Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 14.3s

Alternatives: 15

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Final simplification 99.4%

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

Alternative 2: 81.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log t \cdot \left(a - 0.5\right) + \left(\left(\log z + \log \left(y + x\right)\right) - t\right)\\ t_2 := \log t \cdot a\\ \mathbf{if}\;t\_1 \leq -2000:\\ \;\;\;\;t\_2 - \left(t - \log z\right)\\ \mathbf{elif}\;t\_1 \leq 2000:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log t, \log z\right) + \log y\\ \mathbf{else}:\\ \;\;\;\;\left(t\_2 + \log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (* (log t) (- a 0.5)) (- (+ (log z) (log (+ y x))) t)))
        (t_2 (* (log t) a)))
   (if (<= t_1 -2000.0)
     (- t_2 (- t (log z)))
     (if (<= t_1 2000.0)
       (+ (fma -0.5 (log t) (log z)) (log y))
       (- (+ t_2 (log y)) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (log(t) * (a - 0.5)) + ((log(z) + log((y + x))) - t);
	double t_2 = log(t) * a;
	double tmp;
	if (t_1 <= -2000.0) {
		tmp = t_2 - (t - log(z));
	} else if (t_1 <= 2000.0) {
		tmp = fma(-0.5, log(t), log(z)) + log(y);
	} else {
		tmp = (t_2 + log(y)) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(log(t) * Float64(a - 0.5)) + Float64(Float64(log(z) + log(Float64(y + x))) - t))
	t_2 = Float64(log(t) * a)
	tmp = 0.0
	if (t_1 <= -2000.0)
		tmp = Float64(t_2 - Float64(t - log(z)));
	elseif (t_1 <= 2000.0)
		tmp = Float64(fma(-0.5, log(t), log(z)) + log(y));
	else
		tmp = Float64(Float64(t_2 + log(y)) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Log[z], $MachinePrecision] + N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[t$95$1, -2000.0], N[(t$95$2 - N[(t - N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2000.0], N[(N[(-0.5 * N[Log[t], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 + N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $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))) < -2e3

   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 y around inf

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

   5. Applied rewrites 55.5%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 98.3%

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

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

   1. Initial program 98.5%

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

   3. Taylor expanded in y around inf

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

      1. associate-+r+ N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. remove-double-neg N/A

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

      7. associate--l+ N/A

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

      8. associate-+r+ N/A

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

      9. lower--.f64 N/A

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

   5. Applied rewrites 46.4%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 45.4%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 44.4%

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

   if 2e3 < (+.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 99.7%

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

   3. Taylor expanded in y around inf

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

      1. associate-+r+ N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. remove-double-neg N/A

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

      7. associate--l+ N/A

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

      8. associate-+r+ N/A

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

      9. lower--.f64 N/A

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

   5. Applied rewrites 84.5%

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

   6. Taylor expanded in a around inf

      \[\leadsto \left(\log y + a \cdot \log t\right) - t \]
   7. Step-by-step derivation

   8. Applied rewrites 83.4%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log t \cdot \left(a - 0.5\right) + \left(\left(\log z + \log \left(y + x\right)\right) - t\right) \leq -2000:\\ \;\;\;\;\log t \cdot a - \left(t - \log z\right)\\ \mathbf{elif}\;\log t \cdot \left(a - 0.5\right) + \left(\left(\log z + \log \left(y + x\right)\right) - t\right) \leq 2000:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log t, \log z\right) + \log y\\ \mathbf{else}:\\ \;\;\;\;\left(\log t \cdot a + \log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log t \cdot \left(a - 0.5\right) + \left(\left(\log z + \log \left(y + x\right)\right) - t\right)\\ t_2 := \log t \cdot a\\ \mathbf{if}\;t\_1 \leq -500000000000:\\ \;\;\;\;t\_2 - \left(t - \log z\right)\\ \mathbf{elif}\;t\_1 \leq 1040:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log t, \log \left(z \cdot y\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(t\_2 + \log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (* (log t) (- a 0.5)) (- (+ (log z) (log (+ y x))) t)))
        (t_2 (* (log t) a)))
   (if (<= t_1 -500000000000.0)
     (- t_2 (- t (log z)))
     (if (<= t_1 1040.0)
       (- (fma -0.5 (log t) (log (* z y))) t)
       (- (+ t_2 (log y)) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (log(t) * (a - 0.5)) + ((log(z) + log((y + x))) - t);
	double t_2 = log(t) * a;
	double tmp;
	if (t_1 <= -500000000000.0) {
		tmp = t_2 - (t - log(z));
	} else if (t_1 <= 1040.0) {
		tmp = fma(-0.5, log(t), log((z * y))) - t;
	} else {
		tmp = (t_2 + log(y)) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(log(t) * Float64(a - 0.5)) + Float64(Float64(log(z) + log(Float64(y + x))) - t))
	t_2 = Float64(log(t) * a)
	tmp = 0.0
	if (t_1 <= -500000000000.0)
		tmp = Float64(t_2 - Float64(t - log(z)));
	elseif (t_1 <= 1040.0)
		tmp = Float64(fma(-0.5, log(t), log(Float64(z * y))) - t);
	else
		tmp = Float64(Float64(t_2 + log(y)) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Log[z], $MachinePrecision] + N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[t$95$1, -500000000000.0], N[(t$95$2 - N[(t - N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1040.0], N[(N[(-0.5 * N[Log[t], $MachinePrecision] + N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(t$95$2 + N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $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))) < -5e11

   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 y around inf

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

   5. Applied rewrites 55.6%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 99.3%

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

   if -5e11 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 1040

   1. Initial program 98.3%

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

   3. Taylor expanded in y around inf

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

      1. associate-+r+ N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. remove-double-neg N/A

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

      7. associate--l+ N/A

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

      8. associate-+r+ N/A

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

      9. lower--.f64 N/A

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

   5. Applied rewrites 45.5%

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

   7. Applied rewrites 42.2%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 40.8%

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

   if 1040 < (+.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 99.7%

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

   3. Taylor expanded in y around inf

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

      1. associate-+r+ N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. remove-double-neg N/A

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

      7. associate--l+ N/A

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

      8. associate-+r+ N/A

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

      9. lower--.f64 N/A

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

   5. Applied rewrites 79.4%

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

   6. Taylor expanded in a around inf

      \[\leadsto \left(\log y + a \cdot \log t\right) - t \]
   7. Step-by-step derivation

   8. Applied rewrites 70.5%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log t \cdot \left(a - 0.5\right) + \left(\left(\log z + \log \left(y + x\right)\right) - t\right) \leq -500000000000:\\ \;\;\;\;\log t \cdot a - \left(t - \log z\right)\\ \mathbf{elif}\;\log t \cdot \left(a - 0.5\right) + \left(\left(\log z + \log \left(y + x\right)\right) - t\right) \leq 1040:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log t, \log \left(z \cdot y\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log t \cdot a + \log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log t \cdot \left(a - 0.5\right) + \left(\left(\log z + \log \left(y + x\right)\right) - t\right)\\ t_2 := \log t \cdot a\\ \mathbf{if}\;t\_1 \leq -2000000000000:\\ \;\;\;\;t\_2 - \left(t - \log z\right)\\ \mathbf{elif}\;t\_1 \leq 1040:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log \left(z \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_2 + \log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (* (log t) (- a 0.5)) (- (+ (log z) (log (+ y x))) t)))
        (t_2 (* (log t) a)))
   (if (<= t_1 -2000000000000.0)
     (- t_2 (- t (log z)))
     (if (<= t_1 1040.0)
       (fma (- a 0.5) (log t) (log (* z y)))
       (- (+ t_2 (log y)) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (log(t) * (a - 0.5)) + ((log(z) + log((y + x))) - t);
	double t_2 = log(t) * a;
	double tmp;
	if (t_1 <= -2000000000000.0) {
		tmp = t_2 - (t - log(z));
	} else if (t_1 <= 1040.0) {
		tmp = fma((a - 0.5), log(t), log((z * y)));
	} else {
		tmp = (t_2 + log(y)) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(log(t) * Float64(a - 0.5)) + Float64(Float64(log(z) + log(Float64(y + x))) - t))
	t_2 = Float64(log(t) * a)
	tmp = 0.0
	if (t_1 <= -2000000000000.0)
		tmp = Float64(t_2 - Float64(t - log(z)));
	elseif (t_1 <= 1040.0)
		tmp = fma(Float64(a - 0.5), log(t), log(Float64(z * y)));
	else
		tmp = Float64(Float64(t_2 + log(y)) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Log[z], $MachinePrecision] + N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[t$95$1, -2000000000000.0], N[(t$95$2 - N[(t - N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1040.0], N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 + N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $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))) < -2e12

   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 y around inf

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

   5. Applied rewrites 55.3%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 99.6%

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

   if -2e12 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 1040

   1. Initial program 98.4%

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

   3. Taylor expanded in y around inf

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

      1. associate-+r+ N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. remove-double-neg N/A

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

      7. associate--l+ N/A

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

      8. associate-+r+ N/A

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

      9. lower--.f64 N/A

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

   5. Applied rewrites 46.3%

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

   7. Applied rewrites 43.1%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 40.7%

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

   if 1040 < (+.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 99.7%

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

   3. Taylor expanded in y around inf

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

      1. associate-+r+ N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. remove-double-neg N/A

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

      7. associate--l+ N/A

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

      8. associate-+r+ N/A

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

      9. lower--.f64 N/A

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

   5. Applied rewrites 79.4%

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

   6. Taylor expanded in a around inf

      \[\leadsto \left(\log y + a \cdot \log t\right) - t \]
   7. Step-by-step derivation

   8. Applied rewrites 70.5%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log t \cdot \left(a - 0.5\right) + \left(\left(\log z + \log \left(y + x\right)\right) - t\right) \leq -2000000000000:\\ \;\;\;\;\log t \cdot a - \left(t - \log z\right)\\ \mathbf{elif}\;\log t \cdot \left(a - 0.5\right) + \left(\left(\log z + \log \left(y + x\right)\right) - t\right) \leq 1040:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log \left(z \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log t \cdot a + \log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log t \cdot a\\ t_2 := \log z + \log \left(y + x\right)\\ \mathbf{if}\;t\_2 \leq -800:\\ \;\;\;\;t\_1 - \left(t - \log z\right)\\ \mathbf{elif}\;t\_2 \leq 710:\\ \;\;\;\;\mathsf{fma}\left(\log t, a - 0.5, \log \left(\left(y + x\right) \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 + \log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (log t) a)) (t_2 (+ (log z) (log (+ y x)))))
   (if (<= t_2 -800.0)
     (- t_1 (- t (log z)))
     (if (<= t_2 710.0)
       (- (fma (log t) (- a 0.5) (log (* (+ y x) z))) t)
       (- (+ t_1 (log y)) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log(t) * a;
	double t_2 = log(z) + log((y + x));
	double tmp;
	if (t_2 <= -800.0) {
		tmp = t_1 - (t - log(z));
	} else if (t_2 <= 710.0) {
		tmp = fma(log(t), (a - 0.5), log(((y + x) * z))) - t;
	} else {
		tmp = (t_1 + log(y)) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(t) * a)
	t_2 = Float64(log(z) + log(Float64(y + x)))
	tmp = 0.0
	if (t_2 <= -800.0)
		tmp = Float64(t_1 - Float64(t - log(z)));
	elseif (t_2 <= 710.0)
		tmp = Float64(fma(log(t), Float64(a - 0.5), log(Float64(Float64(y + x) * z))) - t);
	else
		tmp = Float64(Float64(t_1 + log(y)) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]}, Block[{t$95$2 = N[(N[Log[z], $MachinePrecision] + N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -800.0], N[(t$95$1 - N[(t - N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 710.0], N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision] + N[Log[N[(N[(y + x), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(t$95$1 + N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -800

   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 y around inf

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

   5. Applied rewrites 50.6%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 86.4%

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

   if -800 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 710

   1. Initial program 99.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+r- N/A

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

      5. lower--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 99.3

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

      9. lift-+.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. sum-log N/A

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

      13. lower-log.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 99.5

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 99.5

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

   4. Applied rewrites 99.5%

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

   if 710 < (+.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 y around inf

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

      1. associate-+r+ N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. remove-double-neg N/A

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

      7. associate--l+ N/A

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

      8. associate-+r+ N/A

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

      9. lower--.f64 N/A

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

   5. Applied rewrites 66.9%

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

   6. Taylor expanded in a around inf

      \[\leadsto \left(\log y + a \cdot \log t\right) - t \]
   7. Step-by-step derivation

   8. Applied rewrites 57.6%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log z + \log \left(y + x\right) \leq -800:\\ \;\;\;\;\log t \cdot a - \left(t - \log z\right)\\ \mathbf{elif}\;\log z + \log \left(y + x\right) \leq 710:\\ \;\;\;\;\mathsf{fma}\left(\log t, a - 0.5, \log \left(\left(y + x\right) \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log t \cdot a + \log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log t \cdot a\\ t_2 := \log z + \log \left(y + x\right)\\ \mathbf{if}\;t\_2 \leq -800:\\ \;\;\;\;t\_1 - \left(t - \log z\right)\\ \mathbf{elif}\;t\_2 \leq 710:\\ \;\;\;\;\left(\log \left(z \cdot y\right) + \log t \cdot \left(a - 0.5\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 + \log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (log t) a)) (t_2 (+ (log z) (log (+ y x)))))
   (if (<= t_2 -800.0)
     (- t_1 (- t (log z)))
     (if (<= t_2 710.0)
       (- (+ (log (* z y)) (* (log t) (- a 0.5))) t)
       (- (+ t_1 (log y)) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log(t) * a;
	double t_2 = log(z) + log((y + x));
	double tmp;
	if (t_2 <= -800.0) {
		tmp = t_1 - (t - log(z));
	} else if (t_2 <= 710.0) {
		tmp = (log((z * y)) + (log(t) * (a - 0.5))) - t;
	} else {
		tmp = (t_1 + log(y)) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = log(t) * a
    t_2 = log(z) + log((y + x))
    if (t_2 <= (-800.0d0)) then
        tmp = t_1 - (t - log(z))
    else if (t_2 <= 710.0d0) then
        tmp = (log((z * y)) + (log(t) * (a - 0.5d0))) - t
    else
        tmp = (t_1 + log(y)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = Math.log(t) * a;
	double t_2 = Math.log(z) + Math.log((y + x));
	double tmp;
	if (t_2 <= -800.0) {
		tmp = t_1 - (t - Math.log(z));
	} else if (t_2 <= 710.0) {
		tmp = (Math.log((z * y)) + (Math.log(t) * (a - 0.5))) - t;
	} else {
		tmp = (t_1 + Math.log(y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = math.log(t) * a
	t_2 = math.log(z) + math.log((y + x))
	tmp = 0
	if t_2 <= -800.0:
		tmp = t_1 - (t - math.log(z))
	elif t_2 <= 710.0:
		tmp = (math.log((z * y)) + (math.log(t) * (a - 0.5))) - t
	else:
		tmp = (t_1 + math.log(y)) - t
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(t) * a)
	t_2 = Float64(log(z) + log(Float64(y + x)))
	tmp = 0.0
	if (t_2 <= -800.0)
		tmp = Float64(t_1 - Float64(t - log(z)));
	elseif (t_2 <= 710.0)
		tmp = Float64(Float64(log(Float64(z * y)) + Float64(log(t) * Float64(a - 0.5))) - t);
	else
		tmp = Float64(Float64(t_1 + log(y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = log(t) * a;
	t_2 = log(z) + log((y + x));
	tmp = 0.0;
	if (t_2 <= -800.0)
		tmp = t_1 - (t - log(z));
	elseif (t_2 <= 710.0)
		tmp = (log((z * y)) + (log(t) * (a - 0.5))) - t;
	else
		tmp = (t_1 + log(y)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]}, Block[{t$95$2 = N[(N[Log[z], $MachinePrecision] + N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -800.0], N[(t$95$1 - N[(t - N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 710.0], N[(N[(N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(t$95$1 + N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -800

   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 y around inf

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

   5. Applied rewrites 50.6%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 86.4%

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

   if -800 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 710

   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 y around inf

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

      1. associate-+r+ N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. remove-double-neg N/A

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

      7. associate--l+ N/A

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

      8. associate-+r+ N/A

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

      9. lower--.f64 N/A

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

   5. Applied rewrites 69.7%

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

   7. Applied rewrites 65.8%

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

   if 710 < (+.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 y around inf

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

      1. associate-+r+ N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. remove-double-neg N/A

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

      7. associate--l+ N/A

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

      8. associate-+r+ N/A

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

      9. lower--.f64 N/A

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

   5. Applied rewrites 66.9%

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

   6. Taylor expanded in a around inf

      \[\leadsto \left(\log y + a \cdot \log t\right) - t \]
   7. Step-by-step derivation

   8. Applied rewrites 57.6%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log z + \log \left(y + x\right) \leq -800:\\ \;\;\;\;\log t \cdot a - \left(t - \log z\right)\\ \mathbf{elif}\;\log z + \log \left(y + x\right) \leq 710:\\ \;\;\;\;\left(\log \left(z \cdot y\right) + \log t \cdot \left(a - 0.5\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log t \cdot a + \log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log t \cdot a\\ t_2 := \log z + \log \left(y + x\right)\\ \mathbf{if}\;t\_2 \leq -800:\\ \;\;\;\;t\_1 - \left(t - \log z\right)\\ \mathbf{elif}\;t\_2 \leq 710:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log \left(z \cdot y\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 + \log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (log t) a)) (t_2 (+ (log z) (log (+ y x)))))
   (if (<= t_2 -800.0)
     (- t_1 (- t (log z)))
     (if (<= t_2 710.0)
       (- (fma (- a 0.5) (log t) (log (* z y))) t)
       (- (+ t_1 (log y)) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log(t) * a;
	double t_2 = log(z) + log((y + x));
	double tmp;
	if (t_2 <= -800.0) {
		tmp = t_1 - (t - log(z));
	} else if (t_2 <= 710.0) {
		tmp = fma((a - 0.5), log(t), log((z * y))) - t;
	} else {
		tmp = (t_1 + log(y)) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(t) * a)
	t_2 = Float64(log(z) + log(Float64(y + x)))
	tmp = 0.0
	if (t_2 <= -800.0)
		tmp = Float64(t_1 - Float64(t - log(z)));
	elseif (t_2 <= 710.0)
		tmp = Float64(fma(Float64(a - 0.5), log(t), log(Float64(z * y))) - t);
	else
		tmp = Float64(Float64(t_1 + log(y)) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]}, Block[{t$95$2 = N[(N[Log[z], $MachinePrecision] + N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -800.0], N[(t$95$1 - N[(t - N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 710.0], N[(N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(t$95$1 + N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -800

   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 y around inf

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

   5. Applied rewrites 50.6%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 86.4%

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

   if -800 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 710

   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 y around inf

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

      1. associate-+r+ N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. remove-double-neg N/A

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

      7. associate--l+ N/A

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

      8. associate-+r+ N/A

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

      9. lower--.f64 N/A

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

   5. Applied rewrites 69.7%

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

   7. Applied rewrites 65.8%

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

   if 710 < (+.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 y around inf

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

      1. associate-+r+ N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. remove-double-neg N/A

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

      7. associate--l+ N/A

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

      8. associate-+r+ N/A

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

      9. lower--.f64 N/A

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

   5. Applied rewrites 66.9%

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

   6. Taylor expanded in a around inf

      \[\leadsto \left(\log y + a \cdot \log t\right) - t \]
   7. Step-by-step derivation

   8. Applied rewrites 57.6%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log z + \log \left(y + x\right) \leq -800:\\ \;\;\;\;\log t \cdot a - \left(t - \log z\right)\\ \mathbf{elif}\;\log z + \log \left(y + x\right) \leq 710:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log \left(z \cdot y\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log t \cdot a + \log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 0.245:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log \left(y + x\right)\right) + \log z\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) + \log t \cdot \left(a - 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 0.245)
   (+ (fma (- a 0.5) (log t) (log (+ y x))) (log z))
   (+ (- t) (* (log t) (- a 0.5)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 0.245) {
		tmp = fma((a - 0.5), log(t), log((y + x))) + log(z);
	} else {
		tmp = -t + (log(t) * (a - 0.5));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 0.245)
		tmp = Float64(fma(Float64(a - 0.5), log(t), log(Float64(y + x))) + log(z));
	else
		tmp = Float64(Float64(-t) + Float64(log(t) * Float64(a - 0.5)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 0.245], N[(N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision], N[((-t) + N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 0.245

   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 t around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-log.f64 98.0

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

   5. Applied rewrites 98.0%

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

   if 0.245 < 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

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 99.1

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

   5. Applied rewrites 99.1%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 0.245:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log \left(y + x\right)\right) + \log z\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) + \log t \cdot \left(a - 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 0.245:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log z\right) + \log y\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) + \log t \cdot \left(a - 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 0.245)
   (+ (fma (- a 0.5) (log t) (log z)) (log y))
   (+ (- t) (* (log t) (- a 0.5)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 0.245) {
		tmp = fma((a - 0.5), log(t), log(z)) + log(y);
	} else {
		tmp = -t + (log(t) * (a - 0.5));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 0.245)
		tmp = Float64(fma(Float64(a - 0.5), log(t), log(z)) + log(y));
	else
		tmp = Float64(Float64(-t) + Float64(log(t) * Float64(a - 0.5)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 0.245], N[(N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision], N[((-t) + N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 0.245

   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 y around inf

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

      1. associate-+r+ N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. remove-double-neg N/A

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

      7. associate--l+ N/A

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

      8. associate-+r+ N/A

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

      9. lower--.f64 N/A

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

   5. Applied rewrites 60.1%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 59.5%

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

   if 0.245 < 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

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 99.1

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

   5. Applied rewrites 99.1%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 0.245:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log z\right) + \log y\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) + \log t \cdot \left(a - 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 68.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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 y around inf

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

   1. associate-+r+ N/A

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

   2. associate--l+ N/A

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

   3. +-commutative N/A

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

   4. mul-1-neg N/A

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

   5. log-rec N/A

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

   6. remove-double-neg N/A

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

   7. associate--l+ N/A

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

   8. associate-+r+ N/A

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

   9. lower--.f64 N/A

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

5. Applied rewrites 68.3%

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

7. Applied rewrites 52.5%

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

9. Applied rewrites 68.3%

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

10. Final simplification 68.3%

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

Alternative 11: 68.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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 y around inf

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

   1. associate-+r+ N/A

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

   2. associate--l+ N/A

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

   3. +-commutative N/A

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

   4. mul-1-neg N/A

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

   5. log-rec N/A

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

   6. remove-double-neg N/A

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

   7. associate--l+ N/A

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

   8. associate-+r+ N/A

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

   9. lower--.f64 N/A

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

5. Applied rewrites 68.3%

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

6. Final simplification 68.3%

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

Alternative 12: 61.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log t \cdot a\\ \mathbf{if}\;a - 0.5 \leq -2 \cdot 10^{+19}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a - 0.5 \leq -0.4:\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (log t) a)))
   (if (<= (- a 0.5) -2e+19) t_1 (if (<= (- a 0.5) -0.4) (- t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log(t) * a;
	double tmp;
	if ((a - 0.5) <= -2e+19) {
		tmp = t_1;
	} else if ((a - 0.5) <= -0.4) {
		tmp = -t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = log(t) * a
    if ((a - 0.5d0) <= (-2d+19)) then
        tmp = t_1
    else if ((a - 0.5d0) <= (-0.4d0)) then
        tmp = -t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = Math.log(t) * a;
	double tmp;
	if ((a - 0.5) <= -2e+19) {
		tmp = t_1;
	} else if ((a - 0.5) <= -0.4) {
		tmp = -t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = math.log(t) * a
	tmp = 0
	if (a - 0.5) <= -2e+19:
		tmp = t_1
	elif (a - 0.5) <= -0.4:
		tmp = -t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(t) * a)
	tmp = 0.0
	if (Float64(a - 0.5) <= -2e+19)
		tmp = t_1;
	elseif (Float64(a - 0.5) <= -0.4)
		tmp = Float64(-t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = log(t) * a;
	tmp = 0.0;
	if ((a - 0.5) <= -2e+19)
		tmp = t_1;
	elseif ((a - 0.5) <= -0.4)
		tmp = -t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[N[(a - 0.5), $MachinePrecision], -2e+19], t$95$1, If[LessEqual[N[(a - 0.5), $MachinePrecision], -0.4], (-t), t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 a #s(literal 1/2 binary64)) < -2e19 or -0.40000000000000002 < (-.f64 a #s(literal 1/2 binary64))

   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 a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 80.1

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

   5. Applied rewrites 80.1%

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

   if -2e19 < (-.f64 a #s(literal 1/2 binary64)) < -0.40000000000000002

   1. Initial program 99.2%

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

   3. 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 48.2

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

   5. Applied rewrites 48.2%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -2 \cdot 10^{+19}:\\ \;\;\;\;\log t \cdot a\\ \mathbf{elif}\;a - 0.5 \leq -0.4:\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 76.6% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 74.5

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

5. Applied rewrites 74.5%

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

6. Final simplification 74.5%

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

Alternative 14: 73.9% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in y around inf

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

   1. associate-+r+ N/A

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

   2. associate--l+ N/A

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

   3. +-commutative N/A

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

   4. mul-1-neg N/A

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

   5. log-rec N/A

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

   6. remove-double-neg N/A

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

   7. associate--l+ N/A

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

   8. associate-+r+ N/A

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

   9. lower--.f64 N/A

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

5. Applied rewrites 68.3%

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

6. Taylor expanded in a around inf

   \[\leadsto a \cdot \log t - t \]
7. Step-by-step derivation

8. Applied rewrites 71.2%

   \[\leadsto a \cdot \log t - t \]

9. Final simplification 71.2%

   \[\leadsto \log t \cdot a - t \]
10. Add Preprocessing

Alternative 15: 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 99.4%

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

3. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 35.2

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

5. Applied rewrites 35.2%

   \[\leadsto \color{blue}{-t} \]
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 2024243 
(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))))