Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 13.2s

Alternatives: 11

Speedup: N/A×

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

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

FPCore

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

C

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

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((x + y)) - (t - log(z)))
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((x + y)) - (t - Math.log(z)));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate--l+ N/A

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

   4. +-commutative N/A

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

   5. lower-+.f64 N/A

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

   6. lower--.f64 99.6

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Final simplification 99.6%

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

Alternative 2: 92.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log t \cdot \left(a - 0.5\right)\\ t_2 := \log \left(x + y\right)\\ t_3 := \left(\left(t\_2 + \log z\right) - t\right) + t\_1\\ \mathbf{if}\;t\_3 \leq -5 \cdot 10^{+19}:\\ \;\;\;\;\log t \cdot a + \left(-t\right)\\ \mathbf{elif}\;t\_3 \leq 1005:\\ \;\;\;\;\log \left(\left(x + y\right) \cdot z\right) - \mathsf{fma}\left(0.5, \log t, t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-t\right) + t\_2\right) + t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (log t) (- a 0.5)))
        (t_2 (log (+ x y)))
        (t_3 (+ (- (+ t_2 (log z)) t) t_1)))
   (if (<= t_3 -5e+19)
     (+ (* (log t) a) (- t))
     (if (<= t_3 1005.0)
       (- (log (* (+ x y) z)) (fma 0.5 (log t) t))
       (+ (+ (- t) t_2) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log(t) * (a - 0.5);
	double t_2 = log((x + y));
	double t_3 = ((t_2 + log(z)) - t) + t_1;
	double tmp;
	if (t_3 <= -5e+19) {
		tmp = (log(t) * a) + -t;
	} else if (t_3 <= 1005.0) {
		tmp = log(((x + y) * z)) - fma(0.5, log(t), t);
	} else {
		tmp = (-t + t_2) + t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(t) * Float64(a - 0.5))
	t_2 = log(Float64(x + y))
	t_3 = Float64(Float64(Float64(t_2 + log(z)) - t) + t_1)
	tmp = 0.0
	if (t_3 <= -5e+19)
		tmp = Float64(Float64(log(t) * a) + Float64(-t));
	elseif (t_3 <= 1005.0)
		tmp = Float64(log(Float64(Float64(x + y) * z)) - fma(0.5, log(t), t));
	else
		tmp = Float64(Float64(Float64(-t) + t_2) + t_1);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t$95$2 + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + t$95$1), $MachinePrecision]}, If[LessEqual[t$95$3, -5e+19], N[(N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision] + (-t)), $MachinePrecision], If[LessEqual[t$95$3, 1005.0], N[(N[Log[N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision] - N[(0.5 * N[Log[t], $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision], N[(N[((-t) + t$95$2), $MachinePrecision] + t$95$1), $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))) < -5e19

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. flip3-- N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. clear-num N/A

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

      9. flip3-- N/A

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

      10. lift--.f64 N/A

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

      11. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 99.8

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 99.9

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

   10. Applied rewrites 99.9%

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

   if -5e19 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 1005

   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. 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

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower--.f64 99.0

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 99.0

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

   4. Applied rewrites 99.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. flip-- N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lift--.f64 N/A

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

      9. associate-*l/ N/A

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

      10. associate-*r/ N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift--.f64 N/A

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

      14. associate-+r- N/A

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-+.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. lift--.f64 N/A

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

   6. Applied rewrites 91.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(a, a, -0.25\right), \frac{\log t}{a - -0.5}, \log \left(\left(x + y\right) \cdot z\right) - t\right)} \]

   7. Taylor expanded in a around 0

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

      1. associate--l+ N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. distribute-lft-neg-in N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-log.f64 88.5

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

   9. Applied rewrites 88.5%

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

   if 1005 < (+.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 99.5%

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

      1. 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

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower--.f64 99.5

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 89.0

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

   7. Applied rewrites 89.0%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \log t \cdot \left(a - 0.5\right) \leq -5 \cdot 10^{+19}:\\ \;\;\;\;\log t \cdot a + \left(-t\right)\\ \mathbf{elif}\;\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \log t \cdot \left(a - 0.5\right) \leq 1005:\\ \;\;\;\;\log \left(\left(x + y\right) \cdot z\right) - \mathsf{fma}\left(0.5, \log t, t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-t\right) + \log \left(x + y\right)\right) + \log t \cdot \left(a - 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \log t \cdot \left(a - 0.5\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+19}:\\ \;\;\;\;\log t \cdot a + \left(-t\right)\\ \mathbf{elif}\;t\_1 \leq 1005:\\ \;\;\;\;\log \left(\left(x + y\right) \cdot z\right) - \mathsf{fma}\left(0.5, \log t, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, -t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- (+ (log (+ x y)) (log z)) t) (* (log t) (- a 0.5)))))
   (if (<= t_1 -5e+19)
     (+ (* (log t) a) (- t))
     (if (<= t_1 1005.0)
       (- (log (* (+ x y) z)) (fma 0.5 (log t) t))
       (fma (- a 0.5) (log t) (- t))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((log((x + y)) + log(z)) - t) + (log(t) * (a - 0.5));
	double tmp;
	if (t_1 <= -5e+19) {
		tmp = (log(t) * a) + -t;
	} else if (t_1 <= 1005.0) {
		tmp = log(((x + y) * z)) - fma(0.5, log(t), t);
	} else {
		tmp = fma((a - 0.5), log(t), -t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + Float64(log(t) * Float64(a - 0.5)))
	tmp = 0.0
	if (t_1 <= -5e+19)
		tmp = Float64(Float64(log(t) * a) + Float64(-t));
	elseif (t_1 <= 1005.0)
		tmp = Float64(log(Float64(Float64(x + y) * z)) - fma(0.5, log(t), t));
	else
		tmp = fma(Float64(a - 0.5), log(t), Float64(-t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+19], N[(N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision] + (-t)), $MachinePrecision], If[LessEqual[t$95$1, 1005.0], N[(N[Log[N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision] - N[(0.5 * N[Log[t], $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision], N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $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))) < -5e19

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. flip3-- N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. clear-num N/A

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

      9. flip3-- N/A

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

      10. lift--.f64 N/A

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

      11. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 99.8

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 99.9

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

   10. Applied rewrites 99.9%

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

   if -5e19 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 1005

   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. 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

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower--.f64 99.0

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 99.0

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

   4. Applied rewrites 99.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. flip-- N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lift--.f64 N/A

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

      9. associate-*l/ N/A

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

      10. associate-*r/ N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift--.f64 N/A

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

      14. associate-+r- N/A

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-+.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. lift--.f64 N/A

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

   6. Applied rewrites 91.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(a, a, -0.25\right), \frac{\log t}{a - -0.5}, \log \left(\left(x + y\right) \cdot z\right) - t\right)} \]

   7. Taylor expanded in a around 0

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

      1. associate--l+ N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. distribute-lft-neg-in N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-log.f64 88.5

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

   9. Applied rewrites 88.5%

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

   if 1005 < (+.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 99.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. flip3-- N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. clear-num N/A

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

      9. flip3-- N/A

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

      10. lift--.f64 N/A

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

      11. lower-/.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 88.8

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

   7. Applied rewrites 88.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. associate-/r/ N/A

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

      6. lift-/.f64 N/A

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

      7. remove-double-div N/A

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

      8. lower-fma.f64 88.8

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

   9. Applied rewrites 88.8%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \log t \cdot \left(a - 0.5\right) \leq -5 \cdot 10^{+19}:\\ \;\;\;\;\log t \cdot a + \left(-t\right)\\ \mathbf{elif}\;\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \log t \cdot \left(a - 0.5\right) \leq 1005:\\ \;\;\;\;\log \left(\left(x + y\right) \cdot z\right) - \mathsf{fma}\left(0.5, \log t, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, -t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(x + y\right)\\ t_2 := t\_1 + \log z\\ t_3 := \left(\left(-t\right) + t\_1\right) + \log t \cdot \left(a - 0.5\right)\\ \mathbf{if}\;t\_2 \leq -750:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 710:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log \left(\left(x + y\right) \cdot z\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (log (+ x y)))
        (t_2 (+ t_1 (log z)))
        (t_3 (+ (+ (- t) t_1) (* (log t) (- a 0.5)))))
   (if (<= t_2 -750.0)
     t_3
     (if (<= t_2 710.0)
       (fma (- a 0.5) (log t) (- (log (* (+ x y) z)) t))
       t_3))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((x + y));
	double t_2 = t_1 + log(z);
	double t_3 = (-t + t_1) + (log(t) * (a - 0.5));
	double tmp;
	if (t_2 <= -750.0) {
		tmp = t_3;
	} else if (t_2 <= 710.0) {
		tmp = fma((a - 0.5), log(t), (log(((x + y) * z)) - t));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = log(Float64(x + y))
	t_2 = Float64(t_1 + log(z))
	t_3 = Float64(Float64(Float64(-t) + t_1) + Float64(log(t) * Float64(a - 0.5)))
	tmp = 0.0
	if (t_2 <= -750.0)
		tmp = t_3;
	elseif (t_2 <= 710.0)
		tmp = fma(Float64(a - 0.5), log(t), Float64(log(Float64(Float64(x + y) * z)) - t));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + N[Log[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[((-t) + t$95$1), $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -750.0], t$95$3, If[LessEqual[t$95$2, 710.0], N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[(N[Log[N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

      1. lift--.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

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower--.f64 99.8

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 84.1

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

   7. Applied rewrites 84.1%

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

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

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.6

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

      5. lift-+.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. sum-log N/A

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

      9. lower-log.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.6

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

      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 99.6

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

   4. Applied rewrites 99.6%

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

4. Final simplification 96.1%

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

Alternative 5: 81.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- a 0.5) (log t) (- t))))
   (if (<= (- a 0.5) -20.0)
     t_1
     (if (<= (- a 0.5) -0.5)
       (- (fma (log t) -0.5 (log y)) (- t (log z)))
       t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((a - 0.5), log(t), -t);
	double tmp;
	if ((a - 0.5) <= -20.0) {
		tmp = t_1;
	} else if ((a - 0.5) <= -0.5) {
		tmp = fma(log(t), -0.5, log(y)) - (t - log(z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(a - 0.5), log(t), Float64(-t))
	tmp = 0.0
	if (Float64(a - 0.5) <= -20.0)
		tmp = t_1;
	elseif (Float64(a - 0.5) <= -0.5)
		tmp = Float64(fma(log(t), -0.5, log(y)) - Float64(t - log(z)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + (-t)), $MachinePrecision]}, If[LessEqual[N[(a - 0.5), $MachinePrecision], -20.0], t$95$1, If[LessEqual[N[(a - 0.5), $MachinePrecision], -0.5], N[(N[(N[Log[t], $MachinePrecision] * -0.5 + N[Log[y], $MachinePrecision]), $MachinePrecision] - N[(t - N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 a #s(literal 1/2 binary64)) < -20 or -0.5 < (-.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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. flip3-- N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. clear-num N/A

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

      9. flip3-- N/A

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

      10. lift--.f64 N/A

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

      11. lower-/.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 98.4

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

   7. Applied rewrites 98.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. associate-/r/ N/A

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

      6. lift-/.f64 N/A

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

      7. remove-double-div N/A

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

      8. lower-fma.f64 98.4

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

   9. Applied rewrites 98.4%

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

   if -20 < (-.f64 a #s(literal 1/2 binary64)) < -0.5

   1. Initial program 99.6%

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

   3. Taylor expanded in a around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-log.f64 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 63.4%

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

Alternative 6: 81.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- a 0.5) (log t) (- t))))
   (if (<= (- a 0.5) -20.0)
     t_1
     (if (<= (- a 0.5) -0.5)
       (+ (- (fma (log t) -0.5 (log z)) t) (log y))
       t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((a - 0.5), log(t), -t);
	double tmp;
	if ((a - 0.5) <= -20.0) {
		tmp = t_1;
	} else if ((a - 0.5) <= -0.5) {
		tmp = (fma(log(t), -0.5, log(z)) - t) + log(y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(a - 0.5), log(t), Float64(-t))
	tmp = 0.0
	if (Float64(a - 0.5) <= -20.0)
		tmp = t_1;
	elseif (Float64(a - 0.5) <= -0.5)
		tmp = Float64(Float64(fma(log(t), -0.5, log(z)) - t) + log(y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + (-t)), $MachinePrecision]}, If[LessEqual[N[(a - 0.5), $MachinePrecision], -20.0], t$95$1, If[LessEqual[N[(a - 0.5), $MachinePrecision], -0.5], N[(N[(N[(N[Log[t], $MachinePrecision] * -0.5 + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 a #s(literal 1/2 binary64)) < -20 or -0.5 < (-.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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. flip3-- N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. clear-num N/A

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

      9. flip3-- N/A

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

      10. lift--.f64 N/A

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

      11. lower-/.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 98.4

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

   7. Applied rewrites 98.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. associate-/r/ N/A

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

      6. lift-/.f64 N/A

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

      7. remove-double-div N/A

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

      8. lower-fma.f64 98.4

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

   9. Applied rewrites 98.4%

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

   if -20 < (-.f64 a #s(literal 1/2 binary64)) < -0.5

   1. Initial program 99.6%

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

   3. Taylor expanded in a around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-log.f64 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 63.4%

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

4. Final simplification 82.4%

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

Alternative 7: 69.1% 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 99.6%

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

3. Taylor expanded in y around inf

   \[\leadsto \color{blue}{\left(\log z + \left(-1 \cdot \log \left(\frac{1}{y}\right) + \log t \cdot \left(a - \frac{1}{2}\right)\right)\right) - t} \]
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 69.0%

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

6. Final simplification 69.0%

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

Alternative 8: 62.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 3.4 \cdot 10^{+59}:\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 3.4e+59) (* (log t) a) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 3.4e+59) {
		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 <= 3.4d+59) 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 <= 3.4e+59) {
		tmp = Math.log(t) * a;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 3.4e+59:
		tmp = math.log(t) * a
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 3.4e+59)
		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 <= 3.4e+59)
		tmp = log(t) * a;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 3.4e+59], N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if t < 3.40000000000000006e59

   1. Initial program 99.4%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 55.5

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

   5. Applied rewrites 55.5%

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

   if 3.40000000000000006e59 < 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 82.0

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

   5. Applied rewrites 82.0%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.4 \cdot 10^{+59}:\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.4% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift--.f64 N/A

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

   4. flip3-- N/A

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

   5. clear-num N/A

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

   6. un-div-inv N/A

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

   7. lower-/.f64 N/A

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

   8. clear-num N/A

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

   9. flip3-- N/A

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

   10. lift--.f64 N/A

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

   11. lower-/.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 79.8

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

7. Applied rewrites 79.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. clear-num N/A

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

   5. associate-/r/ N/A

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

   6. lift-/.f64 N/A

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

   7. remove-double-div N/A

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

   8. lower-fma.f64 79.8

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

9. Applied rewrites 79.8%

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

Alternative 10: 74.8% accurate, 2.9× speedup

Math

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

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) + Float64(-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 99.6%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift--.f64 N/A

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

   4. flip3-- N/A

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

   5. clear-num N/A

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

   6. un-div-inv N/A

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

   7. lower-/.f64 N/A

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

   8. clear-num N/A

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

   9. flip3-- N/A

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

   10. lift--.f64 N/A

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

   11. lower-/.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 79.8

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

7. Applied rewrites 79.8%

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

8. Taylor expanded in a around inf

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

   1. lower-*.f64 N/A

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

   2. lower-log.f64 77.5

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

10. Applied rewrites 77.5%

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

11. Final simplification 77.5%

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

Alternative 11: 37.7% accurate, 107.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 39.1

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

5. Applied rewrites 39.1%

   \[\leadsto \color{blue}{-t} \]
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 2024270 
(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))))