   11. lower-+.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\log \color{blue}{\left(y + x\right)}}{t} + \frac{\log z}{t}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

   12. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\log \left(y + x\right)}{t} + \color{blue}{\frac{\log z}{t}}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

   13. lower-log.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\log \left(y + x\right)}{t} + \frac{\color{blue}{\log z}}{t}, t, \mathsf{neg}\left(t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

   14. lower-neg.f64 98.8

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

5. Applied rewrites 98.8%

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

6. Taylor expanded in t around inf

   \[\leadsto -1 \cdot \color{blue}{t} + \left(a - \frac{1}{2}\right) \cdot \log t \]
7. Step-by-step derivation

8. Applied rewrites 76.2%

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

9. Final simplification 76.2%

   \[\leadsto \left(-t\right) - \left(0.5 - a\right) \cdot \log t \]
10. Add Preprocessing

Alternative 13: 62.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.02 \cdot 10^{+25}:\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 1.02e+25) (* (log t) a) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 1.02e+25) {
		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 (t <= 1.02d+25) 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 (t <= 1.02e+25) {
		tmp = Math.log(t) * a;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 1.02e+25:
		tmp = math.log(t) * a
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 1.02e+25)
		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 (t <= 1.02e+25)
		tmp = log(t) * a;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 1.02e+25], N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if t < 1.0199999999999999e25

   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. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 54.4

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

   5. Applied rewrites 54.4%

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

   if 1.0199999999999999e25 < 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. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(t\right)} \]

      2. lower-neg.f64 81.4

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

   5. Applied rewrites 81.4%

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

Alternative 14: 37.7% accurate, 107.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(t\right)} \]

   2. lower-neg.f64 35.0

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

5. Applied rewrites 35.0%

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

Developer Target 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024295 
(FPCore (x y z t a)
  :name "Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2"
  :precision binary64

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

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