Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 12.9s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

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

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]

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. Add Preprocessing

Alternative 2: 82.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot \log t\\ t_2 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + t\_1\\ \mathbf{if}\;t\_2 \leq -1000000000000 \lor \neg \left(t\_2 \leq 700\right):\\ \;\;\;\;\left(-t\right) + t\_1\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(z \cdot y\right) \cdot \sqrt{{t}^{-1}}\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) (log t)))
        (t_2 (+ (- (+ (log (+ x y)) (log z)) t) t_1)))
   (if (or (<= t_2 -1000000000000.0) (not (<= t_2 700.0)))
     (+ (- t) t_1)
     (- (log (* (* z y) (sqrt (pow t -1.0)))) t))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (a - 0.5) * log(t);
	double t_2 = ((log((x + y)) + log(z)) - t) + t_1;
	double tmp;
	if ((t_2 <= -1000000000000.0) || !(t_2 <= 700.0)) {
		tmp = -t + t_1;
	} else {
		tmp = log(((z * y) * sqrt(pow(t, -1.0)))) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (a - 0.5d0) * log(t)
    t_2 = ((log((x + y)) + log(z)) - t) + t_1
    if ((t_2 <= (-1000000000000.0d0)) .or. (.not. (t_2 <= 700.0d0))) then
        tmp = -t + t_1
    else
        tmp = log(((z * y) * sqrt((t ** (-1.0d0))))) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (a - 0.5) * Math.log(t);
	double t_2 = ((Math.log((x + y)) + Math.log(z)) - t) + t_1;
	double tmp;
	if ((t_2 <= -1000000000000.0) || !(t_2 <= 700.0)) {
		tmp = -t + t_1;
	} else {
		tmp = Math.log(((z * y) * Math.sqrt(Math.pow(t, -1.0)))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (a - 0.5) * math.log(t)
	t_2 = ((math.log((x + y)) + math.log(z)) - t) + t_1
	tmp = 0
	if (t_2 <= -1000000000000.0) or not (t_2 <= 700.0):
		tmp = -t + t_1
	else:
		tmp = math.log(((z * y) * math.sqrt(math.pow(t, -1.0)))) - t
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(a - 0.5) * log(t))
	t_2 = Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + t_1)
	tmp = 0.0
	if ((t_2 <= -1000000000000.0) || !(t_2 <= 700.0))
		tmp = Float64(Float64(-t) + t_1);
	else
		tmp = Float64(log(Float64(Float64(z * y) * sqrt((t ^ -1.0)))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (a - 0.5) * log(t);
	t_2 = ((log((x + y)) + log(z)) - t) + t_1;
	tmp = 0.0;
	if ((t_2 <= -1000000000000.0) || ~((t_2 <= 700.0)))
		tmp = -t + t_1;
	else
		tmp = log(((z * y) * sqrt((t ^ -1.0)))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + t$95$1), $MachinePrecision]}, If[Or[LessEqual[t$95$2, -1000000000000.0], N[Not[LessEqual[t$95$2, 700.0]], $MachinePrecision]], N[((-t) + t$95$1), $MachinePrecision], N[(N[Log[N[(N[(z * y), $MachinePrecision] * N[Sqrt[N[Power[t, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]]]]

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))) < -1e12 or 700 < (+.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. Step-by-step derivation

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip3-+ N/A

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

      4. log-div N/A

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

      5. lower--.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-rgt-out-- N/A

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

      14. lower-fma.f64 N/A

          \[\leadsto \left(\left(\left(\log \left({y}^{3} + {x}^{3}\right) - \log \color{blue}{\left(\mathsf{fma}\left(y, y - x, x \cdot x\right)\right)}\right) + \log z\right) - t\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      15. lower--.f64 N/A

          \[\leadsto \left(\left(\left(\log \left({y}^{3} + {x}^{3}\right) - \log \left(\mathsf{fma}\left(y, \color{blue}{y - x}, x \cdot x\right)\right)\right) + \log z\right) - t\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      16. lower-*.f64 31.1

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

   4. Applied rewrites 31.1%

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

   5. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(\left(\log z + -3 \cdot \log \left(\frac{1}{y}\right)\right) - \left(t + -2 \cdot \log \left(\frac{1}{y}\right)\right)\right)} + \left(a - \frac{1}{2}\right) \cdot \log t \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\log \left(\frac{1}{y}\right), -3, \log z\right)} - \left(t + -2 \cdot \log \left(\frac{1}{y}\right)\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      5. log-rec N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(\log y\right)}, -3, \log z\right) - \left(t + -2 \cdot \log \left(\frac{1}{y}\right)\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      6. lower-neg.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{-\log y}, -3, \log z\right) - \left(t + -2 \cdot \log \left(\frac{1}{y}\right)\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      7. lower-log.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(-\color{blue}{\log y}, -3, \log z\right) - \left(t + -2 \cdot \log \left(\frac{1}{y}\right)\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      8. lower-log.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(-\log y, -3, \color{blue}{\log z}\right) - \left(t + -2 \cdot \log \left(\frac{1}{y}\right)\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      9. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(-\log y, -3, \log z\right) - \color{blue}{\left(-2 \cdot \log \left(\frac{1}{y}\right) + t\right)}\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      10. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(-\log y, -3, \log z\right) - \left(\color{blue}{\log \left(\frac{1}{y}\right) \cdot -2} + t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      11. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(-\log y, -3, \log z\right) - \color{blue}{\mathsf{fma}\left(\log \left(\frac{1}{y}\right), -2, t\right)}\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      12. log-rec N/A

          \[\leadsto \left(\mathsf{fma}\left(-\log y, -3, \log z\right) - \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(\log y\right)}, -2, t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      13. lower-neg.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(-\log y, -3, \log z\right) - \mathsf{fma}\left(\color{blue}{-\log y}, -2, t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      14. lower-log.f64 71.3

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

   7. Applied rewrites 71.3%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 92.7%

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

   if -1e12 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 700

   1. Initial program 98.9%

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

   3. Taylor expanded in 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. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-out-- N/A

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

      6. metadata-eval N/A

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

      7. fp-cancel-sign-sub-inv N/A

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

      8. distribute-rgt-out N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-log.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-log.f64 47.6

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

   5. Applied rewrites 47.6%

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

   7. Applied rewrites 40.6%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 44.2%

       \[\leadsto \log \left(\left(z \cdot y\right) \cdot \sqrt{\frac{1}{t}}\right) - t \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \leq -1000000000000 \lor \neg \left(\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \leq 700\right):\\ \;\;\;\;\left(-t\right) + \left(a - 0.5\right) \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(z \cdot y\right) \cdot \sqrt{{t}^{-1}}\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot \log t\\ t_2 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + t\_1\\ \mathbf{if}\;t\_2 \leq -1000000000000 \lor \neg \left(t\_2 \leq 940\right):\\ \;\;\;\;\left(-t\right) + t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log t, -0.5, \log \left(z \cdot y\right) - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) (log t)))
        (t_2 (+ (- (+ (log (+ x y)) (log z)) t) t_1)))
   (if (or (<= t_2 -1000000000000.0) (not (<= t_2 940.0)))
     (+ (- t) t_1)
     (fma (log t) -0.5 (- (log (* z y)) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (a - 0.5) * log(t);
	double t_2 = ((log((x + y)) + log(z)) - t) + t_1;
	double tmp;
	if ((t_2 <= -1000000000000.0) || !(t_2 <= 940.0)) {
		tmp = -t + t_1;
	} else {
		tmp = fma(log(t), -0.5, (log((z * y)) - t));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(a - 0.5) * log(t))
	t_2 = Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + t_1)
	tmp = 0.0
	if ((t_2 <= -1000000000000.0) || !(t_2 <= 940.0))
		tmp = Float64(Float64(-t) + t_1);
	else
		tmp = fma(log(t), -0.5, Float64(log(Float64(z * y)) - t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + t$95$1), $MachinePrecision]}, If[Or[LessEqual[t$95$2, -1000000000000.0], N[Not[LessEqual[t$95$2, 940.0]], $MachinePrecision]], N[((-t) + t$95$1), $MachinePrecision], N[(N[Log[t], $MachinePrecision] * -0.5 + N[(N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]]]]

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))) < -1e12 or 940 < (+.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. Step-by-step derivation

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip3-+ N/A

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

      4. log-div N/A

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

      5. lower--.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-rgt-out-- N/A

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

      14. lower-fma.f64 N/A

          \[\leadsto \left(\left(\left(\log \left({y}^{3} + {x}^{3}\right) - \log \color{blue}{\left(\mathsf{fma}\left(y, y - x, x \cdot x\right)\right)}\right) + \log z\right) - t\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      15. lower--.f64 N/A

          \[\leadsto \left(\left(\left(\log \left({y}^{3} + {x}^{3}\right) - \log \left(\mathsf{fma}\left(y, \color{blue}{y - x}, x \cdot x\right)\right)\right) + \log z\right) - t\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      16. lower-*.f64 30.8

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

   4. Applied rewrites 30.8%

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

   5. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(\left(\log z + -3 \cdot \log \left(\frac{1}{y}\right)\right) - \left(t + -2 \cdot \log \left(\frac{1}{y}\right)\right)\right)} + \left(a - \frac{1}{2}\right) \cdot \log t \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\log \left(\frac{1}{y}\right), -3, \log z\right)} - \left(t + -2 \cdot \log \left(\frac{1}{y}\right)\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      5. log-rec N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(\log y\right)}, -3, \log z\right) - \left(t + -2 \cdot \log \left(\frac{1}{y}\right)\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      6. lower-neg.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{-\log y}, -3, \log z\right) - \left(t + -2 \cdot \log \left(\frac{1}{y}\right)\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      7. lower-log.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(-\color{blue}{\log y}, -3, \log z\right) - \left(t + -2 \cdot \log \left(\frac{1}{y}\right)\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      8. lower-log.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(-\log y, -3, \color{blue}{\log z}\right) - \left(t + -2 \cdot \log \left(\frac{1}{y}\right)\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      9. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(-\log y, -3, \log z\right) - \color{blue}{\left(-2 \cdot \log \left(\frac{1}{y}\right) + t\right)}\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      10. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(-\log y, -3, \log z\right) - \left(\color{blue}{\log \left(\frac{1}{y}\right) \cdot -2} + t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      11. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(-\log y, -3, \log z\right) - \color{blue}{\mathsf{fma}\left(\log \left(\frac{1}{y}\right), -2, t\right)}\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      12. log-rec N/A

          \[\leadsto \left(\mathsf{fma}\left(-\log y, -3, \log z\right) - \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(\log y\right)}, -2, t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      13. lower-neg.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(-\log y, -3, \log z\right) - \mathsf{fma}\left(\color{blue}{-\log y}, -2, t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      14. lower-log.f64 72.1

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

   7. Applied rewrites 72.1%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 96.8%

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

   if -1e12 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 940

   1. Initial program 99.0%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   2. 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. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-out-- N/A

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

      6. metadata-eval N/A

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

      7. fp-cancel-sign-sub-inv N/A

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

      8. distribute-rgt-out N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-log.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-log.f64 49.3

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

   5. Applied rewrites 49.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 49.3%

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

   10. Applied rewrites 42.8%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \leq -1000000000000 \lor \neg \left(\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \leq 940\right):\\ \;\;\;\;\left(-t\right) + \left(a - 0.5\right) \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log t, -0.5, \log \left(z \cdot y\right) - t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(x + y\right) + \log z\\ \mathbf{if}\;t\_1 \leq -750 \lor \neg \left(t\_1 \leq 662\right):\\ \;\;\;\;\left(-t\right) + \left(a - 0.5\right) \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log \left(z \cdot \left(y + x\right)\right) - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log (+ x y)) (log z))))
   (if (or (<= t_1 -750.0) (not (<= t_1 662.0)))
     (+ (- t) (* (- a 0.5) (log t)))
     (fma (- a 0.5) (log t) (- (log (* z (+ y x))) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((x + y)) + log(z);
	double tmp;
	if ((t_1 <= -750.0) || !(t_1 <= 662.0)) {
		tmp = -t + ((a - 0.5) * log(t));
	} else {
		tmp = fma((a - 0.5), log(t), (log((z * (y + x))) - t));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(Float64(x + y)) + log(z))
	tmp = 0.0
	if ((t_1 <= -750.0) || !(t_1 <= 662.0))
		tmp = Float64(Float64(-t) + Float64(Float64(a - 0.5) * log(t)));
	else
		tmp = fma(Float64(a - 0.5), log(t), Float64(log(Float64(z * Float64(y + x))) - t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -750.0], N[Not[LessEqual[t$95$1, 662.0]], $MachinePrecision]], N[((-t) + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[(N[Log[N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -750 or 662 < (+.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. Step-by-step derivation

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip3-+ N/A

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

      4. log-div N/A

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

      5. lower--.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-rgt-out-- N/A

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

      14. lower-fma.f64 N/A

          \[\leadsto \left(\left(\left(\log \left({y}^{3} + {x}^{3}\right) - \log \color{blue}{\left(\mathsf{fma}\left(y, y - x, x \cdot x\right)\right)}\right) + \log z\right) - t\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      15. lower--.f64 N/A

          \[\leadsto \left(\left(\left(\log \left({y}^{3} + {x}^{3}\right) - \log \left(\mathsf{fma}\left(y, \color{blue}{y - x}, x \cdot x\right)\right)\right) + \log z\right) - t\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      16. lower-*.f64 8.5

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

   4. Applied rewrites 8.5%

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

   5. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(\left(\log z + -3 \cdot \log \left(\frac{1}{y}\right)\right) - \left(t + -2 \cdot \log \left(\frac{1}{y}\right)\right)\right)} + \left(a - \frac{1}{2}\right) \cdot \log t \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\log \left(\frac{1}{y}\right), -3, \log z\right)} - \left(t + -2 \cdot \log \left(\frac{1}{y}\right)\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      5. log-rec N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(\log y\right)}, -3, \log z\right) - \left(t + -2 \cdot \log \left(\frac{1}{y}\right)\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      6. lower-neg.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{-\log y}, -3, \log z\right) - \left(t + -2 \cdot \log \left(\frac{1}{y}\right)\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      7. lower-log.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(-\color{blue}{\log y}, -3, \log z\right) - \left(t + -2 \cdot \log \left(\frac{1}{y}\right)\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      8. lower-log.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(-\log y, -3, \color{blue}{\log z}\right) - \left(t + -2 \cdot \log \left(\frac{1}{y}\right)\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      9. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(-\log y, -3, \log z\right) - \color{blue}{\left(-2 \cdot \log \left(\frac{1}{y}\right) + t\right)}\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      10. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(-\log y, -3, \log z\right) - \left(\color{blue}{\log \left(\frac{1}{y}\right) \cdot -2} + t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      11. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(-\log y, -3, \log z\right) - \color{blue}{\mathsf{fma}\left(\log \left(\frac{1}{y}\right), -2, t\right)}\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      12. log-rec N/A

          \[\leadsto \left(\mathsf{fma}\left(-\log y, -3, \log z\right) - \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(\log y\right)}, -2, t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      13. lower-neg.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(-\log y, -3, \log z\right) - \mathsf{fma}\left(\color{blue}{-\log y}, -2, t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      14. lower-log.f64 68.8

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

   7. Applied rewrites 68.8%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 84.4%

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

   if -750 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 662

   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

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

      4. lower-fma.f64 99.6

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

      5. lift-+.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. sum-log N/A

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

      9. lower-log.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.4

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 99.4

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

   4. Applied rewrites 99.4%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(x + y\right) + \log z \leq -750 \lor \neg \left(\log \left(x + y\right) + \log z \leq 662\right):\\ \;\;\;\;\left(-t\right) + \left(a - 0.5\right) \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log \left(z \cdot \left(y + x\right)\right) - t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(x + y\right) + \log z\\ \mathbf{if}\;t\_1 \leq -750 \lor \neg \left(t\_1 \leq 662\right):\\ \;\;\;\;\left(-t\right) + \left(a - 0.5\right) \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 + a, \log t, \log \left(z \cdot y\right) - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log (+ x y)) (log z))))
   (if (or (<= t_1 -750.0) (not (<= t_1 662.0)))
     (+ (- t) (* (- a 0.5) (log t)))
     (fma (+ -0.5 a) (log t) (- (log (* z y)) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((x + y)) + log(z);
	double tmp;
	if ((t_1 <= -750.0) || !(t_1 <= 662.0)) {
		tmp = -t + ((a - 0.5) * log(t));
	} else {
		tmp = fma((-0.5 + a), log(t), (log((z * y)) - t));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(Float64(x + y)) + log(z))
	tmp = 0.0
	if ((t_1 <= -750.0) || !(t_1 <= 662.0))
		tmp = Float64(Float64(-t) + Float64(Float64(a - 0.5) * log(t)));
	else
		tmp = fma(Float64(-0.5 + a), log(t), Float64(log(Float64(z * y)) - t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -750.0], N[Not[LessEqual[t$95$1, 662.0]], $MachinePrecision]], N[((-t) + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 + a), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[(N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -750 or 662 < (+.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. Step-by-step derivation

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip3-+ N/A

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

      4. log-div N/A

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

      5. lower--.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-rgt-out-- N/A

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

      14. lower-fma.f64 N/A

          \[\leadsto \left(\left(\left(\log \left({y}^{3} + {x}^{3}\right) - \log \color{blue}{\left(\mathsf{fma}\left(y, y - x, x \cdot x\right)\right)}\right) + \log z\right) - t\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      15. lower--.f64 N/A

          \[\leadsto \left(\left(\left(\log \left({y}^{3} + {x}^{3}\right) - \log \left(\mathsf{fma}\left(y, \color{blue}{y - x}, x \cdot x\right)\right)\right) + \log z\right) - t\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      16. lower-*.f64 8.5

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

   4. Applied rewrites 8.5%

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

   5. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(\left(\log z + -3 \cdot \log \left(\frac{1}{y}\right)\right) - \left(t + -2 \cdot \log \left(\frac{1}{y}\right)\right)\right)} + \left(a - \frac{1}{2}\right) \cdot \log t \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\log \left(\frac{1}{y}\right), -3, \log z\right)} - \left(t + -2 \cdot \log \left(\frac{1}{y}\right)\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      5. log-rec N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(\log y\right)}, -3, \log z\right) - \left(t + -2 \cdot \log \left(\frac{1}{y}\right)\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      6. lower-neg.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{-\log y}, -3, \log z\right) - \left(t + -2 \cdot \log \left(\frac{1}{y}\right)\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      7. lower-log.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(-\color{blue}{\log y}, -3, \log z\right) - \left(t + -2 \cdot \log \left(\frac{1}{y}\right)\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      8. lower-log.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(-\log y, -3, \color{blue}{\log z}\right) - \left(t + -2 \cdot \log \left(\frac{1}{y}\right)\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      9. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(-\log y, -3, \log z\right) - \color{blue}{\left(-2 \cdot \log \left(\frac{1}{y}\right) + t\right)}\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      10. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(-\log y, -3, \log z\right) - \left(\color{blue}{\log \left(\frac{1}{y}\right) \cdot -2} + t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      11. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(-\log y, -3, \log z\right) - \color{blue}{\mathsf{fma}\left(\log \left(\frac{1}{y}\right), -2, t\right)}\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      12. log-rec N/A

          \[\leadsto \left(\mathsf{fma}\left(-\log y, -3, \log z\right) - \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(\log y\right)}, -2, t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      13. lower-neg.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(-\log y, -3, \log z\right) - \mathsf{fma}\left(\color{blue}{-\log y}, -2, t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      14. lower-log.f64 68.8

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

   7. Applied rewrites 68.8%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 84.4%

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

   if -750 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 662

   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. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-out-- N/A

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

      6. metadata-eval N/A

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

      7. fp-cancel-sign-sub-inv N/A

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

      8. distribute-rgt-out N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-log.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-log.f64 66.6

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

   5. Applied rewrites 66.6%

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

   7. Applied rewrites 61.9%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(x + y\right) + \log z \leq -750 \lor \neg \left(\log \left(x + y\right) + \log z \leq 662\right):\\ \;\;\;\;\left(-t\right) + \left(a - 0.5\right) \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 + a, \log t, \log \left(z \cdot y\right) - t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 460:\\ \;\;\;\;\log z + \mathsf{fma}\left(\log t, a - 0.5, \log y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) + \left(a - 0.5\right) \cdot \log t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 460.0)
   (+ (log z) (fma (log t) (- a 0.5) (log y)))
   (+ (- t) (* (- a 0.5) (log t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 460.0) {
		tmp = log(z) + fma(log(t), (a - 0.5), log(y));
	} else {
		tmp = -t + ((a - 0.5) * log(t));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 460.0)
		tmp = Float64(log(z) + fma(log(t), Float64(a - 0.5), log(y)));
	else
		tmp = Float64(Float64(-t) + Float64(Float64(a - 0.5) * log(t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 460.0], N[(N[Log[z], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-t) + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 460

   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. 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. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-out-- N/A

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

      6. metadata-eval N/A

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

      7. fp-cancel-sign-sub-inv N/A

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

      8. distribute-rgt-out N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-log.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-log.f64 59.7

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

   5. Applied rewrites 59.7%

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

   7. Applied rewrites 59.8%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 59.8%

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

   if 460 < t

   1. Initial program 99.9%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip3-+ N/A

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

      4. log-div N/A

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

      5. lower--.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-rgt-out-- N/A

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

      14. lower-fma.f64 N/A

          \[\leadsto \left(\left(\left(\log \left({y}^{3} + {x}^{3}\right) - \log \color{blue}{\left(\mathsf{fma}\left(y, y - x, x \cdot x\right)\right)}\right) + \log z\right) - t\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      15. lower--.f64 N/A

          \[\leadsto \left(\left(\left(\log \left({y}^{3} + {x}^{3}\right) - \log \left(\mathsf{fma}\left(y, \color{blue}{y - x}, x \cdot x\right)\right)\right) + \log z\right) - t\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      16. lower-*.f64 30.1

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

   4. Applied rewrites 30.1%

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

   5. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(\left(\log z + -3 \cdot \log \left(\frac{1}{y}\right)\right) - \left(t + -2 \cdot \log \left(\frac{1}{y}\right)\right)\right)} + \left(a - \frac{1}{2}\right) \cdot \log t \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\log \left(\frac{1}{y}\right), -3, \log z\right)} - \left(t + -2 \cdot \log \left(\frac{1}{y}\right)\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      5. log-rec N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(\log y\right)}, -3, \log z\right) - \left(t + -2 \cdot \log \left(\frac{1}{y}\right)\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      6. lower-neg.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{-\log y}, -3, \log z\right) - \left(t + -2 \cdot \log \left(\frac{1}{y}\right)\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      7. lower-log.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(-\color{blue}{\log y}, -3, \log z\right) - \left(t + -2 \cdot \log \left(\frac{1}{y}\right)\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      8. lower-log.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(-\log y, -3, \color{blue}{\log z}\right) - \left(t + -2 \cdot \log \left(\frac{1}{y}\right)\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      9. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(-\log y, -3, \log z\right) - \color{blue}{\left(-2 \cdot \log \left(\frac{1}{y}\right) + t\right)}\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      10. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(-\log y, -3, \log z\right) - \left(\color{blue}{\log \left(\frac{1}{y}\right) \cdot -2} + t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      11. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(-\log y, -3, \log z\right) - \color{blue}{\mathsf{fma}\left(\log \left(\frac{1}{y}\right), -2, t\right)}\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      12. log-rec N/A

          \[\leadsto \left(\mathsf{fma}\left(-\log y, -3, \log z\right) - \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(\log y\right)}, -2, t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      13. lower-neg.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(-\log y, -3, \log z\right) - \mathsf{fma}\left(\color{blue}{-\log y}, -2, t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

      14. lower-log.f64 74.2

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

   7. Applied rewrites 74.2%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 99.9%

       \[\leadsto \left(-t\right) + \left(a - 0.5\right) \cdot \log t \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 68.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

   \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
2. 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. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. lower-+.f64 N/A

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

   4. +-commutative N/A

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

   5. distribute-rgt-out-- N/A

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

   6. metadata-eval N/A

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

   7. fp-cancel-sign-sub-inv N/A

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

   8. distribute-rgt-out N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. lower-+.f64 N/A

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

   13. lower-log.f64 N/A

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

   14. lower-log.f64 N/A

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

   15. lower--.f64 N/A

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

   16. lower-log.f64 67.2

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

5. Applied rewrites 67.2%

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

Alternative 8: 68.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. associate--l+ N/A

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

   3. lower-+.f64 N/A

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

   4. +-commutative N/A

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

   5. distribute-rgt-out-- N/A

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

   6. metadata-eval N/A

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

   7. fp-cancel-sign-sub-inv N/A

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

   8. distribute-rgt-out N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. lower-+.f64 N/A

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

   13. lower-log.f64 N/A

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

   14. lower-log.f64 N/A

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

   15. lower--.f64 N/A

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

   16. lower-log.f64 67.2

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

5. Applied rewrites 67.2%

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

7. Applied rewrites 67.2%

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

Alternative 9: 76.8% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. lift-log.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. flip3-+ N/A

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

   4. log-div N/A

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

   5. lower--.f64 N/A

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

   6. lower-log.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-+.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-pow.f64 N/A

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

   11. lower-log.f64 N/A

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

   12. +-commutative N/A

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

   13. distribute-rgt-out-- N/A

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

   14. lower-fma.f64 N/A

       \[\leadsto \left(\left(\left(\log \left({y}^{3} + {x}^{3}\right) - \log \color{blue}{\left(\mathsf{fma}\left(y, y - x, x \cdot x\right)\right)}\right) + \log z\right) - t\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

   15. lower--.f64 N/A

       \[\leadsto \left(\left(\left(\log \left({y}^{3} + {x}^{3}\right) - \log \left(\mathsf{fma}\left(y, \color{blue}{y - x}, x \cdot x\right)\right)\right) + \log z\right) - t\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

   16. lower-*.f64 32.2

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

4. Applied rewrites 32.2%

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

5. Taylor expanded in y around inf

   \[\leadsto \color{blue}{\left(\left(\log z + -3 \cdot \log \left(\frac{1}{y}\right)\right) - \left(t + -2 \cdot \log \left(\frac{1}{y}\right)\right)\right)} + \left(a - \frac{1}{2}\right) \cdot \log t \]
6. Step-by-step derivation

   1. lower--.f64 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\log \left(\frac{1}{y}\right), -3, \log z\right)} - \left(t + -2 \cdot \log \left(\frac{1}{y}\right)\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

   5. log-rec N/A

      \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(\log y\right)}, -3, \log z\right) - \left(t + -2 \cdot \log \left(\frac{1}{y}\right)\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

   6. lower-neg.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\color{blue}{-\log y}, -3, \log z\right) - \left(t + -2 \cdot \log \left(\frac{1}{y}\right)\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

   7. lower-log.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(-\color{blue}{\log y}, -3, \log z\right) - \left(t + -2 \cdot \log \left(\frac{1}{y}\right)\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

   8. lower-log.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(-\log y, -3, \color{blue}{\log z}\right) - \left(t + -2 \cdot \log \left(\frac{1}{y}\right)\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

   9. +-commutative N/A

      \[\leadsto \left(\mathsf{fma}\left(-\log y, -3, \log z\right) - \color{blue}{\left(-2 \cdot \log \left(\frac{1}{y}\right) + t\right)}\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

   10. *-commutative N/A

       \[\leadsto \left(\mathsf{fma}\left(-\log y, -3, \log z\right) - \left(\color{blue}{\log \left(\frac{1}{y}\right) \cdot -2} + t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

   11. lower-fma.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(-\log y, -3, \log z\right) - \color{blue}{\mathsf{fma}\left(\log \left(\frac{1}{y}\right), -2, t\right)}\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

   12. log-rec N/A

       \[\leadsto \left(\mathsf{fma}\left(-\log y, -3, \log z\right) - \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(\log y\right)}, -2, t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

   13. lower-neg.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(-\log y, -3, \log z\right) - \mathsf{fma}\left(\color{blue}{-\log y}, -2, t\right)\right) + \left(a - \frac{1}{2}\right) \cdot \log t \]

   14. lower-log.f64 67.2

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

7. Applied rewrites 67.2%

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

8. Taylor expanded in t around inf

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

10. Applied rewrites 79.2%

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

Alternative 10: 62.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 8.5 \cdot 10^{+29}:\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 8.5e+29) (* (log t) a) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 8.5e+29) {
		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 <= 8.5d+29) 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 <= 8.5e+29) {
		tmp = Math.log(t) * a;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 8.5e+29:
		tmp = math.log(t) * a
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 8.5e+29)
		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 <= 8.5e+29)
		tmp = log(t) * a;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 8.5e+29], N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if t < 8.5000000000000006e29

   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. 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 53.8

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

   5. Applied rewrites 53.8%

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

   if 8.5000000000000006e29 < 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 73.4

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

   5. Applied rewrites 73.4%

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

Alternative 11: 37.9% 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 37.5

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

5. Applied rewrites 37.5%

   \[\leadsto \color{blue}{-t} \]
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 2024343 
(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))))