Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 12.1s

Alternatives: 13

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Final simplification 99.7%

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

Alternative 2: 73.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\log z + \log \left(y + x\right)\right) - t\right) - \left(0.5 - a\right) \cdot \log t\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+20}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t\_1 \leq 1000:\\ \;\;\;\;\mathsf{fma}\left(\log t, -0.5, \log \left(z \cdot \left(y + x\right)\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (- (+ (log z) (log (+ y x))) t) (* (- 0.5 a) (log t)))))
   (if (<= t_1 -5e+20)
     (- t)
     (if (<= t_1 1000.0)
       (fma (log t) -0.5 (- (log (* z (+ y x))) t))
       (* (log t) a)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((log(z) + log((y + x))) - t) - ((0.5 - a) * log(t));
	double tmp;
	if (t_1 <= -5e+20) {
		tmp = -t;
	} else if (t_1 <= 1000.0) {
		tmp = fma(log(t), -0.5, (log((z * (y + x))) - t));
	} else {
		tmp = log(t) * a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(log(z) + log(Float64(y + x))) - t) - Float64(Float64(0.5 - a) * log(t)))
	tmp = 0.0
	if (t_1 <= -5e+20)
		tmp = Float64(-t);
	elseif (t_1 <= 1000.0)
		tmp = fma(log(t), -0.5, Float64(log(Float64(z * Float64(y + x))) - t));
	else
		tmp = Float64(log(t) * a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(N[Log[z], $MachinePrecision] + N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] - N[(N[(0.5 - a), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+20], (-t), If[LessEqual[t$95$1, 1000.0], N[(N[Log[t], $MachinePrecision] * -0.5 + N[(N[Log[N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < -5e20

   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 70.1

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

   5. Applied rewrites 70.1%

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

   if -5e20 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 1e3

   1. Initial program 99.4%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-log.f64 94.4

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

   5. Applied rewrites 94.4%

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

   7. Applied rewrites 85.2%

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

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

   1. Initial program 99.6%

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

   3. Taylor expanded in a around 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 76.0

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

   5. Applied rewrites 76.0%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\log z + \log \left(y + x\right)\right) - t\right) - \left(0.5 - a\right) \cdot \log t \leq -5 \cdot 10^{+20}:\\ \;\;\;\;-t\\ \mathbf{elif}\;\left(\left(\log z + \log \left(y + x\right)\right) - t\right) - \left(0.5 - a\right) \cdot \log t \leq 1000:\\ \;\;\;\;\mathsf{fma}\left(\log t, -0.5, \log \left(z \cdot \left(y + x\right)\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 63.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\log z + \log \left(y + x\right)\right) - t\right) - \left(0.5 - a\right) \cdot \log t\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+20}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t\_1 \leq 790:\\ \;\;\;\;\log \left({t}^{\left(a - 0.5\right)} \cdot \left(z \cdot y\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (- (+ (log z) (log (+ y x))) t) (* (- 0.5 a) (log t)))))
   (if (<= t_1 -5e+20)
     (- t)
     (if (<= t_1 790.0)
       (- (log (* (pow t (- a 0.5)) (* z y))) t)
       (* (log t) a)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((log(z) + log((y + x))) - t) - ((0.5 - a) * log(t));
	double tmp;
	if (t_1 <= -5e+20) {
		tmp = -t;
	} else if (t_1 <= 790.0) {
		tmp = log((pow(t, (a - 0.5)) * (z * y))) - t;
	} else {
		tmp = log(t) * a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((log(z) + log((y + x))) - t) - ((0.5d0 - a) * log(t))
    if (t_1 <= (-5d+20)) then
        tmp = -t
    else if (t_1 <= 790.0d0) then
        tmp = log(((t ** (a - 0.5d0)) * (z * y))) - t
    else
        tmp = log(t) * a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = ((Math.log(z) + Math.log((y + x))) - t) - ((0.5 - a) * Math.log(t));
	double tmp;
	if (t_1 <= -5e+20) {
		tmp = -t;
	} else if (t_1 <= 790.0) {
		tmp = Math.log((Math.pow(t, (a - 0.5)) * (z * y))) - t;
	} else {
		tmp = Math.log(t) * a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = ((math.log(z) + math.log((y + x))) - t) - ((0.5 - a) * math.log(t))
	tmp = 0
	if t_1 <= -5e+20:
		tmp = -t
	elif t_1 <= 790.0:
		tmp = math.log((math.pow(t, (a - 0.5)) * (z * y))) - t
	else:
		tmp = math.log(t) * a
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(log(z) + log(Float64(y + x))) - t) - Float64(Float64(0.5 - a) * log(t)))
	tmp = 0.0
	if (t_1 <= -5e+20)
		tmp = Float64(-t);
	elseif (t_1 <= 790.0)
		tmp = Float64(log(Float64((t ^ Float64(a - 0.5)) * Float64(z * y))) - t);
	else
		tmp = Float64(log(t) * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = ((log(z) + log((y + x))) - t) - ((0.5 - a) * log(t));
	tmp = 0.0;
	if (t_1 <= -5e+20)
		tmp = -t;
	elseif (t_1 <= 790.0)
		tmp = log(((t ^ (a - 0.5)) * (z * y))) - t;
	else
		tmp = log(t) * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(N[Log[z], $MachinePrecision] + N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] - N[(N[(0.5 - a), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+20], (-t), If[LessEqual[t$95$1, 790.0], N[(N[Log[N[(N[Power[t, N[(a - 0.5), $MachinePrecision]], $MachinePrecision] * N[(z * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision], N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < -5e20

   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 70.1

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

   5. Applied rewrites 70.1%

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

   if -5e20 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 790

   1. Initial program 99.4%

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

   3. Taylor expanded in t around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-rgt-in N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. div-add-rev N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-log.f64 N/A

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

      14. lower-neg.f64 99.3

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-log.f64 68.0

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

   8. Applied rewrites 68.0%

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

   10. Applied rewrites 57.8%

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

   if 790 < (+.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 99.6%

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

   3. Taylor expanded in a around 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 71.3

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

   5. Applied rewrites 71.3%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\log z + \log \left(y + x\right)\right) - t\right) - \left(0.5 - a\right) \cdot \log t \leq -5 \cdot 10^{+20}:\\ \;\;\;\;-t\\ \mathbf{elif}\;\left(\left(\log z + \log \left(y + x\right)\right) - t\right) - \left(0.5 - a\right) \cdot \log t \leq 790:\\ \;\;\;\;\log \left({t}^{\left(a - 0.5\right)} \cdot \left(z \cdot y\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log z + \log \left(y + x\right)\\ t_2 := \mathsf{fma}\left(\frac{\log \left(\frac{y + x}{z}\right)}{t}, t, -t\right) - \left(0.5 - a\right) \cdot \log t\\ \mathbf{if}\;t\_1 \leq -750:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 685:\\ \;\;\;\;\log \left(z \cdot \left(y + x\right)\right) - \left(t - \log t \cdot \left(a - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log z) (log (+ y x))))
        (t_2
         (- (fma (/ (log (/ (+ y x) z)) t) t (- t)) (* (- 0.5 a) (log t)))))
   (if (<= t_1 -750.0)
     t_2
     (if (<= t_1 685.0)
       (- (log (* z (+ y x))) (- t (* (log t) (- a 0.5))))
       t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log(z) + log((y + x));
	double t_2 = fma((log(((y + x) / z)) / t), t, -t) - ((0.5 - a) * log(t));
	double tmp;
	if (t_1 <= -750.0) {
		tmp = t_2;
	} else if (t_1 <= 685.0) {
		tmp = log((z * (y + x))) - (t - (log(t) * (a - 0.5)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(z) + log(Float64(y + x)))
	t_2 = Float64(fma(Float64(log(Float64(Float64(y + x) / z)) / t), t, Float64(-t)) - Float64(Float64(0.5 - a) * log(t)))
	tmp = 0.0
	if (t_1 <= -750.0)
		tmp = t_2;
	elseif (t_1 <= 685.0)
		tmp = Float64(log(Float64(z * Float64(y + x))) - Float64(t - Float64(log(t) * Float64(a - 0.5))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[z], $MachinePrecision] + N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Log[N[(N[(y + x), $MachinePrecision] / z), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision] * t + (-t)), $MachinePrecision] - N[(N[(0.5 - a), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -750.0], t$95$2, If[LessEqual[t$95$1, 685.0], N[(N[Log[N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(t - N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -750 or 685 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z))

   1. Initial program 99.8%

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

   3. Taylor expanded in t around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-rgt-in N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. div-add-rev N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-log.f64 N/A

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

      14. lower-neg.f64 99.8

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

   5. Applied rewrites 99.8%

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

   7. Applied rewrites 99.8%

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

   9. Applied rewrites 77.0%

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

   if -750 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 685

   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-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. sum-log N/A

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

      9. lower-log.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 99.8

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log z + \log \left(y + x\right) \leq -750:\\ \;\;\;\;\mathsf{fma}\left(\frac{\log \left(\frac{y + x}{z}\right)}{t}, t, -t\right) - \left(0.5 - a\right) \cdot \log t\\ \mathbf{elif}\;\log z + \log \left(y + x\right) \leq 685:\\ \;\;\;\;\log \left(z \cdot \left(y + x\right)\right) - \left(t - \log t \cdot \left(a - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\log \left(\frac{y + x}{z}\right)}{t}, t, -t\right) - \left(0.5 - a\right) \cdot \log t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log z + \log \left(y + x\right)\\ \mathbf{if}\;t\_1 \leq -750:\\ \;\;\;\;-t\\ \mathbf{elif}\;t\_1 \leq 705:\\ \;\;\;\;\log \left(z \cdot \left(y + x\right)\right) - \left(t - \log t \cdot \left(a - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log z) (log (+ y x)))))
   (if (<= t_1 -750.0)
     (- t)
     (if (<= t_1 705.0)
       (- (log (* z (+ y x))) (- t (* (log t) (- a 0.5))))
       (- t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log(z) + log((y + x));
	double tmp;
	if (t_1 <= -750.0) {
		tmp = -t;
	} else if (t_1 <= 705.0) {
		tmp = log((z * (y + x))) - (t - (log(t) * (a - 0.5)));
	} else {
		tmp = -t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = log(z) + log((y + x))
    if (t_1 <= (-750.0d0)) then
        tmp = -t
    else if (t_1 <= 705.0d0) then
        tmp = log((z * (y + x))) - (t - (log(t) * (a - 0.5d0)))
    else
        tmp = -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = Math.log(z) + Math.log((y + x));
	double tmp;
	if (t_1 <= -750.0) {
		tmp = -t;
	} else if (t_1 <= 705.0) {
		tmp = Math.log((z * (y + x))) - (t - (Math.log(t) * (a - 0.5)));
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = math.log(z) + math.log((y + x))
	tmp = 0
	if t_1 <= -750.0:
		tmp = -t
	elif t_1 <= 705.0:
		tmp = math.log((z * (y + x))) - (t - (math.log(t) * (a - 0.5)))
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(z) + log(Float64(y + x)))
	tmp = 0.0
	if (t_1 <= -750.0)
		tmp = Float64(-t);
	elseif (t_1 <= 705.0)
		tmp = Float64(log(Float64(z * Float64(y + x))) - Float64(t - Float64(log(t) * Float64(a - 0.5))));
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = log(z) + log((y + x));
	tmp = 0.0;
	if (t_1 <= -750.0)
		tmp = -t;
	elseif (t_1 <= 705.0)
		tmp = log((z * (y + x))) - (t - (log(t) * (a - 0.5)));
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[z], $MachinePrecision] + N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -750.0], (-t), If[LessEqual[t$95$1, 705.0], N[(N[Log[N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(t - N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-t)]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -750 or 705 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z))

   1. Initial program 99.8%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 43.5

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

   5. Applied rewrites 43.5%

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

   if -750 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 705

   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-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. sum-log N/A

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

      9. lower-log.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 99.8

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log z + \log \left(y + x\right) \leq -750:\\ \;\;\;\;-t\\ \mathbf{elif}\;\log z + \log \left(y + x\right) \leq 705:\\ \;\;\;\;\log \left(z \cdot \left(y + x\right)\right) - \left(t - \log t \cdot \left(a - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log z + \log \left(y + x\right)\\ \mathbf{if}\;t\_1 \leq -750:\\ \;\;\;\;-t\\ \mathbf{elif}\;t\_1 \leq 705:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log \left(z \cdot \left(y + x\right)\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log z) (log (+ y x)))))
   (if (<= t_1 -750.0)
     (- t)
     (if (<= t_1 705.0)
       (fma (- a 0.5) (log t) (- (log (* z (+ y x))) t))
       (- t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log(z) + log((y + x));
	double tmp;
	if (t_1 <= -750.0) {
		tmp = -t;
	} else if (t_1 <= 705.0) {
		tmp = fma((a - 0.5), log(t), (log((z * (y + x))) - t));
	} else {
		tmp = -t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(z) + log(Float64(y + x)))
	tmp = 0.0
	if (t_1 <= -750.0)
		tmp = Float64(-t);
	elseif (t_1 <= 705.0)
		tmp = fma(Float64(a - 0.5), log(t), Float64(log(Float64(z * Float64(y + x))) - t));
	else
		tmp = Float64(-t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[z], $MachinePrecision] + N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -750.0], (-t), If[LessEqual[t$95$1, 705.0], N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[(N[Log[N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], (-t)]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -750 or 705 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z))

   1. Initial program 99.8%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 43.5

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

   5. Applied rewrites 43.5%

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

   if -750 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 705

   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-+.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 99.7

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

      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 99.8

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

      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 99.8

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

   4. Applied rewrites 99.8%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log z + \log \left(y + x\right) \leq -750:\\ \;\;\;\;-t\\ \mathbf{elif}\;\log z + \log \left(y + x\right) \leq 705:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log \left(z \cdot \left(y + x\right)\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 58.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log z + \log \left(y + x\right)\\ \mathbf{if}\;t\_1 \leq -750:\\ \;\;\;\;-t\\ \mathbf{elif}\;t\_1 \leq 705:\\ \;\;\;\;\log \left(z \cdot y\right) - \left(t - \log t \cdot \left(a - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log z) (log (+ y x)))))
   (if (<= t_1 -750.0)
     (- t)
     (if (<= t_1 705.0) (- (log (* z y)) (- t (* (log t) (- a 0.5)))) (- t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log(z) + log((y + x));
	double tmp;
	if (t_1 <= -750.0) {
		tmp = -t;
	} else if (t_1 <= 705.0) {
		tmp = log((z * y)) - (t - (log(t) * (a - 0.5)));
	} else {
		tmp = -t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = log(z) + log((y + x))
    if (t_1 <= (-750.0d0)) then
        tmp = -t
    else if (t_1 <= 705.0d0) then
        tmp = log((z * y)) - (t - (log(t) * (a - 0.5d0)))
    else
        tmp = -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = Math.log(z) + Math.log((y + x));
	double tmp;
	if (t_1 <= -750.0) {
		tmp = -t;
	} else if (t_1 <= 705.0) {
		tmp = Math.log((z * y)) - (t - (Math.log(t) * (a - 0.5)));
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = math.log(z) + math.log((y + x))
	tmp = 0
	if t_1 <= -750.0:
		tmp = -t
	elif t_1 <= 705.0:
		tmp = math.log((z * y)) - (t - (math.log(t) * (a - 0.5)))
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(z) + log(Float64(y + x)))
	tmp = 0.0
	if (t_1 <= -750.0)
		tmp = Float64(-t);
	elseif (t_1 <= 705.0)
		tmp = Float64(log(Float64(z * y)) - Float64(t - Float64(log(t) * Float64(a - 0.5))));
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = log(z) + log((y + x));
	tmp = 0.0;
	if (t_1 <= -750.0)
		tmp = -t;
	elseif (t_1 <= 705.0)
		tmp = log((z * y)) - (t - (log(t) * (a - 0.5)));
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[z], $MachinePrecision] + N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -750.0], (-t), If[LessEqual[t$95$1, 705.0], N[(N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision] - N[(t - N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-t)]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -750 or 705 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z))

   1. Initial program 99.8%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 43.5

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

   5. Applied rewrites 43.5%

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

   if -750 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 705

   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-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. sum-log N/A

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

      9. lower-log.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 99.8

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 67.4

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

   7. Applied rewrites 67.4%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log z + \log \left(y + x\right) \leq -750:\\ \;\;\;\;-t\\ \mathbf{elif}\;\log z + \log \left(y + x\right) \leq 705:\\ \;\;\;\;\log \left(z \cdot y\right) - \left(t - \log t \cdot \left(a - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 58.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log z + \log \left(y + x\right)\\ \mathbf{if}\;t\_1 \leq -750:\\ \;\;\;\;-t\\ \mathbf{elif}\;t\_1 \leq 705:\\ \;\;\;\;\mathsf{fma}\left(-0.5 + a, \log t, \log \left(z \cdot y\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log z) (log (+ y x)))))
   (if (<= t_1 -750.0)
     (- t)
     (if (<= t_1 705.0) (- (fma (+ -0.5 a) (log t) (log (* z y))) t) (- t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log(z) + log((y + x));
	double tmp;
	if (t_1 <= -750.0) {
		tmp = -t;
	} else if (t_1 <= 705.0) {
		tmp = fma((-0.5 + a), log(t), log((z * y))) - t;
	} else {
		tmp = -t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(z) + log(Float64(y + x)))
	tmp = 0.0
	if (t_1 <= -750.0)
		tmp = Float64(-t);
	elseif (t_1 <= 705.0)
		tmp = Float64(fma(Float64(-0.5 + a), log(t), log(Float64(z * y))) - t);
	else
		tmp = Float64(-t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[z], $MachinePrecision] + N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -750.0], (-t), If[LessEqual[t$95$1, 705.0], N[(N[(N[(-0.5 + a), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], (-t)]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -750 or 705 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z))

   1. Initial program 99.8%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 43.5

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

   5. Applied rewrites 43.5%

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

   if -750 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 705

   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-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. sum-log N/A

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

      9. lower-log.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 99.8

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. distribute-rgt-in N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-out N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 67.4

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

   7. Applied rewrites 67.4%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log z + \log \left(y + x\right) \leq -750:\\ \;\;\;\;-t\\ \mathbf{elif}\;\log z + \log \left(y + x\right) \leq 705:\\ \;\;\;\;\mathsf{fma}\left(-0.5 + a, \log t, \log \left(z \cdot y\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in t around inf

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. distribute-rgt-in N/A

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

   4. mul-1-neg N/A

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

   5. lower-fma.f64 N/A

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

   6. div-add-rev N/A

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

   7. lower-/.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 N/A

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

   10. lower-log.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 N/A

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

   13. lower-log.f64 N/A

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

   14. lower-neg.f64 99.7

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

5. Applied rewrites 99.7%

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

6. Taylor expanded in x around 0

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

   1. lower--.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-log.f64 N/A

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

   7. lower--.f64 N/A

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

   8. lower-log.f64 N/A

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

   9. lower-log.f64 68.6

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

8. Applied rewrites 68.6%

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

9. Final simplification 68.6%

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

Alternative 10: 69.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

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

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

   6. lower-fma.f64 N/A

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

   7. lower--.f64 N/A

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

   8. lower-log.f64 N/A

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

   9. lower-log.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lower-log.f64 68.6

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

5. Applied rewrites 68.6%

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

6. Final simplification 68.6%

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

Alternative 11: 62.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 4000000000000:\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 4000000000000.0) (* (log t) a) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 4000000000000.0) {
		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 <= 4000000000000.0d0) 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 <= 4000000000000.0) {
		tmp = Math.log(t) * a;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 4000000000000.0:
		tmp = math.log(t) * a
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 4000000000000.0)
		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 <= 4000000000000.0)
		tmp = log(t) * a;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 4000000000000.0], N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if t < 4e12

   1. Initial program 99.5%

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

   3. 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 55.1

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

   5. Applied rewrites 55.1%

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

   if 4e12 < 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 82.9

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

   5. Applied rewrites 82.9%

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

Alternative 12: 38.3% 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 99.7%

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

3. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 42.6

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

5. Applied rewrites 42.6%

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

Alternative 13: 2.4% accurate, 321.0× 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 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 99.7%

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

3. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 42.6

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

5. Applied rewrites 42.6%

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

7. Applied rewrites 17.9%

   \[\leadsto \frac{0 - t \cdot t}{\color{blue}{0 + t}} \]
8. Step-by-step derivation

9. Applied rewrites 2.1%

   \[\leadsto t \]
10. 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 2024298 
(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))))